GSF-6053 Économétrie Financière I — course materials with detailed README, author headers, and clean-compiling LaTeX sources · b2e79db · gsf6053

GSF-6053 Économétrie Financière I — course materials with detailed README, author headers, and clean-compiling LaTeX sources

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

Simon-Pierre Boucher committed 4 days ago (Aug 6, 2026)

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	1	+# =============================================================================
	2	+# GSF-6053 : Économétrie Financière I
	3	+# Auteur : Simon-Pierre Boucher
	4	+# Contact : contact@spboucher.ai
	5	+# =============================================================================
	6	+# Auto detect text files and perform LF normalization
	7	+* text=auto

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	1	+# =============================================================================
	2	+# GSF-6053 : Économétrie Financière I
	3	+# Auteur : Simon-Pierre Boucher
	4	+# Contact : contact@spboucher.ai
	5	+# =============================================================================
	6	+.DS_Store
	7	+.claude/

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	1	+<!--
	2	+=============================================================================
	3	+GSF-6053 : Économétrie Financière I
	4	+Auteur : Simon-Pierre Boucher
	5	+Contact : contact@spboucher.ai
	6	+=============================================================================
	7	+-->
	8	+
	9	+# GSF-6053 — Financial Econometrics I (Économétrie Financière)
	10	+
	11	+Author / Auteur : Simon-Pierre Boucher
	12	+📧 Contact : [contact@spboucher.ai](mailto:contact@spboucher.ai)
	13	+🏛️ Département de finance, assurance et immobilier — Faculté des sciences de l'administration, Université Laval
	14	+
	15	+[![LaTeX](https://img.shields.io/badge/LaTeX-008080?style=for-the-badge&logo=latex&logoColor=white)](https://www.latex-project.org/)
	16	+[![Stata](https://img.shields.io/badge/Stata-1F4E79?style=for-the-badge&logo=stata&logoColor=white)](https://www.stata.com/)
	17	+[![Course Level](https://img.shields.io/badge/Course%20Level-Graduate-blue)](#)
	18	+[![Credits](https://img.shields.io/badge/Credits-3-green)](#)
	19	+[![Language](https://img.shields.io/badge/Language-Fran%C3%A7ais-lightgrey)](#)
	20	+
	21	+---
	22	+
	23	+## 📚 Course Description
	24	+
	25	+This repository contains the complete teaching material for GSF-6053 — Financial Econometrics I, a graduate course (Hiver 2022) that introduces students to the practical aspects of econometric methods and estimation techniques as applied in finance. All lecture decks are written in LaTeX (Beamer), in French, and each folder ships both the `.tex` source and the compiled `.pdf`.
	26	+
	27	+The course emphasizes modeling challenges specific to financial applications: OLS regression, maximum likelihood, hypothesis testing, heteroskedasticity, autocorrelation, panel data models, and time-series analysis (ARMA/ARIMA, unit roots, cointegration), with hands-on Stata application sessions.
	28	+
	29	+### 📋 Prerequisites
	30	+
	31	+A basic understanding of statistics and linear algebra is recommended to engage fully with the course material.
	32	+
	33	+## 🗂️ Detailed Content by Session
	34	+
	35	+### Section 02 — Régression et Moindres Carrés Ordinaires (MCO / OLS)
	36	+
	37	+\| Session \| Folder \| Topics \|
	38	+\|---\|---\|---\|
	39	+\| Séance 2 \| `Séance 2/` \| Modèle de régression, dérivation des MCO (sommation et format matriciel), exemples numériques, nuages de points et impact de la variance des erreurs \|
	40	+\| Séance 3 \| `Séance 3/` \| Maximum de vraisemblance (MLE), MLE pour le modèle de régression linéaire, estimateur sans biais de la variance, propriétés des estimateurs MLE et MCO \|
	41	+\| Séance 4 \| `Séance 4/` \| Analyse de variance, R², R² ajusté, tests d'hypothèses, contraintes linéaires, tests de Wald, LR (ratio de vraisemblance) et LM (multiplicateur de Lagrange), liens entre F, Wald, LR et LM \|
	42	+\| Séance 5 \| `Séance 5/` \| Suite de la Section 02 (PDF compilé fourni) \|
	43	+
	44	+### Section 03 — Extensions au modèle linéaire simple
	45	+
	46	+\| Session \| Folder \| Topics \|
	47	+\|---\|---\|---\|
	48	+\| Séance 6 \| `Séance 6/` \| Hétéroscédasticité : définition, conséquences et tests diagnostiques (Breusch-Pagan, White) \|
	49	+\| Séance 7 \| `Séance 7/` \| Autocorrélation des erreurs : diagnostic (Durbin-Watson), transformations de Cochrane-Orcutt et Prais-Winsten, estimateur robuste de Newey-West \|
	50	+
	51	+### Section 04 — Les modèles panels
	52	+
	53	+\| Session \| Folder \| Topics \|
	54	+\|---\|---\|---\|
	55	+\| Séance 8 \| `Séance 8/` \| Avantages des données panel, notation générale, modèle pooled, modèle à effets fixes, Least Squares Dummy Variable (LSDV) \|
	56	+\| Séance 9 \| `Séance 9/` \| Modèle empilé, within-group estimator, modèle à effets aléatoires, transformation de Fuller-Battese \|
	57	+
	58	+### Section 05 — Les séries chronologiques
	59	+
	60	+\| Session \| Folder \| Topics \|
	61	+\|---\|---\|---\|
	62	+\| Séance 10 \| `Séance 10/` \| Séries stationnaires et non stationnaires, opérateur de retard, autocovariance et autocorrélation (ACF/PACF), équations de Yule-Walker, modèle AR(1) \|
	63	+\| Séance 11 \| `Séance 11/` \| Estimation des moments, tests de bruit blanc (Box-Pierce, Ljung-Box), tests de racine unitaire (Dickey-Fuller augmenté, Phillips-Perron), cointégration et tests de cointégration \|
	64	+\| Séance 12 \| `Séance 12/` \| Processus autorégressifs (AR), moyennes mobiles (MA), ARMA(p,q), ARIMA(p,d,q), méthodologie Box-Jenkins \|
	65	+
	66	+### 💻 Stata Application Sessions
	67	+
	68	+\| Session \| Folder \| Topics \|
	69	+\|---\|---\|---\|
	70	+\| STATA S02 \| `STATA_S02/` \| Application Stata de la Section 02 : statistiques descriptives, régressions MCO, t-test, F-test, analyse des résidus (avec captures d'écran des sorties Stata) \|
	71	+\| STATA S03 \| `STATA_S03/` \| Application Stata de la Section 03 : détection et correction de l'hétéroscédasticité (tests de Breusch-Pagan et de White, tables du χ²) \|
	72	+
	73	+## 📁 Repository Structure
	74	+
	75	+```
	76	+GSF6053/
	77	+├── README.md ← this file
	78	+├── Séance 2/ … Séance 12/ ← lecture decks (Beamer .tex + compiled .pdf + build files)
	79	+│ └── GSF6053_S<n>.tex/.pdf
	80	+├── STATA_S02/ ← Stata lab, Section 02 (deck + Stata output screenshots .png)
	81	+│ └── GSF6053_STATA_02.tex/.pdf
	82	+└── STATA_S03/ ← Stata lab, Section 03 (deck + Stata output screenshots .png)
	83	+ └── GSF6053_STATA_03.tex/.pdf
	84	+```
	85	+
	86	+Each session folder contains:
	87	+
	88	+- `.tex` — the Beamer source (UTF-8, French)
	89	+- `.pdf` — the compiled slide deck, ready to use
	90	+- `.png` / `.jpg` — figures, regression outputs, statistical tables referenced by the deck
	91	+- Auxiliary LaTeX build files (`.aux`, `.log`, `.nav`, `.out`, `.snm`, `.toc`, `.synctex.gz`)
	92	+
	93	+## 🔨 Building the Slides
	94	+
	95	+All decks compile with a standard TeX distribution (TeX Live / MacTeX). From a session folder:
	96	+
	97	+```bash
	98	+cd "Séance 2"
	99	+pdflatex GSF6053_S2.tex
	100	+pdflatex GSF6053_S2.tex # second pass for the table of contents / navigation
	101	+```
	102	+
	103	+Or compile everything at once from the repository root:
	104	+
	105	+```bash
	106	+for d in "Séance "/ STATA_S0/; do
	107	+ (cd "$d" && f=$(ls *.tex 2>/dev/null) && [ -n "$f" ] && pdflatex -interaction=nonstopmode "$f" && pdflatex -interaction=nonstopmode "$f")
	108	+done
	109	+```
	110	+
	111	+Requirements: `beamer`, `graphicx`, `inputenc` (utf8) — all included in any full TeX Live / MacTeX install. Images must stay next to their `.tex` file (they are referenced by relative path).
	112	+
	113	+## 🎓 Learning Outcomes
	114	+
	115	+By the end of this course, students will be able to:
	116	+
	117	+- Derive and apply the OLS and maximum-likelihood estimators to financial data;
	118	+- Perform and interpret hypothesis tests (t, F, Wald, LR, LM);
	119	+- Diagnose and correct heteroskedasticity and autocorrelation;
	120	+- Estimate panel-data models (pooled, fixed effects, random effects);
	121	+- Model financial time series (AR, MA, ARMA, ARIMA), test for unit roots and cointegration, and apply the Box-Jenkins methodology;
	122	+- Implement all of the above in Stata.
	123	+
	124	+## 🤝 Contributing
	125	+
	126	+This repository is primarily for educational purposes. If you find any errors or have suggestions for improvements, feel free to open an issue or submit a pull request.
	127	+
	128	+## 📄 License & Usage
	129	+
	130	+This material is intended for students enrolled in the GSF-6053 course and for educational use. For any other use, please contact the author.
	131	+
	132	+---
	133	+
	134	+Simon-Pierre Boucher — 📧 [contact@spboucher.ai](mailto:contact@spboucher.ai)
	135	+
	136	+⭐ Star this repository if you find it helpful! ⭐

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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage{graphicx}
	9	+\usepackage{xcolor}
	10	+\usetheme{default}
	11	+\usecolortheme{default}
	12	+
	13	+\title[S02 Régression et MCO]{Section 02 : Régression et Moindres Carrés Ordinaires\\ (Application Stata)}
	14	+\subtitle{GSF-6053: Économétrie Financière}
	15	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	16	+\institute[Université Laval]
	17	+{
	18	+ \inst{1}%
	19	+ Département de finance, assurance et immobilier\\
	20	+ Faculté des sciences de l'administration\\
	21	+ Université Laval}
	22	+\date[Hiver 2022]{1 Février 2022}
	23	+
	24	+\begin{document}
	25	+
	26	+\begin{frame}
	27	+ \titlepage
	28	+\end{frame}
	29	+
	30	+\begin{frame}{Application STATA}
	31	+\begin{itemize}
	32	+\item Dans cette application Stata, nous allons analyser le jeu de données portant le nom \textbf{Wage1}
	33	+\item Il s’agit essentiellement de données collectées auprès de plusieurs travailleurs en 1976.
	34	+\item Dans cette analyse nous chercherons à quantifier et expliquer l’impact de certains facteurs propre au salarié sur son salaire.
	35	+\item Les variables que nous utiliserons dans cette analyse sont les suivantes:
	36	+\begin{itemize}
	37	+\item \textbf{lwage:} log du salaire horaire moyen
	38	+\item \textbf{educ:} années d'études
	39	+\item \textbf{exper:} années d'expérience potentielle
	40	+\item \textbf{tenutre:} années chez l'employeur actuel
	41	+\end{itemize}
	42	+\end{itemize}
	43	+\end{frame}
	44	+
	45	+\begin{frame}{Application STATA}
	46	+
	47	+\begin{block}{Code 1: Pour obtenir jeu de données Wage1 sur Stata}
	48	+\textbf{use http://fmwww.bc.edu/ec-p/data/wooldridge/wage1}
	49	+\end{block}
	50	+
	51	+\begin{block}{Code 2: Vérifier le format des variables utilisées}
	52	+\textbf{describe wage educ exper tenure}
	53	+\end{block}
	54	+
	55	+\begin{block}{Output 2:}
	56	+\begin{center}
	57	+\includegraphics[scale=.5]{describe}
	58	+\end{center}
	59	+\begin{itemize}
	60	+\item On peut voir que nos trois variables sont de type Float, ce qui indique qu'il s'agit d'une valeur numérique entière.
	61	+\item On peut donc l'utiliser pour notre analyse.
	62	+\end{itemize}
	63	+\end{block}
	64	+
	65	+\end{frame}
	66	+
	67	+
	68	+\begin{frame}{Application STATA}
	69	+\begin{block}{Code 3: Calculer et afficher des statistiques descriptives}
	70	+\textbf{summarize wage educ exper tenurex}
	71	+\end{block}
	72	+\begin{block}{Output 3:}
	73	+\begin{center}
	74	+\includegraphics[scale=.5]{summary}
	75	+\end{center}
	76	+\begin{itemize}
	77	+ \item On peut voir que notre jeu de données contient 526 observations
	78	+ \item Que le salaire moyen horraire est de 5.89 et qu'en moyenne les salariés ont 12.56 années de scolarité.
	79	+ \item Pour ce qui est de l'expérience de travail, en moyenne les salariés ont 17.02 années d'expériences et ils sont dans le même emploie depuis 5.10 ans, en moyenne.
	80	+\end{itemize}
	81	+\end{block}
	82	+
	83	+\end{frame}
	84	+
	85	+
	86	+\begin{frame}{Application STATA}
	87	+\begin{block}{Code 4: Relation entre le salaire et l'éducation}
	88	+\textbf{twoway (scatter wage educ) (lfit wage educ)}
	89	+\end{block}
	90	+
	91	+\begin{block}{Output 4:}
	92	+\begin{center}
	93	+\includegraphics[scale=.3]{PLOT_1.png}
	94	+\end{center}
	95	+\end{block}
	96	+\end{frame}
	97	+
	98	+
	99	+\begin{frame}{Application STATA}
	100	+\begin{block}{Code 5: Relation entre le salaire et l'expérience de travail}
	101	+\textbf{twoway (scatter wage exper) (lfit wage exper)}
	102	+\end{block}
	103	+
	104	+\begin{block}{Output 5:}
	105	+\begin{center}
	106	+\includegraphics[scale=.3]{PLOT_2.png}
	107	+\end{center}
	108	+\end{block}
	109	+\end{frame}
	110	+
	111	+\begin{frame}{Application STATA}
	112	+\begin{block}{Code 6: Relation entre le salaire et le nombre d'années chez l'employeur actuel}
	113	+\textbf{twoway (scatter wage tenure) (lfit wage tenure)}
	114	+\end{block}
	115	+
	116	+\begin{block}{Output 6:}
	117	+\begin{center}
	118	+\includegraphics[scale=.3]{PLOT_3.png}
	119	+\end{center}
	120	+\end{block}
	121	+\end{frame}
	122	+
	123	+\begin{frame}{Application STATA}
	124	+\begin{block}{Code 7: Distrbution de la variable \textbf{wage}}
	125	+\textbf{histogram wage, bin(50) normal normopts(lwidth(thick))}
	126	+\end{block}
	127	+
	128	+\begin{block}{Output 7:}
	129	+\begin{center}
	130	+\includegraphics[scale=.3]{PLOT_4.png}
	131	+\end{center}
	132	+\end{block}
	133	+\end{frame}
	134	+
	135	+\begin{frame}{Application STATA}
	136	+\begin{block}{Code 8: Distrbution de la variable \textbf{educ}}
	137	+\textbf{histogram educ, bin(20) normal normopts(lwidth(thick))}
	138	+\end{block}
	139	+
	140	+\begin{block}{Output 8:}
	141	+\begin{center}
	142	+\includegraphics[scale=.3]{PLOT_5.png}
	143	+\end{center}
	144	+\end{block}
	145	+\end{frame}
	146	+
	147	+
	148	+\begin{frame}{Application STATA}
	149	+\begin{block}{Code 9: Distrbution de la variable \textbf{exper}}
	150	+\textbf{histogram exper, bin(50) normal normopts(lwidth(thick))}
	151	+\end{block}
	152	+
	153	+\begin{block}{Output 9:}
	154	+\begin{center}
	155	+\includegraphics[scale=.3]{PLOT_6.png}
	156	+\end{center}
	157	+\end{block}
	158	+\end{frame}
	159	+
	160	+\begin{frame}{Application STATA}
	161	+\begin{block}{Code 10: Distrbution de la variable \textbf{tenure}}
	162	+\textbf{histogram tenure, bin(30) normal normopts(lwidth(thick))}
	163	+\end{block}
	164	+
	165	+\begin{block}{Output 10:}
	166	+\begin{center}
	167	+\includegraphics[scale=.3]{PLOT_7.png}
	168	+\end{center}
	169	+\end{block}
	170	+\end{frame}
	171	+
	172	+\begin{frame}{Application STATA}
	173	+\begin{block}{Régression linéaire simple}
	174	+\begin{equation}
	175	+wage = \beta_0 + \beta_1 educ + \epsilon
	176	+\end{equation}
	177	+\begin{equation}
	178	+wage = \beta_0 + \beta_1 exper + \epsilon
	179	+\end{equation}
	180	+\begin{equation}
	181	+wage = \beta_0 + \beta_1 tenure + \epsilon
	182	+\end{equation}
	183	+\end{block}
	184	+\begin{block}{Régression linéaire multiple}
	185	+\begin{equation}
	186	+wage = \beta_0 + \beta_1 educ +\beta_2 exper + \beta_3 tenure + \epsilon
	187	+\end{equation}
	188	+\end{block}
	189	+\end{frame}
	190	+
	191	+\begin{frame}{Application STATA}
	192	+\begin{block}{Code 11: $wage = \beta_0 + \beta_1 educ + \epsilon$}
	193	+\textbf{reg wage educ}
	194	+\end{block}
	195	+
	196	+\begin{block}{Output 11:}
	197	+\begin{center}
	198	+\includegraphics[scale=.5]{REG1.png}
	199	+\end{center}
	200	+\end{block}
	201	+\end{frame}
	202	+
	203	+
	204	+\begin{frame}{Application STATA}
	205	+\begin{block}{Analyse: $wage = \beta_0 + \beta_1 educ + \epsilon$}
	206	+\begin{itemize}
	207	+\item Avec un $R^2$ de 0.1648, le nombre d’années d’éducation d'un salarié semble expliquer environ 16\% des variations dans le salaire de ce dernier, et ce en moyenne.
	208	+\item La p-value du T-test pour le coefficient de la variable educ est de 0.000, ce qui implique que le coefficient est significativement différent de zéro. (À un seuil de 5\%)
	209	+\item Avec un coefficient estimé de 0.54 pour la variable educ, on peut comprendre qu’en moyenne, une année d’éducation additionnelle amène une augmentation du salaire horaire de 0.54.
	210	+\end{itemize}
	211	+\end{block}
	212	+\end{frame}
	213	+
	214	+
	215	+
	216	+
	217	+
	218	+\begin{frame}{Application STATA}
	219	+\begin{block}{Code 12: $wage = \beta_0 + \beta_1 exper + \epsilon$}
	220	+\textbf{reg wage exper}
	221	+\end{block}
	222	+
	223	+\begin{block}{Output 12:}
	224	+\begin{center}
	225	+\includegraphics[scale=.5]{REG2.png}
	226	+\end{center}
	227	+\end{block}
	228	+\end{frame}
	229	+
	230	+\begin{frame}{Application STATA}
	231	+\begin{block}{Analyse: $wage = \beta_0 + \beta_1 exper + \epsilon$}
	232	+\begin{itemize}
	233	+\item Avec un $R^2$ de 0.0127, le nombre d’années d’expérience d'un salarié semble expliquer environ 1\% des variations dans le salaire de ce dernier, et ce en moyenne.
	234	+\item La p-value du T-test pour le coefficient de la variable exper est de 0.01, ce qui implique que le coefficient est significativement différent de zéro. (À un seuil de 5\%)
	235	+\item Avec un coefficient estimé de 0.0307 pour la variable exper, on peut comprendre qu’en moyenne, une année d’expérience additionnelle amène une augmentation du salaire horaire de 0.0307.
	236	+\end{itemize}
	237	+\end{block}
	238	+\end{frame}
	239	+\begin{frame}{Application STATA}
	240	+\begin{block}{Code 13: $wage = \beta_0 + \beta_1 tenure + \epsilon$}
	241	+\textbf{reg wage tenure}
	242	+\end{block}
	243	+
	244	+\begin{block}{Output 13:}
	245	+\begin{center}
	246	+\includegraphics[scale=.5]{REG3.png}
	247	+\end{center}
	248	+\end{block}
	249	+\end{frame}
	250	+
	251	+
	252	+\begin{frame}{Application STATA}
	253	+\begin{block}{Analyse: $wage = \beta_0 + \beta_1 tenure + \epsilon$}
	254	+\begin{itemize}
	255	+\item Avec un $R^2$ de 0.1203, le nombre d’années chez l'employeur actuel d'un salarié semble expliquer environ 12\% des variations dans le salaire de ce dernier, et ce en moyenne.
	256	+\item La p-value du T-test pour le coefficient de la variable tenure est de 0.000, ce qui implique que le coefficient est significativement différent de zéro. (À un seuil de 5\%)
	257	+\item Avec un coefficient estimé de 0.177 pour la variable tenure, on peut comprendre qu’en moyenne, une année de plus chez l'employeur actuel amène une augmentation du salaire horaire de 0.177.
	258	+\end{itemize}
	259	+\end{block}
	260	+\end{frame}
	261	+
	262	+\begin{frame}{Application STATA}
	263	+\begin{block}{Code 14: $wage = \beta_0 + \beta_1 educ +\beta_2 exper + \beta_3 tenure + \epsilon$}
	264	+\textbf{reg wage educ exper tenure}
	265	+\end{block}
	266	+
	267	+\begin{block}{Output 14:}
	268	+\begin{center}
	269	+\includegraphics[scale=.5]{REG4.png}
	270	+\end{center}
	271	+\end{block}
	272	+\end{frame}
	273	+
	274	+\begin{frame}{Application STATA}
	275	+\begin{block}{Analyse: $wage = \beta_0 + \beta_1 educ +\beta_2 exper + \beta_3 tenure + \epsilon$}
	276	+\begin{itemize}
	277	+\item Avec un $R^2$ de 0.3064, les trois variables combinées ensemble semblent expliquer environ 31\% des variations dans le salaire de ce dernier, et ce en moyenne.
	278	+\item De plus, avec un $R^2$ ajusté de 0.3024 (relativement proche du $R^2$ standard), l'ajout de variables explicatives semble justifier étant donné que la valeur du $R^2$ ajusté ne semble pas avoir diminuer grandement par rapport au $R^2$ standard, suite à la pénalité pour l'ajout de variables explicatives dans le $R^2$ ajusté.
	279	+\item La p-value du F-test est de 0.000, ce qui implique que nous pouvons rejeter l'hypothèse nulle que tous les coefficients sont conjointement non significatifs.
	280	+\end{itemize}
	281	+\end{block}
	282	+\end{frame}
	283	+
	284	+\begin{frame}{Détails sur le T-test}
	285	+
	286	+\begin{itemize}
	287	+\item Dans le T-test que nous venons de faire, nous avons utilisé la P-value pour vérifier si le coefficient est significativement différent de 0.
	288	+\item En effet si cette P-value est inférieur à 0.05, on rejette l’hypothèse nulle que le coefficient est pas significativement différent de 0.
	289	+\item Afin de bien comprendre comment le T-test fonctionne, il est important de bien faire l’analyse sans utiliser la P-value.
	290	+\item Nous allons reprendre la dernière régression que nous venons d’analyser, soit celle avec toutes les variables explicatives incluses.
	291	+\end{itemize}
	292	+
	293	+\end{frame}
	294	+
	295	+\begin{frame}{Détails sur le T-test}
	296	+\textbf{Test de Student:}
	297	+\begin{block}{Two-tailed test}
	298	+\begin{itemize}
	299	+\item Hypothèse nulle est la non-significativité du coefficient de régression
	300	+\item Hypothèses:
	301	+\begin{itemize}
	302	+\item $H_0:$ $\beta_k=0$ $\rightarrow$ \textbf{Hypothèse nule}
	303	+\item $H_1:$ $\beta_k \neq 0$ $\rightarrow$ \textbf{Hypothèse alternative}
	304	+\end{itemize}
	305	+\item Règle de décision:
	306	+Rejeter $H_0$ si:
	307	+\begin{align*}
	308	+t=\frac{\mid \hat{\beta}_k-\beta_0 \mid}{SE_{\hat{\beta}_k}} > t_{n-k,\alpha/2}
	309	+\end{align*}
	310	+\begin{itemize}
	311	+\item Ou $\beta_0$ est la valeur du coefficient sous l'hypothèse nulle, soit 0
	312	+\item $SE_{\hat{\beta}_k}$ est l'écart-type associé à l'estimation de $\hat{\beta}_k$
	313	+\end{itemize}
	314	+\end{itemize}
	315	+\end{block}
	316	+
	317	+\end{frame}
	318	+
	319	+\begin{frame}{Détails sur le T-test}
	320	+
	321	+\begin{itemize}
	322	+\item Le Modèle de régression:
	323	+\begin{align*}
	324	+wage = \beta_0 + \beta_1 educ +\beta_2 exper + \beta_3 tenure + \epsilon
	325	+\end{align*}
	326	+\item Afin de pouvoir faire la comparaison de notre statistique T, nous devons obtenir une valeur critique à un seuil de 5\%, soit
	327	+\begin{align*}
	328	+t_{n-k,\alpha/2}
	329	+\end{align*}
	330	+\item Sachant que nous avons $n=526$ observations et $k=3$ variables explicatives, nous devrons trouve la valeur critique:
	331	+\begin{align*}
	332	+t_{526-3,0.05/2}=t_{523,0.025}
	333	+\end{align*}
	334	+
	335	+\end{itemize}
	336	+
	337	+\end{frame}
	338	+
	339	+\begin{frame}{Détails sur le T-test}
	340	+
	341	+\begin{itemize}
	342	+\item Nous allons maintenant aller regarder dans une table statistique appelée \textbf{T-table} afin de trouver notre valeur critique.
	343	+\item Au niveau de la significativité, on prendra la colonne avec une valeur:
	344	+\begin{itemize}
	345	+\item $\alpha= 0.025$ pour le \textbf{One-tail test}
	346	+\item $\alpha= 0.05$ pour le \textbf{Two-tail test}
	347	+\end{itemize}
	348	+\item Pour ce qui est du nombre de degrés de liberté, nous avons $df=523$, cependant la table ne fournit pas nécessairement une valeur pour tous les df possibles.
	349	+\begin{itemize}
	350	+\item Dans la table qui va suivre, nous avons seulement $df=100$ et $df=1000$ de dispo, nous allons donc prendre $df=1000$ par sécurité.
	351	+\end{itemize}
	352	+\end{itemize}
	353	+
	354	+\end{frame}
	355	+
	356	+\begin{frame}{Détails sur le T-test}
	357	+\begin{center}
	358	+\includegraphics[scale=.4]{tstat}
	359	+\end{center}
	360	+\end{frame}
	361	+
	362	+\begin{frame}{Détails sur le T-test}
	363	+\begin{itemize}
	364	+\item Notre valeur critique est de $t_{526-3,0.05/2}=t_{523,0.025}=1.962$
	365	+\item Pour le coefficient $\beta_1$ attaché à la variable \textbf{educ}:
	366	+\begin{align*}
	367	+t=\frac{\mid 0.5989651-0 \mid}{0.0512835} = 11.69 > 1.962
	368	+\end{align*}
	369	+\begin{itemize}
	370	+\item $t=11.69$ est plus grand que la valeur critique $t_{523,0.025}=1.962$, alors on peut rejetter l'hypothèse nulle et conclure que $\hat{\beta}_1$ est significativement différent de 0, à un seuil de 5\%.
	371	+\end{itemize}
	372	+\end{itemize}
	373	+\end{frame}
	374	+
	375	+\begin{frame}{Détails sur le T-test}
	376	+\begin{itemize}
	377	+\item Pour le coefficient $\beta_2$ attaché à la variable \textbf{exper}:
	378	+\begin{align*}
	379	+t=\frac{\mid 0.0223395-0 \mid}{0.0120568} = 1.85 < 1.962
	380	+\end{align*}
	381	+\begin{itemize}
	382	+\item $t=1.85$ est plus petit que la valeur critique $t_{523,0.025}=1.962$, alors on ne peut pas rejetter l'hypothèse nulle. $\hat{\beta}_2$ n'est pas significativement différent de 0, à un seuil de 5\%.
	383	+\end{itemize}
	384	+\end{itemize}
	385	+\end{frame}
	386	+
	387	+\begin{frame}{Détails sur le T-test}
	388	+\begin{itemize}
	389	+
	390	+\item Pour le coefficient $\beta_3$ attaché à la variable \textbf{tenure}:
	391	+\begin{align*}
	392	+t=\frac{\mid 0.1692687-0 \mid}{0.0216446} = 7.82 > 1.962
	393	+\end{align*}
	394	+\begin{itemize}
	395	+\item $t=7.82$ est plus grand que la valeur critique $t_{523,0.025}=1.962$, alors on peut rejetter l'hypothèse nulle et conclure que $\hat{\beta}_3$ est significativement différent de 0, à un seuil de 5\%.
	396	+\end{itemize}
	397	+\end{itemize}
	398	+\end{frame}
	399	+
	400	+
	401	+\begin{frame}{Détails sur le F-test}
	402	+\begin{itemize}
	403	+\item Comme pour le T-test, on veut également effectuer le F-test sans la P-value.
	404	+\end{itemize}
	405	+\textbf{F-test (significativité conjointe)}
	406	+\begin{block}{Hypothèses:}
	407	+\begin{itemize}
	408	+\item $H_O:$ $\beta_2 =0$ et/ou $\beta_3 = 0$ et/ou $...$ et/ou $\beta_k=0$
	409	+\item $H_1:$ $\beta_2 \neq 0$ et $\beta_3 \neq 0$ et $...$ et/ou $\beta_k \neq 0$
	410	+\end{itemize}
	411	+\end{block}
	412	+\begin{block}{Statistique de test:}
	413	+\begin{align*}
	414	+F=\frac{MS_{group}}{MS_{error}}>F(q,t-k;\alpha)
	415	+\end{align*}
	416	+Sachant
	417	+\begin{itemize}
	418	+\item $MS_{group}=\frac{SS_{group}}{df_{group}}$
	419	+\item $MS_{error}=\frac{SS_{error}}{df_{error}}$
	420	+\end{itemize}
	421	+\end{block}
	422	+\end{frame}
	423	+
	424	+\begin{frame}{Détails sur le F-test}
	425	+\begin{itemize}
	426	+\item On rejette $H_0$ si la statistique $F$ est supérieur à la valeur critique $F(q,t-k;\alpha)$.
	427	+\item Dans le cas qui nous concerne, nous avons 4 coefficients (k=4), soit $\beta_0, \beta_1, \beta_2$ et $\beta_3$
	428	+\item $q=k-1=4-1=3= df_{group}$
	429	+\item $t-k=526-4=522= df_{error}$
	430	+\item On pose un seuil de significativité de $\alpha=0.05$
	431	+\item La valeur critique sera représenté par $F(3,522;0.05)$
	432	+\item Nous allons maintenant regarder dans la table suivante la valeur critique $F(3,522;0.05)$
	433	+\end{itemize}
	434	+\end{frame}
	435	+
	436	+\begin{frame}{Détails sur le F-test}
	437	+\begin{center}
	438	+\includegraphics[scale=.2]{ftable}
	439	+\end{center}
	440	+\end{frame}
	441	+
	442	+\begin{frame}{Détails sur le F-test}
	443	+\begin{itemize}
	444	+\item La valeur de la statistique F est:
	445	+\begin{align*}
	446	+F(3,522;0.05)=2.60
	447	+\end{align*}
	448	+\item Voici maintenant le test:
	449	+\begin{align*}
	450	+F=\frac{\frac{SS_{group}}{df_{group}}}{\frac{SS_{error}}{df_{error}}}>F(q,t-k;\alpha)
	451	+\end{align*}
	452	+\begin{align*}
	453	+F=\frac{\frac{2194.1116}{3}}{\frac{4966.30269}{522}}=76.87>2.60
	454	+\end{align*}
	455	+\end{itemize}
	456	+\end{frame}
	457	+
	458	+
	459	+\begin{frame}{Détails sur le F-test}
	460	+\begin{itemize}
	461	+\item On rejette l'hypothèse nulle étant donné que la statistique $F=76.87$ est supérieur à la valeur critique $F(3,522;0.05)=2.60$.
	462	+\item On rejette donc l'hypothèse que tous les coefficients sont conjointement tous égaux à 0.
	463	+\item En d'autres mots, au moins un coefficient dans cette régression est significativement différent de 0.
	464	+\end{itemize}
	465	+\end{frame}
	466	+
	467	+
	468	+\begin{frame}{Application STATA}
	469	+
	470	+\begin{block}{Code 15: Générer les résidus de la régression}
	471	+\textbf{reg wage educ exper tenure}\\
	472	+\textbf{predict resid, residuals}
	473	+\end{block}
	474	+\begin{itemize}
	475	+\item Sans faire une analyse détaillée des résidus, nous allons créer une variable représentant les résidus de notre régression.
	476	+\item Afin de faire l'analyse, nous allons produire un histogramme de nos résidus et un nuage de points avec les résidus en axe des Y et la variable \textbf{wage} en axe des X.
	477	+\end{itemize}
	478	+\end{frame}
	479	+
	480	+\begin{frame}{Application STATA}
	481	+
	482	+\begin{block}{Code 16: Histogramme des résidus}
	483	+\textbf{histogram resid, bin(50) normal}
	484	+\end{block}
	485	+
	486	+\begin{block}{Output 16:}
	487	+\begin{center}
	488	+\includegraphics[scale=.3]{resid.png}
	489	+\end{center}
	490	+\end{block}
	491	+
	492	+\end{frame}
	493	+
	494	+
	495	+
	496	+\begin{frame}{Application STATA}
	497	+
	498	+\begin{block}{Code 17: Nuage de points des résidus}
	499	+\textbf{twoway (scatter red wage)}
	500	+\end{block}
	501	+
	502	+\begin{block}{Output 17:}
	503	+\begin{center}
	504	+\includegraphics[scale=.3]{scatter_resid.png}
	505	+\end{center}
	506	+\end{block}
	507	+
	508	+\end{frame}
	509	+
	510	+
	511	+\begin{frame}{Application STATA}
	512	+
	513	+\begin{block}{Analyse des résidus}
	514	+\begin{itemize}
	515	+\item Comme caractéristiques souhaitez pour nos résidus, nous souhaitons qu’il ressemble à une loi normale et qu’il n’ait pas de \textbf{patterns} entre ce résidu et la variable \textbf{wage}
	516	+\item Au niveau de l’histogramme des résidus, on peut voir que sans être parfaits les résidus prennent la forme d’une cloche.
	517	+\item Il semble y avoir plus de résidus au niveau du centre (x=0), que ce qui est prévu par la loi normale.
	518	+\item Au niveau du nuage de point, nous avons un problème majeur, il y a clairement une relation entre notre résidu et la variable \textbf{wage}.
	519	+\end{itemize}
	520	+\end{block}
	521	+
	522	+\end{frame}
	523	+
	524	+
	525	+\end{document}
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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage{graphicx}
	9	+\usepackage{xcolor}
	10	+\usetheme{default}
	11	+\usecolortheme{default}
	12	+
	13	+\title[S03]{Section 03: Hétéroscédasticité \\ (Application Stata)}
	14	+\subtitle{GSF-6053: Économétrie Financière}
	15	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	16	+\institute[Université Laval]
	17	+{
	18	+ \inst{1}%
	19	+ Département de finance, assurance et immobilier\\
	20	+ Faculté des sciences de l'administration\\
	21	+ Université Laval}
	22	+\date[Hiver 2022]{15 février 2022}
	23	+
	24	+\begin{document}
	25	+
	26	+\begin{frame}
	27	+ \titlepage
	28	+\end{frame}
	29	+
	30	+\begin{frame}{Application STATA}
	31	+\begin{itemize}
	32	+\item Nous allons maintenant regarder si notre régression présente un problème d’hétéroscédasticité.
	33	+\item Nous allons reprendre les mêmes données que lors de l’application STATA de la section 2.
	34	+\item Afin de détecter s’il y a présence d’hétéroscédasticité, nous allons effectuer les deux tests suivants:
	35	+\begin{itemize}
	36	+\item Test de White
	37	+\item Test de Breusch-Pagan
	38	+\end{itemize}
	39	+\end{itemize}
	40	+\end{frame}
	41	+
	42	+\begin{frame}{Application STATA}
	43	+\textbf{Rappel de la régression (fait dans section 2)}
	44	+\begin{block}{Code 1: $wage = \beta_0 + \beta_1 educ +\beta_2 exper + \beta_3 tenure + \epsilon$}
	45	+\textbf{reg wage educ exper tenure}
	46	+\end{block}
	47	+
	48	+\begin{block}{Output 1:}
	49	+\begin{center}
	50	+\includegraphics[scale=.5]{REG4.png}
	51	+\end{center}
	52	+\end{block}
	53	+
	54	+\end{frame}
	55	+
	56	+\begin{frame}{Application STATA}
	57	+\textbf{Test de White:}
	58	+\begin{itemize}
	59	+\item Régression auxiliaire:
	60	+\begin{align*}
	61	+\hat{\epsilon}^2 & = \alpha_0+\alpha_1 educ+\alpha_2 exper+\alpha_3 tenure \\ & +\alpha_{11} educ^2+\alpha_{22} exper^2 +\alpha_{33} tenure^2 \\ & +\alpha_{12}(educ \times exper) +\alpha_{13}(educ \times tenure) \\ & +\alpha_{23}(exper \times tenure)
	62	+\end{align*}
	63	+\item La statistique de test :
	64	+\begin{align*}
	65	+WHITE = T \times R_{reg.aux}^2
	66	+\end{align*}
	67	+Sachant que $R_{reg.aux}^2$ est le $R^2$ de la régression auxiliaire.
	68	+\end{itemize}
	69	+\end{frame}
	70	+
	71	+\begin{frame}{Application STATA}
	72	+\begin{block}{Code 2: Régression auxiliaire du test de White}
	73	+\begin{itemize}
	74	+ \item \textbf{predict resid, residuals}
	75	+ \item \textbf{gen resid2 = resid\string^2}
	76	+ \item \textbf{gen educ2 = educ\string^2}
	77	+ \item \textbf{gen exper2 = exper\string^2}
	78	+ \item \textbf{gen tenure2 = tenure\string^2}
	79	+ \item \textbf{gen educexper = educ*exper}
	80	+ \item \textbf{gen eductenure = educ*tenure}
	81	+ \item \textbf{gen expertenure = exper*tenure}
	82	+ \item \textbf{reg resid2 educ exper tenure educ2 exper2 tenure2 educexper eductenure expertenure}
	83	+\end{itemize}
	84	+\end{block}
	85	+\end{frame}
	86	+
	87	+\begin{frame}{Application STATA}
	88	+\begin{block}{Output 2:}
	89	+\includegraphics[scale=.5]{aux1.png}
	90	+\end{block}
	91	+\end{frame}
	92	+
	93	+
	94	+\begin{frame}{Application STATA}
	95	+\begin{block}{Test de White}
	96	+\begin{itemize}
	97	+\item Valeur critique de White
	98	+\begin{itemize}
	99	+\item Le nombre de degré de liberté est le nombre de variables dans la régression auxiliaire, soit $df=9$.
	100	+\item Nous trouverons cette valeur critique en utilisant une table chi-carré
	101	+\item On utilise également une significativité de 5\%
	102	+\end{itemize}
	103	+\begin{align*}
	104	+X^2(H)=X^2(9)
	105	+\end{align*}
	106	+\item Hypothèse nulle: la variance est homoscédastique
	107	+\item Hypothèse alternative: la variance est hétéroscédastique
	108	+\end{itemize}
	109	+\end{block}
	110	+
	111	+\end{frame}
	112	+
	113	+
	114	+\begin{frame}{Application STATA}
	115	+\begin{block}{Table Chi-Carré}
	116	+\begin{center}
	117	+\includegraphics[scale=.17]{chitable.png}
	118	+\end{center}
	119	+\end{block}
	120	+\end{frame}
	121	+
	122	+
	123	+\begin{frame}{Application STATA}
	124	+\begin{block}{Valeur critique de White}
	125	+\begin{align*}
	126	+X^2(H)=X^2(9)=16.919
	127	+\end{align*}
	128	+\end{block}
	129	+\begin{block}{Statistique de White}
	130	+\begin{align*}
	131	+WHITE = T \times R_{reg.aux}^2=526 \times 0.1216 = 63.9616
	132	+\end{align*}
	133	+\end{block}
	134	+
	135	+\end{frame}
	136	+
	137	+\begin{frame}{Application STATA}
	138	+\begin{block}{Décision}
	139	+\begin{itemize}
	140	+\item La statistique de White $WHITE = 63.9616$ est supérieur à la valeur critique $X^2(9)=16.919$
	141	+\begin{align*}
	142	+WHITE = 63.9616 > X^2(9)=16.919
	143	+\end{align*}
	144	+\item On rejette donc l'hypothèse nulle que la variance est homoscédastique.
	145	+\item Nous avons possiblement une régression ayant une problème d'hétéroscédastique.
	146	+\end{itemize}
	147	+\end{block}
	148	+\end{frame}
	149	+
	150	+
	151	+\begin{frame}{Application STATA}
	152	+\begin{block}{Code 3: Test de White avec commande rapide}
	153	+\textbf{estat imtest, white}
	154	+\end{block}
	155	+\begin{block}{Output 3:}
	156	+\includegraphics[scale=.5]{white.png}
	157	+\begin{itemize}
	158	+\item On peut voir que la valeur de la statistique est la même que celle obtenue avec la méthode de la régression auxiliaire.
	159	+\item Et la P-value de 0.000 confirme notre décision de rejetter l'hypthèse nulle.
	160	+\end{itemize}
	161	+\end{block}
	162	+\end{frame}
	163	+
	164	+
	165	+\begin{frame}{Application STATA}
	166	+\textbf{Test de Breusch-Pagan:}
	167	+\begin{itemize}
	168	+\item Régression auxiliaire:
	169	+\begin{align*}
	170	+\hat{\epsilon}^2 & = \alpha_0+\alpha_1 educ+\alpha_2 exper+\alpha_3 tenure
	171	+\end{align*}
	172	+\item La statistique de test :
	173	+\begin{align*}
	174	+BP = T \times R_{reg.aux}^2
	175	+\end{align*}
	176	+Sachant que $R_{reg.aux}^2$ est le $R^2$ de la régression auxiliaire.
	177	+\end{itemize}
	178	+\end{frame}
	179	+
	180	+\begin{frame}{Application STATA}
	181	+\begin{block}{Code 4: Régression auxiliaire du test de Breusch-Pagan}
	182	+\begin{itemize}
	183	+ \item \textbf{predict resid, residuals}
	184	+ \item \textbf{gen resid2 = resid\string^2}
	185	+ \item \textbf{reg resid2 educ exper tenure}
	186	+\end{itemize}
	187	+\end{block}
	188	+\end{frame}
	189	+
	190	+
	191	+\begin{frame}{Application STATA}
	192	+\begin{block}{Output 4:}
	193	+\includegraphics[scale=.5]{aux2.png}
	194	+\end{block}
	195	+\end{frame}
	196	+
	197	+
	198	+\begin{frame}{Application STATA}
	199	+\begin{block}{Test de Breusch-Pagan}
	200	+\begin{itemize}
	201	+\item Valeur critique de Breusch-Pagan
	202	+\begin{itemize}
	203	+\item Le nombre de degré de liberté est le nombre de variables dans la régression auxiliaire, soit $df=3$.
	204	+\item Nous trouverons cette valeur critique en utilisant une table chi-carré
	205	+\item On utilise également une significativité de 5\%
	206	+\end{itemize}
	207	+\begin{align*}
	208	+X^2(H)=X^2(3)
	209	+\end{align*}
	210	+\item Hypothèse nulle: la variance est homoscédastique
	211	+\item Hypothèse alternative: la variance est hétéroscédastique
	212	+\end{itemize}
	213	+\end{block}
	214	+
	215	+\end{frame}
	216	+
	217	+\begin{frame}{Application STATA}
	218	+\begin{block}{Table Chi-Carré}
	219	+\begin{center}
	220	+\includegraphics[scale=.17]{chitable_2.png}
	221	+\end{center}
	222	+\end{block}
	223	+\end{frame}
	224	+
	225	+
	226	+\begin{frame}{Application STATA}
	227	+\begin{block}{Valeur critique de Breusch-Pagan}
	228	+\begin{align*}
	229	+X^2(H)=X^2(3)=7.815
	230	+\end{align*}
	231	+\end{block}
	232	+\begin{block}{Statistique de White}
	233	+\begin{align*}
	234	+BP = T \times R_{reg.aux}^2=526 \times 0.0819 = 43.0794
	235	+\end{align*}
	236	+\end{block}
	237	+
	238	+\end{frame}
	239	+
	240	+\begin{frame}{Application STATA}
	241	+\begin{block}{Décision}
	242	+\begin{itemize}
	243	+\item La statistique de Breusch-Pagan $BP = 43.0794$ est supérieur à la valeur critique $X^2(9)=7.815$
	244	+\begin{align*}
	245	+BP = 43.0794 > X^2(3)=7.815
	246	+\end{align*}
	247	+\item On rejette donc l'hypothèse nulle que la variance est homoscédastique.
	248	+\item Nous avons possiblement une régression ayant une problème d'hétéroscédastique.
	249	+\end{itemize}
	250	+\end{block}
	251	+\end{frame}
	252	+
	253	+
	254	+\begin{frame}{Application STATA}
	255	+\begin{block}{Code 5: Test de Breusch-Pagan avec commande rapide}
	256	+\textbf{estat hettest educ exper tenure, iid}
	257	+\end{block}
	258	+\begin{block}{Output 5:}
	259	+\includegraphics[scale=.5]{BP.png}
	260	+\begin{itemize}
	261	+\item On peut voir que la valeur de la statistique est la même que celle obtenue avec la méthode de la régression auxiliaire.
	262	+\item Et la P-value de 0.000 confirme notre décision de rejetter l'hypthèse nulle.
	263	+\end{itemize}
	264	+\end{block}
	265	+\end{frame}
	266	+
	267	+
	268	+
	269	+\begin{frame}{Application STATA}
	270	+\begin{block}{Code 6: Correcteur de White \\ (Correction pour hétéroscédasticité)}
	271	+\textbf{reg wage educ exper tenure, robust}
	272	+\end{block}
	273	+\begin{block}{Output 6:}
	274	+\includegraphics[scale=.5]{correcteur.png}
	275	+\end{block}
	276	+\end{frame}
	277	+
	278	+
	279	+\begin{frame}{Application STATA}
	280	+\begin{block}{Analyse régression corrigée pour hétéroscédasticité}
	281	+\begin{itemize}
	282	+\item Comme on peut voir, les coefficients sont restés identiques.
	283	+\item C’est normal, étant donné que le correcteur de White change simplement la variance des estimateurs et leur valeur.
	284	+\item Le fait que la variance de ses estimateurs soit affectée, implique que la décision de rejet de certains coefficients pourrait changer.
	285	+\item Les coefficients des variables \textbf{educ} et \textbf{tenure} reste significatif.
	286	+\item Cependant, dans la régression OLS, le coefficient de la variable \textbf{exper} n’était pas significatif.
	287	+\item Suite à l’application du correcteur de White, le coefficient de la variable \textbf{exper} devient significatif.
	288	+\end{itemize}
	289	+\end{block}
	290	+\end{frame}
	291	+
	292	+
	293	+
	294	+\end{document}
		\ No newline at end of file

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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage{graphicx}
	9	+\usetheme{default}
	10	+\usecolortheme{default}
	11	+
	12	+\title[S02 Régression et MCO]{Section 05 : Les séries chronologiques\\ (Séance 10)}
	13	+\subtitle{GSF-6053: Économétrie Financière}
	14	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	15	+\institute[Université Laval]
	16	+{
	17	+ \inst{1}%
	18	+ Département de finance, assurance et immobilier\\
	19	+ Faculté des sciences de l'administration\\
	20	+ Université Laval}
	21	+\date[Hiver 2022]{29 Mars 2022}
	22	+
	23	+\begin{document}
	24	+
	25	+\begin{frame}
	26	+ \titlepage
	27	+\end{frame}
	28	+
	29	+
	30	+\begin{frame}{Références}
	31	+\textbf{Obligatoires:}
	32	+\begin{itemize}
	33	+\item \textbf{Notes de cours:} Section 5 (Professeure: Marie-Hélène Gagnon)
	34	+\item \textbf{Woolridge:} chapitres 11, 12 et 19.
	35	+\end{itemize}
	36	+\vspace{0.5cm}
	37	+\textbf{Complémentaires:}
	38	+\begin{itemize}
	39	+\item \textbf{Gujarati et Porter:} chapitres 21 et 22
	40	+\end{itemize}
	41	+\end{frame}
	42	+
	43	+\begin{frame}{Plan de la séance}
	44	+ \tableofcontents
	45	+\end{frame}
	46	+
	47	+\section{Série chronologique}
	48	+\frame{\tableofcontents[current]}
	49	+
	50	+\begin{frame}{Série chronologique}
	51	+\begin{itemize}
	52	+\item \textbf{Définition:} Une \textbf{série chronologique} (ou série temporelle) est toute suite d’observations indexées par un ensemble ordonné $T$ (un intervalle de temps).
	53	+\begin{align*}
	54	+(X_t : t \in T)
	55	+\end{align*}
	56	+\item Les séries chronologiques sont intéressantes, car plusieurs données économiques et financières ne sont pas indépendantes dans le temps.
	57	+\item En effet, il est facile de constater qu’un graphique du prix d’un actif ou du PIB comporte une certaine tendance ou encore une dépendance d’une observation par rapport à la précédente.
	58	+\item Un objectif de l’analyse en série chronologique est de modéliser cette dépendance dans le temps.
	59	+\end{itemize}
	60	+\end{frame}
	61	+
	62	+\begin{frame}{Définitions des séries}
	63	+\begin{block}{Séries continues}
	64	+\begin{itemize}
	65	+\item L’indice t peut prendre toutes les valeurs dans un intervalle de nombres réels.
	66	+\item Rare dans les données économiques ou financières généralement disponibles puisque les relevés se font de façon discrète.
	67	+\item \textbf{Exemples:} Séries de très haute fréquence, relevé de toutes les transactions sur un titre en temps continues
	68	+\end{itemize}
	69	+\end{block}
	70	+\end{frame}
	71	+
	72	+
	73	+\begin{frame}{Définitions des séries}
	74	+\begin{block}{Séries discrètes}
	75	+\begin{itemize}
	76	+\item L’indice t ne prend que des valeurs isolées dans T.
	77	+\item L’ensemble des indices t est un sous-ensemble des nombres entiers.
	78	+\item La durée qui sépare deux observations consécutives d’une série discrète est souvent la même et représente la fréquence des observations (mensuel, annuel, journalière, etc.).
	79	+\item Parmi les séries discrètes, on distingue :
	80	+\begin{itemize}
	81	+\item Les séries en niveau : Observations enregistrées instantanément, telle une photographie. En finance, ce sont les prix.
	82	+\item Les séries en flux : Valeurs cumulées sur un intervalle de temps reflétant de l’histoire d’un phénomène sur cette période de temps. En finance, on parle souvent de rendement.
	83	+\end{itemize}
	84	+\end{itemize}
	85	+\end{block}
	86	+\end{frame}
	87	+
	88	+\begin{frame}{Définitions des séries}
	89	+\begin{itemize}
	90	+\item Afin de lisser les séries chronologiques et d’éliminer les effets des unités de mesure, on utilise souvent une transformation logarithmique.
	91	+\item Une série soumise à cette transformation est souvent plus stable en première différence (flux).
	92	+\item \textcolor{red}{Attention, certaines séries ne doivent pas être prises en log et cette transformation ne règle pas les effets des observations extrêmes.}
	93	+\end{itemize}
	94	+\end{frame}
	95	+
	96	+\section{Première différence}
	97	+\frame{\tableofcontents[current]}
	98	+
	99	+\begin{frame}{Première différence}
	100	+\textbf{La différence première définie par}
	101	+\begin{align*}
	102	+\Delta ln(X_t) = ln(X_t) - ln(X_(t-1)
	103	+\end{align*}
	104	+\begin{itemize}
	105	+\item Elle donne souvent une série plus stable et surtout souvent stationnaire.
	106	+\item Notez que $\Delta ln(X_t)$ est une approximation de la série des taux de croissance de $X_t$:
	107	+\begin{align*}
	108	+\Delta ln(X_t)=ln(X_t)-ln(X_{t-1}) \approx \frac{X_t-X_{t-1}}{X_{t-1}}
	109	+\end{align*}
	110	+\item En finance, les taux de croissance sont des rendements.
	111	+\end{itemize}
	112	+\end{frame}
	113	+
	114	+\section{Saisonnalité et Désaisonalisation}
	115	+\frame{\tableofcontents[current]}
	116	+
	117	+\begin{frame}{Saisonnalité et Désaisonalisation}
	118	+\begin{itemize}
	119	+\item Certaines séries chronologiques présentent des comportements particuliers qui se reproduisent à des intervalles de temps réguliers : même mois, même trimestre, même période de l’année ou le même jour de la semaine.
	120	+\begin{itemize}
	121	+\item On dit que ce sont des évènements saisonniers.
	122	+\end{itemize}
	123	+\item Il faut enlever ces \textbf{effets saisonniers} avant de modéliser la série : \textbf{désaisonnalisation}.
	124	+\item Pour désaisonnaliser une série :
	125	+\begin{itemize}
	126	+\item Identifier les dates correspondant aux différents comportements saisonniers.
	127	+\item Construire des variables dummies pour ces dates.
	128	+\item Estimer un modèle comportant ces variables dummies.
	129	+\end{itemize}
	130	+\end{itemize}
	131	+\end{frame}
	132	+
	133	+
	134	+\section{Processus stochastique}
	135	+\frame{\tableofcontents[current]}
	136	+
	137	+\begin{frame}{Processus stochastique}
	138	+\begin{itemize}
	139	+\item Soit $T$ un ensemble non vide.
	140	+\item Un processus aléatoire sur $T$ est un ensemble de variables aléatoires réelles $X_t$ telles qu’à chaque élément $t \in T$ est associé une variable aléatoire $X_t$.
	141	+\item Notation: $\left\{ X_t: t \in T \right\}$
	142	+\item $T$ peut être continu ou discret, fini ou infini, mais en général, on suppose qu’il est infini.
	143	+\item Un processus stochastique est un ensemble d’observation de variables aléatoires ordonné en $T$.
	144	+\item Chacune des observations du processus stochastique est une variable aléatoire.
	145	+\item Nous utiliserons donc des réalisations de ce processus stochastique pour faire de l’inférence sur cedit processus et l’expliciter.
	146	+\end{itemize}
	147	+\end{frame}
	148	+
	149	+\section{Processus déterministe}
	150	+\frame{\tableofcontents[current]}
	151	+
	152	+\begin{frame}{Processus déterministe}
	153	+\begin{itemize}
	154	+\item Soit $\left\{ X_t : t \in T \right\}$ un processus stochastique
	155	+\begin{itemize}
	156	+\item $T_1 \subseteq T$, $I_t = \left\{X_s : s \leq t\right\}$
	157	+\end{itemize}
	158	+\item On dit que $\left\{X_t : t \in T \right\}$ est un processus déterministe dans $T_1$ si et seulement si:
	159	+\begin{itemize}
	160	+\item Il existe un ensemble de fonctions $\left\{g_t(I_{t-1}) : t \in T_1 \right\}$
	161	+\item Telles que $X_t = g_t(I_{t-1})$ avec probabilité 1, $\forall t \in T_1$.
	162	+\end{itemize}
	163	+\item Un processus déterministe peut être parfaitement prévu à partir de son propre passé aux points où le processus est déterministe $(t \in T_1)$.
	164	+\item Il n’y a pas d’aléa associé à ce processus.
	165	+\end{itemize}
	166	+\end{frame}
	167	+
	168	+\section{Les séries stationnaires}
	169	+\frame{\tableofcontents[current]}
	170	+
	171	+\begin{frame}{Les séries stationnaires}
	172	+\begin{itemize}
	173	+\item Le processus $\left\{X_t : t \in T \right\}$ est stationnaire au second ordre (faiblement stationnaire, ou stationnaire au sens large) si les trois conditions suivantes sont vérifiées:
	174	+\begin{enumerate}
	175	+\item $\forall t \in T, E(X_t^2)< \infty$
	176	+\begin{itemize}
	177	+\item Assure que les moments d’ordre deux existent, car un processus stationnaire au sens large doit avoir une variance finie.
	178	+\end{itemize}
	179	+\item $\forall t \in T, E(X_t)=\mu$ et indépendant de t
	180	+\begin{itemize}
	181	+\item L’espérance des variables aléatoires $X_t$ est la même quelques soit la période $t$.
	182	+\end{itemize}
	183	+\item $\forall t,s \in T \times T, \forall h \geq 0, COV(X_s,X_{s+h})=\gamma(h)$ est indépendant de t.
	184	+\begin{itemize}
	185	+\item Assure que la variance est constante dans le temps.
	186	+\end{itemize}
	187	+\end{enumerate}
	188	+\end{itemize}
	189	+\end{frame}
	190	+
	191	+\section{Les processus non stationnaires}
	192	+\frame{\tableofcontents[current]}
	193	+
	194	+
	195	+\begin{frame}{Les processus non stationnaires}
	196	+\begin{itemize}
	197	+\item Un processus non stationnaire est un processus qui n’est pas faiblement stationnaire; ou encore c’est un processus pour lequel au moins une des trois conditions qui caractérisent la stationnarité faible est violée.
	198	+\item La marche aléatoire est un exemple important à la finance d’un processus non stationnaire.
	199	+\begin{align*}
	200	+X_t=\beta+X_{t-1}+\epsilon_t
	201	+\end{align*}
	202	+Sachant les propriétés suivantes valables $\forall t$
	203	+\begin{itemize}
	204	+\item $E(\epsilon_t)=0$
	205	+\item $V(\epsilon_t)=\sigma_{\epsilon}^2$
	206	+\item $E(\epsilon_t,\epsilon_s)=0$
	207	+\end{itemize}
	208	+\end{itemize}
	209	+\end{frame}
	210	+
	211	+\begin{frame}{Les processus non stationnaires}
	212	+\begin{block}{Considérez la série en commençant à $t=0$}
	213	+\begin{itemize}
	214	+\item On peut exprimer $X_1$ en fonction de $X_0$ dans le cas de la marche aléatoire.
	215	+\begin{align*}
	216	+X_1= \beta+ X_0+\epsilon_{1}
	217	+\end{align*}
	218	+\item On peut également exprimer $X_2$ en fonction de $X_1$.
	219	+\begin{align*}
	220	+X_2= \beta+ X_1+\epsilon_{2}
	221	+\end{align*}
	222	+\end{itemize}
	223	+\end{block}
	224	+\end{frame}
	225	+
	226	+\begin{frame}{Les processus non stationnaires}
	227	+\begin{block}{Considérez la série en commençant à $t=0$}
	228	+\begin{itemize}
	229	+\item On peut voir facilement que nous pouvons incorporer la première équation donnant la solution de $X_1$ dans la deuxième équation donnant la solution de $X_2$.
	230	+\begin{align*}
	231	+X_2 & = \beta+ X_1+\epsilon_{2} \\
	232	+& = \beta+ [\beta+ X_0+\epsilon_{1}]+\epsilon_{2} \\
	233	+& = \beta + \beta +X_0 + \epsilon_1 +\epsilon_2 \\
	234	+& = 2 \beta + X_0 + \epsilon_1 +\epsilon_2
	235	+\end{align*}
	236	+\end{itemize}
	237	+
	238	+\end{block}
	239	+\end{frame}
	240	+
	241	+\begin{frame}{Les processus non stationnaires}
	242	+\begin{block}{Considérez la série en commençant à $t=0$}
	243	+\begin{itemize}
	244	+
	245	+\item Supposons que nous continuons cette petite procédure, mais cette fois nous posons uniquement le processus $X_t$, dans lequel nous allons incorporer $X_{t-1}$ et ensuite $X_{t-2}$, jusqu'à arriver à $X_0$.
	246	+\begin{align*}
	247	+X_t & =\beta + X_{t-1}+\epsilon_t \\
	248	+& = \beta + [\beta + X_{t-2}+\epsilon_{t-1} ] +\epsilon_t \\
	249	+& = 2 \beta + X_{t-2} +\epsilon_{t-1} +\epsilon_t \\
	250	+& = 2 \beta + [\beta + X_{t-3}+\epsilon_{t-2} ] +\epsilon_{t-1} +\epsilon_t \\
	251	+& = 3 \beta + X_{t-3}+\epsilon_{t-2} + \epsilon_{t-1} + \epsilon_t \\
	252	+& = \vdots \hspace{1cm} \vdots \hspace{2cm} \vdots \\
	253	+& = T \beta + X_0 + (\epsilon_0+ \epsilon_1 + \epsilon_2+\cdots +\epsilon_{t-3}+\epsilon_{t-2}+\epsilon_{t-1}+\epsilon_{t})
	254	+\end{align*}
	255	+\end{itemize}
	256	+\end{block}
	257	+\end{frame}
	258	+
	259	+\begin{frame}{Les processus non stationnaires}
	260	+\begin{block}{Considérez la série en commençant à $t=0$}
	261	+\begin{itemize}
	262	+\item Étudions le terme d’erreur
	263	+\begin{align*}
	264	+v_t=\epsilon_1+\epsilon_2+\cdots+\epsilon_t
	265	+\end{align*}
	266	+
	267	+\begin{align*}
	268	+E(v_t) & =E(\epsilon_1)+E(\epsilon_2)+\cdots+E(\epsilon_t) \\
	269	+& =0
	270	+\end{align*}
	271	+
	272	+\begin{align*}
	273	+V(v_t) & =V(\epsilon_1)+V(\epsilon_2)+\cdots+V(\epsilon_t)\\
	274	+& =\sigma_{\epsilon}^2+\sigma_{\epsilon}^2+\cdots+\sigma_{\epsilon}^2\\
	275	+& =T\sigma_{\epsilon}^2
	276	+\end{align*}
	277	+
	278	+\end{itemize}
	279	+\end{block}
	280	+\end{frame}
	281	+
	282	+\begin{frame}{{Les processus non stationnaires}}
	283	+\begin{itemize}
	284	+\item On voit clairement que la variance du processus $V(v_t)$ explose dans le temps étant donné le résultats:
	285	+\begin{align*}
	286	+V(v_t)=T\sigma_{\epsilon}^2
	287	+\end{align*}
	288	+\item Ce que l’on voit dans la variance de ce processus est que plus T sera grand alors plus la variance sera grande. Donc, plus le temps passe et plus nous aurons une variance élevée.
	289	+\item Noter que la série est autorégressive, mais le coefficient de $X_{t-1}$ est égal à $1$.
	290	+\item On parle alors de racine unitaire, de marche aléatoire et d’une série intégrée d’ordre $1, I(1)$.
	291	+
	292	+\end{itemize}
	293	+\end{frame}
	294	+
	295	+\begin{frame}{{Les processus non stationnaires}}
	296	+On considère à nouveau le processus non-stationnaires suivant:
	297	+\begin{align*}
	298	+X_t=\beta+X_{t-1}+\epsilon_t
	299	+\end{align*}
	300	+\begin{itemize}
	301	+\item Si $\beta \neq 0$, ce processus est dit \textbf{marche aléatoire avec intercepte} (random walk with drift)
	302	+\item Lorsque $\beta = 0$, on parle simplement de \textbf{marche aléatoire}. C’est un processus intégré d’ordre 1 également.
	303	+\end{itemize}
	304	+\end{frame}
	305	+
	306	+\begin{frame}{{Les processus non stationnaires}}
	307	+\begin{block}{Racine unitaire et (intégré d’ordre p)}
	308	+\begin{itemize}
	309	+\item Lorsqu’un processus $\left\{Y_t, t \in T \right\}$ peut être représenté sous la forme d’une marche aléatoire avec ou sans tendance, On dit qu’il possède une racine unitaire ou qu’il est intégré d’ordre 1, et noté $Y_t \sim I(1)$.
	310	+\item Si on doit différencier deux fois un processus $\left\{Y_t, t \in T \right\}$ avant d’obtenir un processus stationnaire, on dit que ce processus est intégré d’ordre 2 et on note $Y_t \sim I(2)$.
	311	+\item Plus généralement, un processus $\left\{Y_t, t \in T \right\}$ est intégré d’ordre p s’il faut le différencier p successivement avant d’obtenir un processus stationnaire.
	312	+\end{itemize}
	313	+\end{block}
	314	+\end{frame}
	315	+
	316	+
	317	+\section{Régression fallacieuse}
	318	+\frame{\tableofcontents[current]}
	319	+
	320	+
	321	+\begin{frame}{Régression fallacieuse}
	322	+Nous allons poser deux séries non stationnaires:
	323	+\begin{align*}
	324	+X_t=X_{t-1}+v_t
	325	+\end{align*}
	326	+\begin{align*}
	327	+Y_t=Y_{t-1}+u_t
	328	+\end{align*}
	329	+\begin{itemize}
	330	+\item Nous savons que ces deux séries possèdent une racine unitaire et sont donc non-stationnaire.
	331	+\item Le fait d’inclure dans une régression les variables $X_t$ et $Y_t$ engendrera certainement une régression fallacieuse.
	332	+\item Sans ce préoccuper la stationnarité, nous serions tenté de conclure que X à un pouvoir explicatif sur Y alors que nous avons généré ceux-ci pour n’en avoir aucun.
	333	+\begin{block}{Régression Fallacieuse}
	334	+\begin{align*}
	335	+Y_t = \beta_1 +\beta_2 X_t + \epsilon_t
	336	+\end{align*}
	337	+\end{block}
	338	+\item
	339	+
	340	+\end{itemize}
	341	+\end{frame}
	342	+
	343	+\section{Opérateur de retard}
	344	+\frame{\tableofcontents[current]}
	345	+
	346	+
	347	+\begin{frame}{Opérateur de retard}
	348	+\begin{itemize}
	349	+\item Pour toute série chronologique $\left\{Y_t, t = -\infty, ... + \infty \right\}$, l’opérateur retard, noté $L$, est un opérateur qui transforme toute observation en la valeur qui la précède :
	350	+\begin{align*}
	351	+L(Y_t )= Y_{t-1}, \hspace{0.2cm} \forall t
	352	+\end{align*}
	353	+\item L'utilisation du $L$ sans aucun autre indicatif implique que nous sommes en présence d'un retard, alors que l'utilisation de l'opérateur retard avec un indicatif 2, 3 et p représente respectivement 2, 3 et p retard.
	354	+\begin{align*}
	355	+L^{2}(Y_t)=L(L(Y_t))=L(Y_{t-1})=Y_{t-2}
	356	+\end{align*}
	357	+\begin{align*}
	358	+L^{p}(Y_t)=Y_{t-p}, \hspace{0.2cm} p>1
	359	+\end{align*}
	360	+\item Il s’agit d’un opérateur qui représente une notation en série chronologique. Il a des propriétés comme les autres opérateurs.
	361	+\end{itemize}
	362	+\end{frame}
	363	+
	364	+\section{Introduction au modèle AR(1)}
	365	+\frame{\tableofcontents[current]}
	366	+
	367	+\begin{frame}{Introduction au modèle AR(1)}
	368	+\textbf{Soit le processus:}
	369	+\begin{align*}
	370	+X_t=\alpha X_{t-1}+\epsilon_t
	371	+\end{align*}
	372	+Où le terme d'erreur possède les propriétés suivantes:
	373	+\begin{itemize}
	374	+\item $E(\epsilon_t )=0$
	375	+\item $V(\epsilon_t )=\sigma_{\epsilon}^2$
	376	+\item $E(\epsilon_t,\epsilon_s)=0$
	377	+\begin{itemize}
	378	+\item La série est stationnaire
	379	+\item $\mid \alpha \mid < 1$
	380	+\end{itemize}
	381	+\end{itemize}
	382	+\end{frame}
	383	+
	384	+\begin{frame}{Introduction au modèle AR(1)}
	385	+\textbf{Soit le processus:}
	386	+\begin{align*}
	387	+X_t=\alpha X_{t-1}+\epsilon_t
	388	+\end{align*}
	389	+On pose également, les propriétés suivantes pour notre variable $X_t$:
	390	+\begin{itemize}
	391	+\item $E(X_t )=0$
	392	+\item $V(X_t )= \sigma_X^2$
	393	+\item $E(X_t,X_{t-j})=E(X_s,X_{s-j})$
	394	+\end{itemize}
	395	+\end{frame}
	396	+
	397	+
	398	+\begin{frame}{Introduction au modèle AR(1)}
	399	+\begin{itemize}
	400	+\item Nous allons utiliser une méthode récursive afin de trouver la variance de notre série, soit $V(X_t)=\sigma_{X}^2$.
	401	+\item On sait que la valeur au temps $t$ de notre processus est $X_t$.
	402	+\item Dans le cas du AR(1), on cherche a expliquer les variations de $X_t$ en fonction de la même variable mais avec un retard $X_{t-1}$ et des choc $\epsilon_t$.
	403	+\item Écrivons la série des $X_t$ en fonction des chocs $\epsilon_t$ présents et passés par récurrence (Prochaine diapo)
	404	+\end{itemize}
	405	+\end{frame}
	406	+
	407	+\begin{frame}{Introduction au modèle AR(1)}
	408	+\begin{align*}
	409	+X_t & = \alpha(X_{t-1} )+ \epsilon_t\\
	410	+& = X_t= \alpha (\alpha X_{t-2}+\epsilon_{t-1})+ \epsilon_t \\
	411	+& = \alpha^2 X_{t-2}+ \alpha \epsilon_{t-1}+ \epsilon_t \\
	412	+& = \alpha^2 (\alpha X_{t-3}+\epsilon_{t-2})+\alpha \epsilon_{t-1}+\epsilon_t \\
	413	+& = \alpha^3 X_{t-3}+\alpha^2 \epsilon_{t-2}+\alpha \epsilon_{t-1}+\epsilon_t \\
	414	+& = \alpha^s X_{t-s}+\alpha^{s-1}\epsilon_{t-(s-1)}+\cdots+\alpha^2 \epsilon_{t-2}+\alpha \epsilon_{t-1}+\epsilon_t
	415	+\end{align*}
	416	+\end{frame}
	417	+
	418	+\begin{frame}{Introduction au modèle AR(1)}
	419	+
	420	+\begin{itemize}
	421	+\item \textbf{Considérons les possibilités suivantes:}
	422	+\begin{itemize}
	423	+\item $X_0 = 0$. La valeur initiale du processus $X_t=0$.
	424	+\item Le processus des $X_t$ a commencé au passé infini.
	425	+\end{itemize}
	426	+\item \textbf{Dans les deux cas, je recule assez dans le temps:}
	427	+\begin{align*}
	428	+\alpha^sX_{t-s}=0
	429	+\end{align*}
	430	+\begin{itemize}
	431	+\item $X_{t-s}=0$, car $t-s=0$
	432	+\item $\lim_{\infty}\alpha^s=0$
	433	+\end{itemize}
	434	+\item On peut écrire le processus $X_t$ uniquement avec les chocs $\epsilon$
	435	+\begin{align*}
	436	+X_t \approx \epsilon_t+\alpha \epsilon_{t-1}+\alpha^2\epsilon_{t-2}+\alpha^3\epsilon_{t-3}+\cdots
	437	+\end{align*}
	438	+\end{itemize}
	439	+
	440	+\end{frame}
	441	+
	442	+\begin{frame}{Introduction au modèle AR(1)}
	443	+\begin{block}{Calculons la variance de $X_t$}
	444	+\begin{align*}
	445	+\sigma_X^2 & =V(\epsilon_t)+\alpha^2 V(\epsilon_{t-1})+\alpha^4V(\epsilon_{t-2})+\alpha^6V(\epsilon_{t-3})+\cdots\\
	446	+& = \sigma_{\epsilon}^2+\alpha^2 \sigma_{\epsilon}^2+\alpha^4 \sigma_{\epsilon}^2+\alpha^6 \sigma_{\epsilon}^2+\cdots\\
	447	+& =\sigma_{\epsilon}^2(1+\alpha^2+\alpha^4+\alpha^6+\cdots)\\
	448	+&= \sigma_{\epsilon}^2 \left( \frac{1}{1-\alpha^2} \right)
	449	+\end{align*}
	450	+\end{block}
	451	+\end{frame}
	452	+
	453	+\begin{frame}{Introduction au modèle AR(1)}
	454	+\begin{block}{Calculons $E(X_t X_{t-1})$}
	455	+\begin{align*}
	456	+X_t \approx \epsilon_t+\alpha \epsilon_{t-1}+\alpha^2 \epsilon_{t-2}+\alpha^3 \epsilon_{t-3}+\cdots
	457	+\end{align*}
	458	+\begin{align*}
	459	+X_{t-1} \approx \epsilon_{t-1}+\alpha \epsilon_{t-2}+\alpha^2 \epsilon_{t-3}+\alpha^3 \epsilon_{t-4}+\cdots
	460	+\end{align*}
	461	+\begin{align*}
	462	+E(X_t X_{t-1}) & =\alpha E(\epsilon_{t-1}^2)+\alpha^3E(\epsilon_{t-2}^2)+\alpha^5E(\epsilon_{t-3}^2)+\cdots \\ & = \alpha \sigma_{\epsilon}^2 +\alpha^3 \sigma_{\epsilon}^2 + \alpha^5 \sigma_{\epsilon}^2 \\ & = \alpha\sigma_{\epsilon}^2 (1+\alpha^2+\alpha^4+\cdots) \\ & = \alpha\sigma_{\epsilon}^2 \left( \frac{1}{1-\alpha^2} \right) \\ & = \alpha \sigma_X^2
	463	+\end{align*}
	464	+\end{block}
	465	+\end{frame}
	466	+
	467	+\begin{frame}{Introduction au modèle AR(1)}
	468	+\textbf{On peut montrer de la même façon le cas plus général:}
	469	+\begin{align*}
	470	+E(X_t,X_{t-s})=\alpha^s\sigma_{\epsilon}^2 \left( \frac{1}{1-\alpha^2} \right)=\alpha^s \sigma_X^2
	471	+\end{align*}
	472	+\textbf{Nous avons calculé:}
	473	+\begin{align*}
	474	+E(X_{t-s}^2)=\left( \frac{\sigma_{\epsilon}^2}{1-\alpha^2} \right)=\sigma_X^2 \hspace{0.5cm} \forall \hspace{0.2cm} l'indice
	475	+\end{align*}
	476	+\begin{itemize}
	477	+\item Maintenant que nous avons un exemple de processus stochastique en série chronologique en tête, revenons au concept général.
	478	+\item Nous voulons maintenant définir et caractériser la dépendance et l’évolution dans le temps du processus stochastique.
	479	+\item Pour ce faire, nous définirons certains concepts pour un processus faiblement stationnaire $\left\{ X_t : t \in T \right\}$.
	480	+\end{itemize}
	481	+\end{frame}
	482	+
	483	+
	484	+\section{La fonction d’autocovariance $\gamma$}
	485	+\frame{\tableofcontents[current]}
	486	+
	487	+\begin{frame}{La fonction d’autocovariance $\gamma$}
	488	+\begin{align*}
	489	+\gamma: \mathbb{R} \rightarrow \mathbb{R}
	490	+\end{align*}
	491	+\begin{align*}
	492	+h \rightarrow \gamma (h)=Cov(X_t,X_{t+h}), \forall t \in T.
	493	+\end{align*}
	494	+\begin{itemize}
	495	+\item La fonction d’autocovariance ne dépend que de h et non de t. les observations entre elles dépendent de l’écart qui les séparent et non de leur position dans la série chronologique.
	496	+\item L’autocovariance est donc la variance avec soit même à une même période où $h=0$. Lorsque $h \neq 0$, alors on parle d’autocovariance entre deux périodes.
	497	+\end{itemize}
	498	+\end{frame}
	499	+
	500	+\section{La fonction d’autocorrélation $\rho$}
	501	+\frame{\tableofcontents[current]}
	502	+
	503	+\begin{frame}{La fonction d’autocorrélation $\rho$}
	504	+\begin{align*}
	505	+\gamma: \mathbb{R} \rightarrow \mathbb{R}
	506	+\end{align*}
	507	+\begin{align*}
	508	+h \rightarrow \rho (h) = \frac{Cov(X_t,X_{t+h})}{[Var(X_t) \times Var(X_{t+h})]^{\frac{1}{2}}}=\frac{\gamma (h)}{\gamma (0)}, \forall t \in T
	509	+\end{align*}
	510	+\begin{itemize}
	511	+\item La fonction d’autocorrélation représente donc la corrélation entre deux observations de la série chronologique.
	512	+\item Ces deux fonctions ne dépendent que de $h$ qui existe entre les deux dates auxquelles correspondent les variables aléatoires $X_t$ et $X_{t+h}$ étudiées.
	513	+\end{itemize}
	514	+\end{frame}
	515	+
	516	+
	517	+
	518	+\begin{frame}{La fonction d’autocorrélation $\rho$}
	519	+\begin{align*}
	520	+\forall h, \gamma (h)= \gamma (-h)
	521	+\end{align*}
	522	+\begin{align*}
	523	+\rho (h)= \rho (-h)
	524	+\end{align*}
	525	+\begin{itemize}
	526	+\item Il est clair dans ce cas que pour $h = 0$, la fonction d’autocorrélation est égale à 1.
	527	+\item La fonction d’autocorrélation possède une représentation graphique appelée un corrélogramme ou ACF (autocorrélation fonction).
	528	+\item En pratique, nous aurons seulement une réalisation du processus d’intérêt donc nous regarderons le corrélogramme de l’échantillon (SACF).
	529	+\end{itemize}
	530	+\end{frame}
	531	+
	532	+\begin{frame}{La fonction d’autocorrélation partielle}
	533	+\begin{itemize}
	534	+\item Pour tout processus $\left\{ X_t : t \in T \right\}$, possiblement non stationnaire, on définit la fonction d’autocorrélation partielle (partial autocorrelation function, PACF).
	535	+\item L’autocorrélation entre $X_t$ et $X_{t_h}$ prend en compte les corrélations indirectes éventuelles entre $X_t$ et $X_{t-1}$, $X_{t-1}$ et $X_{t-2}$, $\cdots$ , et entre $X_{t-h+1}$ et $X_{t-h}$.
	536	+\item Par contre, l’autocorrélation partielle entre $X_t$ et $X_{t-h}$ élimine les effets de $X_t$ et $X_{t-1}$, $X_{t-1}$ et $X_{t-2}$,$\cdots$ , $X_{t-h+1}$.
	537	+\end{itemize}
	538	+\end{frame}
	539	+
	540	+\begin{frame}{Processus autorégressif d’ordre 1}
	541	+\begin{align*}
	542	+X_t = \alpha_0 +\alpha_1 X_{t-1}+\epsilon_t, \hspace{0.5cm} \epsilon_t \sim
	543	+ i.i.d
	544	+\end{align*}
	545	+\begin{itemize}
	546	+\item L’autocorrélation partielle de $X_{t}$ et $X_{t-2}$ est l’effet de $X_{t-2}$ sur $X_t$ lorsqu’on a déjà contrôlé l’effet de $X_{t-1}$
	547	+\item En conséquence les autocorrélations partielles d’ordre supérieur ou égal à 2 sont nulles, même si $X_t$ et $X_{t-1}$ sont corrélés et que ce dernier est corrélé avec $X_{t-2}$
	548	+\end{itemize}
	549	+\end{frame}
	550	+
	551	+\begin{frame}{Autocorrélations partielles}
	552	+\textbf{On calcule les autocorrélations partielles d’un processus $\left\{ X_t : t \in T \right\}$ de la façon}
	553	+
	554	+\vspace{0.2cm}
	555	+
	556	+\textbf{1.} Centrer la série : calculer la moyenne $\mu$ de la série et obtenir la série centrée constituée des $X_t^*=X_t-\mu$.
	557	+
	558	+\vspace{0.2cm}
	559	+
	560	+\textbf{2.} Former l’équation autorégressive d’ordre 1:
	561	+\begin{align*}
	562	+X_t^=\phi_{11} X_{t-1}^+e_t
	563	+\end{align*}
	564	+$\phi_{11}$ est l’autocorrélation partielle entre $X_t$ et $X_{t-1}$.
	565	+
	566	+\vspace{0.2cm}
	567	+
	568	+\textbf{3.} Former l’équation autorégressive d’ordre 2:
	569	+\begin{align*}
	570	+X_t^=\phi_{21} X_{t-1}^+\phi_{22} X_{t-2}^*+e_t
	571	+\end{align*}
	572	+$\phi_{22}$ est l’autocorrélation partielle entre $X_t$ et $X_{t-2}$ ou encore la corrélation entre $X_t$ et $X_{t-2}$ lorsque l'effet de $X_{t-1}$ est éliminé.
	573	+
	574	+\end{frame}
	575	+
	576	+
	577	+\begin{frame}{Autocorrélations partielles}
	578	+\textbf{On calcule les autocorrélations partielles d’un processus $\left\{ X_t : t \in T \right\}$ de la façon}
	579	+
	580	+\vspace{0.2cm}
	581	+
	582	+\textbf{4.} Répéter successivement ce procédé pour calculer l’autocorrélation partielle entre $X_t$ et $X_{t-h}$, $h \geq 2$
	583	+
	584	+
	585	+\begin{itemize}
	586	+\item Avec cette méthode, on obtient les résultats de la fonction d’autocorrélation partielle du processus $\left\{ X_t : t \in T \right\}$.
	587	+\item Lorsqu’on utilise un échantillon de données pour faire les calculs, on obtient les autocorrélations partielles d’échantillon.
	588	+\end{itemize}
	589	+
	590	+\end{frame}
	591	+
	592	+\section{Équations de Yule-Walker}
	593	+\frame{\tableofcontents[current]}
	594	+
	595	+\begin{frame}{Équations de Yule-Walker}
	596	+\begin{itemize}
	597	+\item Pour un processus stationnaire, les équations de Yule-Walker permettent d’obtenir les autocorrélations partielles, $\phi_{hh}$, $h=1,2$, $\cdots$ à partir des autocorrélations $\rho(j)$, $j=1,2$, $\cdots,h.$
	598	+\end{itemize}
	599	+\begin{align*}
	600	+\phi_{11}=\rho(1)
	601	+\end{align*}
	602	+\begin{align*}
	603	+\phi_{22}=\frac{\rho(2)-\rho(1)^2}{1-\rho(1)^2}
	604	+\end{align*}
	605	+\begin{align*}
	606	+\phi_{ss}=\frac{\rho(s)-\sum_{j=1}^{s-1}\phi_{s-1,j}\rho(s-j)}{1-\sum_{j=1}^{s-1}\phi_{s-1,j}\rho(j)}
	607	+\end{align*}
	608	+Où $\phi_{sj}=\phi_{s-1,j}-\phi_{ss} \phi_{s-1,s-j}, j=1,2,3,\cdots, s-1$
	609	+\end{frame}
	610	+
	611	+
	612	+\begin{frame}{Équations de Yule-Walker}
	613	+\begin{itemize}
	614	+\item Sachant que les trois définitions présentées ci-haut prennent des formes particulières propres à des processus d’intérêt, ces estimations seront très utiles pour déterminer le processus spécifique le plus probable suivi par un échantillon de données.
	615	+\item De plus, ils peuvent nous documenter intuitivement sur la stationnarité d’un processus. Comment?
	616	+\item Intuitivement, nous savons que les bruits blancs sont stationnaires par construction.
	617	+\item Donc, si notre processus à des caractéristiques proches à un bruit blanc en terme d’autocorrélation et d’autocorrélation partielle, nous pourrons en déduire qu’il est probablement stationnaire.
	618	+\item Ce n’est pas un test formel toutefois et cela contient une part d’arbitraire .
	619	+\end{itemize}
	620	+\end{frame}
	621	+
	622	+
	623	+
	624	+\begin{frame}{Équations de Yule-Walker}
	625	+\begin{itemize}
	626	+\item Comment savoir si cette série est vraiment non stationnaire avec le visuel? Il faudra faire un test formel.
	627	+\item Il y a aussi la question de savoir combien de retard sur l’autocorrélation il convient de calculer pour diagnostiquer un processus.
	628	+\item Une règle du pouce serait de calculer les autocorrélations pour environ un quart du nombre total d’observations.
	629	+\item L’approche la plus souvent adoptée dans ce cas sera de regarder les critères d’information d’Akaike et de Schwarz (définis plus bas).
	630	+\item Il est aussi possible de faire des tests statistiques de significativité sur les autocorrélations.
	631	+\end{itemize}
	632	+\end{frame}
	633	+
	634	+
	635	+
	636	+
	637	+\end{document}

added S\303\251ance 10/GSF6053_S10.toc +13 −0

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	1	+\beamer@sectionintoc {1}{Série chronologique}{4}{0}{1}
	2	+\beamer@sectionintoc {2}{Première différence}{9}{0}{2}
	3	+\beamer@sectionintoc {3}{Saisonnalité et Désaisonalisation}{11}{0}{3}
	4	+\beamer@sectionintoc {4}{Processus stochastique}{13}{0}{4}
	5	+\beamer@sectionintoc {5}{Processus déterministe}{15}{0}{5}
	6	+\beamer@sectionintoc {6}{Les séries stationnaires}{17}{0}{6}
	7	+\beamer@sectionintoc {7}{Les processus non stationnaires}{19}{0}{7}
	8	+\beamer@sectionintoc {8}{Régression fallacieuse}{28}{0}{8}
	9	+\beamer@sectionintoc {9}{Opérateur de retard}{30}{0}{9}
	10	+\beamer@sectionintoc {10}{Introduction au modèle AR(1)}{32}{0}{10}
	11	+\beamer@sectionintoc {11}{La fonction d’autocovariance $\gamma $}{41}{0}{11}
	12	+\beamer@sectionintoc {12}{La fonction d’autocorrélation $\rho $}{43}{0}{12}
	13	+\beamer@sectionintoc {13}{Équations de Yule-Walker}{50}{0}{13}

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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage{graphicx}
	9	+\usetheme{default}
	10	+\usecolortheme{default}
	11	+
	12	+\title[S02 Régression et MCO]{Section 05 : Les séries chronologiques\\ (Séance 11)}
	13	+\subtitle{GSF-6053: Économétrie Financière}
	14	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	15	+\institute[Université Laval]
	16	+{
	17	+ \inst{1}%
	18	+ Département de finance, assurance et immobilier\\
	19	+ Faculté des sciences de l'administration\\
	20	+ Université Laval}
	21	+\date[Hiver 2022]{5 avril 2022}
	22	+
	23	+\begin{document}
	24	+
	25	+\begin{frame}
	26	+ \titlepage
	27	+\end{frame}
	28	+
	29	+
	30	+\begin{frame}{Références}
	31	+\textbf{Obligatoires:}
	32	+\begin{itemize}
	33	+\item \textbf{Notes de cours:} Section 5 (Professeure: Marie-Hélène Gagnon)
	34	+\item \textbf{Woolridge:} chapitres 11, 12 et 19.
	35	+\end{itemize}
	36	+\vspace{0.5cm}
	37	+\textbf{Complémentaires:}
	38	+\begin{itemize}
	39	+\item \textbf{Gujarati et Porter:} chapitres 21 et 22
	40	+\end{itemize}
	41	+\end{frame}
	42	+
	43	+\begin{frame}{Plan de la séance}
	44	+ \tableofcontents
	45	+\end{frame}
	46	+
	47	+\section{Estimation des moments de processus stationnaire}
	48	+\frame{\tableofcontents[current]}
	49	+
	50	+\begin{frame}{Estimation des moments de processus stationnaire}
	51	+\begin{itemize}
	52	+\item Si le processus est stationnaire, on sait que ces T variables aléatoires ont la même moyenne, la même variance, et que leur autocovariance ne dépend que du retard qui sépare les dates auxquelles elles correspondent.
	53	+\item On peut estimer ces paramètres du processus à partir de ces T observations
	54	+\end{itemize}
	55	+\textbf{Estimation de la moyenne:}
	56	+\begin{align*}
	57	+\hat{\mu}=\frac{1}{T}\sum_{t=1}^T X_t
	58	+\end{align*}
	59	+\textbf{L’autocovariance de délai h:}
	60	+\begin{align*}
	61	+\hat{\gamma}(h)=\frac{1}{T-h}\sum_{t=1}^{t-h}(X_t-\hat{\mu})(X_{t+h}-\hat{\mu})
	62	+\end{align*}
	63	+\end{frame}
	64	+
	65	+
	66	+\begin{frame}{Estimation des moments de processus stationnaire}
	67	+
	68	+\textbf{Pour $h=0$, on aura un estimateur de la variance du processus :}
	69	+\begin{align*}
	70	+\hat{\gamma}(0)=\frac{1}{T}\sum_{t=1}^{T}(X_t-\hat{\mu})(X_t-\hat{\mu})
	71	+\end{align*}
	72	+\textbf{Le coefficient d’autocorrélation de retard h est estimé:}
	73	+\begin{align*}
	74	+\hat{\rho}(h)=\frac{\hat{\gamma}(h)}{\hat{\gamma}(0)}
	75	+\end{align*}
	76	+\end{frame}
	77	+
	78	+
	79	+\section{Tests de bruits blancs}
	80	+\frame{\tableofcontents[current]}
	81	+
	82	+\begin{frame}{Tests de bruits blancs}
	83	+\begin{block}{Cas particulier: Le bruit blanc}
	84	+\begin{itemize}
	85	+\item Si le processus est un bruit blanc, les autocovariances et autocorrélations de délai $h > 0$ sont nulles. On peut alors montrer que
	86	+\begin{align*}
	87	+\hat{\rho}(h) \rightarrow N \left(0,\frac{1}{T}\right)
	88	+\end{align*}
	89	+\begin{align*}
	90	+\frac{\hat{\rho}(h)}{\sqrt{\frac{1}{T}}} \rightarrow N(0,1) \hspace{1cm} \textbf{stat 1}
	91	+\end{align*}
	92	+\end{itemize}
	93	+\end{block}
	94	+\end{frame}
	95	+
	96	+\begin{frame}{Tests de bruits blancs}
	97	+\begin{block}{Cas particulier: Le bruit blanc}
	98	+\begin{itemize}
	99	+\item Ou que
	100	+\begin{align*}
	101	+\hat{\rho}(h) \rightarrow N \left(0,\frac{T-h}{T(T+2)}\right)
	102	+\end{align*}
	103	+\begin{align*}
	104	+\frac{\hat{\rho}(h)}{\sqrt{\frac{T-h}{T(T+2)}}} \rightarrow N(0,1) \hspace{1cm} \textbf{stat 2}
	105	+\end{align*}
	106	+\end{itemize}
	107	+\end{block}
	108	+\end{frame}
	109	+
	110	+\begin{frame}{Tests de bruits blancs}
	111	+\begin{block}{Cas particulier: Le bruit blanc}
	112	+\begin{itemize}
	113	+\item On peut utiliser ces résultats distributionnels pour tester empiriquement la nullité de l’autocorrélation d’ordre h.
	114	+\begin{align*}
	115	+H_0: \rho (h) = 0
	116	+\end{align*}
	117	+\begin{align*}
	118	+H_A: \rho(h)\neq 0
	119	+\end{align*}
	120	+\item On rejette l’hypothèse nulle si la statistique choisie (1 ou 2) a une valeur plus élevée que le point critique d’une loi normale centrée réduite.
	121	+\end{itemize}
	122	+\end{block}
	123	+\end{frame}
	124	+
	125	+\begin{frame}{Tests Portmanteau}
	126	+\begin{itemize}
	127	+\item Une façon plus pratique de tester les autocorrélations est un test Portmanteau.
	128	+\item Un test Portmanteau teste conjointement la nullité des autocorrélations de délai $1, 2, 3,\cdots , J$. (on pense qu’après $J$ retard, le processus s’apparente à un bruit blanc avec des autocorrélations nulles)
	129	+\end{itemize}
	130	+\begin{align*}
	131	+H_0: \rho(1) = \rho(2) = \cdots =\rho(h)=0
	132	+\end{align*}
	133	+\begin{align*}
	134	+H_A: \exists h: \rho(h) \neq 0, h \in \left\{ 1,2,3,\cdots,J \right\}
	135	+\end{align*}
	136	+
	137	+\begin{itemize}
	138	+\item Les deux tests présentés ci-bas permettent de détecter si le processus est autocorrélé à un ordre $h \leq J$.
	139	+\item Si l’on pense qu’il ne peut y avoir d’autocorrélation d’ordre supérieur à $J$ et si le test d’ordre $J$ ne rejette pas, alors on peut dire que le processus est un bruit blanc.
	140	+\end{itemize}
	141	+\end{frame}
	142	+
	143	+\begin{frame}{Test de Box Pierce}
	144	+\begin{itemize}
	145	+\item On utilise la statistique 1 comme résultat distributionnel, on obtient la statistique de test conjoint suivant :
	146	+\begin{align*}
	147	+Q_1=\sum_{h=1}^J \left(\frac{\hat{\rho}(h)}{\sqrt{\frac{1}{T}}}\right)^2=T\sum_{h=1}^J(\hat{\rho}(h))^2
	148	+\end{align*}
	149	+\item Le test rejette pour de grandes valeurs de la statistique $Q1$. On rejette $H_0$ si la statistique $Q1$ calculée est plus grande que le point critique donné par une table du chi-deux à $J$ degrés de liberté.
	150	+\end{itemize}
	151	+\end{frame}
	152	+
	153	+\begin{frame}{Test de Ljung-Box}
	154	+\begin{itemize}
	155	+\item Le test de Ljung-Box utilise plutôt le résultat distributionnel associé à la statistique 2.
	156	+\begin{align*}
	157	+Q_2=\sum_{h=1}^J \left(\frac{\hat{\rho}(h)}{\sqrt{\frac{T-h}{T(T+2)}}}\right)^2=T\sum_{h=1}^J \frac{(\hat{\rho}(h))^2}{T-h}
	158	+\end{align*}
	159	+\item Ce test se compare aussi au point critique d’une khi-carré avec $J$ degrés de liberté
	160	+\item En général, le Ljung-Box est plus utilisé, car on lui reconnait de meilleures propriétés en petit échantillon que le Box-Pierce.
	161	+\end{itemize}
	162	+\end{frame}
	163	+
	164	+
	165	+\section{Tests de racine unitaire}
	166	+\frame{\tableofcontents[current]}
	167	+
	168	+\begin{frame}{Tests de racine unitaire}
	169	+\begin{itemize}
	170	+\item Pour connaitre le degré d’intégration d’une variable, il faut se baser sur des procédures formelles de test statistique.
	171	+\item Les tests de racine unitaire sont considérés comme des tests de stationnarité.
	172	+\item Soit $\left\{ Y_t, t \in R \right\}$ un processus stochastique. Nous voulons savoir si ce processus possède une racine unitaire.
	173	+\end{itemize}
	174	+\end{frame}
	175	+
	176	+\begin{frame}{Test de Dickey-Fuller augmenté}
	177	+\begin{itemize}
	178	+\item Hypothèse nulle $H_0$ : le processus $\left\{ Y_t, t \in R \right\}$ possède une racine unitaire.
	179	+\item Dans sa forme la plus complète, le test est basé sur l’estimation du modèle suivant avec tendance déterministe et constante:
	180	+\begin{align*}
	181	+(1-L)Y_t=\alpha+\beta t + \delta Y_{t-1}+\sum_{t=1}^m \pi_i(1-L)Y_{t-i}+u_t
	182	+\end{align*}
	183	+\item En transformant $\Delta Y_t=Y_t-Y_{t-1}$
	184	+\begin{align*}
	185	+Y_t=\alpha+\beta t + \delta Y_{t-1}+\sum_{t=1}^m \pi_iY_{t-i}+u_t
	186	+\end{align*}
	187	+\item Où $u_t$ $\sim$ $i.i.d$ $N(0,\sigma^2)$ et $m$ est choisi selon le degré d’autocorrélation potentiel du processus $Y_t$.
	188	+\end{itemize}
	189	+\end{frame}
	190	+
	191	+\begin{frame}{Test de Dickey-Fuller augmenté}
	192	+\textbf{Dans ce modèle, l’hypothèse nulle s’écrit:}
	193	+\begin{align*}
	194	+H_0:\delta =0
	195	+\end{align*}
	196	+\textbf{Contre l’alternative unilatérale:}
	197	+\begin{align*}
	198	+H_A:\delta<0
	199	+\end{align*}
	200	+\begin{itemize}
	201	+\item L’intuition du test. Prenez le test de Dickey Fuller non augmenté (pour un seul retard sans tendance ni constante), si l’hypothèse nulle n’est pas rejetée, cela implique que $\Delta Y_t= u_t$.
	202	+\item Comme le terme d’erreur est bruit blanc et stationnaire, on sait que s’il y a une racine unitaire dans $Y_t$, sa première différence elle est stationnaire.
	203	+\item Sous l’hypothèse nulle, il est impossible d’utiliser un ratio de Student comparé à une distribution t à cause de la racine unitaire.
	204	+\end{itemize}
	205	+\end{frame}
	206	+
	207	+\begin{frame}{Test de Dickey-Fuller augmenté}
	208	+\begin{itemize}
	209	+\item Il faudra donc utiliser une table statistique propre au test et dérivée par Dickey et Fuller
	210	+\item La région critique du test est obtenue en comparant le ratio t pour la variable $Y_{t-1}$ à une valeur critique obtenue à partir d’une distribution non standard qui est très sensible à la spécification de l’équation estimée.
	211	+\end{itemize}
	212	+\begin{align*}
	213	+DK=\frac{(\hat{\rho}-1)}{SE(\hat{\rho})}
	214	+\end{align*}
	215	+\begin{itemize}
	216	+\item la distribution t de Student ne peut pas être utilisée, car les valeurs critiques appropriées sont inférieures à celles de la distribution t. (pour un seuil de significativité de 5\%, elle est inférieure à la valeur 1.96 obtenue avec la distribution t).
	217	+\end{itemize}
	218	+\end{frame}
	219	+
	220	+\begin{frame}{Test de Dickey-Fuller augmenté}
	221	+\begin{itemize}
	222	+\item Il est important de noter que la spécification du test (avec ou sans constante ou avec ou sans tendance) va affecter les points critiques de la statistique.
	223	+\item Il faut donc utiliser les points critiques distincts pour ces trois spécifications.
	224	+\item Il n’y a pas de procédure ferme pour déterminer quel type de test de Dickey-Fuller il convient de faire.
	225	+\item Cependant, on peut utiliser les critères de sélections Aikaiké et Schwarz pour déterminer s’il convient d’ajouter des retards pour les erreurs.
	226	+\item On peut aussi faire un test F de significativité conjointe sur les coefficients de la constante et de la tendance afin de savoir s’ils sont pertinents.
	227	+\item Par contre, la valeur critique n’est pas comparée à une distribution F, mais plutôt à une table particulière dérivée par Dickey et Fuller.
	228	+\end{itemize}
	229	+\end{frame}
	230	+
	231	+\begin{frame}{Test de Philips-Perron}
	232	+\begin{itemize}
	233	+\item Le test de Philipps-Perron tient aussi compte de la possible autocorrélation des erreurs, mais sans ajouter des lags des erreurs dans le modèle.
	234	+\item Tester $H_0: \rho =1$ contre l’alternative $H_A: \rho < 1$ dans le modèle:
	235	+\begin{align*}
	236	+Y_t=\alpha +\beta t +\rho Y_{t-1}+u_t
	237	+\end{align*}
	238	+\item Supposons un échantillon de taille $T$, soit $\hat{\rho}$ l'estimateur de $\rho$
	239	+\item La statistique de test $Z_{\rho}$ est définie par:
	240	+\begin{align*}
	241	+Z_{\rho}=(\hat{\rho}-1)T-\frac{1}{2}\frac{T \sigma^2}{S_T^2}(\hat{\lambda}^2-\hat{\gamma}_0)
	242	+\end{align*}
	243	+Avec
	244	+\begin{align*}
	245	+\lambda^2=\gamma_0+2\sum_{j=1}^{\infty} \gamma_j
	246	+\end{align*}
	247	+où $\gamma_j$ est l’autocovariance d’ordre $j$ du processus, $j = 0, 1, 2,\cdots$
	248	+\end{itemize}
	249	+\end{frame}
	250	+
	251	+\begin{frame}{Test de Philips-Perron}
	252	+\begin{itemize}
	253	+\item Pour les calculs dans un échantillon de taille T, on a:
	254	+\begin{align*}
	255	+\hat{\lambda}^2=\hat{\gamma}_0+2 \sum_{j=1}^{l} \left[1-\frac{j}{1+l} \right]\hat{\gamma}_j
	256	+\end{align*}
	257	+\begin{align*}
	258	+\hat{\gamma}_j=\frac{1}{T}\sum_{t=j+1}^T\hat{u}_t \hat{u}_{t-j}, \hspace{0.5cm} j=0,1,\cdots,T
	259	+\end{align*}
	260	+\begin{align*}
	261	+S_T^2=(T-K)\sum_{t=1}^T\hat{u}_t^2
	262	+\end{align*}
	263	+\item où K est le nombre de paramètres estimés du modèle de régression et et l le paramètre de troncature de la sommation infinie.
	264	+\item l est choisi à partir de l’ordre à partir duquel les autocovariances sont suffisamment faibles pour être assimilées à zéro.
	265	+\end{itemize}
	266	+\end{frame}
	267	+
	268	+
	269	+\section{Transformation d'une séries non stationnaire en série stationnaire}
	270	+\frame{\tableofcontents[current]}
	271	+
	272	+\begin{frame}{Transformation d'une séries non stationnaire en série stationnaire}
	273	+\begin{itemize}
	274	+\item On a vu que si une série n’est pas stationnaire, cela peut être causé par une tendance
	275	+\begin{align*}
	276	+X_t=\delta+\beta t+\epsilon_t
	277	+\end{align*}
	278	+\item ou par une racine unitaire
	279	+\begin{align*}
	280	+X_t=\beta+ X_{t-1}+\epsilon_t
	281	+\end{align*}
	282	+\item Lorsque l’on veut ramener une série chronologique non stationnaire à une série stationnaire, il existe deux possibilités.
	283	+\item Si la série possède une racine unitaire, nous savons que par définition leur première différence sera stationnaire.
	284	+\end{itemize}
	285	+\end{frame}
	286	+
	287	+
	288	+\begin{frame}{Transformation d'une séries non stationnaire en série stationnaire}
	289	+\begin{itemize}
	290	+\item Donc, la solution est de les différencier. Si la série a une tendance déterministe, la solution est d’utiliser le résidu ($\hat{u}_t$) de régression de la série chronologique sur une constante et une tendance:
	291	+\begin{align*}
	292	+\hat{u}=[\hat{Y}_t-\hat{\beta}_1+\hat{\beta}_2 t]
	293	+\end{align*}
	294	+\item Parfois, la tendance est non-linéaire et il faudra ajouter à la régression $t^2$.
	295	+\item Avant de choisir une des deux méthodes, il convient de bien diagnostiquer ce qui rend la série non stationnaire, puisqu’utiliser la mauvaise méthode pour stationnariser la série peut avoir des conséquences importantes.
	296	+\end{itemize}
	297	+\end{frame}
	298	+
	299	+
	300	+\section{La conintégration}
	301	+\frame{\tableofcontents[current]}
	302	+
	303	+\begin{frame}{La conintégration}
	304	+\begin{itemize}
	305	+\item Lorsque les deux variables d’une régression ne sont pas stationnaires, la régression sera fallacieuse.
	306	+\item Il est aussi possible que ces variables (ou d’un système d’équations) partagent une même tendance: \textbf{c’est le concept de cointégration}.
	307	+\item Des séries de données non stationnaires sont dites cointégrées s’il existe une combinaison linéaire de ces séries qui est stationnaire.
	308	+\end{itemize}
	309	+\end{frame}
	310	+
	311	+\begin{frame}{La conintégration}
	312	+\begin{block}{Démonstration dans le cas de deux variables cointégrées, $X_t$ et $Y_t$:}
	313	+\textbf{Puisqu’elles sont intégrées d’ordre 1:}
	314	+\begin{align*}
	315	+X_t=\mu_{xt}+e_{xt}
	316	+\end{align*}
	317	+\begin{align*}
	318	+Y_t=\mu_{yt}+e_{x=yt}
	319	+\end{align*}
	320	+\begin{itemize}
	321	+\item Où $\mu_{xt}$ et $\mu_{yt}$ sont les composantes marche aléatoire associée à $X_t$ et $Y_t$ respectivement, et $e_{xt}$ et $e_{yt}$ sont leur composante stationnaire.
	322	+\end{itemize}
	323	+\end{block}
	324	+\end{frame}
	325	+
	326	+\begin{frame}{La conintégration}
	327	+\begin{itemize}
	328	+\item Si $X_t$ et $Y_t$ sont cointégrés, il existe un vecteur de cointégration $(\beta_1, \beta_2)$ tel que $\beta_1X_t + \beta_2Y_t$ soit stationnaire.
	329	+\begin{align*}
	330	+\beta_1 X_t+\beta_2 Y_t & =\beta_1(\mu_{xt}+e_{xt})+\beta_2(\mu_{yt}+e_{x=yt})\\
	331	+& = (\beta_1\mu_{xt}+\beta_2 \mu_{yt})+(\beta_1e_{xt}+\beta_2 e_{yt})
	332	+\end{align*}
	333	+\item Ainsi, on voit que puisque $(\beta_1e_{xt}+\beta_2 e_{yt})$ est stationnaire comme somme de deux processus stationnaires.
	334	+\item Alors pour que $\beta_1 X_t+\beta_2 Y_t$ soit stationnaire, le terme $(\beta_1\mu_{xt}+\beta_2 \mu_{yt})$ doit être nul :
	335	+\begin{align*}
	336	+(\beta_1\mu_{xt}+\beta_2 \mu_{yt})=0
	337	+\end{align*}
	338	+\end{itemize}
	339	+\end{frame}
	340	+
	341	+
	342	+
	343	+\begin{frame}{La conintégration}
	344	+\begin{itemize}
	345	+\item Mais $\beta_1 \neq 0$ et $\beta_2 \neq 0$, et on doit avoir:
	346	+\begin{align*}
	347	+\mu_{xt}=\frac{\beta_2}{\beta_1} \mu_{yt}
	348	+\end{align*}
	349	+\item Ainsi les deux tendances stochastiques (composantes contenant la racine unitaire) $\mu_{xt}$ et $\mu_{yt}$ sont identiques à une constante multiplicative près. Les deux processus cointégrés possèdent donc une tendance stochastique commune.
	350	+\item Le concept peut être étendu à plusieurs variables k, auquel cas il y aurait k paramètres de cointégration.
	351	+\item Lorsque des variables sont cointégrées, toute déviation systématique de la relation d’équilibre de long terme a des répercussions sur l’évolution de court terme de chacune d’elles.
	352	+\end{itemize}
	353	+\end{frame}
	354	+
	355	+\begin{frame}{Test de cointégration}
	356	+\begin{itemize}
	357	+\item Supposons un échantillon pour deux séries $\left\{ X_t \right\}$ et $\left\{ Y_t \right\}$ qui seraient intégrées d’ordre 1. La procédure décrite ci-dessous s’applique aussi dans le cas d’un nombre quelconque de variables.
	358	+\item Nous voulons déterminer s’il existe une relation d’équilibre de long terme entre ces variables, c’est-à-dire si ces variables sont cointégrées.
	359	+\item La procédure à quatre étapes de Engle et Granger (1987) qui se base sur les tests de stationnarité.
	360	+\end{itemize}
	361	+\end{frame}
	362	+
	363	+\begin{frame}{Test de cointégration}
	364	+\begin{block}{Étape 1:}
	365	+Prétester les variables $\left\{ X_t \right\}$ et $\left\{ Y_t \right\}$ pour leur ordre d’intégration; utiliser pour cela le test de racine unitaire de Dickey-Fuller Augmenté.
	366	+\begin{enumerate}
	367	+\item Si les variables sont toutes stationnaires, alors on doit utiliser les méthodes usuelles des séries stationnaires pour les modéliser.
	368	+\item Si les variables sont intégrées de différents ordres, alors on
	369	+peut conclure qu’elles ne sont pas cointégrées. Dans notre contexte, les variables seront presque toujours $I(1)$.
	370	+\item Si les variables sont intégrées de même ordre, passer à l’étape 2.
	371	+\end{enumerate}
	372	+\end{block}
	373	+\end{frame}
	374	+
	375	+
	376	+\begin{frame}{Test de cointégration}
	377	+\begin{block}{Étape 2:}
	378	+\begin{itemize}
	379	+\item Si les deux variables sont I(1), il faut estimer la relation d’équilibre de long terme, sous la forme :
	380	+\begin{align*}
	381	+Y_t=\beta_0+\beta_1 X_t+e_t
	382	+\end{align*}
	383	+\item Pour déterminer si les variables sont cointégrées, considérons la série $\hat{e}_t$ des résidus de l’estimation de (1).
	384	+\item Si la série des résidus est stationnaire, alors $Y_t$ et $X_t$ sont cointégrées d’ordre $(1,1)$.
	385	+\item Pour tester la stationnarité des résidus, on peut utiliser un test de racine unitaire (une version adaptée du test de Dickey-Fuller augmenté). Soit la spécification suivante :
	386	+\begin{align*}
	387	+\Delta \hat{e}_t=\alpha_1 \hat{e}_{t-1}+\epsilon_t
	388	+\end{align*}
	389	+\item On doit tester l’hypothèse nulle $H_0: \alpha_1 = 0$. Si on ne rejette pas $H_0$, alors la série des résidus possède une racine unitaire, et les variables $Y_t$ et $X_t$ ne sont pas cointégrées.
	390	+\end{itemize}
	391	+\end{block}
	392	+\end{frame}
	393	+
	394	+\begin{frame}{Test de cointégration}
	395	+\begin{block}{Étape 2:}
	396	+\begin{itemize}
	397	+\item Si par contre on rejette $H_0$, alors la série des résidus est stationnaire et les variables $Y_t$ et $X_t$ sont cointégrées d’ordre $(1,1)$.
	398	+\item Les points critiques utilisés pour faire ce test ne sont pas exactement ceux du test de DFA habituel, car la série des résidus a été construite sous certaines contraintes, notamment celles qui visent à minimiser la somme des carrés résiduels, ce qui rend la variance de cette série la plus faible possible.
	399	+\item Pour tenir compte de ce fait, on a construit des points critiques modifiés du test de DFA qui sont appropriés à ce type de situation.
	400	+\end{itemize}
	401	+\end{block}
	402	+\end{frame}
	403	+
	404	+\begin{frame}{Test de cointégration}
	405	+\begin{block}{Étape 2:}
	406	+\begin{itemize}
	407	+\item Si l’on constate que les résidus sont corrélés, alors on peut spécifier une version augmentée où l’on introduit des retards de la variable expliquée, ce qui peut résoudre ce problème d’autocorrélation :
	408	+\begin{align*}
	409	+\Delta \hat{e}_t=\alpha_1 \hat{e}_{t-1}+\sum_{j=1}^p b_j \Delta \hat{e}_{t-j}+\epsilon_t
	410	+\end{align*}
	411	+\item Le paramètre d’intérêt est toujours $\alpha_1$ et on procède de la même façon que celle décrite précédemment.
	412	+\item Si les variables ne sont pas cointégrées alors qu’elles sont intégrées d’ordre 1, alors on peut les étudier en appliquant les méthodes habituelles aux séries des différences premières.
	413	+\item Si les variables sont cointégrées, alors passer à l’étape 3.
	414	+\end{itemize}
	415	+\end{block}
	416	+\end{frame}
	417	+
	418	+
	419	+\begin{frame}{Étape 3:}
	420	+\begin{itemize}
	421	+\item Si les variables sont cointégrées, les résidus (stationnaires) de la relation de long terme peuvent être utilisés pour estimer le modèle de correction d’erreur
	422	+\item À cette étape, on remplace le terme de correction d’erreur, $Y_{t-1}-\beta_0+\beta_1 X_{t-1}$ par le terme correspondant estimé, les résidus,soit $\hat{e}_{t-1}$.
	423	+\item On obtient le modèle de correction d’erreur :
	424	+\begin{align*}
	425	+\Delta Y_t=a_{y0}+c_y(\hat{e}_{t-1})+\sum a_{yyi}\Delta Y_{t-i}+\sum a_{yxi} \Delta X_{t-i}+\epsilon_{yt}
	426	+\end{align*}
	427	+\begin{align*}
	428	+\Delta X_t=a_{x0}+c_x(\hat{e}_{t-1})+\sum a_{xyi}\Delta Y_{t-i}+\sum a_{xxi} \Delta X_{t-i}+\epsilon_{xt}
	429	+\end{align*}
	430	+\item Comme les deux équations ont le même ensemble de régresseurs, et que toutes les variables du modèle sont stationnaires, on peut estimer ce modèle par les moindres carrés ordinaires.
	431	+\end{itemize}
	432	+\end{frame}
	433	+
	434	+
	435	+\begin{frame}{Étape 4:}
	436	+\begin{itemize}
	437	+\item Vérifier que le modèle est bien spécifié. Les deux séries de résidus du modèle de correction d’erreur doivent être des processus bruit blanc.
	438	+\item Si les résidus sont autocorrélés, alors c’est que l’on n’a pas introduit suffisamment de retards dans le modèle ; il faut réestimer le modèle en augmentant le nombre de retards.
	439	+\end{itemize}
	440	+\end{frame}
	441	+
	442	+
	443	+\end{document}
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	2	+\beamer@sectionintoc {2}{Tests de bruits blancs}{7}{0}{2}
	3	+\beamer@sectionintoc {3}{Tests de racine unitaire}{14}{0}{3}
	4	+\beamer@sectionintoc {4}{Transformation d'une séries non stationnaire en série stationnaire}{22}{0}{4}
	5	+\beamer@sectionintoc {5}{La conintégration}{25}{0}{5}

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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage{graphicx}
	9	+\usepackage{booktabs}
	10	+\usetheme{default}
	11	+\usecolortheme{default}
	12	+
	13	+\title[S02 Régression et MCO]{Section 05 : Les séries chronologiques\\ (Séance 12)}
	14	+\subtitle{GSF-6053: Économétrie Financière}
	15	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	16	+\institute[Université Laval]
	17	+{
	18	+ \inst{1}%
	19	+ Département de finance, assurance et immobilier\\
	20	+ Faculté des sciences de l'administration\\
	21	+ Université Laval}
	22	+\date[Hiver 2022]{12 avril 2022}
	23	+
	24	+\begin{document}
	25	+
	26	+
	27	+\begin{frame}
	28	+ \titlepage
	29	+\end{frame}
	30	+
	31	+
	32	+
	33	+\begin{frame}{Références}
	34	+\textbf{Obligatoires:}
	35	+\begin{itemize}
	36	+\item \textbf{Notes de cours:} Section 5 (Professeure: Marie-Hélène Gagnon)
	37	+\item \textbf{Woolridge:} chapitres 11, 12 et 19.
	38	+\end{itemize}
	39	+\vspace{0.5cm}
	40	+\textbf{Complémentaires:}
	41	+\begin{itemize}
	42	+\item \textbf{Gujarati et Porter:} chapitres 21 et 22
	43	+\end{itemize}
	44	+\end{frame}
	45	+
	46	+\begin{frame}{Plan de la séance}
	47	+ \tableofcontents
	48	+\end{frame}
	49	+
	50	+\section{Processus moyennes mobiles}
	51	+\frame{\tableofcontents[current]}
	52	+\begin{frame}{Processus moyennes mobiles}
	53	+\begin{itemize}
	54	+\item Soit $\left\{ \epsilon_t \right\}$ un processus bruit blanc.
	55	+\item Considérons le processus $\left\{Y_t \right\}$ défini à partir de $\left\{ \epsilon_t \right\}$ comme suit :
	56	+\begin{align*}
	57	+Y_t=\mu+\epsilon_t+\theta \epsilon_{t-1}
	58	+\end{align*}
	59	+où $\mu$, $\theta$ sont des paramètres constants
	60	+\item En d’autres termes, le processus $\left\{Y_t \right\}$ est une moyenne pondérée du présent et du passé immédiat de $\left\{ \epsilon_t \right\}$ , à une constante additive près.
	61	+\item On dit que le processus $\left\{Y_t \right\}$ est un processus moyenne mobile d’ordre 1, et on note MA (1).
	62	+\end{itemize}
	63	+\end{frame}
	64	+
	65	+\begin{frame}{Processus moyennes mobiles}
	66	+\begin{block}{Processus moyenne mobile d’ordre 1}
	67	+\textbf{La moyenne non conditionnelle du processus est}
	68	+\begin{align*}
	69	+E(Y_t)&=\mu+E(\epsilon_t)+\theta E(\epsilon_{t-1})\\ & = \mu
	70	+\end{align*}
	71	+Sachant:
	72	+\begin{itemize}
	73	+\item $Var(\epsilon_t)=Var(\epsilon_{t-1})=\sigma^2$
	74	+\item $Cov(\epsilon_t,\epsilon_{t-1})=Cov(\epsilon_t,\epsilon_{t-2})=0$
	75	+\end{itemize}
	76	+\textbf{La variance du processus est}
	77	+\begin{align*}
	78	+Var(Y_t)&=E(Y_t-\mu)^2 \\& = E(\epsilon_t+\theta \epsilon_{t-1})^2 \\ & = (1+\theta^2)\sigma^2
	79	+\end{align*}
	80	+\end{block}
	81	+\end{frame}
	82	+
	83	+\begin{frame}{Processus moyennes mobiles}
	84	+\begin{block}{Processus moyenne mobile d’ordre 1}
	85	+\textbf{L’autocovariance du processus est}
	86	+\begin{align*}
	87	+\gamma(1)&=E(Y_t-\mu)(Y_{t-1}-\mu)\\ & =E(\epsilon_t+\theta \epsilon_{t-1})(\epsilon_{t-1}+\theta \epsilon_{t-2})\\ & = \theta \sigma^2
	88	+\end{align*}
	89	+\begin{itemize}
	90	+\item Les autocovariances d’ordre supérieur à 1 sont nulles comme on peut le vérifier en utilisant le fait que
	91	+\begin{align*}
	92	+Cov(\epsilon_t,\epsilon_{t-h})=Cov(\epsilon_t,\epsilon_{t-h-1})=0, \forall h >1
	93	+\end{align*}
	94	+
	95	+\end{itemize}
	96	+\end{block}
	97	+\end{frame}
	98	+
	99	+\begin{frame}{Processus moyennes mobiles}
	100	+\begin{block}{Processus moyenne mobile d’ordre 1}
	101	+\begin{itemize}
	102	+\item Tout processus MA(1) est faiblement stationnaire, car c’est une combinaison linéaire de bruits blancs et ces deux premiers moments satisfont les conditions de stationnarité peu importe la valeur prise par les coefficients estimés.
	103	+\item La fonction d’autocorrélation $\rho$ est définie par :
	104	+\begin{align*}
	105	+\rho(h)=\frac{\rho(h)}{\rho(0)}=\left\{\begin{matrix}
	106	+1 \hspace{0.5cm} si \hspace{0.5cm} h=0 \\
	107	+\frac{\theta}{1+\theta^2} \hspace{0.5cm} si \hspace{0.5cm} h=1 \\
	108	+0 \hspace{0.5cm} si \hspace{0.5cm} h>1
	109	+\end{matrix}\right.
	110	+\end{align*}
	111	+\item La fonction d’autocorrélation devient nulle après une période; le signe de $\theta$ détermine le sens (négatif ou positif) de l’autocorrélation d’ordre 1.
	112	+\item La fonction d’autocorrélation ne change pas si $\theta$ est remplacé par $1/\theta$.
	113	+\end{itemize}
	114	+\end{block}
	115	+\end{frame}
	116	+
	117	+\begin{frame}{Processus moyennes mobiles}
	118	+\begin{block}{Processus moyenne mobile d’ordre q}
	119	+\textbf{Considérons maintenant le processus $\left\{Y_t \right\}$ défini par:}
	120	+\begin{align*}
	121	+Y_t=\mu+\sum_{j=0}^q \theta_j \epsilon_{t-j}
	122	+\end{align*}
	123	+avec $\theta_0=1$ et $\left\{ \epsilon_t \right\}$ est un processus bruit blanc.
	124	+\begin{itemize}
	125	+\item Alors, $\left\{Y_t \right\}$ est un processus moyenne mobile d’ordre q, on note MA(q).
	126	+\end{itemize}
	127	+\end{block}
	128	+\end{frame}
	129	+
	130	+\begin{frame}{Processus moyennes mobiles}
	131	+\begin{block}{Processus moyenne mobile d’ordre q}
	132	+\begin{itemize}
	133	+\item En utilisant les propriétés du processus bruit blanc, on obtient :
	134	+\begin{align*}
	135	+E(Y_t)=\mu
	136	+\end{align*}
	137	+\begin{align*}
	138	+\gamma(h)=\left\{\begin{matrix}
	139	+(1+\theta_1^2+\theta_2^2+\cdots+\theta_q^2)\sigma^2 \hspace{0.5cm} si \hspace{0.5cm} h=0 \\
	140	+(\theta_h+\theta_{h+1}\theta_1+\theta_{h+2}\theta_2+\cdots+\theta_q\theta_{q-h})\sigma^2 \hspace{0.1cm} si\hspace{0.1cm}h=1,2,..,q \\
	141	+0 \hspace{0.5cm} si \hspace{0.5cm} h>q
	142	+\end{matrix}\right.
	143	+\end{align*}
	144	+\end{itemize}
	145	+\end{block}
	146	+\end{frame}
	147	+
	148	+
	149	+
	150	+\begin{frame}{Processus moyennes mobiles}
	151	+\begin{block}{Processus moyenne mobile d’ordre q}
	152	+\begin{itemize}
	153	+\item Le processus est donc stationnaire quelles que soit les valeurs des paramètres $\theta_j$, $j = 1, 2, \cdots, q$.
	154	+\item La fonction d’autocorrélation devient nulle après un délai de q périodes.
	155	+\item Plus généralement, définissons le processus limite du processus mobile d’ordre $q$ lorsque $q \rightarrow \infty$. Cette limite s’écrit :
	156	+\begin{align*}
	157	+Y_t=\mu+\sum_{j=0}^{\infty} \psi_j \epsilon_{t-j}
	158	+\end{align*}
	159	+\item Et est appelé moyenne mobile d’ordre infini, note $MA(\infty)$.
	160	+\end{itemize}
	161	+
	162	+\end{block}
	163	+\end{frame}
	164	+
	165	+
	166	+\begin{frame}{Processus moyennes mobiles}
	167	+\begin{block}{Processus moyenne mobile d’ordre q}
	168	+Si
	169	+\begin{align*}
	170	+\sum_{j=0}^{\infty} \psi_j^2 <\infty
	171	+\end{align*}
	172	+\begin{itemize}
	173	+\item Alors le processus moyenne mobile d’ordre infini est faiblement stationnaire.
	174	+\item Sa moyenne, variance et autocovariance sont obtenues comme limites de la moyenne, de la variance et des autocovariances du processus moyenne mobile d’ordre q lorsque $q \rightarrow \infty$
	175	+\end{itemize}
	176	+\end{block}
	177	+\end{frame}
	178	+
	179	+
	180	+\section{Processus autoregressifs}
	181	+\frame{\tableofcontents[current]}
	182	+\begin{frame}{Processus autoregressifs}
	183	+\begin{itemize}
	184	+\item Un processus autorégressif d’ordre p est un processus qui satisfait une équation linéaire en différence stochastique d’ordre p de la forme :
	185	+\begin{align*}
	186	+Y_t=c+\sum_{j=1}^p\psi_jY_{t-j}+\epsilon_t
	187	+\end{align*}
	188	+où $\left\{ \epsilon_t \right\}$ est un bruit blanc $(0,\sigma^2)$.
	189	+\end{itemize}
	190	+\end{frame}
	191	+
	192	+
	193	+
	194	+\begin{frame}{Processus autoregressifs}
	195	+\begin{block}{Processus autorégressif d’ordre 1}
	196	+\begin{align*}
	197	+Y_t=c+\phi Y_{t-1}+\epsilon_t
	198	+\end{align*}
	199	+\begin{itemize}
	200	+\item Sait que si $\mid \phi \mid \ge 1$, le processus explose ; le processus ne peut pas être stationnaire.
	201	+\item Par contre, si $\mid \phi \mid < 1$, le processus est stable et on montre qu’il est faiblement stationnaire.
	202	+\end{itemize}
	203	+\end{block}
	204	+\end{frame}
	205	+
	206	+
	207	+\begin{frame}{Processus autoregressifs}
	208	+\begin{block}{Processus autorégressif d’ordre 1}
	209	+\begin{itemize}
	210	+\item En effet, si $\phi < 1$, on obtient en faisant de la substitution récursive :
	211	+\begin{align*}
	212	+Y_t & =(c+\epsilon_t)+\phi(c+\epsilon_t+\phi Y_{t-2})\\ & = (c+\epsilon_t) +\phi (c+\epsilon_{t-1})+\phi^2( c+\epsilon_{t-2}+\phi Y_{t-3})\\ & = (c+\epsilon_t) +\phi (c+\epsilon_{t-1})+ \phi^2 (c+\epsilon_{t-2})+\phi^3 (c+\epsilon_{t-3})+\cdots \\ & = c(1+\phi+\phi^2+\phi^3+\cdots)+\epsilon_t+\phi \epsilon_{t-1}+\phi^2\epsilon_{2}+\phi^3\epsilon_{3}+\cdots \\ & = \frac{c}{1-\phi}+\epsilon_t+\phi \epsilon_{t-1}+\phi^2\epsilon_{2}+\phi^3\epsilon_{3}+\cdots
	213	+\end{align*}
	214	+\item Ce résultat montre que le processus AR(1) peut s’écrire comme un processus $MA(\infty)$
	215	+\end{itemize}
	216	+\end{block}
	217	+\end{frame}
	218	+
	219	+
	220	+\begin{frame}{Processus autoregressifs}
	221	+\begin{block}{Processus autorégressif d’ordre 1}
	222	+\textbf{Son espérance :}
	223	+\begin{align*}
	224	+\mu = E(Y_t)=\frac{c}{1-\phi}
	225	+\end{align*}
	226	+\textbf{Sa variance :}
	227	+\begin{align*}
	228	+\gamma(0)&=E(Y_t-\mu)^2 \\ & = E(\epsilon_t+\phi \epsilon_{t-1}+\phi^2 \epsilon_{t-2}+\phi^3 \epsilon_{t-3}+\cdots)^2 \\ & = (1+\phi^2+\phi^4+\phi^6+\cdots)\sigma^2 \\ & = \frac{\sigma^2}{1-\phi^2}
	229	+\end{align*}
	230	+\end{block}
	231	+\end{frame}
	232	+
	233	+\begin{frame}{Processus autoregressifs}
	234	+\begin{block}{Processus autorégressif d’ordre 1}
	235	+\textbf{L’autocovariance d’ordre h :}
	236	+\begin{align*}
	237	+\gamma(h) & =E(Y_t-\mu)(Y_{t-h}-\mu) \\ & = E(\epsilon_t+\phi \epsilon_{t-1}+\phi^2 \epsilon_{t-2}+\phi^3 \epsilon_{t-3} \\ & +\cdots+\phi^h\epsilon_{t-h}+\phi^{h+1}\epsilon_{t-h-1}+\cdots) \\ & \times (\epsilon_{t-h} +\phi \epsilon_{t-h-1} +\phi^2\epsilon_{t-h-2}+\phi^3\epsilon_{t-h-3}+\cdots) \\ & = E(\phi^h\epsilon_{t-h}+\phi^{h+1}\epsilon_{t-h-1}) \\ & \times(\epsilon_{t-h} +\phi \epsilon_{t-h-1} +\phi^2\epsilon_{t-h-2}+\phi^3\epsilon_{t-h-3}) \\ & =\phi^hE(\epsilon_{t-h} +\phi \epsilon_{t-h-1} +\phi^2\epsilon_{t-h-2}+\phi^3\epsilon_{t-h-3}+\cdots)^2 \\ & = \frac{\phi^h\sigma^2}{1-\phi^2}
	238	+\end{align*}
	239	+\end{block}
	240	+\end{frame}
	241	+
	242	+
	243	+\begin{frame}{Processus autoregressifs}
	244	+\begin{block}{Processus autorégressif d’ordre 1}
	245	+\textbf{La fonction d’autocorrélation $\rho$ :}
	246	+\begin{align*}
	247	+\rho(h)=\frac{\gamma(h)}{\gamma(0)}=\phi^h
	248	+\end{align*}
	249	+et puisque $\mid \phi \mid < 1$, l’autocorrélation $\rho(h)$ décroit lorsque le retard h augmente.
	250	+\end{block}
	251	+\end{frame}
	252	+
	253	+
	254	+\begin{frame}{Processus autoregressifs}
	255	+\begin{block}{Processus autorégressif d’ordre 2}
	256	+
	257	+\begin{align*}
	258	+Y_t=c+\phi_1 Y_{t-1}+\phi_2 Y_{t-2}+\epsilon_t
	259	+\end{align*}
	260	+\begin{itemize}
	261	+\item Ou encore sous la forme avec l’opération retard :
	262	+\end{itemize}
	263	+\begin{align*}
	264	+(1-\phi_1L-\phi_2L^2)Y_t=c+\epsilon_t
	265	+\end{align*}
	266	+\begin{itemize}
	267	+\item Il n’est plus évident de savoir si le processus est stationnaire seulement en regardant la valeur d’un coefficient comme dans le cas AR(1).
	268	+\end{itemize}
	269	+\end{block}
	270	+\end{frame}
	271	+
	272	+
	273	+\begin{frame}{Processus autoregressifs}
	274	+\begin{block}{Processus autorégressif d’ordre 2}
	275	+
	276	+\begin{itemize}
	277	+\item \textbf{Résultat connu que nous exploitons:} On sait que cette équation en différence est stable si les racines $\lambda_1$ et $\lambda_2$ de l’équation caractéristique suivante
	278	+\end{itemize}
	279	+\begin{align*}
	280	+\lambda^2-\phi_1\lambda-\phi_2=0
	281	+\end{align*}
	282	+Sont de module inférieur à l’unité.
	283	+\end{block}
	284	+\end{frame}
	285	+
	286	+\begin{frame}{Processus autoregressifs}
	287	+\begin{block}{Processus autorégressif d’ordre 2}
	288	+\begin{itemize}
	289	+\item Lorsque les racines sont réelles, elles sont données par la formule habituelle :
	290	+
	291	+\begin{align*}
	292	+\lambda_1 = \frac{\phi_1+\sqrt{\phi_1^2+4\phi_2}}{2}
	293	+\end{align*}
	294	+\begin{align*}
	295	+\lambda_2 = \frac{\phi_1+\sqrt{\phi_1^2+4\phi_2}}{2}
	296	+\end{align*}
	297	+\item Alors $(1-\phi_1 L-\phi_2 L^2 )=(1-\lambda_1L)(1-\lambda_2L)$
	298	+\end{itemize}
	299	+
	300	+\end{block}
	301	+\end{frame}
	302	+
	303	+
	304	+
	305	+\begin{frame}{Processus autoregressifs}
	306	+\begin{block}{Processus autorégressif d’ordre 2}
	307	+\textbf{La moyenne du AR(2) est donnée par}
	308	+\begin{align*}
	309	+\mu=E(Y_t)=\frac{c}{1-\phi_1-\phi_2}
	310	+\end{align*}
	311	+\begin{itemize}
	312	+\item Ce qui revient à en prenant le processus en déviation par rapport à la moyenne:
	313	+\begin{align*}
	314	+(Y_t-\mu)=\phi_1(Y_{t-1}-\mu)+\phi_2(Y_{t-2}-\mu)+\epsilon_t
	315	+\end{align*}
	316	+\end{itemize}
	317	+\end{block}
	318	+\end{frame}
	319	+
	320	+
	321	+
	322	+
	323	+\begin{frame}{Processus autoregressifs}
	324	+\begin{block}{Processus autorégressif d’ordre 2}
	325	+\begin{itemize}
	326	+\item Multiplier les deux membres de cette dernière relation par $(Y_{t-h}-\mu),h \geq 0$ et prendre l’espérance pour obtenir la fonction pour les autocovariances suivantes :
	327	+\begin{align*}
	328	+\gamma(h)=\left\{\begin{matrix}
	329	+\phi_1 \gamma(h-1)+\phi_2\gamma(h-2) \hspace{0.5cm} si \hspace{0.5cm} h\ge0 \\
	330	+\phi_1 \gamma(1)+\phi_2 \gamma(2)+\sigma^2 \hspace{0.5cm} si \hspace{0.5cm} h=0
	331	+\end{matrix}\right.
	332	+\end{align*}
	333	+\item Les autocorrélations obéissent également à une suite récurrente d’ordre 2 :
	334	+\begin{align*}
	335	+\rho(h)=\phi_1 \rho(h-1)+\phi_2 \rho(h-2) \hspace{0.5cm} si \hspace{0.5cm} h\ge1
	336	+\end{align*}
	337	+\item On peut alors calculer $\rho(1)$ , $\rho(2)$ , puis obtenir
	338	+\begin{align*}
	339	+\gamma(0)=\frac{(1-\phi_2)\sigma^2}{(1+\phi_2)[(1-\phi_2)^2-\phi_1^2]}
	340	+\end{align*}
	341	+\end{itemize}
	342	+\end{block}
	343	+\end{frame}
	344	+
	345	+
	346	+
	347	+\begin{frame}{Processus autoregressifs}
	348	+\begin{block}{Processus autorégressif d’ordre p}
	349	+\begin{align*}
	350	+Y_t=c+\sum_{j=1}^p\phi_jY_{t-j}+\epsilon_t
	351	+\end{align*}
	352	+\textbf{Les racines de l'équation:}
	353	+\begin{align*}
	354	+\lambda^p-\phi_1\lambda^{p-1}-\phi_2\lambda^{p-2}-\cdots -\phi_{p-1}\lambda-\phi_p
	355	+\end{align*}
	356	+\begin{itemize}
	357	+\item Sont de module inférieur à l’unité, le processus est stable et faiblement stationnaire
	358	+\end{itemize}
	359	+\end{block}
	360	+\end{frame}
	361	+
	362	+
	363	+\begin{frame}{Processus autoregressifs}
	364	+\begin{block}{Processus autorégressif d’ordre p}
	365	+\begin{align*}
	366	+\mu=E(Y_t)=\frac{c}{1-\phi_1+\phi_2-\cdots-\phi_p}
	367	+\end{align*}
	368	+\textbf{On obtient de même une relation de récurrence d’ordre p pour les autocovariances:}
	369	+\begin{align*}
	370	+\gamma(h)=\left\{\begin{matrix}
	371	+\phi_1 \gamma(h-1)+\phi_2\gamma(h-2)+\cdots+\phi_p\gamma(h-p) \hspace{0.5cm} si \hspace{0.5cm} h\ge 1 \\
	372	+\sum_{i=1}^{p}\phi_i\gamma (i)+\sigma^2 \hspace{0.5cm} si \hspace{0.5cm} h=0
	373	+\end{matrix}\right.
	374	+\end{align*}
	375	+\textbf{La même relation de récurrence vérifiée par les autocorrélations:}
	376	+\begin{align*}
	377	+\rho(h)= \phi_1 \rho(h-1)+\phi_2\rho(h-2)+\cdots+\phi_p\rho(h-p)
	378	+\end{align*}
	379	+\end{block}
	380	+\end{frame}
	381	+
	382	+\section{Processus ARMA (P,Q)}
	383	+\frame{\tableofcontents[current]}
	384	+\begin{frame}{Processus ARMA (P,Q)}
	385	+\begin{itemize}
	386	+\item Un processus $\left\{ Y_t \right\}$ qui contient à la fois des termes autorégressifs à l’ordre p et des termes moyennes mobiles à l’ordre q est dit processus ARMA$(p,q)$.
	387	+\item Le processus ARMA (p, q) vérifie une équation en différence stochastique du type :
	388	+\begin{align*}
	389	+Y_t=c+\phi_1 Y_{t-1}+ \phi_2 Y_{t-2}+ \cdots +\phi_p Y_{t-p} \\ +\epsilon_t+\theta \epsilon_{t-1}+ \theta_2 \epsilon_{t-2}+\cdots+\theta_q \epsilon_{t-q}
	390	+\end{align*}
	391	+\begin{align*}
	392	+(1-L\phi_1-L^2\phi_2-\cdots- L^p)Y_t \\ =c+(1+\theta_1L +\theta_2L^2+\cdots+\theta_qL^q)\epsilon_t
	393	+\end{align*}
	394	+\end{itemize}
	395	+\end{frame}
	396	+
	397	+\begin{frame}{Processus ARMA (P,Q)}
	398	+\begin{itemize}
	399	+\item La partie moyenne mobile du processus ARMA(p, q) étant toujours stationnaire, le processus est stationnaire si la partie autorégressive est stationnaire.
	400	+\item Donc, la condition de stationnarité du processus ARMA(p, q) est que les racines de l’équation
	401	+\begin{align*}
	402	+\lambda^p-\phi_1\lambda^{p-1}-\phi_2\lambda^{p-2}-\cdots -\phi_{p-1}\lambda-\phi_p
	403	+\end{align*}
	404	+sont de module inférieur à l’unité.
	405	+\item Dans ce cas, la moyenne est
	406	+\begin{align*}
	407	+\mu=E(Y_t)=\frac{c}{1-\phi_1+\phi_2-\cdots-\phi_p}
	408	+\end{align*}
	409	+\end{itemize}
	410	+\end{frame}
	411	+
	412	+
	413	+\begin{frame}{Processus ARMA (P,Q)}
	414	+\begin{itemize}
	415	+\item Pour identifier les ordres $p$ et/ou $q$ d’un processus $ARMA(p,q)$ stationnaire, on peut fort utilement se servir des
	416	+\begin{itemize}
	417	+\item Fonctions d’autocorrélation d’échantillon (graphiques corrélogramme et ACF).
	418	+\item Fonctions d’autocorrélation partielle d’échantillon (graphiques corrélogramme et PACF).
	419	+
	420	+\end{itemize}
	421	+\end{itemize}
	422	+\end{frame}
	423	+
	424	+\begin{frame}{Processus ARMA (P,Q)}
	425	+\begin{itemize}
	426	+\item On utilise les propriétés suivantes:
	427	+\begin{enumerate}
	428	+\item Les autocorrélations d’un processus $ARMA(p,q)$ commencent à décroître et tendre vers 0 après q retards.
	429	+\item Les autocorrélations partielles d’un processus $ARMA(p,q)$ commencent à décroître et tendre vers 0 après p retards.
	430	+\item Pour un processus $AR(p)$ , les autocorrélations partielles deviennent abruptement nulles (donc très faibles et non significatives) après p retards.
	431	+\item Pour un processus $MA(q)$, les autocorrélations deviennent abruptement nulles (donc très faibles et non significatives) après q retards.
	432	+\end{enumerate}
	433	+\end{itemize}
	434	+\end{frame}
	435	+
	436	+\begin{frame}{Processus ARMA (P,Q)}
	437	+\begin{itemize}
	438	+\item Lorsqu’on sort de cette étape d’identification avec plus d’un modèle, on peut utiliser des critères de sélection de modèles pour choisir un modèle parmi ceux qui sont potentiellement candidats pour expliquer les données.
	439	+\item Les deux critères de sélection de modèles les plus utilisés sont
	440	+\begin{itemize}
	441	+\item Le critère d’information d’Akaike \textbf{(Akaike Information Criterion, AIC)}
	442	+\item Le critère bayésien de Schwartz \textbf{(Schwartz Bayesian Criterion, SBC)}.
	443	+
	444	+\end{itemize}
	445	+\end{itemize}
	446	+\end{frame}
	447	+
	448	+
	449	+\begin{frame}{Processus ARMA (P,Q)}
	450	+\begin{itemize}
	451	+\item Pour un modèle donné, on procède à l’estimation et on calcule à l’aide des formules suivantes :
	452	+\begin{align*}
	453	+AIC = T \times ln\textbf{(somme des carrés des résidus)} + 2n
	454	+\end{align*}
	455	+\begin{align*}
	456	+SBC = T \times ln\textbf{(somme des carrés des résidus)} + n ln(T)
	457	+\end{align*}
	458	+\begin{itemize}
	459	+\item Où $T$ est le nombre d’observations utilisées pour faire l’estimation, n est le nombre de paramètres estimés
	460	+\item Pour un $ARMA(p,q)$ , $n = p + q+ \textbf{possible intercepte}$.
	461	+\end{itemize}
	462	+\item Les valeurs de AIC et SBC devraient être les plus faibles possible (à meilleur ajustement, une plus grande proximité entre observations et ajustements, donc plus faibles résidus).
	463	+\item Donc, en comparant deux modèles à l’aide de l’un de ces critères, on choisira le modèle pour lequel le critère utilisé est le plus petit.
	464	+\end{itemize}
	465	+\end{frame}
	466	+
	467	+\section{Processus ARIMA (P,D,Q)}
	468	+\frame{\tableofcontents[current]}
	469	+\begin{frame}{Processus ARIMA (P,D,Q)}
	470	+\begin{itemize}
	471	+\item Si un processus ARMA(P,Q) n’est pas stationnaire, on peut
	472	+\begin{itemize}
	473	+\item Introduire une fonction du temps pour enlever une éventuelle tendance.
	474	+\item Le différencier.
	475	+\end{itemize}
	476	+\item En supposant que la série non stationnaire est intégrée d’ordre d, la série obtenue après d différences premières est stationnaire.
	477	+\item Si cette nouvelle série peut être modélisée comme réalisation d’un processus $ARMA(p,q)$, on dit que la série initiale suit un processus $ARIMA(p, d, q)$.
	478	+\item Dans cette représentation, le $I$ est pour intégré, et le $d$ est pour l’ordre d’intégration.
	479	+\item Le processus $ARIMA(p,d,q)$ s’étudie de la même manière qu’un processus $ARMA(p,q)$ après l’avoir différencié $d$ fois.
	480	+\end{itemize}
	481	+\end{frame}
	482	+
	483	+\section{L’analyse Box-Jenkins}
	484	+\frame{\tableofcontents[current]}
	485	+\begin{frame}{L’analyse Box-Jenkins}
	486	+\begin{itemize}
	487	+\item L’analyse Box-Jenkins fournit une méthodologie pour identifier le type du processus qui nous intéresse. Les quatre étapes de la méthode de Box-Jenkins sont :
	488	+\begin{enumerate}
	489	+\item Identification
	490	+\item Estimation
	491	+\item Diagnostique
	492	+\item Prévision
	493	+\end{enumerate}
	494	+\end{itemize}
	495	+\end{frame}
	496	+
	497	+
	498	+\begin{frame}{L’analyse Box-Jenkins}
	499	+\begin{block}{Identification}
	500	+\begin{itemize}
	501	+\item La première chose à faire est de représenter la série par un graphique temporel pour voir son évolution dans le temps.
	502	+\begin{itemize}
	503	+\item C’est une étape qui vise à identifier visuellement si la série a des bris, une tendance, des clusters de volatilité, etc.
	504	+\end{itemize}
	505	+\item Testez la stationnarité: On doit avoir une série stationnaire pour procéder à l’analyse Box-Jenkins.
	506	+\begin{itemize}
	507	+\item On utilise donc le test de Dickey-Fuller (ou Philipps-Perron) approprié pour investiguer la stationnarité.
	508	+\item Si la série n’est pas stationnaire, il faut la différencier.
	509	+\end{itemize}
	510	+\item Il faut ensuite estimer et évaluer les autocorrélations et les autocorrélations partielles.
	511	+\begin{itemize}
	512	+\item Elles peuvent difficile à étudier puisqu’il faut se rappeler que vous observez une réalisation du processus et non le processus lui-même et que les autocorrélations et autocorrélations partielles sont estimées elle aussi.
	513	+\end{itemize}
	514	+\end{itemize}
	515	+\end{block}
	516	+\end{frame}
	517	+
	518	+
	519	+\begin{frame}{L’analyse Box-Jenkins}
	520	+\begin{block}{Identification}
	521	+\begin{itemize}
	522	+
	523	+\item Il faut ensuite estimer et évaluer les autocorrélations et les autocorrélations partielles.
	524	+\begin{itemize}
	525	+\item En général, on reconnait les évolutions suivantes sur les processus:
	526	+\begin{table}[]
	527	+\begin{tabular}{@{}\|l\|l\|l\|@{}}
	528	+\toprule
	529	+\textbf{Type de modèle} & \textbf{Autocorrélations} & \textbf{\begin{tabular}[c]{@{}l@{}}Autocorrélations \\ partielles\end{tabular}} \\ \midrule
	530	+\textbf{AR(p)} & \begin{tabular}[c]{@{}l@{}}Décrois \\ exponentiellement \\ ou en oscillant\end{tabular} & \begin{tabular}[c]{@{}l@{}}Significatif \\ jusqu’à p\end{tabular} \\ \midrule
	531	+\textbf{MA(q)} & \begin{tabular}[c]{@{}l@{}}Significatif \\ jusqu’à q\end{tabular} & \begin{tabular}[c]{@{}l@{}}Décrois \\ exponentiellement\end{tabular} \\ \midrule
	532	+\textbf{ARMA (p,q)} & \begin{tabular}[c]{@{}l@{}}Décrois \\ exponentiellement \\ ou en oscillant \\ après le lag q.\end{tabular} & \begin{tabular}[c]{@{}l@{}}Décrois \\ exponentiellement \\ ou en oscillant \\ après le lag p.\end{tabular} \\ \bottomrule
	533	+\end{tabular}
	534	+\end{table}
	535	+\end{itemize}
	536	+\end{itemize}
	537	+\end{block}
	538	+\end{frame}
	539	+
	540	+
	541	+\begin{frame}{L’analyse Box-Jenkins}
	542	+\begin{block}{Identification}
	543	+\begin{itemize}
	544	+\item Box-Jenkins suggère d’avoir les modèles les plus parcimonieux possible.
	545	+\item Dans l’analyse des autocorrélations et des autocorrélations partielles, il convient de prioriser les choix de modèle le plus parcimonieux sans toutefois négliger des retards significatifs.
	546	+\end{itemize}
	547	+\end{block}
	548	+\end{frame}
	549	+
	550	+\begin{frame}{L’analyse Box-Jenkins}
	551	+\begin{block}{Estimation}
	552	+\begin{itemize}
	553	+\item Estimez le modèle retenu.
	554	+\item Lorsque plusieurs modèles ont été retenus à l’étape précédente, il faut tous les estimer.
	555	+\item On choisit le meilleur modèle avec les critères de décision disponibles : $R^2$, test $F$ ou test $LR$ si vous tester des modèles emboités, $AIC/SBC$.
	556	+\end{itemize}
	557	+\end{block}
	558	+\end{frame}
	559	+
	560	+
	561	+\begin{frame}{L’analyse Box-Jenkins}
	562	+\begin{block}{Diagnostique}
	563	+\begin{itemize}
	564	+\item Faire un graphique des résidus du modèle choisi.
	565	+\begin{itemize}
	566	+\item Il peut être plus facile de travailler avec des résidus standardisés par l’écart type pour voir les observations extrêmes.
	567	+\end{itemize}
	568	+\item Tester si les résidus présentent : des autocorrélations, des effets ARCH ou de l’hétéroscédasticité.
	569	+\begin{itemize}
	570	+\item Testez que les résidus sont bruits blancs avec un test portemanteau.
	571	+\end{itemize}
	572	+\end{itemize}
	573	+\end{block}
	574	+\end{frame}
	575	+
	576	+
	577	+\begin{frame}{L’analyse Box-Jenkins}
	578	+\begin{block}{Prévision}
	579	+\begin{itemize}
	580	+\item À partir du processus identifié, on peut faire de la prévision (souvent utilisé pour e court terme).
	581	+\item Cette partie est présentée plus en détail dans les notes de la section 5.
	582	+\end{itemize}
	583	+\end{block}
	584	+\end{frame}
	585	+
	586	+\end{document}
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	4	+\beamer@sectionintoc {4}{Processus ARIMA (P,D,Q)}{34}{0}{4}
	5	+\beamer@sectionintoc {5}{L’analyse Box-Jenkins}{36}{0}{5}

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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage{graphicx}
	9	+\usetheme{default}
	10	+\usecolortheme{default}
	11	+
	12	+\title[S02 Régression et MCO]{Section 02 : Régression et Moindres Carrés Ordinaires\\ (Séance 2)}
	13	+\subtitle{GSF-6053: Économétrie Financière}
	14	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	15	+\institute[Université Laval]
	16	+{
	17	+ \inst{1}%
	18	+ Département de finance, assurance et immobilier\\
	19	+ Faculté des sciences de l'administration\\
	20	+ Université Laval}
	21	+\date[Hiver 2022]{18 Janvier 2022}
	22	+
	23	+\begin{document}
	24	+
	25	+\begin{frame}
	26	+ \titlepage
	27	+\end{frame}
	28	+
	29	+\begin{frame}{Références}
	30	+\textbf{Obligatoires:}
	31	+\begin{itemize}
	32	+\item \textbf{Notes de cours:} Section 2 (Professeure: Marie-Hélène Gagnon)
	33	+\item \textbf{Woolridge:} chapitres 2 à 7
	34	+\end{itemize}
	35	+\vspace{0.5cm}
	36	+\textbf{Complémentaires:}
	37	+\begin{itemize}
	38	+\item \textbf{Gujarati et Porter:} chapitres 1 à 9.
	39	+\item \textbf{Greene:} chapitres 2, 3, 4, 5, 9, 14, 20, appendices C et D
	40	+\end{itemize}
	41	+\end{frame}
	42	+
	43	+\begin{frame}{Plan de la séance}
	44	+ \tableofcontents
	45	+\end{frame}
	46	+
	47	+\section{Modèle de Régression}
	48	+\frame{\tableofcontents[current]}
	49	+
	50	+\begin{frame}{Modèle de Régression}
	51	+\begin{itemize}
	52	+\item Espérance conditionnelle d'une variable indépendante \textbf{(X)} conditionnelle à la valeur connue des régresseurs \textbf{(Y)}
	53	+
	54	+\begin{align*}
	55	+E(Y \vert X_i)=f(X_i)
	56	+\end{align*}
	57	+Où $f(X_i)$ est une fonction des variables explicatives $\textbf{X}$
	58	+\vspace{0.5cm}
	59	+\item Relation conditionnelle linéaire
	60	+
	61	+\begin{align*}
	62	+E(Y \vert X_i)=\beta_1 + \beta_2 X_i
	63	+\end{align*}
	64	+\item Composate de $Y$:
	65	+\begin{itemize}
	66	+\item Partie systématique ou déterministe, $E(Y\vert X_i)$
	67	+\item Partie aléatoire et non systématique, $\mu_i$
	68	+\end{itemize}
	69	+\end{itemize}
	70	+\end{frame}
	71	+
	72	+\begin{frame}{Modèle de Régression}
	73	+\begin{itemize}
	74	+\item Modèle de Régression Linéaire Simple
	75	+\begin{align*}
	76	+Y_i = \beta_1 + \beta_2 X_i + \mu_i
	77	+\end{align*}
	78	+\item Prendre espérance des deux cotés
	79	+\begin{align*}
	80	+E(Y_i \vert X_i)=E(Y \vert X_i)+E(\mu_i)
	81	+\end{align*}
	82	+Sachant $E(Y_i \vert X_i)=E(Y \vert X_i)$ alors $E(\mu_i)=0$
	83	+\item Les paramètres obtenus à partir d'un échantillon sont des estimateurs (notés avec \hspace{0.1cm} $\hat{}$ \hspace{0.1cm})
	84	+\begin{align*}
	85	+\hat{Y}_i=\hat{\beta}_1+\hat{\beta}_2 X_i
	86	+\end{align*}
	87	+\begin{align*}
	88	+Y_i = \hat{\beta}_1+\hat{\beta}_2 X_i+\hat{\mu}_i
	89	+\end{align*}
	90	+$\hat{\mu}_i$ représente l'estimateur du terme d'erreur, le résidu
	91	+\end{itemize}
	92	+\end{frame}
	93	+
	94	+\section{Moindres Carrés Ordinaires (En sommation)}
	95	+
	96	+\frame{\tableofcontents[current]}
	97	+
	98	+\begin{frame}{Moindres Carrés Ordinaires (En sommation)}
	99	+\begin{itemize}
	100	+\item Définissons les résidus
	101	+\begin{align*}
	102	+\hat{\mu}_i = Y_i - \hat{Y}_i = Y_i - \hat{\beta}_1-\hat{\beta}_2 X_i
	103	+\end{align*}
	104	+\item Choisir estimateurs de $\hat{\beta}_1$ et $\hat{\beta_2}$ de facon à minimiser la somme des carrés des résidus
	105	+\begin{align*}
	106	+\sum_{i=1}^{N} \hat{\mu}_i^2 & = \left[\hat{\mu}_1^2 +\hat{\mu}_2^2+...+\hat{\mu}_N^2 \right] \\ & = \sum_{i=1}^{N}(Y_i-\hat{\beta}_1-\hat{\beta}_2 X_i)^2
	107	+\end{align*}
	108	+\end{itemize}
	109	+\end{frame}
	110	+
	111	+\begin{frame}{Moindres Carrés Ordinaires (En sommation)}
	112	+\begin{itemize}
	113	+\item Minimiser la somme des carrés des résidus par rapport à $\hat{\beta}_1$ et $\hat{\beta}_2$
	114	+\begin{align*}
	115	+\min_{\hat{\beta}_1,\hat{\beta}_2} \sum_{i=1}^{N}(Y_i-\hat{\beta}_1-\hat{\beta}_2 X_i)^2=\min_{\hat{\beta}_1,\hat{\beta}_2} \sum_{i=1}^{N} \hat{\mu}_i^2
	116	+\end{align*}
	117	+\item Équation normales
	118	+\begin{equation}
	119	+\frac{\partial \sum_{i=1}^{N} \hat{\mu}_i^2}{\partial \hat{\beta_1}}=-2\sum_{i=1}^N(Y_i-\hat{\beta}_1-\hat{\beta}_2 X_i)=0
	120	+\end{equation}
	121	+\begin{equation}
	122	+\frac{\partial \sum_{i=1}^{N} \hat{\mu}_i^2}{\partial \hat{\beta_2}}=-2\sum_{i=1}^NX_i(Y_i-\hat{\beta}_1-\hat{\beta}_2 X_i)=0
	123	+\end{equation}
	124	+\end{itemize}
	125	+\end{frame}
	126	+
	127	+\begin{frame}{Moindres Carrés Ordinaires (En sommation)}
	128	+\begin{itemize}
	129	+\item Solution de $\hat{\beta}_1$ à l'aide de l'équation (1)
	130	+\begin{align*}
	131	+-2\sum_{i=1}^N(Y_i-\hat{\beta}_1-\hat{\beta}_2 X_i)=0
	132	+\end{align*}
	133	+\begin{align*}
	134	+\sum_{i=1}^N Y_i - N\hat{\beta}_1-\hat{\beta}_2 \sum_{i=1}^N X_i=0
	135	+\end{align*}
	136	+\begin{align*}
	137	+N\hat{\beta}_1=\sum_{i=1}^N Y_i - \hat{\beta}_2 \sum_{i=1}^N X_i
	138	+\end{align*}
	139	+\end{itemize}
	140	+\end{frame}
	141	+
	142	+\begin{frame}{Moindres Carrés Ordinaires (En sommation)}
	143	+\begin{itemize}
	144	+\item Solution de $\hat{\beta}_1$ à l'aide de l'équation (1)
	145	+
	146	+\vspace{0.5cm}
	147	+On pose la propriété suivante:
	148	+\begin{itemize}
	149	+\item $\sum_{i=1}^N Y_i=N\overline{Y}$
	150	+\item $\sum_{i=1}^N X_i=N\overline{X}$
	151	+\end{itemize}
	152	+\begin{align*}
	153	+N\hat{\beta}_1 = N\overline{Y} - N \hat{\beta}_2 \overline{X}
	154	+\end{align*}
	155	+\begin{align*}
	156	+\hat{\beta}_1=\frac{N\overline{Y}}{N}-\frac{N\hat{\beta}_2\overline{X}}{N}
	157	+\end{align*}
	158	+
	159	+\begin{block}{Solution de $\hat{\beta}_1$}
	160	+\begin{align*}
	161	+\hat{\beta}_1=\overline{Y}-\hat{\beta}_2 \overline{X}
	162	+\end{align*}
	163	+\end{block}
	164	+\end{itemize}
	165	+\end{frame}
	166	+
	167	+\begin{frame}{Moindres Carrés Ordinaires (En sommation)}
	168	+\begin{itemize}
	169	+\item Solution de $\hat{\beta}_2$ à l'aide de l'équation (2)
	170	+\begin{align*}
	171	+-2\sum_{i=1}^NX_i(Y_i-\hat{\beta}_1-\hat{\beta}_2 X_i)=0
	172	+\end{align*}
	173	+\begin{align*}
	174	+\sum_{i=1}^N X_i Y_i - \hat{\beta}_1 \sum_{i=1}^N X_i -\hat{\beta}_2\sum_{i=1}^N X_i^2=0
	175	+\end{align*}
	176	+Sachant $\overline{Y}-\hat{\beta}_2 \overline{X}$
	177	+\begin{align*}
	178	+\sum_{i=1}^N X_i Y_i - [\overline{Y}-\hat{\beta}_2 \overline{X}] \sum_{i=1}^N X_i -\hat{\beta}_2\sum_{i=1}^N X_i^2=0
	179	+\end{align*}
	180	+\end{itemize}
	181	+\end{frame}
	182	+
	183	+\begin{frame}{Moindres Carrés Ordinaires (En sommation)}
	184	+\begin{itemize}
	185	+\item Solution de $\hat{\beta}_2$ à l'aide de l'équation (2)
	186	+\begin{align*}
	187	+\sum_{i=1}^N X_i Y_i-\overline{Y}\sum_{i=1}^N X_i \hat{\beta}_2 \overline{X} \sum_{i=1}^N X_i -\hat{\beta}_2 \sum_{i=1}^N X_i^2=0
	188	+\end{align*}
	189	+\begin{align*}
	190	+\sum_{i=1}^N X_i Y_i-N \overline{X} \overline{Y}+\hat{\beta}_2 N \overline{X}^2-\hat{\beta}_2 \sum_{i=1}^N X_i^2=0
	191	+\end{align*}
	192	+\begin{align*}
	193	+\sum_{i=1}^N X_i Y_i-N \overline{X} \overline{Y}=\hat{\beta}_2 \sum_{i=1}^N X_i^2-\hat{\beta}_2 N \overline{X}^2
	194	+\end{align*}
	195	+\begin{align*}
	196	+\sum_{i=1}^N X_i Y_i-N \overline{X} \overline{Y}= \hat{\beta}_2 \left[ \sum_{i=1}^N X_i^2- N \overline{X}^2 \right]
	197	+\end{align*}
	198	+\end{itemize}
	199	+\end{frame}
	200	+
	201	+\begin{frame}{Moindres Carrés Ordinaires (En sommation)}
	202	+\begin{itemize}
	203	+\item Solution de $\hat{\beta}_2$ à l'aide de l'équation (2)
	204	+\begin{align*}
	205	+\hat{\beta}_2=\frac{\sum_{i=1}^N X_i Y_i-N \overline{X} \overline{Y}}{\sum_{i=1}^N X_i^2- N \overline{X}^2}
	206	+\end{align*}
	207	+Sachant que
	208	+\begin{itemize}
	209	+\item $\sum_{i=1}^N X_i Y_i-N \overline{X} \overline{Y}=\sum_{i=1}^N(X_i-\overline{X})(Y_i-\overline{Y})$
	210	+\item $\sum_{i=1}^N X_i^2 - N \overline{X}^2 =\sum_{i=1}^N(X_i-\overline{X})^2$
	211	+\end{itemize}
	212	+\begin{block}{Solution de $\hat{\beta}_2$}
	213	+\begin{align*}
	214	+\hat{\beta}_2=\frac{\sum_{i=1}^N(X_i-\overline{X})(Y_i-\overline{Y})}{\sum_{i=1}^N(X_i-\overline{X})^2}
	215	+\end{align*}
	216	+\end{block}
	217	+\end{itemize}
	218	+\end{frame}
	219	+
	220	+\section{Moindres Carrés Ordinaires (Format Matricielle)}
	221	+
	222	+\frame{\tableofcontents[current]}
	223	+
	224	+\begin{frame}{Moindres Carrés Ordinaires (Format Matricielle)}
	225	+\begin{itemize}
	226	+
	227	+\item Vecteur de la variable à expliquer \textbf{(n $\times$ 1)}
	228	+$$Y=\begin{pmatrix}
	229	+y_1\\
	230	+y_2\\
	231	+\vdots\\
	232	+y_n\\
	233	+\end{pmatrix}$$
	234	+\item Matrice de variables explicatives \textbf{(n $\times$ (k+1))}
	235	+$$X=\begin{bmatrix}
	236	+ 1&x_{11} &x_{21} &\cdots &x_{k1} \\
	237	+ 1&x_{12} &\cdots & \cdots &x_{k2} \\
	238	+ 1& \vdots & \vdots & \ddots & \vdots\\
	239	+ 1&x_{1n} & \cdots & \cdots &x_{kn}
	240	+\end{bmatrix}$$
	241	+\end{itemize}
	242	+\end{frame}
	243	+
	244	+\begin{frame}{Moindres Carrés Ordinaires (Format Matricielle)}
	245	+\begin{itemize}
	246	+\item Vecteur de coefficients associés \textbf{(k+1 $\times$ 1)}
	247	+$$\beta=\begin{pmatrix}
	248	+\beta_0\\
	249	+\beta_1\\
	250	+\beta_2\\
	251	+\vdots\\
	252	+\beta_k\\
	253	+\end{pmatrix}$$
	254	+\item Vecteur associé au terme d'erreur \textbf{(n $\times$ 1)}
	255	+$$u=\begin{pmatrix}
	256	+u_1\\
	257	+u_2\\
	258	+\vdots\\
	259	+u_n\\
	260	+\end{pmatrix}$$
	261	+\end{itemize}
	262	+\end{frame}
	263	+
	264	+\begin{frame}{Moindres Carrés Ordinaires (Format Matricielle)}
	265	+\begin{itemize}
	266	+\item Modèle Univarié
	267	+\begin{align*}
	268	+\begin{pmatrix}
	269	+y_1\\
	270	+y_2\\
	271	+\vdots\\
	272	+y_n\\
	273	+\end{pmatrix}=\begin{bmatrix}
	274	+ 1&x_{11} &x_{21} &\cdots &x_{k1} \\
	275	+ 1&x_{12} &\cdots & \cdots &x_{k2} \\
	276	+ 1& \vdots & \vdots & \ddots & \vdots\\
	277	+ 1&x_{1n} & \cdots & \cdots &x_{kn}
	278	+\end{bmatrix} \begin{pmatrix}
	279	+\beta_0\\
	280	+\beta_1\\
	281	+\beta_2\\
	282	+\vdots\\
	283	+\beta_k\\
	284	+\end{pmatrix}+\begin{pmatrix}
	285	+u_1\\
	286	+u_2\\
	287	+\vdots\\
	288	+u_n\\
	289	+\end{pmatrix}
	290	+\end{align*}
	291	+Que nous réécrirons comme suit:
	292	+\begin{align*}
	293	+Y=X \beta + u
	294	+\end{align*}
	295	+\end{itemize}
	296	+\end{frame}
	297	+
	298	+\begin{frame}{Moindres Carrés Ordinaires (Format Matricielle)}
	299	+\begin{itemize}
	300	+\item Hypothèses distributionnelles sur $u$
	301	+\begin{itemize}
	302	+\item $u \sim iid$ et $E(u)=0$
	303	+\item $V(u)=\sigma^2 I$
	304	+\item $u \sim$ iid $N(0,\sigma^2)$
	305	+\end{itemize}
	306	+\begin{align*}
	307	+E(u)=\begin{pmatrix}
	308	+E(u_1)=0\\
	309	+E(u_2)=0\\
	310	+\vdots\\
	311	+E(u_n)=0\\
	312	+\end{pmatrix}
	313	+\end{align*}
	314	+\item Il n'y a pas de corrélation sérielle des erreurs
	315	+\begin{align*}
	316	+V(u)=E(uu')=\sigma^2 I
	317	+\end{align*}
	318	+\item Découle de l'indépendance
	319	+\begin{align*}
	320	+COV=[u_i, u_j]=0
	321	+\end{align*}
	322	+Pour $i \ne j$ $\forall i$ et $\forall j$
	323	+\end{itemize}
	324	+
	325	+\end{frame}
	326	+
	327	+\begin{frame}{Moindres Carrés Ordinaires (Format Matricielle)}
	328	+\begin{align*}
	329	+E(uu')=E\begin{pmatrix}
	330	+u_1\\
	331	+u_2\\
	332	+\vdots\\
	333	+u_n\\
	334	+\end{pmatrix} \begin{pmatrix}
	335	+u_1 &u_2 & \cdots &u_n\\
	336	+\end{pmatrix}
	337	+\end{align*}
	338	+\begin{align*}
	339	+E(uu')=E\begin{pmatrix}
	340	+u_1^2 & u_1 u_2 & \cdots &u_1u_n\\
	341	+u_1u_2 &u_2^2 & \cdots & u_2u_n\\
	342	+\vdots & \vdots & \ddots & \vdots \\
	343	+u_1u_n & u_2u_n & \cdots & u_n^2
	344	+\end{pmatrix}
	345	+\end{align*}
	346	+\begin{align*}
	347	+E(uu')=\begin{pmatrix}
	348	+E(u_1^2) & E(u_1 u_2) & \cdots &E(u_1u_n)\\
	349	+E(u_1u_2) &E(u_2^2) & \cdots & E(u_2u_n)\\
	350	+\vdots & \vdots & \ddots & \vdots \\
	351	+E(u_1u_n) & E(u_2u_n) & \cdots & E(u_n^2)
	352	+\end{pmatrix}
	353	+\end{align*}
	354	+\end{frame}
	355	+
	356	+\begin{frame}{Moindres Carrés Ordinaires (Format Matricielle)}
	357	+\textbf{Rappel:}
	358	+\begin{align*}
	359	+Var(u)=E(u)^2-[E(u)]^2
	360	+\end{align*}
	361	+\textbf{Avec Hypothèse d'indépendance:}
	362	+\begin{align*}
	363	+V(u)= \begin{pmatrix}
	364	+\sigma^2 &\cdots &0 \\
	365	+\vdots &\sigma^2 & \vdots \\
	366	+0 & \cdots & \sigma^2
	367	+\end{pmatrix} = \sigma^2\begin{pmatrix}
	368	+1 &\cdots &0 \\
	369	+\vdots &1 & \vdots \\
	370	+0 & \cdots & 1
	371	+\end{pmatrix}
	372	+\end{align*}
	373	+\end{frame}
	374	+
	375	+
	376	+\begin{frame}{Dérivation MCO (Format Matricielle)}
	377	+\begin{align*}
	378	+\hat{\beta} = ARGMIN_{\beta} (\hat{u}'\hat{u})
	379	+\end{align*}
	380	+\begin{align*}
	381	+\hat{\beta} = ARGMIN_{\beta} ((Y-X \hat{\beta})'(Y-X\hat{\beta}))
	382	+\end{align*}
	383	+Distribution des termes de la multiplication matricielle:
	384	+\begin{align*}
	385	+(Y-X \beta)' (Y-X \beta)=Y'Y-Y'X \hat{\beta}-\hat{\beta}'X'Y +\hat{\beta}'X'X \hat{\beta}
	386	+\end{align*}
	387	+Sachant
	388	+\begin{align*}
	389	+Y'X \hat{\beta}=\hat{\beta}'X'Y
	390	+\end{align*}
	391	+Alors:
	392	+\begin{align*}
	393	+(Y-X \beta)' (Y-X \beta)=Y'Y -2Y'X \hat{\beta}+\hat{\beta}'X'X \hat{\beta}
	394	+\end{align*}
	395	+\end{frame}
	396	+\begin{frame}{Dérivation MCO (Format Matricielle)}
	397	+\begin{itemize}
	398	+\item Dérivé de la somme des résidus au carrés par rapport à $\beta$
	399	+
	400	+\begin{align*}
	401	+\frac{\partial (RSS)}{\partial \hat{\beta}}=\frac{\partial (Y'Y -2Y'X \hat{\beta}+\hat{\beta}'X'X \hat{\beta})}{\partial \hat{\beta}}
	402	+\end{align*}
	403	+Sachant
	404	+\begin{align*}
	405	+\frac{\partial ([Y'X]\hat{\beta})}{\partial \hat{\beta}}=[Y'X]=X'Y
	406	+\end{align*}
	407	+\begin{align*}
	408	+\frac{\partial (\hat{\beta}'[X'X]\hat{\beta})}{\partial \hat{\beta}}=2[X'X]\hat{\beta}
	409	+\end{align*}
	410	+\end{itemize}
	411	+\end{frame}
	412	+
	413	+\begin{frame}{Dérivation MCO (Format Matricielle)}
	414	+\begin{align*}
	415	+\frac{\partial (RSS)}{\partial \hat{\beta}}=-2X'Y+2X'X \hat{\beta}=0
	416	+\end{align*}
	417	+\begin{align*}
	418	+X'Y=X'X\hat{\beta}
	419	+\end{align*}
	420	+\begin{align*}
	421	+\hat{\beta}=(X'X)^{-1}X'Y
	422	+\end{align*}
	423	+\begin{block}{Estimateur de $\hat{\beta}$ par MCO}
	424	+\begin{align*}
	425	+\hat{\beta}=(X'X)^{-1}X'Y
	426	+\end{align*}
	427	+\end{block}
	428	+\end{frame}
	429	+
	430	+\begin{frame}{Dérivation MCO (Format Matricielle)}
	431	+\begin{itemize}
	432	+\item Il n'existe pas d'estimateur de $\sigma^2$ pour la méthode des Moindres carrés ordinaires
	433	+\item On peut estimer empiriquement comme suit
	434	+\begin{align*}
	435	+\hat{\sigma^2}= \frac{1}{T}\sum_{i=1}^n \hat{u}_i=\frac{\hat{u}'\hat{u}}{n}
	436	+\end{align*}
	437	+\end{itemize}
	438	+\end{frame}
	439	+
	440	+\section{Exemple Numérique}
	441	+
	442	+\frame{\tableofcontents[current]}
	443	+
	444	+\begin{frame}{Exemple Numérique}
	445	+\begin{itemize}
	446	+\item Nous allons générer des variables aléatoires à l’aide d’un modèle dont nous allons déterminer les paramètres.
	447	+\item Nous allons utiliser un modèle linéaire simple ayant le format suivant:
	448	+\begin{align*}
	449	+y=\beta_1+\beta_2 x_i + \epsilon_i
	450	+\end{align*}
	451	+\begin{itemize}
	452	+\item L'intercepte de notre modèle sera de $\beta_1 = 60$.
	453	+\item La pente de notre modèle sera de $\beta_2=2$.
	454	+\item Nous allons faire l'hypothèse que notre terme d'erreur suit une loi normale de moyenne 0 et variance 4 ($\epsilon \sim N(0,4)$).
	455	+\end{itemize}
	456	+\item Il nous est maintenant possible de déterminer une valeur pour chaque $y_i$ en utilisant les $x_i$.
	457	+\item Pour les $x_i$ nous allons générer uniformément 40 observations entre 0 et 40.
	458	+\end{itemize}
	459	+\end{frame}
	460	+
	461	+\begin{frame}{Nuage de points et régression linéaire ($\epsilon \sim N(0,2)$)}
	462	+
	463	+\includegraphics[width=10cm, height=8cm]{ols_2.png}
	464	+\end{frame}
	465	+
	466	+\begin{frame}{Nuage de points et régression linéaire ($\epsilon \sim N(0,4)$)}
	467	+
	468	+\includegraphics[width=10cm, height=8cm]{ols_4.png}
	469	+\end{frame}
	470	+
	471	+\begin{frame}{Nuage de points et régression linéaire ($\epsilon \sim N(0,6)$)}
	472	+
	473	+\includegraphics[width=10cm, height=8cm]{ols_6.png}
	474	+\end{frame}
	475	+
	476	+\begin{frame}{Nuage de points et régression linéaire ($\epsilon \sim N(0,8)$)}
	477	+
	478	+\includegraphics[width=10cm, height=8cm]{ols_8.png}
	479	+\end{frame}
	480	+
	481	+\begin{frame}{Exemple Numérique}
	482	+\begin{table}
	483	+\begin{center}
	484	+\begin{tabular}{l c c c c}
	485	+\hline
	486	+ & $\epsilon \sim N(0,2)$ & $\epsilon \sim N(0,4)$ & $\epsilon \sim N(0,6)$ & $\epsilon \sim N(0,8)$ \\
	487	+\hline
	488	+(Intercept) & $59.30^{*}$ & $59.28^{}$ & $60.89^{}$ & $55.68^{*}$ \\
	489	+ & $(0.68)$ & $(1.35)$ & $(2.43)$ & $(2.31)$ \\
	490	+x & $2.02^{*}$ & $2.03^{}$ & $2.00^{}$ & $2.13^{*}$ \\
	491	+ & $(0.03)$ & $(0.06)$ & $(0.10)$ & $(0.10)$ \\
	492	+\hline
	493	+R$^2$ & $0.99$ & $0.96$ & $0.89$ & $0.91$ \\
	494	+Adj. R$^2$ & $0.99$ & $0.96$ & $0.89$ & $0.91$ \\
	495	+Num. obs. & $50$ & $50$ & $50$ & $50$ \\
	496	+\hline
	497	+\multicolumn{5}{l}{\scriptsize{$^{*}p<0.001$; $^{}p<0.01$; $^{*}p<0.05$}}
	498	+\end{tabular}
	499	+\caption{Table de régression}
	500	+\label{table:coefficients}
	501	+\end{center}
	502	+\end{table}
	503	+\end{frame}
	504	+
	505	+\end{document}
	506	+
	507	+
	508	+
	509	+

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	2	+\beamer@sectionintoc {2}{Moindres Carrés Ordinaires (En sommation)}{7}{0}{2}
	3	+\beamer@sectionintoc {3}{Moindres Carrés Ordinaires (Format Matricielle)}{15}{0}{3}
	4	+\beamer@sectionintoc {4}{Exemple Numérique}{26}{0}{4}

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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage{graphicx}
	9	+\usetheme{default}
	10	+\usecolortheme{default}
	11	+
	12	+\title[S02 Régression et MCO]{Section 02 : Régression et Moindres Carrés Ordinaires\\ (Séance 3)}
	13	+\subtitle{GSF-6053: Économétrie Financière}
	14	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	15	+\institute[Université Laval]
	16	+{
	17	+ \inst{1}%
	18	+ Département de finance, assurance et immobilier\\
	19	+ Faculté des sciences de l'administration\\
	20	+ Université Laval}
	21	+\date[Hiver 2022]{25 Janvier 2022}
	22	+
	23	+\begin{document}
	24	+
	25	+\begin{frame}
	26	+ \titlepage
	27	+\end{frame}
	28	+
	29	+
	30	+\begin{frame}{Références}
	31	+\textbf{Obligatoires:}
	32	+\begin{itemize}
	33	+\item \textbf{Notes de cours:} Section 2 (Professeure: Marie-Hélène Gagnon)
	34	+\item \textbf{Woolridge:} chapitres 2 à 7
	35	+\end{itemize}
	36	+\vspace{0.5cm}
	37	+\textbf{Complémentaires:}
	38	+\begin{itemize}
	39	+\item \textbf{Gujarati et Porter:} chapitres 1 à 9.
	40	+\item \textbf{Greene:} chapitres 2, 3, 4, 5, 9, 14, 20, appendices C et D
	41	+\end{itemize}
	42	+\end{frame}
	43	+
	44	+
	45	+\begin{frame}{Plan de la séance}
	46	+ \tableofcontents
	47	+\end{frame}
	48	+
	49	+\section{Maximum de Vraissemblance}
	50	+
	51	+\frame{\tableofcontents[current]}
	52	+
	53	+\begin{frame}{Maximum de Vraissemblance}
	54	+\begin{itemize}
	55	+\item Le maximum de vraisemblance est une méthode générale pour estimer les paramàtres d’un modèle statistique.
	56	+\item Nous aurons une série d’observations d’une variable aléatoire y et un modèle statistique potentiel pour cette variable.
	57	+\begin{itemize}
	58	+\item Ce modèle peut inclure la dépendance de y sur d’autres variables prédictrices.
	59	+\item Ainsi qu’une distribution statistique pour la portion non-expliquée de la variation de y.
	60	+\end{itemize}
	61	+\item Selon le maximum de vraisemblance, les meilleurs estimés des paramètres d’un modèle sont ceux qui maximisent la probabilité des valeurs observées de la variable
	62	+\end{itemize}
	63	+\end{frame}
	64	+
	65	+\begin{frame}{Maximum de Vraissemblance}
	66	+\begin{itemize}
	67	+\item Fonction de densité pour une variable aléatoire $Y$ conditionné sur un ensemble de paramètres $\theta$
	68	+\begin{align*}
	69	+f(Y \mid \theta)
	70	+\end{align*}
	71	+\item La fonction de densité jointe de $n$ observations est simplement le produit des densités individuels
	72	+
	73	+\begin{align*}
	74	+f(y_1,y_2,...,y_n \mid \theta)=\prod_{i=1}^{n} f(y_i \mid \theta)=L(\theta \mid Y)
	75	+\end{align*}
	76	+\begin{itemize}
	77	+\item Ensemble de paramètres $\theta$ est inconnu.
	78	+\item Nous voulons les valeurs de $Y$ provenant de l'échantillon.
	79	+\end{itemize}
	80	+\item Donner à $\theta$ la valeur qui maximise la probabilité d'obtenir un échantillon identique à celui qu'on dispose.
	81	+\end{itemize}
	82	+\end{frame}
	83	+
	84	+\section{MLE: Modèle de régression linéaire}
	85	+
	86	+\frame{\tableofcontents[current]}
	87	+
	88	+
	89	+\begin{frame}{MLE: Modèle de régression linéaire}
	90	+\begin{itemize}
	91	+\item Hypothèse distributionnelle sur les $u_t$
	92	+\begin{align*}
	93	+u_t \sim \hspace{0.1cm} iid \hspace{0.1cm} N(0,\sigma^2)
	94	+\end{align*}
	95	+\item Loi conjointe des $u_1, u_2, ..., u_T$
	96	+\begin{align*}
	97	+f(u_1,u_2,...,u_T)=\prod_{t=1}^{T}f(u_t)=\left(\frac{1}{\sqrt{2 \pi \sigma^2}}\right)^T e^{-\frac{\sum_{t=1}^T(u_t)^2}{2 \sigma^2}}
	98	+\end{align*}
	99	+\item Interessés à la loi coinjointes des $y_t$
	100	+\item Effectué un changement de variable:
	101	+\begin{align*}
	102	+u_t=Y_t-X_t^{'} \beta
	103	+\end{align*}
	104	+\end{itemize}
	105	+\end{frame}
	106	+
	107	+\begin{frame}{MLE: Modèle de régression linéaire}
	108	+\begin{itemize}
	109	+\item Loi conjointe des $Y_t$ (Vraissemblance)
	110	+\begin{align*}
	111	+f(Y_1,Y_2,...,Y_T)=&\prod_{t=1}^{T}g(Y_t)=\left(\frac{1}{\sqrt{2 \pi \sigma^2}}\right)^T e^{-\frac{\sum_{t=1}^T(Y_t-X_t^{'} \beta)^2}{2 \sigma^2}} \\ & = (2 \pi \sigma^2)^{-T/2} e^{-\frac{(Y-X \beta)'(Y-X \beta)}{2\sigma^2}}
	112	+\end{align*}
	113	+\item On veut enseuite obtenir la Log-Vraissemblance en prenant le logarithme de la fonction de vraissemblance.
	114	+\begin{align*}
	115	+L= -\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\sigma^2)-\frac{1}{2} \times \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2}
	116	+\end{align*}
	117	+\end{itemize}
	118	+\end{frame}
	119	+
	120	+\begin{frame}{MLE: Modèle de régression linéaire}
	121	+\begin{itemize}
	122	+\item Maximiser la log-vraissemblance en fonction de $\beta$ et $\sigma^2$
	123	+\item \textbf{En fonction de $\beta$}
	124	+\begin{align*}
	125	+\frac{\partial L}{\partial \beta} = \frac{1}{\sigma^2} \times \frac{\partial (Y'Y =2Y'X\beta +\beta'X'X \beta)}{\partial \beta}=0
	126	+\end{align*}
	127	+\textbf{Condition de première ordre:}
	128	+\begin{align*}
	129	+-2X'Y+2X'X\beta=0
	130	+\end{align*}
	131	+\begin{align*}
	132	+-X'Y+X'X\beta=0
	133	+\end{align*}
	134	+\begin{align*}
	135	+X'X\beta = X'Y
	136	+\end{align*}
	137	+\textbf{Estimateur $\beta$ par MLE:}
	138	+\begin{align*}
	139	+\hat{\beta}= (X'X)^{-1}X'Y
	140	+\end{align*}
	141	+\end{itemize}
	142	+\end{frame}
	143	+
	144	+\begin{frame}{MLE: Modèle de régression linéaire}
	145	+\begin{itemize}
	146	+\item \textbf{En fonction de $\sigma^2$}
	147	+\begin{align*}
	148	+\frac{\partial L}{\partial \sigma^2}=\frac{\partial \left[-\frac{T}{2} \log (\sigma^2)-\frac{1}{2} \times \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2} \right]}{\partial \sigma^2}=0
	149	+\end{align*}
	150	+\item Nous allons dériver par rapport à $\sigma^2$ en deux parties
	151	+\begin{itemize}
	152	+\item 1er terme:
	153	+\begin{align*}
	154	+\frac{\partial \left[-\frac{T}{2} \log (\sigma^2) \right]}{\partial \sigma^2}=-\frac{T}{2} \times \frac{1}{\sigma^2}
	155	+\end{align*}
	156	+\end{itemize}
	157	+\begin{itemize}
	158	+\item 2e terme:
	159	+\begin{align*}
	160	+\frac{\partial \left[-\frac{1}{2} \times \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2} \right]}{\partial \sigma^2}=\frac{\partial \left[-\frac{1}{2} \times (\sigma^2)^{-1}(Y-X \beta)'(Y-X \beta) \right]}{\partial \sigma^2}
	161	+\end{align*}
	162	+\end{itemize}
	163	+\end{itemize}
	164	+\end{frame}
	165	+
	166	+\begin{frame}{MLE: Modèle de régression linéaire}
	167	+\begin{itemize}
	168	+\item \textbf{En fonction de $\sigma^2$}
	169	+\begin{itemize}
	170	+\item Suite pour le 2e terme:
	171	+\end{itemize}
	172	+\begin{align*}
	173	+\frac{\partial \left[-\frac{1}{2} \times (\sigma^2)^{-1}(Y-X \beta)'(Y-X \beta) \right]}{\partial \sigma^2}\\ = \frac{T}{2} \times (\sigma^2)^{-2}(Y-X \beta)'(Y-X \beta) \\ =\frac{1}{2} \times \sigma^{-4}(Y-X\beta)'(Y-X \beta) \\ = \frac{T}{2} \times \frac{(Y-X\beta)'(Y-X \beta)}{\sigma^4}
	174	+\end{align*}
	175	+\end{itemize}
	176	+\end{frame}
	177	+
	178	+\begin{frame}{MLE: Modèle de régression linéaire}
	179	+\begin{itemize}
	180	+\item \textbf{En fonction de $\sigma^2$}
	181	+\begin{itemize}
	182	+\item On additionne les deux termes et nous avons notre condition de première ordre
	183	+\end{itemize}
	184	+\begin{align*}
	185	+\frac{\partial L}{\partial \sigma^2}= -\frac{T}{2} \times \frac{1}{\sigma^2}+ \frac{1}{2} \times \frac{(Y-X\beta)'(Y-X \beta)}{\sigma^4}
	186	+\end{align*}
	187	+\item On veut maintenant isoler $\sigma^2$ pour obtenir l'estimateur $\hat{\sigma}^2$
	188	+\begin{align*}
	189	+\frac{1}{2} \times \frac{(Y-X\beta)'(Y-X \beta)}{\sigma^4}=\frac{T}{2} \times \frac{1}{\sigma^2}
	190	+\end{align*}
	191	+\item On peut multiplier par $2 \sigma^2$ de chaque coté pour simplifier
	192	+\begin{align*}
	193	+\frac{(Y-X\beta)'(Y-X \beta)}{\sigma^2}=T
	194	+\end{align*}
	195	+\textbf{Estimateur $\hat{\sigma}^2$ pour la méthode des MLE}
	196	+\begin{align*}
	197	+\hat{\sigma}^2=\frac{(Y-X \beta)'(Y-X \beta)}{T}
	198	+\end{align*}
	199	+\end{itemize}
	200	+\end{frame}
	201	+
	202	+
	203	+\section{Propriétés Estimateur MLE ET MCO}
	204	+
	205	+\frame{\tableofcontents[current]}
	206	+
	207	+
	208	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	209	+\begin{itemize}
	210	+\item Pour $\hat{\beta}$ des MCO et MLE, leurs résultats coincident
	211	+\end{itemize}
	212	+\begin{block}{Estimateur \textbf{BLUE}}
	213	+\begin{itemize}
	214	+\item Best linear unbiased estimator
	215	+\begin{itemize}
	216	+\item Estimateur sans biais
	217	+\item Estimateur ayant une variance minimal
	218	+\end{itemize}
	219	+\item L'estimateur des moindres carrés ordinaires est BLUE
	220	+\item L'estimateur $\hat{\sigma}^2$ du Maximum de vraissemblance est biaisé vers le bas
	221	+\item On peut trouver une alternative sans biais
	222	+\begin{align*}
	223	+\hat{S}^2=\frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{T-K}
	224	+\end{align*}
	225	+\end{itemize}
	226	+\end{block}
	227	+\end{frame}
	228	+
	229	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	230	+\begin{block}{Estimateur \textbf{BLUE}}
	231	+\begin{itemize}
	232	+\item \textbf{Sans biais} : Espérance de l'estimateur égale à la vraie valeur du paramètre
	233	+\begin{align*}
	234	+E(\hat{\theta})=\theta
	235	+\end{align*}
	236	+\item \textbf{Efficace:} Si l'estimateur atteint la borne de Crameur-Rao (autrement dit, l'inverse de la matrice d'information de fisher)
	237	+\item On veut un estimateur ayant la variance la plus petite possible
	238	+\begin{itemize}
	239	+\item Cela donne une meilleur pécision
	240	+\end{itemize}
	241	+\end{itemize}
	242	+\end{block}
	243	+\end{frame}
	244	+
	245	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	246	+\begin{itemize}
	247	+\item Borne de Cramer-Rao : Pour tout estimateur régulier et sans biais, sa variance
	248	+est bornée par l’inverse de la matrice d’information.
	249	+\item La matrice d’information est quant à elle une façon de mesurer la quantité
	250	+d’information sur les paramètres dans $\theta$ contenue dans $X$.
	251	+\item Une définition équivalente serait que la variance d’un estimateur sans biais
	252	+sera toujours au moins aussi grande que l’inverse de la matrice d’information :
	253	+\end{itemize}
	254	+\begin{block}{Inverse de la matrice d'information}
	255	+\begin{align*}
	256	+[I(\theta)]^{-1} &=\left( -E \left[\frac{\partial^2 \log L(\theta)}{\partial \theta^2} \right] \right)^{-1} \\ &=\left( E \left[ \left(\frac{\partial \log L(\theta)}{\partial \theta}\right)^2 \right] \right)^{-1}
	257	+\end{align*}
	258	+\end{block}
	259	+\end{frame}
	260	+
	261	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	262	+
	263	+\begin{block}{Matrice d'information et Hessienne}
	264	+\begin{itemize}
	265	+\item La matrice d'information est simplement une matrice hessienne d'une d'une fonction.
	266	+\item Il s'agit essentiellement d'une matrice de dérivé seconde:
	267	+\item On suppose une fonction $f(x_1, x_2)$ et on représente la hessienne de cette fonction par $H_{i,j}(f)$
	268	+\item On doit représenter $H_{i,j}(f)$ comme étant l'ensemble des dérivés secondes partiels possible.
	269	+\begin{align*}
	270	+H_{i,j}=\frac{\partial^2 f}{\partial x_i \partial x_j}
	271	+\end{align*}
	272	+\item Il y aura donc 4 dérivés secondes partiels possibles
	273	+\begin{itemize}
	274	+\item $i=1$ et $j=1$, alors $\partial x_1^2$
	275	+\item $i=1$ et $j=2$, alors $\partial x_1 \partial x_2$
	276	+\item $i=2$ et $j=1$, alors $\partial x_2 \partial x_1$
	277	+\item $i=2$ et $j=2$, alors $\partial x_2^2$
	278	+\end{itemize}
	279	+\end{itemize}
	280	+\end{block}
	281	+\end{frame}
	282	+
	283	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	284	+\begin{block}{Matrice d'information}
	285	+\textbf{Dans le cas du modèle linéaire estimé par MLE}
	286	+\begin{itemize}
	287	+\item On aura 4 dérivés seconde étant donnée que nous avons deux paramètres à estimer, soit $\beta$ et $\sigma^2$.
	288	+\begin{itemize}
	289	+\item \textbf{En haut à gauche:} $\partial \beta \partial \beta$
	290	+\item \textbf{En haut à droite:} $\partial \beta \partial \sigma^2$
	291	+\item \textbf{En bas à gauche:} $\partial \sigma^2 \partial \beta$
	292	+\item \textbf{En bas à droite:} $(\partial \sigma^2)^2$
	293	+\end{itemize}
	294	+
	295	+\begin{align*}
	296	+I(\beta, \sigma^2) = -E\begin{bmatrix}
	297	+\left( \frac{\partial^2 L}{\partial \beta \partial \beta'}\right) & \left( \frac{\partial^2 L}{\partial \beta \partial \sigma^2}\right) \\
	298	+\left( \frac{\partial^2 L}{\partial \sigma^2 \partial \beta'}\right) & \left( \frac{\partial^2 L}{(\partial \sigma^2)^2}\right)
	299	+\end{bmatrix}
	300	+\end{align*}
	301	+\item Nous allons maintenant résoudre les 4 dérivés secondes partiels possibles:
	302	+\end{itemize}
	303	+\end{block}
	304	+\end{frame}
	305	+
	306	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	307	+\begin{block}{En haut à gauche}
	308	+\begin{align*}
	309	+\frac{\partial^2 L}{\partial \beta \partial \beta'} & =\frac{\partial^2 \left[ -\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\sigma^2)-\frac{1}{2} \times \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2} \right]}{\partial \beta \partial \beta'} \\ & = \frac{\partial \left[-\frac{1}{2 \sigma^2} \times (-2X'Y+2X'X \hat{\beta}) \right]}{\partial \beta} \\ & = -\frac{1}{\sigma^2}(X'X)
	310	+\end{align*}
	311	+\end{block}
	312	+
	313	+\end{frame}
	314	+
	315	+
	316	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	317	+\begin{block}{En haut à droite}
	318	+\begin{align*}
	319	+\frac{\partial^2 L}{\partial \beta \partial \sigma^2} & =\frac{\partial^2 \left[ -\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\sigma^2)-\frac{1}{2} \times \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2} \right]}{\partial \beta \partial \sigma^2} \\ & = \frac{\partial \left[-\frac{1}{2 \sigma^2} \times (-2X'Y+2X'X \hat{\beta}) \right]}{\partial \sigma^2} \\ & = \frac{1}{2 \sigma^4} \times (-2X'Y + 2X'X\hat{\beta}) \\ & = \frac{1}{\sigma^4} \times (X'Y -X'X \hat{\beta}) \\ &= -\frac{1}{\sigma^4} \times (X'[Y-X \hat{\beta}]) \\ & = -\frac{1}{\sigma^4}(X'u)
	320	+\end{align*}
	321	+Sachant $Y-X \hat{\beta}=u$
	322	+
	323	+\end{block}
	324	+
	325	+\end{frame}
	326	+
	327	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	328	+\begin{block}{En bas à gauche}
	329	+\begin{align*}
	330	+\frac{\partial^2 L}{\partial \sigma^2 \partial \beta'} & = \left( \frac{\partial^2 L}{\partial \beta \partial \sigma^2}\right)^{'} \\ & = -\frac{1}{\sigma^4}(X'u)' \\ & = -\frac{1}{\sigma^4}(u'X)
	331	+\end{align*}
	332	+\end{block}
	333	+\end{frame}
	334	+
	335	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	336	+\begin{block}{En bas à droite}
	337	+\begin{align*}
	338	+\frac{\partial^2 L}{(\partial \sigma^2)^2} & = \frac{\partial^2 \left[ -\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\sigma^2)-\frac{1}{2} \times \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2} \right]}{(\partial \sigma^2)^2} \\ & = \frac{\partial \left[-\frac{-T}{2 \sigma^2}+\frac{(Y-X \beta)'(Y-X \beta)}{2\sigma^4} \right]}{\partial \sigma^2} \\ & = \frac{T}{2 \sigma^4} - \frac{(Y-X\beta)'(Y-X \beta)}{\sigma^6}
	339	+\end{align*}
	340	+\end{block}
	341	+\end{frame}
	342	+
	343	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	344	+\begin{itemize}
	345	+\item Espérance mathématique de chacune des dérivés
	346	+\end{itemize}
	347	+\begin{block}{En haut à gauche}
	348	+\begin{align*}
	349	+E \left(-\frac{\partial^2 L}{\partial \beta \partial \beta'} \right) &= E \left(- \left[-\frac{1}{\sigma^2} (X'X) \right]\right) \\ & = \frac{1}{\sigma^2}(X'X)
	350	+\end{align*}
	351	+\end{block}
	352	+\end{frame}
	353	+
	354	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	355	+
	356	+\begin{block}{En haut à droite}
	357	+\begin{align*}
	358	+E \left(-\frac{\partial^2 L}{\partial \beta \partial \sigma^2} \right) &= E \left(- \left[-\frac{1}{\sigma^4} (X'Y-X'X \beta) \right]\right) \\ & = \frac{1}{\sigma^4}(X'E(Y)-X'X \beta) \\ & = \frac{1}{\sigma^4} (X'X \beta - X'X \beta) \\ & = 0
	359	+\end{align*}
	360	+Sachant $E(Y)=X \beta$
	361	+\end{block}
	362	+\end{frame}
	363	+
	364	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	365	+
	366	+\begin{block}{En bas à gauche}
	367	+\begin{align*}
	368	+E \left(-\frac{\partial^2 L}{\partial \sigma^2 \partial \beta'} \right) = 0
	369	+\end{align*}
	370	+\end{block}
	371	+\begin{block}{En bas à droite}
	372	+\begin{align*}
	373	+E \left(-\frac{\partial^2 L}{(\partial \sigma^2)^2} \right) & =E \left(- \left[\frac{T}{2 \sigma^4}-\frac{(Y-X \beta)'(Y-X \beta)}{\sigma^6} \right] \right) \\ & = -\frac{T}{2 \sigma^4}+\frac{E[(Y-X \beta)'(Y-X \beta)]}{\sigma^6}
	374	+\end{align*}
	375	+\end{block}
	376	+\end{frame}
	377	+
	378	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	379	+\begin{block}{En bas à droite}
	380	+\textbf{Utilisons la trace:}
	381	+\begin{align*}
	382	+E[(Y-X \beta)'(Y-X \beta)] & = E(u'u) \\ & = E(Trace(u'u)) \\ & = E(Trace(uu')) \\ & = Trace(E(uu')) \\ & = Trace(\sigma^2 I_T) \\ & = T\sigma^2
	383	+\end{align*}
	384	+\end{block}
	385	+\end{frame}
	386	+
	387	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	388	+\begin{block}{En bas à droite}
	389	+\textbf{Donc:}
	390	+\begin{align*}
	391	+E \left(-\frac{\partial^2 L}{(\partial \sigma^2)^2} \right) & = -\frac{T}{2 \sigma^4}+\frac{T \sigma^2}{\sigma^6} \\ & = -\frac{T}{2 \sigma^4}+\frac{T}{\sigma^4} \\ & = -\frac{T}{2 \sigma^4}+\frac{2T}{2\sigma^4} \\ & = \frac{T}{2\sigma^4}
	392	+\end{align*}
	393	+\end{block}
	394	+\end{frame}
	395	+
	396	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	397	+\begin{block}{Matrice d'information}
	398	+\begin{align*}
	399	+I(\beta, \sigma^2)=\begin{bmatrix}
	400	+\frac{1}{\sigma^2} (X'X) & 0 \\
	401	+0 & \frac{T}{2\sigma^4}
	402	+\end{bmatrix}
	403	+\end{align*}
	404	+\end{block}
	405	+
	406	+\begin{block}{Inverse matrice d'information}
	407	+\begin{align*}
	408	+I^{-1}(\beta, \sigma^2)=\begin{bmatrix}
	409	+ \sigma^2(X'X) & 0 \\
	410	+0 & \frac{2\sigma^4}{T}
	411	+\end{bmatrix}
	412	+\end{align*}
	413	+\end{block}
	414	+
	415	+\end{frame}
	416	+
	417	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	418	+\begin{block}{Espérance}
	419	+On sait déja que l'estimateur $\hat{\beta}$ possède la solution suivante:
	420	+\begin{align*}
	421	+\hat{\beta} = (X'X)^{-1}X'Y
	422	+\end{align*}
	423	+Sachant $Y= X\beta +u$
	424	+\begin{align*}
	425	+\hat{\beta} = (X'X)^{-1}X'[X\beta +u]
	426	+\end{align*}
	427	+\begin{align*}
	428	+\hat{\beta} = (X'X)^{-1}X'X\beta +(X'X)^{-1}X'u
	429	+\end{align*}
	430	+Sachant également $(X'X)^{-1}X'X = I$
	431	+\begin{align*}
	432	+\hat{\beta} = \beta +(X'X)^{-1}X'u
	433	+\end{align*}
	434	+\end{block}
	435	+\end{frame}
	436	+
	437	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	438	+\begin{block}{Espérance}
	439	+On applique l'espérance de chaque coté de l'équation
	440	+\begin{align*}
	441	+E(\hat{\beta}) & =E(\beta +(X'X)^{-1}X'u) \\ & = \beta +(X'X)^{-1}X'E(u)
	442	+\end{align*}
	443	+Sachant $E(u)=0$
	444	+\begin{align*}
	445	+E(\hat{\beta})=\beta
	446	+\end{align*}
	447	+\textbf{On peut donc maintenant affirmer que $\hat{\beta}$ est un estimateur sans biais de $\beta$}
	448	+\end{block}
	449	+\end{frame}
	450	+
	451	+
	452	+
	453	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	454	+\begin{block}{Variance}
	455	+On peut exprimer la variance de $\hat{\beta}$ comme suit:
	456	+\begin{align*}
	457	+Var(\hat{\beta})=E[(\hat{\beta}-\beta)(\hat{\beta}-\beta)']
	458	+\end{align*}
	459	+Sachant l'éqation que nous avons déja obtenus dans le calcule de l'espérance:
	460	+\begin{align*}
	461	+\hat{\beta}=\beta +(X'X)^{-1}X'u
	462	+\end{align*}
	463	+Alors il nous est possible d'exprimer la déviation de l'estimateur $\hat{\beta}$ par rapport à sa vrai valeur $\beta$.
	464	+\begin{align*}
	465	+\hat{\beta}-\beta = (X'X)^{-1}X'u
	466	+\end{align*}
	467	+\end{block}
	468	+\end{frame}
	469	+
	470	+
	471	+\begin{frame}{Propriétés Estimateur MLE ET MCO}
	472	+\begin{block}{Variance}
	473	+On peut donc incorporer l'équation de $(\hat{\beta}-\beta)$ dans l'équation de la variance de $\hat{\beta}$
	474	+\begin{align*}
	475	+Var(\hat{\beta})&=E[((X'X)^{-1}X'u)((X'X)^{-1}X'u)'] \\ & = E[(X'X)^{-1}X'uu'X(X'X)^{-1}] \\ & = (X'X)^{-1}X'E(uu')X(X'X)^{-1}
	476	+\end{align*}
	477	+Sachant $E(uu')=\sigma^2I$
	478	+\begin{align*}
	479	+Var(\hat{\beta})=(X'X)^{-1}X'\sigma^2IX(X'X)^{-1}
	480	+\end{align*}
	481	+Sachant $(X'X)^{-1}X'X)=I$
	482	+\begin{align*}
	483	+Var(\hat{\beta})=\sigma^2(X'X)^{-1}
	484	+\end{align*}
	485	+\end{block}
	486	+\textbf{On voit donc que la variance de $\hat{\beta}$ atteint la borne de Crameur-Rao ou l'inverse de la matrice d'information}
	487	+\end{frame}
	488	+
	489	+\section{Propriétés de $\hat{\sigma}^2$}
	490	+
	491	+\frame{\tableofcontents[current]}
	492	+
	493	+
	494	+\begin{frame}{Propriétés de $\hat{\sigma}^2$}
	495	+On sait que:
	496	+\begin{align*}
	497	+\frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{\sigma^2} \sim X^2(T-K)
	498	+\end{align*}
	499	+\begin{itemize}
	500	+\item Sachant $\hat{u}_t=Y-X \hat{\beta}$
	501	+\begin{itemize}
	502	+\item $\hat{u}_t$ sont normales par hypothèses
	503	+\item $\hat{u}_t^{'}\hat{u}_t$ suit une loi chi carré
	504	+\end{itemize}
	505	+\end{itemize}
	506	+\end{frame}
	507	+
	508	+\begin{frame}{Propriétés de $\hat{\sigma}^2$}
	509	+\textbf{On peut montrer que l'espérance de ce terme est la suivante:}
	510	+\begin{align*}
	511	+E \left[ \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2}\right]=(T-K)
	512	+\end{align*}
	513	+\textbf{On peut montrer que la variance de ce terme est la suivante:}
	514	+\begin{align*}
	515	+V \left[ \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2}\right]=2(T-K)
	516	+\end{align*}
	517	+\end{frame}
	518	+
	519	+\begin{frame}{Propriétés de $\hat{\sigma}^2$}
	520	+\textbf{On sait que l'estimateur $\hat{\sigma}^2$ est le suivant:}
	521	+\begin{align*}
	522	+\hat{\sigma}^2=\frac{(Y-X \beta)'(Y-X \beta)}{T}
	523	+\end{align*}
	524	+\textbf{Espérance de $\hat{\sigma}^2$}
	525	+\begin{align*}
	526	+E(\hat{\sigma}^2)=E \left( \frac{(Y-X \beta)'(Y-X \beta)}{T}\right)
	527	+\end{align*}
	528	+\end{frame}
	529	+
	530	+\begin{frame}{Propriétés de $\hat{\sigma}^2$}
	531	+\begin{itemize}
	532	+\item On peut multiplier le numérateur et le dénominateur par $\sigma^2$ afin d'écrire l'équation de l'espérance comme suit:
	533	+\begin{align*}
	534	+E(\hat{\sigma}^2)=E \left( \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2} \times \frac{\sigma^2}{T} \right)
	535	+\end{align*}
	536	+\item Sachant $E \left[ \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2}\right]=(T-K)$ on peut formuler à l'équation de $\hat{\sigma}^2$ comme suit:
	537	+\begin{align*}
	538	+\frac{\sigma^2}{T}(T-K)
	539	+\end{align*}
	540	+\end{itemize}
	541	+\end{frame}
	542	+
	543	+
	544	+\begin{frame}{Propriétés de $\hat{\sigma}^2$}
	545	+\begin{itemize}
	546	+\item On voit que cette estimateur est biaisé et la borne de Crameur-Rao ne peu s'appliquer dans le ce cas.
	547	+\item Cependant, si $T$ devient suffisament grand, alors:
	548	+\begin{align*}
	549	+T-K \approx T
	550	+\end{align*}
	551	+\item On voit clairement que le biais s'annule
	552	+\begin{align*}
	553	+E(\hat{\sigma}^2)=\frac{\sigma^2}{T}(T)=\sigma^2
	554	+\end{align*}
	555	+\end{itemize}
	556	+\end{frame}
	557	+
	558	+
	559	+\section{Estimateur de la variance sans biais}
	560	+
	561	+\frame{\tableofcontents[current]}
	562	+
	563	+
	564	+\begin{frame}{Estimateur de la variance sans biais}
	565	+\begin{itemize}
	566	+\item L'estimateur de la variance sans biais est représenté par $\hat{S}^2$
	567	+\begin{align*}
	568	+\hat{S}^2=\frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{T-K}
	569	+\end{align*}
	570	+\textbf{Espérance de $\hat{S}^2$}
	571	+\begin{align*}
	572	+E(\hat{S}^2)=E \left[ \frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{T-K}\right]
	573	+\end{align*}
	574	+\item On peut multiplier le numérateur et le dénominateur par $\sigma^2$ afin d'écrire l'équation de l'espérance de $\hat{S}^2$ comme suit:
	575	+\begin{align*}
	576	+E(\hat{S}^2)=E \left[ \frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{\sigma^2} \times \frac{\sigma^2}{T-K}\right]
	577	+\end{align*}
	578	+\end{itemize}
	579	+\end{frame}
	580	+
	581	+\begin{frame}{Estimateur de la variance sans biais}
	582	+\begin{itemize}
	583	+\item Sachant $E \left[ \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2}\right]=(T-K)$ on peut formuler à l'équation de l'espérance de $\hat{S}^2$ comme suit:
	584	+\begin{align*}
	585	+E(\hat{S}^2) & =(T-K) \times \frac{\sigma^2}{T-K} \\ & = \sigma^2
	586	+\end{align*}
	587	+\item On voit donc que l'estimateur de la variance $\hat{S}^2$ est sans biais étant donnée que sont espérance égale la vrai valeur de la variance $\sigma^2$
	588	+\end{itemize}
	589	+\end{frame}
	590	+
	591	+
	592	+\begin{frame}{Estimateur de la variance sans biais}
	593	+
	594	+\textbf{Variance de $\hat{S}^2$}
	595	+\begin{align*}
	596	+V(\hat{S}^2)=V \left[ \frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{T-K}\right]
	597	+\end{align*}
	598	+\begin{itemize}
	599	+\item On peut multiplier le numérateur et le dénominateur par $\sigma^2$ afin d'écrire l'équation de la variance de $\hat{S}^2$ comme suit:
	600	+\begin{align*}
	601	+V(\hat{S}^2)=V \left[ \frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{\sigma^2} \times \frac{\sigma^2}{T-K}\right]
	602	+\end{align*}
	603	+\item On peut sortir $\frac{\sigma^2}{T-K}$ de l'opérateur variance en élevant ce terme à la puissance 2.
	604	+\begin{align*}
	605	+V(\hat{S}^2)= \frac{\sigma^4}{(T-K)^2} V\left[ \frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{\sigma^2} \right]
	606	+\end{align*}
	607	+\end{itemize}
	608	+\end{frame}
	609	+
	610	+
	611	+\begin{frame}{Estimateur de la variance sans biais}
	612	+
	613	+\textbf{Variance de $\hat{S}^2$}
	614	+\begin{itemize}
	615	+\item Sachant $V \left[ \frac{(Y-X \beta)'(Y-X \beta)}{\sigma^2}\right]=2(T-K)$, on peut écrire la variance de $\hat{S}^2$ comme suit:
	616	+\begin{align*}
	617	+V(\hat{S}^2) & = \frac{\sigma^4}{(T-K)^2} \times [2(T-K)] \\ & = \frac{2 \sigma^4}{T-K}
	618	+\end{align*}
	619	+\item On voit clairement que $V(\hat{S}^2)$ n'atteint pas la borne de Cramer-Rao étant donnée que cette variance est plus grande que celle donnée par la borne.
	620	+\begin{align*}
	621	+\frac{2 \sigma^4}{T-K} > \frac{2 \sigma^4}{T}
	622	+\end{align*}
	623	+\end{itemize}
	624	+\end{frame}
	625	+\end{document}
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	5	+\beamer@sectionintoc {5}{Estimateur de la variance sans biais}{40}{0}{5}

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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage[T1]{fontenc}
	9	+\usepackage{graphicx}
	10	+\usepackage{booktabs}
	11	+\usepackage{xcolor}
	12	+\usetheme{default}
	13	+\usecolortheme{default}
	14	+
	15	+\title[S02 Régression et MCO]{Section 02 : Régression et Moindres Carrés Ordinaires\\ (Séance 4)}
	16	+\subtitle{GSF-6053: Économétrie Financière}
	17	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	18	+\institute[Université Laval]
	19	+{
	20	+ \inst{1}%
	21	+ Département de finance, assurance et immobilier\\
	22	+ Faculté des sciences de l'administration\\
	23	+ Université Laval}
	24	+\date[Hiver 2022]{1 Février 2022}
	25	+
	26	+\begin{document}
	27	+
	28	+\begin{frame}
	29	+ \titlepage
	30	+\end{frame}
	31	+
	32	+
	33	+\begin{frame}{Références}
	34	+\textbf{Obligatoires:}
	35	+\begin{itemize}
	36	+\item \textbf{Notes de cours:} Section 2 (Professeure: Marie-Hélène Gagnon)
	37	+\item \textbf{Woolridge:} chapitres 2 à 7
	38	+\end{itemize}
	39	+\vspace{0.5cm}
	40	+\textbf{Complémentaires:}
	41	+\begin{itemize}
	42	+\item \textbf{Gujarati et Porter:} chapitres 1 à 9.
	43	+\item \textbf{Greene:} chapitres 2, 3, 4, 5, 9, 14, 20, appendices C et D
	44	+\end{itemize}
	45	+\end{frame}
	46	+
	47	+
	48	+\begin{frame}{Plan de la séance}
	49	+ \tableofcontents
	50	+\end{frame}
	51	+
	52	+\section{Analyse de Variance}
	53	+
	54	+\frame{\tableofcontents[current]}
	55	+
	56	+\begin{frame}{Analyse de Variance}
	57	+\begin{itemize}
	58	+\item Variation dans la variable expliquée \textbf{(Y)}
	59	+\begin{align*}
	60	+Y_t=\hat{Y}_t+\hat{u}_t
	61	+\end{align*}
	62	+\item Composantes de la variation de \textbf{Y}
	63	+\begin{itemize}
	64	+\item $\hat{Y}_t$ est la variation due à la partie expliquée
	65	+\item $\hat{u}_t$ est induite par la partie de \textbf{Y} non expliquée
	66	+\end{itemize}
	67	+\end{itemize}
	68	+\end{frame}
	69	+
	70	+\begin{frame}{Analyse de Variance}
	71	+\begin{itemize}
	72	+\item Variation totale de \textbf{Y}
	73	+\begin{align*}
	74	+SS_{tot}=\sum_{t=1}^{T}(Y_t-\overline{Y})^2
	75	+\end{align*}
	76	+\item Variation partie expliquée de \textbf{Y}
	77	+\begin{align*}
	78	+SS_{reg}=\sum_{t=1}^{T}(\hat{Y}_t-\overline{Y})^2
	79	+\end{align*}
	80	+\item Variation non expliquée de \textbf{Y} ou Résidus
	81	+\begin{align*}
	82	+SS_{err}=\sum_{t=1}^{T}(Y_t-\hat{Y}_t)^2=\sum_{t=1}^{T} \hat{u}_t^2
	83	+\end{align*}
	84	+\end{itemize}
	85	+\end{frame}
	86	+
	87	+\begin{frame}{Analyse de Variance}
	88	+\begin{itemize}
	89	+\item À l'aide des 3 dernières définitions, soit $SS_{tot}, SS_{reg}$ et $SS_{err}$, nous allons maintenant prouver l'équation suivante:
	90	+\begin{align*}
	91	+SS_{tot}=SS_{reg}+SS_{err}
	92	+\end{align*}
	93	+\item On pose la définition de $SS_{tot}$ qu'on sait déja:
	94	+\begin{align*}
	95	+SS_{tot}=\sum_{t=1}^{T}(Y_t-\overline{Y})^2
	96	+\end{align*}
	97	+Sachant $Y_t= \hat{Y}_t + \hat{u}_t$
	98	+
	99	+\begin{align*}
	100	+SS_{tot}=\sum_{t=1}^{T}(\hat{Y}_t + \hat{u}_t-\overline{Y})^2
	101	+\end{align*}
	102	+\end{itemize}
	103	+\end{frame}
	104	+
	105	+
	106	+\begin{frame}{Analyse de Variance}
	107	+\begin{itemize}
	108	+\item Afin d'éliminer et de transformer certains termes, il nous faut distribuer les termes de notre parenthèse élevé au carré.
	109	+\begin{align*}
	110	+SS_{tot} =\sum_{t=1}^{T}(\hat{Y}_t + \hat{u}_t-\overline{Y})\times(\hat{Y}_t + \hat{u}_t-\overline{Y})
	111	+\end{align*}
	112	+
	113	+\begin{align*}
	114	+=\sum_{t=1}^{T} \left[ \hat{Y}_t^2 +\hat{Y}_t\hat{u}_t -\hat{Y}_t\overline{Y} +\hat{u}_t\hat{Y}_t + \hat{u}_t^2-\hat{u}_t \overline{Y} - \overline{Y} \hat{Y}_t -\overline{Y} \hat{u}_t +\overline{Y}^2 \right]
	115	+\end{align*}
	116	+
	117	+\end{itemize}
	118	+\end{frame}
	119	+
	120	+
	121	+\begin{frame}{Analyse de Variance}
	122	+\begin{itemize}
	123	+\item Nous allons maintenant simplifier notre équation $SS_{tot}$, en utilisant l'identité remarquable.
	124	+\begin{itemize}
	125	+\item Rappel: Si nous avons $(a^2+2ab+b^2)$, alors on peut écrire les 3 termes sous une forme polynomial $(a+b)^2$
	126	+\end{itemize}
	127	+\item Dans le cas qui nous interesse, notre indentité remarquable sera composé de $\hat{Y}_t$ et $\overline{Y}$. On peut donc isoler, dans notre dernière équation les termes nécessaires:
	128	+\begin{align*}
	129	+SS_{tot} =\sum_{t=1}^{T} \left[(\hat{Y}_t^2+\overline{Y}^2-2\hat{Y}_t \overline{Y})+2\hat{Y}_t\hat{u}_t + \hat{u}_t^2-2 \hat{u}_t \overline{Y}\right]
	130	+\end{align*}
	131	+Sachant que selon l'identité remarquable $(\hat{Y}_t^2+\overline{Y}^2-2\hat{Y}_t \overline{Y})=(\hat{Y}_t-\overline{Y})^2$
	132	+\begin{align*}
	133	+SS_{tot} =\sum_{t=1}^{T} \left[(\hat{Y}_t-\overline{Y})^2+2\hat{Y}_t\hat{u}_t + \hat{u}_t^2-2 \hat{u}_t \overline{Y}\right]
	134	+\end{align*}
	135	+\end{itemize}
	136	+
	137	+ \end{frame}
	138	+
	139	+
	140	+\begin{frame}{Analyse de Variance}
	141	+\begin{itemize}
	142	+\item On doit maintenant distribuer notre opérateur sommation à tous les termes de l'équation exprimant $SS_{tot}$.
	143	+\begin{align*}
	144	+SS_{tot}=\sum_{t=1}^{T} (\hat{Y}_t-\overline{Y})^2 + \sum_{t=1}^{T} \hat{u}_t^2 +2\sum_{t=1}^{T}\hat{Y}_t\hat{u}_t -2 \overline{Y}\sum_{t=1}^{T}\hat{u}_t
	145	+\end{align*}
	146	+Sachant:
	147	+\begin{itemize}
	148	+\item $SS_{reg}=\sum_{t=1}^{T}(\hat{Y}_t-\overline{Y})^2$
	149	+\item $SS_{err}=\sum_{t=1}^{T} \hat{u}_t^2$
	150	+\end{itemize}
	151	+\item Alors on peut écrire l'équation exprimant $SS_{tot}$ comme suit:
	152	+\begin{align*}
	153	+SS_{tot}=SS_{reg} + SS_{err} +2\sum_{t=1}^{T}\hat{Y}_t\hat{u}_t -2 \overline{Y}\sum_{t=1}^{T}\hat{u}_t
	154	+\end{align*}
	155	+\end{itemize}
	156	+
	157	+\end{frame}
	158	+
	159	+
	160	+\begin{frame}{Analyse de Variance}
	161	+\begin{itemize}
	162	+\item Les résidus sont orthogonaux au sous-espace engendré par les colonnes de $X$
	163	+\begin{align*}
	164	+2\sum_{t=1}^T \hat{Y}_t \hat{u}_t=0
	165	+\end{align*}
	166	+\item La somme des résidus est nulle
	167	+\begin{align*}
	168	+2 \overline{Y} \sum_{t=1}^T \hat{u}_t=0
	169	+\end{align*}
	170	+\item Avec les deux dernières propriétés on très bien que dans l'éuation du $SS_{tot}$, les deux derniers termes s’annulent:
	171	+\begin{align*}
	172	+SS_{tot}=SS_{reg}+SS_{err}
	173	+\end{align*}
	174	+\end{itemize}
	175	+\end{frame}
	176	+
	177	+\section{Coefficient de détermination $(R^2)$}
	178	+
	179	+\frame{\tableofcontents[current]}
	180	+
	181	+\begin{frame}{Coefficient de détermination $(R^2)$}
	182	+\begin{itemize}
	183	+\item Indique le pouvoir explicatif de la régression
	184	+\item Proportion des variations de $Y$ expliquées par les régresseurs du modèle par rapport aux variations total.
	185	+\item On peut donc exprimer le $R^2$ comme étant le ratio entre la somme des résidus au carrés et somme des variations total au carrés.
	186	+\begin{align*}
	187	+R^2=\frac{SS_{reg}}{SS_{tot}}=1-\frac{SS_{err}}{SS_{tot}}
	188	+\end{align*}
	189	+\item On voit clairement que notre $R^2$ sera compris entre $0$ et $1$
	190	+\begin{align*}
	191	+0 \le R^2 \le 1
	192	+\end{align*}
	193	+\end{itemize}
	194	+\end{frame}
	195	+
	196	+
	197	+\section{Coefficient de détermination ajusté $(R^2$ ajusté)}
	198	+
	199	+\frame{\tableofcontents[current]}
	200	+
	201	+\begin{frame}{Coefficient de détermination ajusté $(R^2$ ajusté)}
	202	+\begin{itemize}
	203	+\item Contrairement au $R^2$ standard, le $R^2$ ajusté vient pénélisé l'ajout de variables explicatives inutiles.
	204	+\item $R^2$ $\rightarrow$ Si on ajoute un variable explicative à note modèle, il augmentera forcément
	205	+\item $R^2$ ajusté $\rightarrow$ Si on ajoute un variable explicative à note modèle, il peut augmenté, mais si l'ajout de cette variable est inutile il pourra également diminuer.
	206	+
	207	+\end{itemize}
	208	+\end{frame}
	209	+
	210	+
	211	+\begin{frame}{Coefficient de détermination ajusté $(R^2$ ajusté)}
	212	+\begin{itemize}
	213	+\item L'équation exprimant le $R^2$ ajusté contient comme pour le $R^2$, le $SS_{err}$ et le $SS_{tot}$.
	214	+\item Cependant afin de tenir compte du nombre de régresseurs ajoutés au modèle, on aura besoin de la constante $k$, étant le nombre de variable indépendantes.
	215	+\item Nous aurons également besoin du nombres d'observation dans notre échantillons, représenté par $n$.
	216	+\end{itemize}
	217	+\end{frame}
	218	+
	219	+
	220	+\begin{frame}{Coefficient de détermination ajusté $(R^2$ ajusté)}
	221	+\begin{itemize}
	222	+\item \textbf{Équation exprimation le $R^2$ ajusté:}
	223	+\begin{align*}
	224	+R_{ADJ}^2 & =1-\frac{\left( \frac{SS_{err}}{n-k}\right)}{\left( \frac{SS_{tot}}{n-1}\right)} \\ & = 1-\frac{SS_{err}(n-1)}{SS_{tot}(n-k)}
	225	+\end{align*}
	226	+Ou \textbf{k} représente le nombre de régresseurs
	227	+\end{itemize}
	228	+\end{frame}
	229	+
	230	+\begin{frame}{Coefficient de détermination ajusté $(R^2$ ajusté)}
	231	+\begin{itemize}
	232	+\item On peut également montrer qu'il est possible d'exprimer le $R^2$ ajusté en fonction du $R^2$ standard
	233	+\item Si $\frac{SS_{reg}}{SS_{tot}}=R^2$ alors $\frac{SS_{err}}{SS_{tot}}=1-R^2$
	234	+Comme montré ci-haut, le $R^2$ ajusté peut être exprimé comme suit:
	235	+\begin{align*}
	236	+R_{ADJ}^2=1-\frac{SS_{err}}{SS_{tot}} \times \frac{(n-1)}{n-k}
	237	+\end{align*}
	238	+Sachant $\frac{SS_{err}}{SS_{tot}}=1-R^2$, on peut écrire à nouveau le $R^2$ ajusté en fonction du $R^2$
	239	+\begin{align*}
	240	+R_{ADJ}^2=1-(1-R^2) \times \frac{(n-1)}{n-k}
	241	+\end{align*}
	242	+\end{itemize}
	243	+\end{frame}
	244	+
	245	+\begin{frame}{Coefficient de détermination ajusté $(R^2$ ajusté)}
	246	+\textbf{Coefficient de détermination ajusté:}
	247	+
	248	+\begin{itemize}
	249	+\item Si nous avons uniquement 1 variables explicatives dans notre modèle, alors le $R^2$ sera égale au $R_{ADJ}^2$.
	250	+\begin{itemize}
	251	+\item Dans ce cas, nous avons 1 régresseur, soit $k=1$
	252	+\end{itemize}
	253	+\begin{align*}
	254	+R_{ADJ}^2 & =1-(1-R^2) \times \frac{(n-1)}{n-k} \\ & = 1-(1-R^2) \times \frac{(n-1)}{n-1} \\ & =1-(1-R^2) \\ & = R^2
	255	+\end{align*}
	256	+\end{itemize}
	257	+\end{frame}
	258	+
	259	+\begin{frame}{Coefficient de détermination ajusté $(R^2$ ajusté)}
	260	+\textbf{Coefficient de détermination ajusté:}
	261	+
	262	+\begin{itemize}
	263	+\item Si nous avons uniquement plusieurs variables explicatives dans notre modèle, alors le $R^2$ sera plus grand ou égale au $R_{ADJ}^2$.
	264	+\begin{itemize}
	265	+\item Dans ce cas, nous avons 2 régresseur et plus, soit $k \ge 2$
	266	+\end{itemize}
	267	+\item S'il y a plusieurs variables explicatives alors
	268	+\begin{align*}
	269	+n-1 > n-k
	270	+\end{align*}
	271	+De facon équivalente:
	272	+\begin{align*}
	273	+\frac{(n-1)}{n-k} > 1
	274	+\end{align*}
	275	+\item Le terme $\frac{(n-1)}{n-k}$ supérieurs à 1 est multiplié à ce qui n'est pas expliqué par le modèle $(1-R^2)$
	276	+\end{itemize}
	277	+\textbf{Ce qui nous permet d'affirmer:}
	278	+\begin{align*}
	279	+R_{ADJ}^2 \leq R^2 \hspace{0.2cm} \textbf{et} \hspace{0.2cm} R_{ADJ}^2 \leq 1
	280	+\end{align*}
	281	+\end{frame}
	282	+
	283	+
	284	+
	285	+\section{Test d'hypothèse}
	286	+
	287	+\frame{\tableofcontents[current]}
	288	+
	289	+\begin{frame}{Test d'hypothèse}
	290	+\textbf{Un test d’hypothèse sur les paramètres d’un modèle économétrique requiert les
	291	+ étapes suivantes:}
	292	+
	293	+\begin{itemize}
	294	+\item \textbf{Étape 1:}
	295	+\begin{itemize}
	296	+\item Écrire l’hypothèse nulle ($H_0$) et l’alternative ($H_1$).
	297	+\item La plupart du temps, $H_0$ est l’hypothèse que l’on veut rejeter.
	298	+\item Par exemple, qu’un paramètre est non significatif ou égal à une certaine valeur.
	299	+\item L’hypothèse alternative peut être unilatérale ($<$ et $>$) ou bilatérale ($\neq$).
	300	+\end{itemize}
	301	+\end{itemize}
	302	+\end{frame}
	303	+
	304	+
	305	+\begin{frame}{Test d'hypothèse}
	306	+
	307	+\begin{itemize}
	308	+\item \textbf{Étape 2:}
	309	+\begin{itemize}
	310	+\item Définir la statistique de test (student, F, Wald, LR, LM etc..)
	311	+\item Déterminer si possible, la distribution loi de cette statistique sous $H_0$.
	312	+\end{itemize}
	313	+\item \textbf{Étape 3:}
	314	+\begin{itemize}
	315	+\item Choisir le niveau de significativité du test $\alpha$.
	316	+\item On fixe donc la possibilité d’erreur de type 1 soit la probabilité de rejeter $H_0$ lorsque celle-ci est vraie.
	317	+\item Généralement, on fixe le niveau à 5 \% ou 1 \% et plus rarement à 10 \%.
	318	+
	319	+\end{itemize}
	320	+\end{itemize}
	321	+\end{frame}
	322	+
	323	+
	324	+\begin{frame}{Test d'hypothèse}
	325	+
	326	+\begin{itemize}
	327	+\item \textbf{Étape 4:}
	328	+\begin{itemize}
	329	+\item Déterminer la règle de décision du test avec la région critique de confiance $CR_{\alpha}$.
	330	+\item La plupart du temps, cela demande de savoir la distribution de la statistique sous $H_0$.
	331	+\item Lorsque la valeur calculée pour la statistique de test se trouve dans la région critique: on rejette $H_0$ en faveur de $H_1$ au seuil de confiance $\alpha$.
	332	+\end{itemize}
	333	+\item \textbf{Étape 5:}
	334	+\begin{itemize}
	335	+\item Utiliser les valeurs obtenues de la régression pour calculer la statistique de test.
	336	+\end{itemize}
	337	+\item \textbf{Étape 6:}
	338	+\begin{itemize}
	339	+\item Appliquer la règle de décision vue dans l'étape 4.
	340	+\end{itemize}
	341	+\end{itemize}
	342	+\end{frame}
	343	+
	344	+\begin{frame}{Test d'hypothèse}
	345	+\begin{block}{Note sur la p-value}
	346	+\begin{itemize}
	347	+\item Parfois, on regarde la p-value au lieu de comparer $CR_{\alpha}$ à la valeur de la statistique.
	348	+\item P-value : C’est le plus petit niveau auquel on petit rejeter l’hypothèse nulle.
	349	+\item En d’autres mots, c’est la probabilité d’avoir un évènement aussi extrême que celle observée en assumant que $H_0$ est vraie.
	350	+\item Plus la p-value est faible, plus la probabilité que l’évènement observé soit faible, étant donné l’hypothèse nulle.
	351	+\begin{itemize}
	352	+\item plus il y a de chances que l’hypothèse nulle est rejetée.
	353	+\end{itemize}
	354	+\end{itemize}
	355	+\end{block}
	356	+\end{frame}
	357	+
	358	+\begin{frame}{Test d'hypothèse}
	359	+\textbf{Test de significativité:}
	360	+\begin{block}{Significativité individuelle d'un paramètre}
	361	+\begin{itemize}
	362	+\item On regarde si le régresseur est statistiquement non nul
	363	+\item Utilise le test de student
	364	+\end{itemize}
	365	+\end{block}
	366	+\begin{block}{Significativité conjointe des paramètres}
	367	+\begin{itemize}
	368	+\item On regarde si au moins un des régresseurs est statistiquement non nul (un effet)
	369	+\item Utilise le F-test
	370	+\end{itemize}
	371	+\end{block}
	372	+
	373	+\end{frame}
	374	+
	375	+
	376	+
	377	+
	378	+\begin{frame}{Test d'hypothèse}
	379	+\textbf{Test de Student:}
	380	+\begin{block}{Two-tailed test}
	381	+\begin{itemize}
	382	+\item Hypothèse nulle est la non-significativité du coefficient de régression
	383	+\item Hypothèses:
	384	+\begin{itemize}
	385	+\item $H_0:$ $\beta_k=0$ $\rightarrow$ \textbf{Hypothèse nule}
	386	+\item $H_1:$ $\beta_k \neq 0$ $\rightarrow$ \textbf{Hypothèse alternative}
	387	+\end{itemize}
	388	+\item Règle de décision:
	389	+Rejeter $H_0$ si:
	390	+\begin{align*}
	391	+t=\frac{\mid \hat{\beta}_k-\beta_0 \mid}{SE_{\hat{\beta}_k}} > t_{n-k,\alpha/2}
	392	+\end{align*}
	393	+\begin{itemize}
	394	+\item Ou $\beta_0$ est la valeur du coefficient sous l'hypothèse nulle, soit 0
	395	+\item $SE_{\hat{\beta}_k}$ est l'écart-type associé à l'estimation de $\hat{\beta}_k$
	396	+\end{itemize}
	397	+\end{itemize}
	398	+\end{block}
	399	+\end{frame}
	400	+
	401	+\begin{frame}{Test d'hypothèse}
	402	+\textbf{Test de Student:}
	403	+\begin{block}{Right-tailed test}
	404	+\begin{itemize}
	405	+\item Hypothèses:
	406	+\begin{itemize}
	407	+\item $H_0:$ $\beta_k \leq 0$ $\rightarrow$ \textbf{Hypothèse nule}
	408	+\item $H_1:$ $\beta_k > 0$ $\rightarrow$ \textbf{Hypothèse alternative}
	409	+\end{itemize}
	410	+\item Règle de décision:
	411	+Rejeter $H_0$ si:
	412	+\begin{align*}
	413	+t=\frac{\hat{\beta}_k-\beta_0}{SE_{\hat{\beta}_k}} > t_{n-k,\alpha}
	414	+\end{align*}
	415	+\begin{itemize}
	416	+\item Ou $\beta_0$ est la valeur du coefficient sous l'hypothèse nulle, soit 0
	417	+\item $SE_{\hat{\beta}_k}$ est l'écart-type associé à l'estimation de $\hat{\beta}_k$
	418	+\end{itemize}
	419	+\end{itemize}
	420	+\end{block}
	421	+\end{frame}
	422	+
	423	+
	424	+\begin{frame}{Test d'hypothèse}
	425	+\textbf{Test de Student:}
	426	+\begin{block}{Left-tailed test}
	427	+\begin{itemize}
	428	+\item Hypothèses:
	429	+\begin{itemize}
	430	+\item $H_0:$ $\beta_k \geq 0$ $\rightarrow$ \textbf{Hypothèse nule}
	431	+\item $H_1:$ $\beta_k < 0$ $\rightarrow$ \textbf{Hypothèse alternative}
	432	+\end{itemize}
	433	+\item Règle de décision:
	434	+Rejeter $H_0$ si:
	435	+\begin{align*}
	436	+t=\frac{\hat{\beta}_k-\beta_0}{SE_{\hat{\beta}_k}} < -t_{n-k,\alpha}
	437	+\end{align*}
	438	+\begin{itemize}
	439	+\item Ou $\beta_0$ est la valeur du coefficient sous l'hypothèse nulle, soit 0
	440	+\item $SE_{\hat{\beta}_k}$ est l'écart-type associé à l'estimation de $\hat{\beta}_k$
	441	+\end{itemize}
	442	+\end{itemize}
	443	+\end{block}
	444	+\end{frame}
	445	+
	446	+
	447	+\begin{frame}{Test d'hypothèse}
	448	+\textbf{F-test (significativité conjointe)}
	449	+\begin{itemize}
	450	+\item Comparer la différence entre les résidus au carrés d'un modèle contraint et d'un modèle non contraint.
	451	+\begin{itemize}
	452	+\item Si la différence est grande
	453	+\item Plus les résidus du modèle contraint sont grand par rapport au modèle contraint
	454	+\item Plus la contrainte coûte cher à appliquer en terme de performance
	455	+\item Plus la statistique F est grande
	456	+\item Passé un certain coût critique $\rightarrow$ rejette $H_0$
	457	+\end{itemize}
	458	+\end{itemize}
	459	+\end{frame}
	460	+
	461	+\begin{frame}{Test d'hypothèse}
	462	+\textbf{F-test (significativité conjointe)}
	463	+\begin{block}{Hypothèses:}
	464	+\begin{itemize}
	465	+\item $H_0:$ $\beta_2 = \beta_3 = ... = \beta_k=0$
	466	+\item $H_1:$ $\beta_2 \neq 0$ et $\beta_3 \neq 0$ et $...$ et/ou $\beta_k \neq 0$
	467	+\end{itemize}
	468	+\end{block}
	469	+\begin{block}{Statistique de test:}
	470	+\begin{itemize}
	471	+\item Modèle non contraint $\rightarrow$ $(\hat{u}'\hat{u})$
	472	+\item Modèle contraint $\rightarrow$ $(\hat{u}_0'\hat{u}_0)$
	473	+\end{itemize}
	474	+\begin{align*}
	475	+F = \frac{\hat{u}_0'\hat{u}_0-\hat{u}'\hat{u}}{\hat{u}'\hat{u}}\times \frac{t-k}{q} \sim (q,t-k)
	476	+\end{align*}
	477	+\end{block}
	478	+\end{frame}
	479	+
	480	+\begin{frame}{Test d'hypothèse}
	481	+\textbf{F-test (significativité conjointe)}
	482	+
	483	+\begin{block}{Décision:}
	484	+On rejette $H_0$ à un seuil de $\alpha$ si
	485	+\begin{align*}
	486	+F = \frac{\hat{u}_0'\hat{u}_0-\hat{u}'\hat{u}}{\hat{u}'\hat{u}} \times \frac{t-k}{q} > F(q,t-k;\alpha)
	487	+\end{align*}
	488	+\end{block}
	489	+\end{frame}
	490	+
	491	+
	492	+\section{Les contraintes linéaires}
	493	+
	494	+\frame{\tableofcontents[current]}
	495	+
	496	+\begin{frame}{Les contraintes linéaires}
	497	+
	498	+\textbf{Format général:}
	499	+\begin{align*}
	500	+H_0: R \beta = r
	501	+\end{align*}
	502	+\begin{itemize}
	503	+\item $R=$ matrice de sélection contenant une ligne pour chaque contrainte non redondante $q$
	504	+\item Vecteur de réel constants
	505	+\end{itemize}
	506	+
	507	+
	508	+\end{frame}
	509	+
	510	+\begin{frame}{Les contraintes linéaires}
	511	+\begin{itemize}
	512	+\item On cherche à maximiser notre log-vraisemblance afin de trouver une solution pour $\beta$, tout en respectant une contrainte linéaire.
	513	+\item Il s'agit essentiellement d'une optimization sous contrainte à l'aide d'un multiplicateur de lagrange.
	514	+\item On représente l'estimateur $\beta$ obtenus avec la méthode des moindres carrés ordinaire est représenté par $\beta^{OLS}$.
	515	+\item L'estimateur $\beta$ obtenus sous une contrainte linéaire est représenté par $\beta^{LC}$
	516	+
	517	+\end{itemize}
	518	+\end{frame}
	519	+
	520	+\begin{frame}{Les contraintes linéaires}
	521	+\begin{itemize}
	522	+\item Nous voulons minimiser la somme des carrés des résidus, mais cette fois, nous posons la contrainte : $R \beta = r$
	523	+\item Cela conduit à la fonction de Lagrange suivante:
	524	+\begin{align*}
	525	+L(\beta,\lambda) & =(Y-X\beta)'(Y-X\beta)+2 \lambda'(R \beta-r)\\ & = Y'Y-2Y'X\beta +\beta' X'X \beta +2 \lambda' R\beta -2 \lambda'r
	526	+\end{align*}
	527	+\item On dérive ensuite la fonction $L(\beta,\lambda)$ par rapport à $\beta$ et $\lambda$
	528	+\end{itemize}
	529	+
	530	+\end{frame}
	531	+
	532	+\begin{frame}{Les contraintes linéaires}
	533	+\begin{block}{Dérivé par rapport à $\beta$}
	534	+\begin{align*}
	535	+\frac{\partial L(\beta, \lambda)}{\partial \beta}=-2 X'Y+2X'X \beta+2 \lambda'R=0
	536	+\end{align*}
	537	+\begin{align*}
	538	+X'X \beta + R'\lambda = X'Y
	539	+\end{align*}
	540	+\end{block}
	541	+\begin{block}{Dérivé par rapport à $\lambda$}
	542	+\begin{align*}
	543	+\frac{\partial L(\beta, \lambda)}{\partial \lambda}=2R\beta-2r=0
	544	+\end{align*}
	545	+\begin{align*}
	546	+R \beta=r
	547	+\end{align*}
	548	+\end{block}
	549	+\end{frame}
	550	+
	551	+\begin{frame}{Les contraintes linéaires}
	552	+\textbf{Format matricielle}
	553	+\begin{itemize}
	554	+\item Équation 1:
	555	+\begin{align*}
	556	+X'X \times \textcolor{red}{\beta} + R' \times \textcolor{red}{\lambda} = X'Y
	557	+\end{align*}
	558	+\item Équation 2:
	559	+\begin{align*}
	560	+R \times \textcolor{red}{\beta} + 0 \times \textcolor{red}{\lambda} = r
	561	+\end{align*}
	562	+\end{itemize}
	563	+Ce qui nous permet d'obtenir la représentation suivante:
	564	+\begin{align*}
	565	+\begin{bmatrix}
	566	+X'X & R' \\
	567	+R & 0
	568	+\end{bmatrix}
	569	+\begin{bmatrix}
	570	+\textcolor{red}{\beta} \\
	571	+\textcolor{red}{\lambda}
	572	+\end{bmatrix}
	573	+=
	574	+\begin{bmatrix}
	575	+X'Y \\
	576	+r
	577	+\end{bmatrix}
	578	+\end{align*}
	579	+\end{frame}
	580	+
	581	+
	582	+\begin{frame}{Les contraintes linéaires}
	583	+
	584	+\begin{itemize}
	585	+\item On peut ensuite obtenir une solution pour $\beta$ et $\lambda$
	586	+
	587	+\begin{align*}
	588	+\begin{bmatrix}
	589	+\textcolor{red}{\beta} \\
	590	+\textcolor{red}{\lambda}
	591	+\end{bmatrix}
	592	+=
	593	+\begin{bmatrix}
	594	+X'X & R' \\
	595	+R & 0
	596	+\end{bmatrix}^{-1}
	597	+\begin{bmatrix}
	598	+X'Y \\
	599	+r
	600	+\end{bmatrix}
	601	+\end{align*}
	602	+
	603	+\item Pour obtenir une solution de $\beta^{LC}$, il nous suffit de trouver la solution pour $\beta$ dans l'équation 1.
	604	+\begin{align*}
	605	+X'X \beta^{LC} + R' \lambda = X'Y
	606	+\end{align*}
	607	+\begin{align*}
	608	+X'X \beta^{LC} = X'Y-R' \lambda
	609	+\end{align*}
	610	+\begin{align*}
	611	+\beta^{LC} = (X'X)^{-1}X'Y-(X'X)^{-1}R' \lambda
	612	+\end{align*}
	613	+\end{itemize}
	614	+\end{frame}
	615	+
	616	+\begin{frame}{Les contraintes linéaires}
	617	+\begin{block}{Estimateur sous contrainte linéaire ($\beta^{LC})$}
	618	+\begin{align*}
	619	+\beta^{LC} = (X'X)^{-1}X'Y-(X'X)^{-1}R' \lambda
	620	+\end{align*}
	621	+\end{block}
	622	+\begin{block}{Estimateur OLS ($\beta^{OLS}$)}
	623	+\begin{align*}
	624	+\beta^{OLS} = (X'X)^{-1}X'Y
	625	+\end{align*}
	626	+\end{block}
	627	+On voit donc facilement qu'il nous est possible d'exprimer l'estimateur $\beta^{LC}$ en fonction de l'estimateur $\beta^{OLS}$
	628	+\begin{align*}
	629	+\beta^{LC} = \beta^{OLS}-(X'X)^{-1}R' \lambda
	630	+\end{align*}
	631	+\end{frame}
	632	+
	633	+\begin{frame}{Les contraintes linéaires}
	634	+\begin{itemize}
	635	+\item On doit également trouver une solution pour $\lambda$ afin de pouvoir l'incorporer dans la solution de $\beta^{LC}$
	636	+\item On commence par multiplier chaque coté l'équation de la solution de $\beta^{LC}$ par $R$
	637	+\begin{align*}
	638	+R\beta^{LC} = R\beta^{OLS}-R(X'X)^{-1}R' \lambda
	639	+\end{align*}
	640	+\item On sait déja que $R\beta^{LC}=r$, étant donné la contrainte linéaire posée
	641	+\begin{align*}
	642	+r = R\beta^{OLS}-R(X'X)^{-1}R' \lambda
	643	+\end{align*}
	644	+\item Pour finalement isoler $\beta^{OLS}$
	645	+\begin{align*}
	646	+\lambda=(R(X'X)^{-1}R')^{-1}(R\beta^{OLS}-r)
	647	+\end{align*}
	648	+\end{itemize}
	649	+\end{frame}
	650	+
	651	+\begin{frame}{Les contraintes linéaires}
	652	+\begin{itemize}
	653	+\item On substitut la solution de $\lambda$ dans l'équation de la solution de $\beta^{LC}$
	654	+\begin{align*}
	655	+\beta^{LC} = \beta^{OLS}-(X'X)^{-1}R'(R(X'X)^{-1}R')^{-1}(R\beta^{OLS}-r)
	656	+\end{align*}
	657	+\item On voit que $\beta^{LC}$ (Contraint) est exprimé en fonction de $\beta^{OLS}$ (Non-Contraint).
	658	+\item Le test de Fisher (F-Test) est une spécification d'un contrainte linéaire.
	659	+\begin{itemize}
	660	+\item Le modèle non-contraint est le modèle que nous souhaitons estimer et vérifier la significativité.
	661	+\item Le modèle contraint est simplement un modèle dans lequel nous allons contraindre tous les coefficients d'être égale à 0.
	662	+\end{itemize}
	663	+\end{itemize}
	664	+\end{frame}
	665	+
	666	+\begin{frame}{Les contraintes linéaires}
	667	+\begin{block}{Représentation du F-test sous le format de contrainte linéaire}
	668	+\begin{itemize}
	669	+\item On suppose un modèle avec un intercept $\beta_0$ et une pente $\beta_1$.
	670	+\item On cherche à tester une contrainte linéaire ayant le format $R\beta=r$
	671	+\item La matrice $R$:
	672	+\begin{align*}
	673	+R = \begin{bmatrix}
	674	+1 & 0 \\
	675	+0 & 1
	676	+\end{bmatrix}
	677	+\end{align*}
	678	+\item La matrice $\beta$:
	679	+\begin{align*}
	680	+\beta= \begin{bmatrix}
	681	+\beta_0 \\
	682	+\beta_1
	683	+\end{bmatrix}
	684	+\end{align*}
	685	+\end{itemize}
	686	+\end{block}
	687	+\end{frame}
	688	+
	689	+\begin{frame}{Les contraintes linéaires}
	690	+\begin{block}{Représentation du F-test sous le format de contrainte linéaire}
	691	+\begin{itemize}
	692	+\item La matrice $r$
	693	+\begin{align*}
	694	+r= \begin{bmatrix}
	695	+0 \\
	696	+0
	697	+\end{bmatrix}
	698	+\end{align*}
	699	+\item Ce qui nous donne la contrainte linéaire $(R \beta=r)$ dans le cas du F-Test:
	700	+\begin{align*}
	701	+\begin{bmatrix}
	702	+1 & 0 \\
	703	+0 & 1
	704	+\end{bmatrix}
	705	+\begin{bmatrix}
	706	+\beta_0 \\
	707	+\beta_1
	708	+\end{bmatrix}
	709	+=
	710	+\begin{bmatrix}
	711	+0 \\
	712	+0
	713	+\end{bmatrix}
	714	+\end{align*}
	715	+\end{itemize}
	716	+\end{block}
	717	+\end{frame}
	718	+
	719	+\begin{frame}{Les contraintes linéaires}
	720	+\begin{itemize}
	721	+\item Nous regardons maintenant trois classes de tests qui peuvent être utilisées dans des contextes plus généraux:
	722	+\begin{itemize}
	723	+\item Test d’hypothèse de restrictions non linéaires sur les paramètres
	724	+\item Test d’hypothèse sur la matrice de variance-covariance
	725	+\item Tests dans des modèles sans normalités des erreurs
	726	+\end{itemize}
	727	+\item Les tests présentés sont
	728	+\begin{itemize}
	729	+\item Multiplicateur de Lagrange (LM)
	730	+\item Test de Wald
	731	+\item Ratio de vraisemblance (LR)
	732	+\end{itemize}
	733	+\end{itemize}
	734	+\end{frame}
	735	+
	736	+\section{Test de Wald}
	737	+
	738	+\frame{\tableofcontents[current]}
	739	+
	740	+\begin{frame}{Test de Wald}
	741	+\begin{itemize}
	742	+\item Le test de Wald est une forme quadratique basé sur la distance entre $(R \beta -r)$ et zéro.
	743	+\item C’est-à-dire si l’estimé non contraint vérifie la contrainte.
	744	+\item On rejette l’hypothèse nulle si la statistique est suffisamment grande.
	745	+\begin{block}{Hypothèses}
	746	+\begin{itemize}
	747	+\item $H_0:$ $R \beta = r$
	748	+\item $H_1:$ $R \beta \neq r$
	749	+\end{itemize}
	750	+\end{block}
	751	+\begin{block}{Statistique $W$}
	752	+\begin{align*}
	753	+W=(R \beta -r)'[V(R \beta -r)]^{-1}(R \beta-r)
	754	+\end{align*}
	755	+Où $V(R \beta -r)$ est la variance entre les deux termes.
	756	+\end{block}
	757	+\end{itemize}
	758	+\end{frame}
	759	+
	760	+\begin{frame}{Test de Wald}
	761	+\begin{block}{Statistique $W$}
	762	+\begin{itemize}
	763	+\item En décomposant la variance, on trouve une solution pour la statistique de Wald.
	764	+\begin{align*}
	765	+W = \frac{1}{\hat{\sigma}^2}(R \beta -r)'[R(X'X)^{-1}R']^{-1}(R \beta -r)
	766	+\end{align*}
	767	+\end{itemize}
	768	+\end{block}
	769	+\begin{itemize}
	770	+\item Sous l’hypothèse nulle, W suit aussi une khi carré (q).
	771	+\item Le Wald est intéressant, car tous les estimés nécessaires sont non contraints.
	772	+\item On ne fait pas d’estimation contrainte, car on s’intéresse à savoir si le modèle non contraint rejette l’hypothèse nulle d’être « assez prêt » de la contrainte.
	773	+\end{itemize}
	774	+\end{frame}
	775	+
	776	+\section{Ratio de vraisemblance (LR)}
	777	+
	778	+\frame{\tableofcontents[current]}
	779	+
	780	+\begin{frame}{Ratio de vraisemblance (LR)}
	781	+\begin{itemize}
	782	+\item \textbf{Intuition:} comparer la valeur de la vraisemblance aux maximum contraint et non contraint.
	783	+\item Si ces deux valeurs sont proches l’une de l’autre, cela implique que la contrainte est aisément satisfaite par les observations et non couteuse en termes de maximisation de la vraisemblance.
	784	+\item La vraissemblance du modèle non-contraint est représenté par $\hat{L}$, alors que la vraissemblance du modèle contraint est représenté par $\hat{L}_c$
	785	+\item L'estimateur $\hat{L}$ sera une fonction des estimateurs $\hat{\beta}$ et $\hat{\sigma}^2$ du modèle non contraint $\rightarrow$ $\hat{L}(\hat{\beta},\hat{\sigma}^2)$
	786	+\item L'estimateur $\hat{L}_c$ sera une fonction des estimateurs $\hat{\beta}_c$ et $\hat{\sigma}_c^2$ du modèle non contraint $\rightarrow$ $\hat{L}_c(\hat{\beta}_c,\hat{\sigma}_c^2)$
	787	+\end{itemize}
	788	+\end{frame}
	789	+
	790	+\begin{frame}{Ratio de vraisemblance (LR)}
	791	+\begin{itemize}
	792	+\item Afin de pouvoir simplifier la vraissemblance, nous poserons l'hypothèse de normalité des résidus.
	793	+\begin{align*}
	794	+\hat{L} & =-\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\hat{\sigma}^2)-\frac{1}{2} \times \frac{(Y-X \hat{\beta})'(Y-X \hat{\beta})}{\hat{\sigma}^2} \\ & =-\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\hat{\sigma}^2)-\frac{1}{2} \times \frac{\hat{u}'\hat{u}}{\hat{\sigma}^2} \\ & = -\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\hat{\sigma}^2)-\frac{1}{2} \times \frac{T \hat{\sigma}^2}{\hat{\sigma}^2} \\ & = -\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\hat{\sigma}^2)-\frac{T}{2}
	795	+\end{align*}
	796	+\end{itemize}
	797	+\end{frame}
	798	+
	799	+\begin{frame}{Ratio de vraisemblance (LR)}
	800	+\begin{block}{Vraissemblance du modèle non-contraint}
	801	+\begin{align*}
	802	+\hat{L}=-\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\hat{\sigma}^2)-\frac{T}{2}
	803	+\end{align*}
	804	+\end{block}
	805	+\begin{block}{Vraissemblance du modèle contraint}
	806	+\begin{align*}
	807	+\hat{L}_c=-\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\hat{\sigma}_c^2)-\frac{T}{2}
	808	+\end{align*}
	809	+\end{block}
	810	+\end{frame}
	811	+
	812	+\begin{frame}{Ratio de vraisemblance (LR)}
	813	+\begin{itemize}
	814	+\item On veut comparer la valeur de la vraisemblance $\hat{L}$ et $\hat{L}_c$
	815	+\item La statistique LR s’écrit donc comme la différence entre les estimés contraints et non-contraints de $\sigma^2$
	816	+\begin{align*}
	817	+LR & =2[\hat{L}-\hat{L}_c]\\ & = 2\left[ -\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\hat{\sigma}^2)-\frac{T}{2} \right]\\ & -2\left[ -\frac{T}{2} \log (2 \pi) -\frac{T}{2} \log (\hat{\sigma}_c^2)-\frac{T}{2} \right] \\ & =T \log (\hat{\sigma}_c^2)-T \log (\hat{\sigma}^2) \\ & = T \log \left( \frac{\hat{\sigma}_c^2}{\hat{\sigma}^2}\right) = T \log \left( \frac{\hat{u}_c^{'}\hat{u}_c}{\hat{u}^{'} \hat{u}}\right)
	818	+\end{align*}
	819	+\item La statistique suit donc une loi Khi carré (q) asymptotiquement
	820	+\end{itemize}
	821	+\end{frame}
	822	+
	823	+\section{Multiplicateur de Lagrange}
	824	+
	825	+\frame{\tableofcontents[current]}
	826	+
	827	+
	828	+\begin{frame}{Multiplicateur de Lagrange}
	829	+\begin{itemize}
	830	+\item Il s’agit d’une régression auxiliaire.
	831	+\item La statistique de test sera :$T \times R^2$
	832	+\item Cette statistique suit alors une khi-carré avec q degré de liberté.
	833	+\item Intuition de ce test : vérifier si le score (la dérivé première du Lagrangien en fonction des paramètres) est proche de zéro évalué en $\hat{\beta}_c$.
	834	+\item Si oui, la contrainte n’est pas très couteuse en termes d’optimisation et il est probable que les paramètres prennent les valeurs définies par l’hypothèse nulle.
	835	+\end{itemize}
	836	+\end{frame}
	837	+
	838	+\begin{frame}{Multiplicateur de Lagrange}
	839	+\begin{block}{Statistique LM}
	840	+\begin{align*}
	841	+LM= \left[ \frac{\partial L}{\partial \beta} (\hat{\beta}_c)\right]^{'}\left[V \frac{\partial L}{\partial \beta} (\hat{\beta}_c)\right]^{-1}\left[ \frac{\partial L}{\partial \beta} (\hat{\beta}_c)\right]
	842	+\end{align*}
	843	+En effectuant les dérivés premières et quelques manipulations, on obtient la solution pour la statistique LM.
	844	+\begin{align*}
	845	+LM=\frac{1}{\sigma^2} (\hat{\beta}-\hat{\beta}_c)'R'CR(\hat{\beta}-\hat{\beta}_c)
	846	+\end{align*}
	847	+
	848	+\end{block}
	849	+\end{frame}
	850	+
	851	+\begin{frame}{Multiplicateur de Lagrange}
	852	+\begin{block}{Statistique LM}
	853	+\begin{itemize}
	854	+\item On peut aussi réécrire cette statistique comme une fonction des résidus contraints et non contraints.
	855	+\begin{align*}
	856	+LM = \frac{\hat{u}_c^{'}\hat{u}_c-\hat{u}^{'}\hat{u}}{\frac{\hat{u}_c^{'}\hat{u}_c}{T}}
	857	+\end{align*}
	858	+\item Cette statistique suit une loi Chi carré avec q degré de liberté asymptotiquement sous l’hypothèse nulle.
	859	+\item Le LM exploite le fait que la maximisation du log vraisemblance sous la contrainte de l’hypothèse nulle revient à maximiser sans contrainte la fonction Lagrangienne associée.
	860	+\item Le test est alors basé sur le fait que si la contrainte sous $H_0$ est respectée par les données, le vecteur de Lagrange devrait être nul.
	861	+
	862	+\end{itemize}
	863	+\end{block}
	864	+\end{frame}
	865	+
	866	+\section{Lien entre les Statistiques F, WALD, LR et LM}
	867	+
	868	+\frame{\tableofcontents[current]}
	869	+
	870	+\begin{frame}{Lien entre les Statistiques F, WALD, LR et LM}
	871	+\begin{itemize}
	872	+\item Toutes ces statistiques se calculent à partir de la somme des résidus au carré.
	873	+\item Il est possible d’exprimer les trois statistiques présentées comme une transformation de la statistique F dans le cas de contraintes linéaires et du modèle linéaire simple.
	874	+\end{itemize}
	875	+\begin{block}{Statistique F}
	876	+
	877	+\begin{align*}
	878	+F= \frac{\hat{u}_0'\hat{u}_0-\hat{u}'\hat{u}}{\hat{u}'\hat{u}} \times \frac{t-k}{q}>F(q,t-k;\alpha)
	879	+\end{align*}
	880	+\begin{itemize}
	881	+\item On rejette $H_0$ si la statistique de test est plus grande que le point critique associé
	882	+\end{itemize}
	883	+\end{block}
	884	+\end{frame}
	885	+
	886	+\begin{frame}{Lien entre les Statistiques F, WALD, LR et LM}
	887	+\begin{block}{Statistique Wald}
	888	+\begin{align*}
	889	+Wald=T \frac{\hat{u}_c'\hat{u}_c-\hat{u}'\hat{u}}{\hat{u}'\hat{u}} =\frac{Tq}{T-K}F
	890	+\end{align*}
	891	+\end{block}
	892	+\begin{block}{Statistique LR}
	893	+\begin{align*}
	894	+LR=T \log \left[ \frac{\hat{u}_c'\hat{u}_c}{\hat{u}'\hat{u}} \right]= T \log \left[F \frac{q}{T-K}+1 \right]
	895	+\end{align*}
	896	+\end{block}
	897	+\begin{block}{Statistique LM}
	898	+\begin{align*}
	899	+LM=\frac{\hat{u}_c^{'}\hat{u}_c-\hat{u}^{'}\hat{u}}{\frac{\hat{u}_c^{'}\hat{u}_c}{T}}=\frac{T}{\left[\frac{1}{F}\right] \left[\frac{T-K}{q}\right]+1}
	900	+\end{align*}
	901	+\end{block}
	902	+
	903	+\end{frame}
	904	+
	905	+\begin{frame}{Lien entre les Statistiques F, WALD, LR et LM}
	906	+\begin{itemize}
	907	+\item Rejette si la statistique de test est plus grande que le point critique d’une Chi-carré avec q degrés de liberté.
	908	+\item Bien que toutes ses statistiques soient maintenant exprimées en fonction des résidus contraints et non contraints, elles ne donnent pas la même valeur.
	909	+\item L’inférence pourrait donc potentiellement être différente.
	910	+\end{itemize}
	911	+
	912	+\end{frame}
	913	+\end{document}
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	1	+\beamer@sectionintoc {1}{Analyse de Variance}{4}{0}{1}
	2	+\beamer@sectionintoc {2}{Coefficient de détermination $(R^2)$}{12}{0}{2}
	3	+\beamer@sectionintoc {3}{Coefficient de détermination ajusté $(R^2$ ajusté)}{14}{0}{3}
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	7	+\beamer@sectionintoc {7}{Ratio de vraisemblance (LR)}{49}{0}{7}
	8	+\beamer@sectionintoc {8}{Multiplicateur de Lagrange}{54}{0}{8}
	9	+\beamer@sectionintoc {9}{Lien entre les Statistiques F, WALD, LR et LM}{58}{0}{9}

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	1078	+
	1079	+I treat `x^1^2' essentially like `x^1{}^2'.
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	1081	+! Double superscript.
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	1085	+
	1086	+I treat `x^1^2' essentially like `x^1{}^2'.
	1087	+
	1088	+! Double superscript.
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	1092	+
	1093	+I treat `x^1^2' essentially like `x^1{}^2'.
	1094	+
	1095	+! Double superscript.
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	1099	+
	1100	+I treat `x^1^2' essentially like `x^1{}^2'.
	1101	+
	1102	+! Double superscript.
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	1105	+l.422 \end{frame}
	1106	+
	1107	+I treat `x^1^2' essentially like `x^1{}^2'.
	1108	+
	1109	+! Double superscript.
	1110	+<recently read> ^
	1111	+
	1112	+l.422 \end{frame}
	1113	+
	1114	+I treat `x^1^2' essentially like `x^1{}^2'.
	1115	+
	1116	+! Double superscript.
	1117	+<recently read> ^
	1118	+
	1119	+l.422 \end{frame}
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	1121	+I treat `x^1^2' essentially like `x^1{}^2'.
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	1123	+
	1124	+Overfull \vbox (9.69542pt too high) detected at line 422
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	1134	+
	1135	+I treat `x^1^2' essentially like `x^1{}^2'.
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	1137	+! Double superscript.
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	1139	+
	1140	+l.432 \end{frame}
	1141	+
	1142	+I treat `x^1^2' essentially like `x^1{}^2'.
	1143	+
	1144	+! Double superscript.
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	1146	+
	1147	+l.432 \end{frame}
	1148	+
	1149	+I treat `x^1^2' essentially like `x^1{}^2'.
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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage{graphicx}
	9	+\usetheme{default}
	10	+\usecolortheme{default}
	11	+
	12	+\title[S03 Extensions MLS]{Section 03 : Extensions au modèle linéaire simple\\ (Séance 6)}
	13	+\subtitle{GSF-6053: Économétrie Financière}
	14	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	15	+\institute[Université Laval]
	16	+{
	17	+ \inst{1}%
	18	+ Département de finance, assurance et immobilier\\
	19	+ Faculté des sciences de l'administration\\
	20	+ Université Laval}
	21	+\date[Hiver 2022]{15 février 2022}
	22	+
	23	+\begin{document}
	24	+
	25	+\begin{frame}
	26	+ \titlepage
	27	+\end{frame}
	28	+
	29	+\begin{frame}{Références}
	30	+\textbf{Obligatoires:}
	31	+\begin{itemize}
	32	+\item \textbf{Notes de cours:} Section 3 (Professeure: Marie-Hélène Gagnon)
	33	+\item \textbf{Woolridge:} chapitres 3, 8, 12.
	34	+\end{itemize}
	35	+\vspace{0.5cm}
	36	+\textbf{Complémentaires:}
	37	+\begin{itemize}
	38	+\item \textbf{Gujarati et Porter:} chapitres 10, 11, 12, 13 et appendice C.
	39	+\item \textbf{Greene:} chapitres 2, 3, 4, 5, 9, 14, 20 C et D
	40	+\end{itemize}
	41	+\end{frame}
	42	+
	43	+
	44	+\begin{frame}{Plan de la séance}
	45	+ \tableofcontents
	46	+\end{frame}
	47	+
	48	+\section{Hétéroscédasticité}
	49	+
	50	+\frame{\tableofcontents[current]}
	51	+
	52	+
	53	+\begin{frame}{Hétéroscédasticité}
	54	+\begin{block}{Homoscédasticité}
	55	+\begin{itemize}
	56	+\item OLS makes the assumption that the variance of the error term is constant.
	57	+\begin{align*}
	58	+V(u_i)= \sigma^2
	59	+\end{align*}
	60	+\end{itemize}
	61	+\end{block}
	62	+
	63	+\begin{block}{Hétéroscédasticité}
	64	+\begin{itemize}
	65	+\item Si les résidus n'ont pas une variance constante, on dit qu'ils sont hétéroscédastiques.
	66	+\begin{align*}
	67	+V(u_i)= \sigma_i^2
	68	+\end{align*}
	69	+\end{itemize}
	70	+\end{block}
	71	+\end{frame}
	72	+
	73	+
	74	+\begin{frame}{Hétéroscédasticité}
	75	+\begin{block}{Forme de $\Omega$}
	76	+\begin{itemize}
	77	+\item On connait rarement $\Omega$, il faut spécifier une forme pour cette matrice qui est gérable.
	78	+\item Estimation de tous les $\sigma_i^2$ pour chaque observations.
	79	+\begin{align*}
	80	+\Omega= \begin{bmatrix} \sigma_1^2 & 0 & \cdots & 0 \\
	81	+0 & \sigma_{2}^2 & \cdots & 0 \\
	82	+\vdots & \vdots & \ddots & \vdots \\
	83	+0 & 0 & \cdots & \sigma_T^2
	84	+\end{bmatrix}
	85	+\end{align*}
	86	+\begin{itemize}
	87	+\item N’est pas une forme estimable, car le nombre de paramètres à estimer $T+K>T$.
	88	+\end{itemize}
	89	+\item Il faut donc spécifier un modèle (une paramétrisation) pour la forme de matrice $\Omega$ pour réduire le nombre de paramètres inconnus.
	90	+\end{itemize}
	91	+\end{block}
	92	+\end{frame}
	93	+
	94	+\begin{frame}{Hétéroscédasticité}
	95	+\begin{block}{Hétéroscédasticité groupée}
	96	+\begin{itemize}
	97	+\item On a une variance hétéroscédastique, mais en sous-groupe.
	98	+\item À l’intérieur du sous-groupe, la variance est constante.
	99	+\end{itemize}
	100	+\begin{align*}
	101	+\sigma_t^2=\sigma_{10}^2, \hspace{0.5cm} t=1,...,T_1
	102	+\end{align*}
	103	+
	104	+\begin{align*}
	105	+\sigma_t^2=\sigma_{20}^2, \hspace{0.5cm} t=T_{1+1},...,T
	106	+\end{align*}
	107	+\end{block}
	108	+\end{frame}
	109	+
	110	+
	111	+\begin{frame}{Hétéroscédasticité}
	112	+\begin{block}{Hétéroscédasticité groupée}
	113	+\begin{align*}
	114	+\Omega=\begin{bmatrix}
	115	+\begin{bmatrix}
	116	+\sigma_{10}^2 & 0 & \cdots & 0 \\
	117	+0 & \sigma_{10}^2 & \cdots & 0 \\
	118	+\vdots & \vdots & \ddots & \vdots \\
	119	+0 & 0 & \cdots & \sigma_{10}^2
	120	+\end{bmatrix} & \begin{bmatrix}
	121	+0 & 0 & \cdots & 0 \\
	122	+0 & 0 & \cdots & 0 \\
	123	+\vdots & \vdots & \ddots & \vdots \\
	124	+0 & 0 & \cdots & 0
	125	+\end{bmatrix} \\ \\
	126	+ \begin{bmatrix}
	127	+0 & 0 & \cdots & 0 \\
	128	+0 & 0 & \cdots & 0 \\
	129	+\vdots & \vdots & \ddots & \vdots \\
	130	+0 & 0 & \cdots & 0
	131	+\end{bmatrix} & \begin{bmatrix}
	132	+\sigma_{20}^2 & 0 & \cdots & 0 \\
	133	+0 & \sigma_{20}^2 & \cdots & 0 \\
	134	+\vdots & \vdots & \ddots & \vdots \\
	135	+0 & 0 & \cdots & \sigma_{20}^2
	136	+\end{bmatrix}
	137	+\end{bmatrix}
	138	+\end{align*}
	139	+\end{block}
	140	+\end{frame}
	141	+
	142	+\begin{frame}{Hétéroscédasticité}
	143	+\begin{block}{Hétéroscédasticité groupée}
	144	+On peut réécrire la matrice $\Omega$, avec hétéroscédasticité groupée comme suit:
	145	+\begin{align*}
	146	+\Omega=
	147	+\begin{bmatrix}
	148	+\sigma_{10}^2 I_{T_1} & 0\\
	149	+ 0 & \sigma_{20}^2 I_{T_2}
	150	+\end{bmatrix}
	151	+\end{align*}
	152	+Sachant:
	153	+\begin{align*}
	154	+\sigma_{10}^2 I_{T_1}=\begin{bmatrix}
	155	+\sigma_{10}^2 & 0 & \cdots & 0 \\
	156	+0 & \sigma_{10}^2 & \cdots & 0 \\
	157	+\vdots & \vdots & \ddots & \vdots \\
	158	+0 & 0 & \cdots & \sigma_{10}^2
	159	+\end{bmatrix} \textbf{et}\hspace{0.2cm} \sigma_{20}^2 I_{T_2}=\begin{bmatrix}
	160	+\sigma_{20}^2 & 0 & \cdots & 0 \\
	161	+0 & \sigma_{20}^2 & \cdots & 0 \\
	162	+\vdots & \vdots & \ddots & \vdots \\
	163	+0 & 0 & \cdots & \sigma_{20}^2
	164	+\end{bmatrix}
	165	+\end{align*}
	166	+\end{block}
	167	+\end{frame}
	168	+
	169	+\begin{frame}{Hétéroscédasticité}
	170	+\begin{block}{Hétéroscédasticité groupée}
	171	+Dans un cadre plus général:
	172	+\begin{align*}
	173	+\Omega= \begin{bmatrix} \sigma_{10}^2I_{T_1} & 0 & \cdots & 0 \\
	174	+0 & \sigma_{20}^2I_{T_2} & \cdots & 0 \\
	175	+\vdots & \vdots & \ddots & \vdots \\
	176	+0 & 0 & \cdots & \sigma_{G0}^2I_{T_G}
	177	+\end{bmatrix}
	178	+\end{align*}
	179	+\begin{align*}
	180	+\Omega^{-1}= \begin{bmatrix} \frac{1}{\sigma_{10}^2I_{T_1}} & 0 & \cdots & 0 \\
	181	+0 & \frac{1}{\sigma_{20}^2I_{T_2}} & \cdots & 0 \\
	182	+\vdots & \vdots & \ddots & \vdots \\
	183	+0 & 0 & \cdots & \frac{1}{\sigma_{G0}^2I_{T_G}}
	184	+\end{bmatrix}
	185	+\end{align*}
	186	+\end{block}
	187	+\end{frame}
	188	+
	189	+\begin{frame}{Hétéroscédasticité}
	190	+\begin{block}{Hétéroscédasticité groupée}
	191	+Dans un cadre plus général:
	192	+
	193	+\begin{align*}
	194	+P^{-1}= \begin{bmatrix} \frac{1}{\sigma_{10}I_{T_1}} & 0 & \cdots & 0 \\
	195	+0 & \frac{1}{\sigma_{20}I_{T_2}} & \cdots & 0 \\
	196	+\vdots & \vdots & \ddots & \vdots \\
	197	+0 & 0 & \cdots & \frac{1}{\sigma_{G0}I_{T_G}}
	198	+\end{bmatrix}
	199	+\end{align*}
	200	+\begin{itemize}
	201	+\item Les moindres carrés sont donc pondérés par la matrice $P^{-1}$, ce qui revient à pondérer par l’écart type qui est hétéroscédastique par groupe $G$.
	202	+\end{itemize}
	203	+\end{block}
	204	+\end{frame}
	205	+
	206	+
	207	+\begin{frame}{Hétéroscédasticité}
	208	+\begin{block}{Hétéroscédasticité groupée}
	209	+Modèle transformé:
	210	+
	211	+\begin{align*}
	212	+\begin{bmatrix}
	213	+\frac{Y_1}{\sqrt{Z_1}}\\
	214	+\frac{Y_2}{\sqrt{Z_2}}\\
	215	+\vdots \\
	216	+\frac{Y_T}{\sqrt{Z_T}}
	217	+\end{bmatrix}
	218	+=
	219	+\begin{bmatrix}
	220	+\frac{1}{\sqrt{Z_1}}\\
	221	+\frac{1}{\sqrt{Z_2}}\\
	222	+\vdots \\
	223	+\frac{1}{\sqrt{Z_T}}
	224	+\end{bmatrix}
	225	+\beta_0
	226	++
	227	+\begin{bmatrix}
	228	+\frac{X_{11}}{\sqrt{Z_1}}\\
	229	+\frac{X_{12}}{\sqrt{Z_2}}\\
	230	+\vdots \\
	231	+\frac{X_{1T}}{\sqrt{Z_T}}
	232	+\end{bmatrix}
	233	+\beta_1
	234	++ \cdots +
	235	+\begin{bmatrix}
	236	+\frac{X_{K-11}}{\sqrt{Z_1}}\\
	237	+\frac{X_{K-12}}{\sqrt{Z_2}}\\
	238	+\vdots \\
	239	+\frac{X_{K-1T}}{\sqrt{Z_T}}
	240	+\end{bmatrix}
	241	+\beta_K
	242	++
	243	+\begin{bmatrix}
	244	+\frac{u_1}{\sqrt{Z_1}}\\
	245	+\frac{u_2}{\sqrt{Z_2}}\\
	246	+\vdots \\
	247	+\frac{u_T}{\sqrt{Z_T}}
	248	+\end{bmatrix}
	249	+\end{align*}
	250	+\begin{itemize}
	251	+\item Remarquez que le modèle devient sans constante. Ceci a des implications entre autres pour le $R^2$ du modèle.
	252	+\end{itemize}
	253	+\end{block}
	254	+\end{frame}
	255	+
	256	+\begin{frame}{Hétéroscédasticité}
	257	+\begin{block}{Variance varie en fonction de plusieurs variables exogènes}
	258	+Soit le modèle suivant:
	259	+\begin{align*}
	260	+Y_t=\beta_0 + \beta_1 X_{1t}+\cdots+ \beta_{K-1} X_{k-1t}+u_t
	261	+\end{align*}
	262	+\begin{align*}
	263	+\sigma_t^2=\alpha_0+\alpha_1 Z_{1t}+\cdots+\alpha_H Z_{Ht}+\epsilon_t
	264	+\end{align*}
	265	+\end{block}
	266	+\end{frame}
	267	+
	268	+
	269	+\begin{frame}{Hétéroscédasticité}
	270	+\begin{block}{Variance varie en fonction de plusieurs variables exogènes}
	271	+\begin{itemize}
	272	+\item Cette paramétrisation de la variance demande d’estimer $\hat{\sigma}_t^2$ par une régression artificielle.
	273	+\begin{itemize}
	274	+\item On utilise les résidus au carré, $\hat{u}^2$, pour avoir une proxy de $\sigma_t^1$.
	275	+\item On régresse les $\hat{u}^2$ sur les variables exogènes pour trouver les
	276	+paramètres $\alpha_h$ définissant la variance.
	277	+\item On trouve les valeurs des $\hat{\sigma}_t^2$ estimés par le modèle avec les résultats de
	278	+la régression auxiliaire :
	279	+\begin{align*}
	280	+\hat{\sigma}_t^2=\hat{\alpha}_0+\hat{\alpha}_1 Z_{1t}+\cdots+\hat{\alpha}_HZ_{Ht}
	281	+\end{align*}
	282	+\end{itemize}
	283	+\item Cette technique demande des données supplémentaires sur les exogènes.
	284	+\end{itemize}
	285	+\end{block}
	286	+\end{frame}
	287	+
	288	+
	289	+\begin{frame}{Hétéroscédasticité}
	290	+\begin{block}{Variance varie en fonction de plusieurs variables exogènes}
	291	+Le modèle transformé sera:
	292	+\begin{align*}
	293	+\frac{Y_t}{\hat{\sigma}_T}=\beta_0 \frac{1}{\hat{\sigma}_t}+\beta_1 \frac{X_{1t}}{\hat{\sigma}_t}+\cdots+\beta_{K-1} \frac{X_{K-1t}}{\hat{\sigma}_t}+\frac{u_t}{\hat{\sigma}_t}
	294	+\end{align*}
	295	+Où
	296	+\begin{align*}
	297	+\hat{\sigma}=\sqrt{\hat{\alpha}_0+\hat{\alpha}_1 Z_{1t}+\cdots+\hat{\alpha}_H Z_{Ht}}
	298	+\end{align*}
	299	+En estimant par OLS
	300	+\begin{itemize}
	301	+\item Contrairement au cas à une seule variable exogène, les estimateurs des $\beta_K$ sont obtenus par FGLS et nom par GLS, car nous avons estimé la matrice $\Omega$ à partir de variable exogène.
	302	+\end{itemize}
	303	+\end{block}
	304	+\end{frame}
	305	+
	306	+\section{Tests diagnostiques d'hétéroscédasticité}
	307	+
	308	+\frame{\tableofcontents[current]}
	309	+
	310	+
	311	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	312	+\begin{itemize}
	313	+\item Nous introduisons des tests pour savoir si les régressions considérées souffrent de problème d’hétéroscédasticité des erreurs (et résidus).
	314	+\item Une approche intuitive serait de regarder un graphique des résidus de régression dans le but de trouver des anomalies.
	315	+\item Une approche par analyse de graphique n’est pas un test statistique formel de la présence d’hétéroscédasticité.
	316	+\end{itemize}
	317	+\end{frame}
	318	+
	319	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	320	+\begin{block}{Test de Goldfeld-Quandt (GQ)}
	321	+\textbf{Modèle:}
	322	+\begin{align*}
	323	+Y_i=\beta_0+\beta_1 X_{1i}+\cdots+\beta_{K-1}X_{K-1i}+u_i
	324	+\end{align*}
	325	+\begin{itemize}
	326	+\item On teste l’hypothèse nulle d’une variance constante dans le temps contre l’alternative que la variance augmente de manière monotone selon un exogène $Z_{1i}$.
	327	+\item Hypothèse nulle:
	328	+\begin{align*}
	329	+H_0: \sigma_i^2 = \sigma^2 \forall i
	330	+\end{align*}
	331	+\item Hypothèse alternative:
	332	+\begin{align*}
	333	+H_A: \sigma_i^2 \hspace{0.2cm} \textbf{augmente de manière monotone selon} \hspace{0.2cm} Z_{1i}
	334	+\end{align*}
	335	+\end{itemize}
	336	+\end{block}
	337	+\end{frame}
	338	+
	339	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	340	+\begin{block}{Test de Goldfeld-Quandt (GQ)}
	341	+\textbf{Marche à suivre:}
	342	+\begin{enumerate}
	343	+\item Ordonner les observations selon la variable Z.
	344	+\item Omettre c observation centrale
	345	+\item Effectuer deux régressions séparées sur les $\frac{N-c}{2}$ premières et les $\frac{N-c}{2}$ dernières observations et calculer $\hat{\sigma}_1^2$ et $\hat{\sigma}_2^2$
	346	+\item La statistique de test est:
	347	+\begin{align*}
	348	+GQ = \frac{\hat{\sigma}_2^2}{\hat{\sigma}_1^2} \sim F \left( \frac{N-c-2K}{2}, \frac{N-c-2K}{2}\right)
	349	+\end{align*}
	350	+\end{enumerate}
	351	+\end{block}
	352	+\end{frame}
	353	+
	354	+
	355	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	356	+\begin{block}{Test de Goldfeld-Quandt (GQ)}
	357	+\begin{itemize}
	358	+\item les estimateurs de $\hat{\sigma}_1^2$ et de $\hat{\sigma}_2^2$ sont calculés à partir des deux régressions en sous-groupe.
	359	+\item Le nombre d’observations sera $\frac{N-c}{2}$ et le nombre de degrés de liberté sera $\frac{N-c}{2-k}$.
	360	+\item Ceci est équivalent à $\frac{N-c-2K}{2}$ qui représente les degrés de liberté du numérateur et du dénominateur de la statistique.
	361	+\end{itemize}
	362	+\end{block}
	363	+\end{frame}
	364	+
	365	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	366	+\begin{block}{Test de Goldfeld-Quandt (GQ)}
	367	+\begin{itemize}
	368	+\item Le test est exact et non asymptotique.
	369	+\item On perd les observations du centre, car on veut voir ce qui se passe aux extrémités.
	370	+\item Si les erreurs sont normales, le test suit exactement une loi F et non asymptotiquement.
	371	+\item On n’a donc pas besoin d’une grande taille d’échantillon pour ce test.
	372	+\end{itemize}
	373	+\end{block}
	374	+\end{frame}
	375	+
	376	+
	377	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	378	+\begin{block}{Test de Breusch-Pagan (BP)}
	379	+\begin{itemize}
	380	+\item Le test est plus général que le précédent et plus utilisé
	381	+\item En présence d'exogènes:
	382	+\begin{align*}
	383	+Y_t=\beta_0+\beta_1 X_{1t}+ \cdots+\beta_{K-1} X_{K-1t}+u_t
	384	+\end{align*}
	385	+\begin{align*}
	386	+\sigma_t^2=\alpha_0+\alpha_1 Z_{1t}+\cdots+\alpha_H Z_{Ht}+\epsilon_t
	387	+\end{align*}
	388	+\begin{align*}
	389	+H_0: \alpha_1=\cdots=\alpha_H=0 \forall h
	390	+\end{align*}
	391	+\begin{align*}
	392	+H_A: \exists \alpha_h \neq 0
	393	+\end{align*}
	394	+\end{itemize}
	395	+\end{block}
	396	+\end{frame}
	397	+
	398	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	399	+\begin{block}{Test de Breusch-Pagan (BP)}
	400	+\begin{itemize}
	401	+\item Le test est basé sur une régression artificielle suivante :
	402	+
	403	+\begin{align*}
	404	+u_t^2=\hat{\alpha_0}+\hat{\alpha_1} Z_{1t}+\cdots+\hat{\alpha_H} Z_{Ht}
	405	+\end{align*}
	406	+\item Le test porte sur tous les coefficients de la régression, sauf la constante.
	407	+\item Il est donc naturel qu’il soit basé sur le R2 de la régression artificielle.
	408	+\end{itemize}
	409	+\end{block}
	410	+\end{frame}
	411	+
	412	+
	413	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	414	+\begin{block}{Test de Breusch-Pagan (BP)}
	415	+\textbf{Statistique de test:}
	416	+\begin{align*}
	417	+BP=T \times R_{reg artificielle}^2
	418	+\end{align*}
	419	+\begin{align*}
	420	+BP^{asy} \sim X^{2}(H)
	421	+\end{align*}
	422	+\begin{itemize}
	423	+\item Le $R^2$ de la régression artificielle sera petit, mais cela ne veut pas dire qu’il n’y a pas d’hétéroscédasticité, car le test est basé sur $(T \times R^2)$.
	424	+\end{itemize}
	425	+\end{block}
	426	+\end{frame}
	427	+
	428	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	429	+\begin{block}{Test d’effet ARCH sur les résidus de régression.}
	430	+\begin{itemize}
	431	+\item \textbf{ARCH:} autoregressive conditionnal heteroscedasticity
	432	+\item On veut tester l’hypothèse de \textbf{volatility clostering}, la variabilité d'aujourd’hui est reliée à la variabilité d'hier.
	433	+
	434	+\begin{align*}
	435	+u_t = \epsilon_t \sqrt{\alpha_0+\alpha_1 u_{t-1}^2}
	436	+\end{align*}
	437	+Sachant
	438	+\begin{align*}
	439	+\epsilon_t \sim N(0,1)
	440	+\end{align*}
	441	+\end{itemize}
	442	+\end{block}
	443	+\end{frame}
	444	+
	445	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	446	+\begin{block}{Test d’effet ARCH sur les résidus de régression.}
	447	+\begin{itemize}
	448	+\item \textbf{Variance conditionnelle au passé:}
	449	+\begin{align*}
	450	+E(u_t^2 \mid u_{t-1})=\alpha_0+\alpha_1 u_{t-1}^2
	451	+\end{align*}
	452	+\item La variance conditionnelle au passé est hétéroscédastique (varie dans le temps ici)
	453	+\item $V(u_t)$ (la variance inconditionnelle) est invariante dans le temps si $\mid \alpha_1 \mid < 1$
	454	+\end{itemize}
	455	+\end{block}
	456	+\end{frame}
	457	+
	458	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	459	+\begin{block}{Test d’effet ARCH sur les résidus de régression.}
	460	+\begin{itemize}
	461	+\item \textbf{Variance inconditionnelle:}
	462	+\begin{align*}
	463	+E(u_t^2) = E(\epsilon_t^2) \times E(\alpha_0+\alpha_1 u_{t-1}^2)+cov(\epsilon_t,u_{t-1})
	464	+\end{align*}
	465	+\begin{align*}
	466	+V(u_t)=V(u)= \frac{\alpha_0}{1-\alpha_1}
	467	+\end{align*}
	468	+\item Pour évaluer cette hypothèse, on peut appliquer un test de Breusch-Pagan basé sur une régression artificielle des $\hat{u}_t^2$sur une constante et $\hat{u}_{t-1}^2$
	469	+\end{itemize}
	470	+\end{block}
	471	+\end{frame}
	472	+
	473	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	474	+\begin{block}{Test de White}
	475	+\begin{itemize}
	476	+\item Ce test s’applique pour tester l’hétéroscédasticité de forme générale.
	477	+\item On ne spécifie pas de forme pour la variance.
	478	+\item L’hypothèse nulle est que la variance est homoscédastique contre une alternative hétéroscédastique générale.
	479	+\end{itemize}
	480	+Soit le modèle:
	481	+\begin{align*}
	482	+Y_t=\beta_0+\beta_1 X_{1t}+\beta_2 X_{2t}+u_t
	483	+\end{align*}
	484	+Le test est basé sur une régression artificielle:
	485	+\begin{align*}
	486	+\hat{u}_t^2=\alpha_0+\alpha_1 X_{1t}+\alpha_{11} X_{1t}^2+\alpha_2 X_{2t}+\alpha_{22} X_{2t}^2+\alpha_{12} X_{1t}X_{2t}+\epsilon_t
	487	+\end{align*}
	488	+\end{block}
	489	+\end{frame}
	490	+
	491	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	492	+\begin{block}{Test de White}
	493	+\textbf{Statistique de test:}
	494	+\begin{align*}
	495	+WHITE = T \times R_{reg.art}^2
	496	+\end{align*}
	497	+\begin{align*}
	498	+WHITE^{asy} \sim X^2(H)
	499	+\end{align*}
	500	+\begin{itemize}
	501	+\item L’avantage du test de White est qu’il est très général.
	502	+\item Par contre, on perd des degrés de liberté de la régression artificielle, car on doit estimer les produits croisés et les régresseurs au carré.
	503	+\end{itemize}
	504	+\end{block}
	505	+\end{frame}
	506	+
	507	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	508	+\begin{block}{Le Correction de White}
	509	+\begin{itemize}
	510	+\item Lorsque q’on ne peut spécifier une forme paramétrique pour la variance hétéroscédastique des erreurs, on ne peut pas utiliser les estimateurs GLS et FGLS définis plus haut.
	511	+\item Une approche alternative est d’utiliser l’estimateur de $\hat{\beta}_{OLS}$ , mais de corriger l’estimateur de sa variance pour tenir compte de l’hétéroscédasticité.
	512	+\item On se souvient que $\hat{\beta}_{OLS}$ était toujours sans biais pour les cas hétéroscédastiques, mais que sa variance était fausse. On trouve donc une correction pour la variance des OLS.
	513	+\end{itemize}
	514	+\end{block}
	515	+\end{frame}
	516	+
	517	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	518	+\begin{block}{Estimateur de la variance suggéré}
	519	+\begin{align*}
	520	+\hat{V}(\hat{\beta}_{OLS})=T(X'X)^{-1} \hat{\Lambda} (X'X)^{-1}
	521	+\end{align*}
	522	+Où
	523	+\begin{align*}
	524	+\hat{\Lambda}=\frac{1}{T} \sum_{t=1}^T \hat{u}_t^2 (x_t'x_t)
	525	+\end{align*}
	526	+Où $x_t$ est la $t^{\text{ième}}$ ligne de la matrice X. Cet estimateur est difficile à visualiser.
	527	+\begin{align*}
	528	+Var(\hat{\beta}_j)=\frac{\sum \hat{w}_{ji}^2\hat{u}_i^2}{(\sum \hat{w}_{ji}^2)^2}
	529	+\end{align*}
	530	+\end{block}
	531	+\end{frame}
	532	+
	533	+\begin{frame}{Tests diagnostiques d'hétéroscédasticité}
	534	+\begin{block}{Estimateur de la variance suggéré}
	535	+\begin{itemize}
	536	+\item Les $\hat{u}_i$ sont les résidus de la régression originale et les $\hat{w}_j$ sont les résidus obtenus d'une régression auxiliaire du régresseur j sur tous les autres.
	537	+\item Le correcteur de White est donc une approximation de la variance qui pondère les observations des résidus en fonction des régresseurs dans le calcul de la variance.
	538	+\item En général, si on a des indices sur la structure de la matrice $\Omega$, il est préférable de passer par l’estimateur FGLS si la qualité de l’information ajoutée est bonne.
	539	+\item Le correcteur de White est très utile lorsqu’on n’a aucune idée de la forme fonctionnelle à donner à $\Omega$.
	540	+\end{itemize}
	541	+\end{block}
	542	+\end{frame}
	543	+\end{document}

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	1	+% =============================================================================
	2	+% GSF-6053 : Économétrie Financière I
	3	+% Auteur : Simon-Pierre Boucher
	4	+% Contact : contact@spboucher.ai
	5	+% =============================================================================
	6	+\documentclass{beamer}
	7	+\usepackage[utf8]{inputenc}
	8	+\usepackage[T1]{fontenc}
	9	+\usepackage{graphicx}
	10	+\usetheme{default}
	11	+\usecolortheme{default}
	12	+
	13	+\title[S03 Extensions MLS]{Section 03 : Extensions au modèle linéaire simple\\ (Séance 7)}
	14	+\subtitle{GSF-6053: Économétrie Financière}
	15	+\author[SP. Boucher]{Simon-Pierre Boucher\inst{1}}
	16	+\institute[Université Laval]
	17	+{
	18	+ \inst{1}%
	19	+ Département de finance, assurance et immobilier\\
	20	+ Faculté des sciences de l'administration\\
	21	+ Université Laval}
	22	+\date[Hiver 2022]{22 février 2022}
	23	+
	24	+\begin{document}
	25	+
	26	+\begin{frame}
	27	+ \titlepage
	28	+\end{frame}
	29	+
	30	+\begin{frame}{Références}
	31	+\textbf{Obligatoires:}
	32	+\begin{itemize}
	33	+\item \textbf{Notes de cours:} Section 3 (Professeure: Marie-Hélène Gagnon)
	34	+\item \textbf{Woolridge:} chapitres 3, 8, 12.
	35	+\end{itemize}
	36	+\vspace{0.5cm}
	37	+\textbf{Complémentaires:}
	38	+\begin{itemize}
	39	+\item \textbf{Gujarati et Porter:} chapitres 10, 11, 12, 13 et appendice C.
	40	+\item \textbf{Greene:} chapitres 2, 3, 4, 5, 9, 14, 20 C et D
	41	+\end{itemize}
	42	+\end{frame}
	43	+
	44	+
	45	+\begin{frame}{Plan de la séance}
	46	+ \tableofcontents
	47	+\end{frame}
	48	+
	49	+\section{Autocorrélation des erreurs}
	50	+
	51	+\frame{\tableofcontents[current]}
	52	+
	53	+
	54	+\begin{frame}{Autocorrélation des erreurs}
	55	+\begin{itemize}
	56	+\item L’autocorrélation ou la corrélation sérielle des erreurs dans les séries chronologiques est assez fréquente.
	57	+\item Elle découle souvent du fait qu’il y a une certaine « inertie » dans les données économiques et financières, c’est-à-dire que les observations passées se reflètent souvent dans les observations présentes et futures.
	58	+\item Cela peut amener des problèmes de corrélation sérielle des erreurs si une variable autocorrelée est omise du modèle, par exemple.
	59	+\item Il est important de s’assurer lorsque vous détecter de l’autocorrélation qu’elle n’est pas dû à la non-stationnarité de Y et X.
	60	+\item En effet, si ces deux quantités sont non stationnaires, les erreurs le seront possiblement aussi et seront possiblement autocorrelées.
	61	+
	62	+\end{itemize}
	63	+\end{frame}
	64	+
	65	+\begin{frame}{Autocorrélation des erreurs}
	66	+\begin{itemize}
	67	+\item Même si le modèle est bien spécifié, on peut penser que les erreurs (les chocs) affectant les marchés boursiers aujourd’hui ont des chances d’influencer la magnitude et le signe des chocs affectant les marchés boursiers demain aussi.
	68	+\item Dans la plupart des cas en finance, l’autocorrélation sera positive, mais il est possible en théorie d’avoir de l’autocorrélation négative.
	69	+\end{itemize}
	70	+\end{frame}
	71	+
	72	+\begin{frame}{Autocorrélation des erreurs}
	73	+\textbf{Soit le modèle de base :}
	74	+\begin{align*}
	75	+y=X \beta +u
	76	+\end{align*}
	77	+\begin{align*}
	78	+Y_t=\beta_0+\beta_1 X_{1t}+\beta_2 X_{2t}+\cdots+\beta_{K-1} X_{K-1t}+u_t
	79	+\end{align*}
	80	+\begin{align*}
	81	+u_t=\rho u_{t-1}+\epsilon_t
	82	+\end{align*}
	83	+Où $\epsilon_t$ est un bruit blanc de moyenne nulle et de variance constante dans le temps.
	84	+\end{frame}
	85	+
	86	+\begin{frame}{Autocorrélation des erreurs}
	87	+\begin{itemize}
	88	+\item La matrice de variance covariance des erreurs sera affectée, ce qui impliquera comme dans le cas hétéroscédasticitique que l’estimateur OLS standard ne sera pas BLUE et que la variance des estimateurs OLS fausse, bien que l’estimateur sera sans biais.
	89	+\item La démonstration de ceci est du même ressort que celle faite pour l’hétéroscédasticité.
	90	+\item En effet, bien que l’autocorrélation implique des erreurs homoscédastiques (la variance est constante), les termes hors diagonale ne sont pas nuls comme dans le cas de base du modèle linéaire.
	91	+\end{itemize}
	92	+\end{frame}
	93	+
	94	+\begin{frame}{Autocorrélation des erreurs}
	95	+\textbf{Regardons la matrice de variance-covariance des erreurs:}
	96	+\begin{align*}
	97	+E(uu')=\begin{bmatrix}
	98	+E(u_1^2) & E(u_1u_2) & E(u_1u_3) & \cdots & E(u_1u_T)\\
	99	+E(u_2u_1) & E(u_2^2) & E(u_2u_3) & \cdots & E(u_2u_T)\\
	100	+E(u_3u_1) & E(u_3u_2) & E(u_3^2) & \cdots & E(u_3u_T)\\
	101	+\vdots & \vdots & \vdots & \ddots & \vdots \\
	102	+E(u_Tu_1) & E(u_Tu_2) & E(u_Tu_3) & \cdots & E(u_T^2)
	103	+\end{bmatrix}
	104	+\end{align*}
	105	+\begin{align*}
	106	+E(uu')=\begin{bmatrix}
	107	+\sigma_u^2 & \rho \sigma_u^2 & \rho^2 \sigma_u^2 & \cdots & \rho^{T-1} \sigma_u^2 \\
	108	+\rho \sigma_u^2 & \sigma_u^2 & \rho \sigma_u^2 & \cdots & \rho^{T-2} \sigma_u^2 \\
	109	+\rho^2 \sigma_u^2 & \rho \sigma_u^2 & \sigma_u^2 & \cdots & \rho^{T-3} \sigma_u^2 \\
	110	+\vdots & \vdots & \vdots & \ddots & \vdots \\
	111	+\rho^{T-1} \sigma_u^2 & \rho^{T-2} \sigma_u^2 & \rho^{T-3} \sigma_u^2 & \cdots & \sigma_u^2
	112	+\end{bmatrix}
	113	+\end{align*}
	114	+\end{frame}
	115	+
	116	+
	117	+\begin{frame}{Autocorrélation des erreurs}
	118	+\textbf{Regardons la matrice de variance-covariance des erreurs:}
	119	+\begin{align*}
	120	+E(uu')=\sigma_u^2\begin{bmatrix}
	121	+1 & \sigma & \sigma^2 & \cdots & \rho^{T-1}\\
	122	+\rho & 1 & \rho & \cdots & \rho^{T-2} \\
	123	+\rho^2 & \rho & 1 & \cdots & \rho^{T-3} \\
	124	+\vdots & \vdots & \vdots & \ddots & \vdots\\
	125	+\rho^{T-1} & \rho^{T-2} & \rho^{T-3} & \cdots & 1
	126	+\end{bmatrix}
	127	+\end{align*}
	128	+\begin{align*}
	129	+E(uu')= \frac{\sigma_{\epsilon}^2}{1-\rho^2} \begin{bmatrix}
	130	+1 & \sigma & \sigma^2 & \cdots & \rho^{T-1}\\
	131	+\rho & 1 & \rho & \cdots & \rho^{T-2} \\
	132	+\rho^2 & \rho & 1 & \cdots & \rho^{T-3} \\
	133	+\vdots & \vdots & \vdots & \ddots & \vdots\\
	134	+\rho^{T-1} & \rho^{T-2} & \rho^{T-3} & \cdots & 1
	135	+\end{bmatrix}=\sigma_{\epsilon}^2 \Omega
	136	+\end{align*}
	137	+\end{frame}
	138	+
	139	+
	140	+
	141	+\begin{frame}{Autocorrélation des erreurs}
	142	+\textbf{Composition de $\Omega$:}
	143	+\begin{align*}
	144	+\Omega = \frac{1}{1-\rho^2}\begin{bmatrix}
	145	+1 & \sigma & \sigma^2 & \cdots & \rho^{T-1}\\
	146	+\rho & 1 & \rho & \cdots & \rho^{T-2} \\
	147	+\rho^2 & \rho & 1 & \cdots & \rho^{T-3} \\
	148	+\vdots & \vdots & \vdots & \ddots & \vdots\\
	149	+\rho^{T-1} & \rho^{T-2} & \rho^{T-3} & \cdots & 1
	150	+\end{bmatrix}
	151	+\end{align*}
	152	+\end{frame}
	153	+
	154	+\begin{frame}{Autocorrélation des erreurs}
	155	+\textbf{Hypothèses de départ:}
	156	+\begin{block}{$1-\epsilon_t$ est un bruit blanc}
	157	+\begin{itemize}
	158	+\item $E(\epsilon_t)=0$
	159	+\item $V(\epsilon_t)=\sigma_{\epsilon}^2 \forall t$
	160	+\item $E(\epsilon_t\epsilon_s)=0 \forall t \neq s$
	161	+\end{itemize}
	162	+\end{block}
	163	+
	164	+\begin{block}{$2-\mid \rho \mid < 1$}
	165	+\begin{itemize}
	166	+\item 1 sera une valeur critique. (Racine unitaire)
	167	+\item Ceci implique que $u_t$ est un processus stationnaire
	168	+\begin{enumerate}
	169	+\item $E(u_t)=0$
	170	+\item $V(u)=\sigma_u^2, \forall t$
	171	+\item $E(u_tu_{t-j})=E(u_su_{s-j}) \forall t,s$
	172	+\end{enumerate}
	173	+\end{itemize}
	174	+\end{block}
	175	+\end{frame}
	176	+
	177	+\begin{frame}{Autocorrélation des erreurs}
	178	+\textbf{Hypothèses de départ:}
	179	+\begin{block}{$3-\epsilon_t$ n'est pas corrélé avec les $u_s$ donc l'indice (s) est antérieur à t. $\rightarrow \epsilon_t$ n'est pas corrélé avec $u_{t-1}$}
	180	+Écrivons la série des $u_t$ en fonction des chocs $\varepsilon_t$ présents et passés:
	181	+\begin{align*}
	182	+u_t & =\rho u_{t-1}+\epsilon_t \\ & = \rho [\rho u_{t-2}+\epsilon_{t-1}]+\epsilon_t \\ & = \rho^2 u_{t-2}+\rho \epsilon_{t-1}+\epsilon_t \\ & = \rho^2[\rho u_{t-3}+\epsilon_{t-2}]+\rho \epsilon_{t-1}+\epsilon_t \\ & = \rho^3 u_{t-3}+\rho^2 \epsilon_{t-2}+\rho \epsilon_{t-1}+\epsilon_t \\ & = \rho^su_{t-s}+\epsilon_t+\rho \epsilon_{t-1}+\rho^2 \epsilon_{t-2}+\cdots + \rho^{s-1}\epsilon_{t-(s-1)}
	183	+\end{align*}
	184	+\end{block}
	185	+\end{frame}
	186	+
	187	+\begin{frame}{Autocorrélation des erreurs}
	188	+Considérons les possibilités suivantes :
	189	+\begin{itemize}
	190	+\item $u_0=0$ $\rightarrow$ la valeur initiale du processus des $u_t=0$
	191	+\begin{align*}
	192	+(\rightarrow u_{t-s}=0 \hspace{0.2cm} \textbf{si} \hspace{0.2cm} t-s=0)
	193	+\end{align*}
	194	+\item Le processus des $u_t$ a commencé au passé infini.
	195	+\begin{align*}
	196	+(\lim_{s \rightarrow \infty} \rho^s=0)
	197	+\end{align*}
	198	+\item Dans les deux cas, si je recule assez dans le temps, la contribution de la valeur initiale est :
	199	+\begin{align*}
	200	+\rho^s u_{t-s}=0
	201	+\end{align*}
	202	+\end{itemize}
	203	+\end{frame}
	204	+
	205	+
	206	+\begin{frame}{Autocorrélation des erreurs}
	207	+\begin{itemize}
	208	+\item Les $u_t$ peuvent donc être réécrit comme une somme de bruits blancs pondérés :
	209	+\begin{align*}
	210	+u_t = \epsilon_t+\rho \epsilon_{t-1} + \rho^2 \epsilon_{t-2} + \rho^3 \epsilon_{t-3}+\cdots+\rho^{s-1}\epsilon_{t-(s-1)}
	211	+\end{align*}
	212	+\item On voit que l’effet des chocs les plus proches dans le passé est le plus important.
	213	+\end{itemize}
	214	+\end{frame}
	215	+
	216	+\begin{frame}{Autocorrélation des erreurs}
	217	+\begin{itemize}
	218	+\item Calculons la variance de $u_t$
	219	+\begin{align*}
	220	+\sigma_u^2 & =V(\epsilon_t)+\rho^2V(\epsilon_{t-1})+\rho^4V(\epsilon_{t-2})+...+\rho^{2(s-1)}V(\epsilon_{t-(s-1)}) \\ & = \sigma_{\epsilon}^2+\rho^2\sigma_{\epsilon}^2+\rho^4\sigma_{\epsilon}^2+\rho^6\sigma_{\epsilon}^2+\rho^{2(s-1)}\sigma_{\epsilon}^2 \\ & = \sigma_{\epsilon}^2(1+\rho^2+\rho^4+\rho^6+\rho^{2(s-1)}) \\ & = \sigma_{\epsilon}^2 \left( \frac{1}{1-\rho^2} \right)
	221	+\end{align*}
	222	+La stationnarité implique:
	223	+\begin{align*}
	224	+E(u_{t}^2)=E(u_{t-1}^2)
	225	+\end{align*}
	226	+Ainsi $\sigma_{u}^2=\rho^2 \sigma_{u}^2+\sigma_{\epsilon}^2$ et $\sigma_{u}^2=\left(\frac{\sigma_{\epsilon}^2}{1-\rho^2} \right)$
	227	+\end{itemize}
	228	+\end{frame}
	229	+
	230	+\begin{frame}{Autocorrélation des erreurs}
	231	+\begin{itemize}
	232	+\item On a donc démontré d’où viennent les coefficients sur la diagonale de la matrice $\Omega$.
	233	+\begin{block}{Calculez $E(u_tu_{t-1})$}
	234	+\begin{align*}
	235	+u_t = \epsilon_t +\rho \epsilon_{t-1}+\rho^{2}\epsilon_{t-2}+\rho^{3} \epsilon_{t-3}+\cdots
	236	+\end{align*}
	237	+\begin{align*}
	238	+u_{t-1} = \epsilon_{t-1} +\rho \epsilon_{t-2}+\rho^{2}\epsilon_{t-3}+\rho^{3} \epsilon_{t-4}+\cdots
	239	+\end{align*}
	240	+\end{block}
	241	+\end{itemize}
	242	+\end{frame}
	243	+
	244	+\begin{frame}{Autocorrélation des erreurs}
	245	+\begin{block}{Rappel: $E(\epsilon_t \epsilon_s)=0 \forall t \neq s$}
	246	+\begin{align*}
	247	+E(u_t u_{t-1}) & = \rho E(\epsilon_{t-1}^2)+\rho^3 E(\epsilon_{t-2}^2)+\rho^5E(\epsilon_{t-3}^2)+...\\ & = \rho\sigma_{\epsilon}^2+\rho^3 \sigma_{\epsilon}^2+\rho^5 \sigma_{\epsilon}^2+... \\ & = \rho \sigma_{\epsilon}^2(1+\rho^2+\rho^4+...) \\ & = \rho \sigma_{\epsilon}^2 \frac{1}{1-\rho^2} \\ & = \rho \sigma_u^2
	248	+\end{align*}
	249	+\end{block}
	250	+
	251	+\end{frame}
	252	+
	253	+\begin{frame}{Autocorrélation des erreurs}
	254	+\begin{block}{Rappel: $E(\epsilon_t \epsilon_s)=0 \forall t \neq s$}
	255	+Par le même raisonnement, on peut démontrer que
	256	+\begin{align*}
	257	+E(u_tu_{t-s})=\rho^s\sigma_{\epsilon}^2\frac{1}{1-\rho^2}=\rho^s\sigma_u^2
	258	+\end{align*}
	259	+\end{block}
	260	+
	261	+\end{frame}
	262	+
	263	+\begin{frame}{Autocorrélation des erreurs}
	264	+Preuve alternative qui ne passe pas par l’infini, mais par l’hypothèse que $\epsilon_t$ n’est pas corrélé avec les $u_t$ antérieurs.
	265	+\begin{align*}
	266	+u_t=\rho^s u_{t-s} +\epsilon_{t}+\rho \epsilon_{t-1} \rho^2 \epsilon_{t-2}+ \cdots+ \rho^{s-1} \epsilon_{t-(s-1)}
	267	+\end{align*}
	268	+\begin{align*}
	269	+u_tu_{t-s}=\rho^s u_{t-s}^2 +\epsilon_{t}u_{t-s}+\rho \epsilon_{t-1}u_{t-s} \rho^2 \epsilon_{t-2}u_{t-s} \\ + \cdots+ \rho^{s-1} \epsilon_{t-(s-1)}u_{t-s}
	270	+\end{align*}
	271	+\begin{align*}
	272	+E(u_tu_{t-s})=\rho^s E(u_{t-s}^2) +E(\epsilon_{t}u_{t-s})+\rho E(\epsilon_{t-1}u_{t-s}) \rho^2 E(\epsilon_{t-2}u_{t-s}) \\ + \cdots+ \rho^{s-1} E(\epsilon_{t-(s-1)}u_{t-s})
	273	+\end{align*}
	274	+
	275	+\end{frame}
	276	+
	277	+\begin{frame}{Autocorrélation des erreurs}
	278	+\begin{align*}
	279	+E(u_tu_{t-s})=\rho^s E(u_{t-s}^2)
	280	+\end{align*}
	281	+Sachant que
	282	+\begin{align*}
	283	+E(u_{t-s}^2)=\sigma_u^2= \sigma_{\epsilon}^2 \frac{1}{1-\rho^2}
	284	+\end{align*}
	285	+\begin{align*}
	286	+E(u_tu_{t-s})= \rho^s \left( \frac{\sigma_{\epsilon}^2}{1-\rho^2} \right)= \rho^s \sigma_u^2
	287	+\end{align*}
	288	+\end{frame}
	289	+
	290	+\begin{frame}{Autocorrélation des erreurs}
	291	+\begin{itemize}
	292	+\item Nous venons donc de démontrer comment la matrice de variance-covariance est obtenue selon différentes hypothèses de départ.
	293	+\item On peut démontrer que la matrice $P^{-1}$ associée sera :
	294	+\end{itemize}
	295	+\begin{align*}
	296	+P^{-1}=\begin{bmatrix}
	297	+\sqrt{1-\rho^2} & 0 & 0 & \cdots & 0 & 0 \\
	298	+-\rho & 1 & 0 \cdots & 0 & 0 \\
	299	+0 & -\rho & 1 & \cdots & 0 & 0 \\
	300	+\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
	301	+0 & 0 & 0 & \cdots & 1 & 0 \\
	302	+0 & 0 & 0 & \cdots & -\rho & 1
	303	+\end{bmatrix}
	304	+\end{align*}
	305	+\end{frame}
	306	+
	307	+\section{Transformation de Prais-Waisten}
	308	+
	309	+\frame{\tableofcontents[current]}
	310	+
	311	+
	312	+\begin{frame}{Transformation de Prais-Waisten}
	313	+\begin{itemize}
	314	+\item La transformation de Prais-Waisten consiste à multiplier les éléments de la régression par la matrice $P^{-1}$.
	315	+\begin{align*}
	316	+P^{-1} Y = \begin{bmatrix}
	317	+\sqrt{1-\rho^2} & 0 & 0 & \cdots & 0 & 0 \\
	318	+-\rho & 1 & 0 \cdots & 0 & 0 \\
	319	+0 & -\rho & 1 & \cdots & 0 & 0 \\
	320	+\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
	321	+0 & 0 & 0 & \cdots & 1 & 0 \\
	322	+0 & 0 & 0 & \cdots & -\rho & 1
	323	+\end{bmatrix} \begin{bmatrix}
	324	+Y_1 \\
	325	+Y_2 \\
	326	+Y_3 \\
	327	+\vdots \\
	328	+Y_{T-1}\\
	329	+Y_T
	330	+\end{bmatrix}
	331	+\end{align*}
	332	+\end{itemize}
	333	+\end{frame}
	334	+
	335	+
	336	+\begin{frame}{Transformation de Prais-Waisten}
	337	+\begin{itemize}
	338	+\item La transformation de Prais-Waisten consiste à multiplier les éléments de la régression par la matrice $P^{-1}$
	339	+\end{itemize}
	340	+
	341	+\begin{align*}
	342	+P^{-1} Y = \begin{bmatrix}
	343	+\sqrt{1-\rho^2}Y_1 \\
	344	+-\rho Y_1 + Y_2 \\
	345	+-\rho Y_2 + Y_2 \\
	346	+\vdots \\
	347	+-\rho Y_{T-2}+Y_{T-1}\\
	348	+=\rho Y_{T-1} + Y_T
	349	+\end{bmatrix}
	350	+\end{align*}
	351	+
	352	+\end{frame}
	353	+
	354	+\begin{frame}{Transformation de Prais-Waisten}
	355	+\begin{itemize}
	356	+\item La transformation sera la même pour chaque régresseur X.
	357	+\item Le modèle transformé s’écrit :
	358	+\begin{align*}
	359	+\sqrt{1-\rho^2}Y_1=\sqrt{1-\rho^2}\beta_0 + \beta_1 \sqrt{1-\rho^2}X_{11}+\beta_2 \sqrt{1-\rho^2}X_{21}\\ + \cdots+ \sqrt{1-\rho^2}u_1
	360	+\end{align*}
	361	+\begin{align*}
	362	+Y_t-\rho Y_{t-1}=(1-\rho) \beta_0+\beta_1 (X_{1t}-\rho X_{1,T-1})+\beta_2 (X_ {2t}-\rho X_{2,t-1}) \\ + \cdots +\beta_{k-1} (X_{kt}-\rho X_{k-1,t-1})+u_t-\rho u_{t-1}
	363	+\end{align*}
	364	+\item Il faut porter attention à la constante dans ce modèle qui n’est plus la constante d’origine.
	365	+\item Si on a $\rho$ inconnu, ce modèle n’est plus linéaire, mais on peut toujours l’estimer pas OLS.
	366	+\end{itemize}
	367	+\end{frame}
	368	+
	369	+
	370	+\section{Transformation de Cochran-Orcutt}
	371	+
	372	+\frame{\tableofcontents[current]}
	373	+
	374	+
	375	+\begin{frame}{Transformation de Cochran-Orcutt}
	376	+\begin{itemize}
	377	+\item Une autre façon de transformer le modèle serait le laisser tomber la première observation et de transformer les Y comme suit :
	378	+\begin{align*}
	379	+Y_t=\rho Y_{t-1}, t=2,...,T
	380	+\end{align*}
	381	+\item Les X sont transformés de la même facon:
	382	+\begin{align*}
	383	+X_{it}-\rho X_{i,t-1}, t=2,...,T
	384	+\end{align*}
	385	+\item Le modèle transformé de Cochran-Orcutt s’écrit de la même façon que Prais-Winsten, mais sans la première observation :
	386	+\begin{align*}
	387	+Y_t-\rho Y_{t-1}=(1-\rho) \beta_0+\beta_1 (X_{1t}-\rho X_{1,T-1})+\beta_2 (X_ {2t}-\rho X_{2,t-1}) \\ + \cdots +\beta_{k-1} (X_{kt}-\rho X_{k-1,t-1})+u_t-\rho u_{t-1}
	388	+\end{align*}
	389	+\end{itemize}
	390	+\end{frame}
	391	+
	392	+\section{Le problème de $\rho$}
	393	+
	394	+\frame{\tableofcontents[current]}
	395	+
	396	+
	397	+\begin{frame}{Le problème de $\rho$}
	398	+\begin{itemize}
	399	+\item $\rho$ est inconnu.
	400	+\item Il faut donc trouver un estimateur.
	401	+\item Plusieurs sont suggérés dans la littérature économétrique.
	402	+\end{itemize}
	403	+\begin{block}{Un estimateur usuel:}
	404	+\begin{align*}
	405	+\hat{\rho}=\frac{\sum_{t=2}^T \hat{u}_t \hat{u}_{t-1}}{\sum_{t=2}^T \hat{u}_{t-1}^2}
	406	+\end{align*}
	407	+Où $\hat{u}_t$ est le résidu OLS de la régression de Y sur X. On peut montrer que $PLIM$ $\hat{\rho}=\rho$
	408	+\end{block}
	409	+\end{frame}
	410	+
	411	+\begin{frame}{Le problème de $\rho$}
	412	+\begin{block}{Estimateur de Hildreth-LU:}
	413	+\begin{itemize}
	414	+\item Cet estimateur ne passe pas par les OLS.
	415	+\item En balayant l’intervalle $]-1, 1 [$, on peut choisir la valeur de $\rho$ qui minimise la somme des carrés des erreurs du modèle transformé.
	416	+
	417	+\end{itemize}
	418	+\end{block}
	419	+\end{frame}
	420	+
	421	+\begin{frame}{Le problème de $\rho$}
	422	+\begin{block}{Estimateur de Hildreth-LU:}
	423	+Les étapes sont :
	424	+\begin{enumerate}
	425	+\item Choisir $\rho^{(1)} = -1$ (ou -0.99999)
	426	+\item Obtenir le modèle transformé correspondant.
	427	+\item Appliquer les OLS et sauvegarder $\hat{u}'\hat{u}_{\rho(1)}$.
	428	+\item Choisir une nouvelle valeur : $\rho^{(2)} = \rho^{(1)} + \textbf{step}$
	429	+\item Obtenir le modèle transformé correspondant
	430	+\item Appliquer les OLS et sauvegarder $\hat{u}'\hat{u}_{\rho (2)}$
	431	+\item Continuer jusqu’à ce que $]-1, 1 [$ est couvert.

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