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GSF-6053 Financial Econometrics I — matériel de cours (Université Laval).

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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 (Aug 6, 2026) Browse files

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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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