spb/wp46_uqo

1\section{A unified taxonomy and mathematical framework}\label{sec:taxonomy}23Figure~\ref{fig:genealogy} places the family on a single map: sixty-four architectures from 1982 to 2026, arranged by4family and by the year of their seminal reference, with the nesting or extension relations drawn as arrows; filled markers5are the architectures scored in the pre-registered benchmark, half-filled markers the seven post-registration extensions, and6hollow markers the variants that are inventoried but not scored (with the reason recorded in the encyclopedic table of7Appendix~\ref{app:encyclopedia}). The map makes two things visible at once: almost everything descends from GARCH(1,1)8through a handful of operations---asymmetric news impact, power transformation, fractional differencing, component9decomposition, regime switching, score-driven updating, and the replacement of the squared return by a realized measure---and10the multivariate branch grew in parallel from the same operations applied to covariance matrices.1112\begin{figure}[tp]\centering13\includegraphics[width=\textwidth]{figures/fig_genealogy.pdf}14\caption{Genealogy of the ARCH/GARCH family, 1982--2026. Rows are families, the horizontal position is the year of the seminal15reference, arrows run from a parent to the model that nests or extends it. Filled: scored in the pre-registered benchmark;16half-filled (thick edge): post-registration extension; hollow: inventoried, not scored. Generated from17\texttt{results/manifests/garch\_genealogy.yaml} (64 entries; references in Appendix~\ref{app:encyclopedia}).}\label{fig:genealogy}18\end{figure}1920Throughout, $r_t$ is the daily log return in percent, $\F_{t}$ the information set at the close of day $t$,21$r_t=\mu+\varepsilon_t$ with $\varepsilon_t=\sigma_t z_t$, $z_t\,|\,\F_{t-1}\sim D(0,1;\bm\eta)$ a standardized22innovation with distribution parameters $\bm\eta$, and $\sigma_t^2=\Var(r_t\,|\,\F_{t-1})$ in percent$^2$. Every model23below specifies $\sigma_t^2$ as a function of $\F_{t-1}$; a \emph{specification} is an architecture combined with an24innovation law and an order. Table~\ref{tab:taxonomy} lists the architectures with their defining recursion, admissibility25conditions and the multi-step rule actually implemented; Appendix~\ref{app:formulas} and the model cards in the26replication package give the complete parameter spaces and the reparameterizations used by the optimizer.2728\paragraph{Classical.} ARCH($q$) \citep{engle1982autoregressive}: $\sigma_t^2=\omega+\sum_{i=1}^q\alpha_i\varepsilon_{t-i}^2$.29GARCH($p,q$) \citep{bollerslev1986generalized}: $\sigma_t^2=\omega+\sum_{i=1}^p\alpha_i\varepsilon_{t-i}^2+\sum_{j=1}^q\beta_j\sigma_{t-j}^2$,30with $\omega>0$, $\alpha_i,\beta_j\ge0$ and covariance stationarity iff $\sum\alpha_i+\sum\beta_j<1$. IGARCH imposes31$\sum\alpha_i+\sum\beta_j=1$ \citep{engle1986modelling}: it is strictly but not covariance stationary32\citep{nelson1990stationarity}, and its $k$-step forecasts grow linearly, $\sigma_{t+k|t}^2=\omega+\sigma_{t+k-1|t}^2$.33GARCH-M \citep{engle1987estimating} adds $\delta\sigma_t^2$ to the conditional mean; its variance forecasts coincide with34those of GARCH(1,1). The constant-variance model, the exponentially weighted moving average with $\lambda=0.94$35\citep{jpmorgan1996riskmetrics}---an IGARCH(1,1) with $\omega=0$---the RiskMetrics~2006 multi-horizon filter36\citep{zumbach2009riskmetrics} and rolling historical variances over 22, 66 and 252 days are the baselines.3738\paragraph{Asymmetric and power.} The GJR model \citep{glosten1993relation} adds $\gamma\varepsilon_{t-1}^2\ind{\varepsilon_{t-1}<0}$ to39GARCH(1,1); its multi-step recursion uses $\kappa=\E[z^2\ind{z<0}]$, which equals $1/2$ for symmetric laws and is computed40numerically for the skewed Student-$t$, so that $\sigma_{t+k|t}^2=\omega+(\alpha+\gamma\kappa+\beta)\sigma_{t+k-1|t}^2$. What41several software packages call ``TGARCH'' is the \citet{zakoian1994threshold} model, in which the threshold acts on the42conditional \emph{standard deviation}: $\sigma_t=\omega+\alpha^{+}\varepsilon_{t-1}^{+}-\alpha^{-}\varepsilon_{t-1}^{-}+\beta\sigma_{t-1}$;43we keep both and never equate them. EGARCH \citep{nelson1991conditional} models $\log\sigma_t^2=\omega+\alpha(|z_{t-1}|-\E|z|)+\gamma z_{t-1}+\beta\log\sigma_{t-1}^2$,44which needs no positivity constraints and is stationary iff $|\beta|<1$. The nonlinear asymmetric GARCH of45\citet{engle1993measuring}, $\sigma_t^2=\omega+\alpha(\varepsilon_{t-1}-\theta\sigma_{t-1})^2+\beta\sigma_{t-1}^2$, has persistence46$\alpha(1+\theta^2)+\beta$; the quadratic GARCH of \citet{sentana1995quadratic} adds a linear term47$\gamma\varepsilon_{t-1}$ whose positivity requires $\gamma^2<4\alpha\omega$. APARCH \citep{ding1993long},48$\sigma_t^{\delta}=\omega+\alpha(|\varepsilon_{t-1}|-\gamma\varepsilon_{t-1})^{\delta}+\beta\sigma_{t-1}^{\delta}$, nests the absolute-value49GARCH ($\delta=1,\gamma=0$) of \citet{taylor2008modelling} and \citet{schwert1989why}; for $\delta\neq2$ the conditional50expectation of $\sigma_{t+k}^2$ has no closed form and we use simulation (2{,}000 paths, fixed seed), as for EGARCH and TGARCH.5152\paragraph{Fractional persistence.} FIGARCH($1,d,1$) \citep{baillie1996fractionally} writes53$\sigma_t^2=\omega+\{1-\beta L-(1-\phi L)(1-L)^d\}\varepsilon_t^2+\beta\sigma_{t-1}^2$, with hyperbolically decaying weights54truncated at 1{,}000 lags in the implementation; the sufficient positivity conditions of \citet{conrad2006inequality}55and the arch package's constraints are not identical to those of \texttt{rugarch}, which explains a log-likelihood gap of561.5 units on the shared verification fixture (Appendix~\ref{app:verification}). FIEGARCH, FIAPARCH and HYGARCH have no57vetted implementation in the version-locked libraries and are inventoried but not scored.5859\paragraph{Components and regimes.} The component model of \citet{engle1999long} decomposes60$\sigma_t^2=q_t+\alpha(\varepsilon_{t-1}^2-q_{t-1})+\beta(\sigma_{t-1}^2-q_{t-1})$ with a slowly mean-reverting trend61$q_t=\omega+\rho q_{t-1}+\phi(\varepsilon_{t-1}^2-\sigma_{t-1}^2)$, $0<\alpha+\beta<\rho<1$, $0<\phi<\min(\alpha,\beta)$. The62Markov-switching GARCH of \citet{haas2004new} runs one GARCH(1,1) recursion per regime $k\in\{1,2\}$, driven by the common63innovation and mixed with the filtered regime probabilities of a two-state chain $(p_{11},p_{22})$; regimes are64identified by ordering their unconditional variances, the likelihood is evaluated with the Hamilton filter, and the65$k$-step forecast uses the exact recursion $\E[\sigma_{k,t+j+1}^2\,|\,\F_t]=\omega_k+\alpha_k\E[\varepsilon_{t+j}^2\,|\,\F_t]+\beta_k\E[\sigma_{k,t+j}^2\,|\,\F_t]$66with $\E[\varepsilon_{t+j}^2\,|\,\F_t]=\sum_k\Prob(S_{t+j}=k\,|\,\F_t)\E[\sigma_{k,t+j}^2\,|\,\F_t]$.6768\paragraph{Score-driven comparator.} The GAS model with Student-$t$ innovations and inverse-information scaling69\citep{creal2013generalized,harvey2013dynamic} updates $f_t=\sigma_t^2$ by70$f_{t}=\omega+\alpha s_{t-1}+\beta f_{t-1}$, $s_t=\tfrac{\nu+3}{\nu}\big[\tfrac{(\nu+1)f_t\varepsilon_t^2}{(\nu-2)f_t+\varepsilon_t^2}-f_t\big]$,71which reduces to GARCH(1,1) as $\nu\to\infty$ and otherwise bounds the impact of outliers; it is labelled a non-GARCH72comparator.7374\paragraph{Realized-measure tier.} With $x_t$ the realized measure of day $t$ observed at its close, the75realized tier uses $x_{t-1}\in\F_{t-1}$. HAR \citep{corsi2009simple} projects the $h$-day average of the target proxy on76$(1,x_t,\bar x_t^{(5)},\bar x_t^{(22)})$ by least squares, one direct regression per horizon (in levels and, with Duan's77smearing correction, in logs). GARCH-X \citep{engle2002new} adds $\gamma x_{t-1}$ to GARCH(1,1). HEAVY78\citep{shephard2010realising} replaces the squared return by the realized measure in the return equation,79$h_t=\omega+\alpha x_{t-1}+\beta h_{t-1}$, and adds a measurement equation $\mu_t=\omega_R+\alpha_Rx_{t-1}+\beta_R\mu_{t-1}$ for80$\E[x_t\,|\,\F_{t-1}]$, estimated by the multiplicative-error quasi-likelihood $-\sum(\log\mu_t+x_t/\mu_t)$81\citep{engle2006multiple}; multi-step forecasts follow from the two linear recursions. Realized GARCH82\citep{hansen2012realized}, in its log-linear form, couples $\log\sigma_t^2=\omega+\beta\log\sigma_{t-1}^2+\gamma\log x_{t-1}$83with $\log x_t=\xi+\varphi\log\sigma_t^2+\tau_1z_t+\tau_2(z_t^2-1)+u_t$, $u_t\sim N(0,\sigma_u^2)$, estimated jointly;84stationarity requires $|\beta+\varphi\gamma|<1$ and multi-step forecasts are simulated.8586\paragraph{Multivariate tier.} For a basket of $N$ daily returns, the covariance models are the rolling sample covariance,87the EWMA covariance, CCC \citep{bollerslev1990modelling}, DCC \citep{engle2002dynamic} with correlation targeting,88asymmetric DCC \citep{cappiello2006asymmetric} and the scalar BEKK \citep{engle1995multivariate} with covariance targeting;89all produce symmetric positive-definite one-step forecasts by construction and are scored against the realized covariance90with the matrix losses of Section~\ref{sec:design}.9192\begin{table}[tp]\centering\footnotesize93\caption{Architectures in the registry. ``Implementation'' is the version-locked library (arch 8.0.0) or the validated custom94code of the replication package; ``multi-step'' is the rule used for $h>1$.}\label{tab:taxonomy}95\input{tables/tab_taxonomy}96\end{table}97