1\section{The GARCH family: an encyclopedic table}\label{app:encyclopedia}23Table~\ref{tab:encyclopedia} lists every architecture of Figure~\ref{fig:genealogy} with its seminal reference, its defining4variance equation in the notation of Section~\ref{sec:taxonomy}, what it adds to its parent, and its status in this paper5(scored, extension, unsupported, or out of scope with the reason). Nomenclature traps---which packages call the Zakoïan model6``TGARCH'' and which call GJR ``TARCH'', the three EGARCH parameterizations, NGARCH versus NAGARCH, APARCH versus PGARCH---are7documented in the replication package (\texttt{docs/genealogy\_notes.md}).89{\footnotesize\input{tables/tab_encyclopedia}}1011\section{Complete specifications, admissibility conditions and forecast recursions}\label{app:formulas}1213This appendix states, for every scored architecture, the recursion, the parameter space enforced by the optimizer, the14multi-step rule and the nesting relations, exactly as implemented (file references are to the replication package;15the model cards in \texttt{model\_cards/} add starting values, retry policy, tests and cross-check evidence). Notation:16$r_t=\mu+\varepsilon_t$, $\varepsilon_t=\sigma_tz_t$, $z_t\sim D(0,1;\bm\eta)$ with $D\in\{$normal, Student-$t(\nu)$, GED($\nu$),17Hansen skew-$t(\eta,\lambda)\}$ in the parameterization of the arch package (\texttt{distributions.py}, log-densities verified18to $10^{-15}$ against arch); $\sigma^2_{t+k|t}=\E[\sigma^2_{t+k}\,|\,\F_t]$; the cumulative forecast is $\sum_{k\le h}\sigma^2_{t+k|t}$.19The initial variance of every recursion is the exponentially weighted backcast $\sigma_0^2=\sum_{i<75}w_i\varepsilon_i^2$,20$w_i\propto0.94^i$, as in arch.2122\begin{description}[style=nextline,leftmargin=0pt]23\item[GARCH($p,q$) (arch).] $\sigma_t^2=\omega+\sum_{i\le p}\alpha_i\varepsilon_{t-i}^2+\sum_{j\le q}\beta_j\sigma_{t-j}^2$; $\omega>0$, $\alpha_i,\beta_j\ge0$;24 admissible iff $\sum\alpha_i+\sum\beta_j<1$ (flagged otherwise); $\sigma^2_{t+k|t}=\omega+\sum_i\alpha_i\E[\varepsilon^2_{t+k-i}|\F_t]+\sum_j\beta_j\sigma^2_{t+k-j|t}$ with25 $\E[\varepsilon^2_{t+m}|\F_t]=\sigma^2_{t+m|t}$ for $m\ge1$. The order labels follow arch ($p$ = ARCH lags, $q$ = GARCH lags).26\item[IGARCH(1,1) (custom).] $\beta=1-\alpha$, $\omega>0$, $\alpha\in(0,1)$ (logistic); $\sigma^2_{t+k|t}=\omega+\sigma^2_{t+k-1|t}$. EWMA: $\omega=0$, $\alpha=0.06$ fixed.27\item[GARCH-M(1,1) (custom).] $r_t=\mu+\delta\sigma_t^2+\varepsilon_t$ with GARCH(1,1) variance; the likelihood is evaluated on28 $\varepsilon_t=r_t-\mu-\delta\sigma_t^2$ recursively; variance forecasts as GARCH(1,1); mean forecast $\mu+\delta\sigma^2_{t+1|t}$.29\item[GJR(1,1) (arch).] $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\gamma\varepsilon_{t-1}^2\ind{\varepsilon_{t-1}<0}+\beta\sigma_{t-1}^2$; arch enforces30 $\alpha+\beta+\gamma/2<1$ and uses $\kappa=1/2$ in the multi-step recursion for every innovation law, which is exact for symmetric31 laws and an approximation for the skew-$t$ (the custom twin \texttt{gjr11\_custom}, used only for verification, computes32 $\kappa=\E[z^2\ind{z<0}]$ numerically). $\sigma^2_{t+k|t}=\omega+(\alpha+\gamma\kappa+\beta)\sigma^2_{t+k-1|t}$, $k\ge2$.33\item[TGARCH(1,1) (arch, power 1).] $\sigma_t=\omega+\alpha|\varepsilon_{t-1}|+\gamma|\varepsilon_{t-1}|\ind{\varepsilon_{t-1}<0}+\beta\sigma_{t-1}$34 (Zakoïan's threshold on the standard deviation, equivalent to $\alpha^+=\alpha$, $\alpha^-=\alpha+\gamma$); no closed-form35 $\E[\sigma^2_{t+k}]$ for $k\ge2$: simulation. No stationarity flag is computed (arch's $\alpha+\beta<1$ is not the relevant condition).36\item[AVGARCH(1,1) (arch, power 1, no asymmetry).] $\sigma_t=\omega+\alpha|\varepsilon_{t-1}|+\beta\sigma_{t-1}$; simulation for $k\ge2$.37\item[APARCH(1,1) (arch).] $\sigma_t^{\delta}=\omega+\alpha(|\varepsilon_{t-1}|-\gamma\varepsilon_{t-1})^{\delta}+\beta\sigma_{t-1}^{\delta}$, $\delta\in(0.05,4)$, $|\gamma|<1$;38 simulation for $k\ge2$.39\item[EGARCH(1,1) (arch).] $\log\sigma_t^2=\omega+\alpha\big(|z_{t-1}|-\sqrt{2/\pi}\big)+\gamma z_{t-1}+\beta\log\sigma_{t-1}^2$, $\beta\in[0,1)$ in arch40 (the centring constant $\sqrt{2/\pi}$ is used for every innovation law and is absorbed by $\omega$); simulation for $k\ge2$.41\item[NAGARCH(1,1) (custom).] $\sigma_t^2=\omega+\alpha(\varepsilon_{t-1}-\theta\sigma_{t-1})^2+\beta\sigma_{t-1}^2$; reparameterized with persistence42 $s=\alpha(1+\theta^2)+\beta\in(0,1)$ and share $w=\alpha(1+\theta^2)/s$; $\sigma^2_{t+k|t}=\omega+s\,\sigma^2_{t+k-1|t}$.43\item[QGARCH(1,1) (custom).] $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\gamma\varepsilon_{t-1}+\beta\sigma_{t-1}^2$ with44 $\gamma=2\sqrt{\alpha\omega}\tanh(\vartheta)$ so that $\gamma^2<4\alpha\omega$ (positivity) and $\alpha+\beta<1$;45 $\sigma^2_{t+1|t}$ uses $\varepsilon_t$ and, for $k\ge2$, $\E[\varepsilon_{t+k-1}|\F_t]=0$ gives the GARCH recursion.46\item[FIGARCH(1,$d$,1) (arch).] $\sigma_t^2=\omega+\big[1-\beta L-(1-\phi L)(1-L)^d\big]\varepsilon_t^2+\beta\sigma_{t-1}^2$, $d\in[0,1]$, with the47 ARCH($\infty$) weights truncated at 1{,}000 lags and arch's sufficient positivity constraints; analytic multi-step through the48 truncated weights. The oldest lags are backcast inside a 1{,}000-day window.49\item[CGARCH(1,1) (custom).] $\sigma_t^2=q_t+\alpha(\varepsilon_{t-1}^2-q_{t-1})+\beta(\sigma_{t-1}^2-q_{t-1})$, $q_t=\omega+\rho q_{t-1}+\phi(\varepsilon_{t-1}^2-\sigma_{t-1}^2)$;50 reparameterized so that $0<\alpha+\beta<\rho<1$ and $0<\phi<\min(\alpha,\beta)$; $q_{t+k|t}=\omega+\rho q_{t+k-1|t}$,51 $(\sigma^2-q)_{t+k|t}=(\alpha+\beta)(\sigma^2-q)_{t+k-1|t}$.52\item[MS-GARCH(2) (custom).] Regime $k$: $\sigma_{k,t}^2=\omega_k+\alpha_k\varepsilon_{t-1}^2+\beta_k\sigma_{k,t-1}^2$; chain53 $\Prob(S_t=k\,|\,S_{t-1}=k)=p_{kk}$; density mixture $f(r_t|\F_{t-1})=\sum_k\Prob(S_t=k|\F_{t-1})f_k(r_t)$ evaluated by the Hamilton54 filter; regime 1 has the lower unconditional variance (smooth penalty); innovations normal or Student-$t$ with common $\nu$.55 Multi-step: $\pi_{t+j}=\pi_{t+j-1}P$, $\sigma^2_{t+j|t}=\sum_k\pi_{t+j,k}\sigma^2_{k,t+j|t}$,56 $\sigma^2_{k,t+j+1|t}=\omega_k+\alpha_k\sigma^2_{t+j|t}+\beta_k\sigma^2_{k,t+j|t}$---the recursion of \citet{haas2004new} as57 implemented in \texttt{MSGARCH}, which treats the chain and the regime variances as uncorrelated beyond the origin58 (verified numerically against \texttt{MSGARCH::predict}, Appendix~\ref{app:verification}).59\item[GAS-$t$(1,1) (custom).] As in Section~\ref{sec:taxonomy}; $\omega>0$, $\alpha>0$, $\beta\in[0,1)$; $f_{t+k|t}=\omega+\beta f_{t+k-1|t}$ since $\E[s_t]=0$.60\item[HAR / log-HAR (OLS).] For $h\in\{1,5,22\}$: $\bar y_{t,h}=c_h+b_{h,d}y_t+b_{h,w}\bar y_t^{(5)}+b_{h,m}\bar y_t^{(22)}+e_{t,h}$ where61 $\bar y_{t,h}=h^{-1}\sum_{k\le h}y_{t+k}$; the cumulative forecast is $h\hat{\bar y}_{t,h}$ (floored at $10^{-8}$); log-HAR fits62 the regression in logs and multiplies by Duan's smearing factor $n^{-1}\sum\exp(\hat e)$.63\item[GARCH-X(1,1) (custom).] $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\gamma x_{t-1}+\beta\sigma_{t-1}^2$, $\gamma\ge0$, $\alpha+\beta<1$; the realized64 measure follows the HEAVY-RM equation below for $k\ge2$ (the package's choice, not part of \citealp{engle2002new}).65\item[HEAVY(1,1) (custom).] $h_t=\omega+\alpha x_{t-1}+\beta h_{t-1}$, $\mu_t=\omega_R+\alpha_Rx_{t-1}+\beta_R\mu_{t-1}$; objective = Gaussian66 (or $t$) log-likelihood of $r_t$ plus the multiplicative-error quasi-likelihood $-\sum_t(\log\mu_t+x_t/\mu_t)$, maximized jointly in67 one optimizer call (the two equations share no parameters, so this equals separate estimation); $\mu_0$ = mean of the68 first 75 measures; $h_{t+k|t}=\omega+\alpha\mu_{t+k-1|t}+\beta h_{t+k-1|t}$, $\mu_{t+k|t}=\omega_R+(\alpha_R+\beta_R)\mu_{t+k-1|t}$.69\item[Realized GARCH(1,1), log-linear (custom).] As in Section~\ref{sec:taxonomy}; $|\beta+\varphi\gamma|<1$ flagged; one-step forecast70 $\exp(\omega+\beta\log\sigma_t^2+\gamma\log x_t)$; $k\ge2$ by simulation of $(z,u)$ with 2{,}000 paths.71\item[Tier C.] DCC: $Q_t=(1-a-b)\bar S+az_{t-1}z_{t-1}'+bQ_{t-1}$, $R_t=\mathrm{diag}(Q_t)^{-1/2}Q_t\mathrm{diag}(Q_t)^{-1/2}$, $a,b\ge0$, $a+b<1$ (logistic72 shares); aDCC adds $g\,n_{t-1}n_{t-1}'$ with $n=z\odot\ind{z<0}$ and targeting $(1-a-b)\bar S-g\bar N$; scalar BEKK: $H_t=(1-a-b)\bar\Sigma+a\varepsilon_{t-1}\varepsilon_{t-1}'+bH_{t-1}$;73 CCC: $R_t=\bar S$; univariate stage GARCH(1,1)-normal; positive definiteness of $H_{t+1|t}$ checked (minimum eigenvalue recorded).74\end{description}75