spb/wp46_uqo

1% Generated by scripts/make_encyclopedia.py from results/manifests/garch_genealogy.yaml -- do not edit by hand.2{\footnotesize\setlength{\tabcolsep}{3pt}\renewcommand{\arraystretch}{1.05}3\begin{longtable}{@{}>{\raggedright\arraybackslash}p{2.15cm}>{\raggedright\arraybackslash}p{0.75cm}>{\raggedright\arraybackslash}p{2.0cm}>{\raggedright\arraybackslash}p{6.25cm}>{\raggedright\arraybackslash}p{2.85cm}>{\raggedright\arraybackslash}p{1.45cm}@{}}4\caption{Encyclopedia of the ARCH/GARCH family and its closest relatives, 1982--2026.}\label{tab:encyclopedia}\\5\toprule6Model & Year & Reference & Variance equation & Adds & Status \\7\midrule8\endfirsthead9\multicolumn{6}{l}{\footnotesize\textit{Table~\ref{tab:encyclopedia} (continued)}}\\10\toprule11Model & Year & Reference & Variance equation & Adds & Status \\12\midrule13\endhead14\midrule15\multicolumn{6}{r}{\footnotesize\textit{continued on next page}}\\16\endfoot17\bottomrule18\addlinespace[2pt]19\multicolumn{6}{@{}p{\dimexpr\textwidth-2pt\relax}@{}}{\scriptsize \emph{Notes.} One line per materially distinct architecture; orders and innovation laws are not separate entries. Notation: $r_t=\mu+\varepsilon_t$, $\varepsilon_t=\sigma_tz_t$, $z_t\sim$ i.i.d.\ $D(0,1)$; $x_t$ is a realized measure, $H_t$ a conditional covariance matrix, $S_t$ a Markov chain, $L$ the lag operator. Status: \emph{scored} = in the binding registry (or the Tier~C set) and evaluated; \emph{extension} = being implemented, registry status \emph{unsupported}; \emph{unsupported} = no feasible implementation at the benchmark cadence; \emph{out of scope} = not in the registry (reason in \texttt{results/manifests/garch\_genealogy.yaml}). Registry identifiers in brackets. Year = year of the seminal reference as verified (EWMA: 4th edition of the Technical Document, method of 1994; RiskMetrics 2006: 2007 document, verified SSRN record 2009; AVGARCH and SV: Taylor's 1986 book, verified record = 2008 edition). Seminal references absent from the bibliography (verified by DOI on 2026-11-09, listed in \texttt{results/metrics/references\_candidates\_genealogy.json}): }\\20\endlastfoot21\midrule22\multicolumn{6}{l}{\textit{Classical ARCH/GARCH and filters}}\\23\addlinespace[1pt]24ARCH($q$) {\scriptsize[\texttt{arch5}]} & 1982 & \citet{engle1982autoregressive} & $\sigma_t^2=\omega+\sum_{i=1}^{q}\alpha_i\varepsilon_{t-i}^2$ & conditional variance as an autoregression in squared shocks & scored \\25GARCH($p,q$) {\scriptsize[\texttt{garch11}]} & 1986 & \citet{bollerslev1986generalized} & $\sigma_t^2=\omega+\sum_{i=1}^{p}\alpha_i\varepsilon_{t-i}^2+\sum_{j=1}^{q}\beta_j\sigma_{t-j}^2$ & lagged conditional variances: parsimonious ARMA-type dynamics (ARCH($\infty$)) & scored \\26IGARCH {\scriptsize[\texttt{igarch11}]} & 1986 & \citet{engle1986modelling} & $\sigma_t^2=\omega+\sum_{i}\alpha_i\varepsilon_{t-i}^2+\sum_{j}\beta_j\sigma_{t-j}^2$ with $\sum_i\alpha_i+\sum_j\beta_j=1$ & unit persistence: shocks to the variance never die out & scored \\27log-GARCH & 1986 & \citet{geweke1986comment} & $\log\sigma_t^2=\omega+\sum_{i=1}^{p}\alpha_i\log\varepsilon_{t-i}^2+\sum_{j=1}^{q}\beta_j\log\sigma_{t-j}^2$ & multiplicative (log-linear) variance dynamics; no positivity constraints; $\log\varepsilon_t^2$ is an ARMA & out of scope \\28ARCH-M / GARCH-M {\scriptsize[\texttt{garchm11}]} & 1987 & \citet{engle1987estimating} & $r_t=\mu+\delta\,g(\sigma_t^2)+\varepsilon_t$ with $g(\sigma_t^2)\in\{\sigma_t^2,\sigma_t,\log\sigma_t^2\}$ \newline $\sigma_t^2$ as GARCH & conditional variance enters the conditional mean (time-varying risk premium) & scored \\29EWMA / RiskMetrics (1994) {\scriptsize[\texttt{ewma094}]} & 1996 & \citet{jpmorgan1996riskmetrics} & $\sigma_t^2=(1-\lambda)\varepsilon_{t-1}^2+\lambda\sigma_{t-1}^2$, $\lambda=0.94$ (daily) & zero-intercept IGARCH(1,1) with a fixed decay; no estimation & scored \\30RiskMetrics 2006 {\scriptsize[\texttt{rm2006}]} & 2007 & \citet{zumbach2009riskmetrics} & $\sigma_t^2=\sum_{k=1}^{K}w_k\sigma_{k,t}^2$, $\sigma_{k,t}^2=\mu_k\sigma_{k,t-1}^2+(1-\mu_k)\varepsilon_{t-1}^2$, $\mu_k=e^{-1/\tau_k}$, $\tau_k=\tau_1\rho^{k-1}$, $w_k\propto1-\log\tau_k/\log\tau_0$ & mixture of EWMAs over geometrically spaced horizons with logarithmic weights (long-memory-like kernel, no estimation) & scored \\31\midrule32\multicolumn{6}{l}{\textit{Asymmetric (news-impact) GARCH}}\\33\addlinespace[1pt]34AGARCH (Engle 1990) & 1990 & \citet{engle1990discussion} & $\sigma_t^2=\omega+\alpha(\varepsilon_{t-1}-\theta)^2+\beta\sigma_{t-1}^2$ & additive shift of the news-impact curve in return units (asymmetric GARCH) & out of scope \\35EGARCH {\scriptsize[\texttt{egarch11}]} & 1991 & \citet{nelson1991conditional} & $\log\sigma_t^2=\omega+\sum_{j}\beta_j\log\sigma_{t-j}^2+\sum_{i}\alpha_i\,g(z_{t-i})$, $g(z)=\theta z+\gamma(|z|-\E|z|)$ & log variance driven by sign and size of the standardized shock; no positivity constraints & scored \\36GJR-GARCH {\scriptsize[\texttt{gjr11}]} & 1993 & \citet{glosten1993relation} & $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\gamma\varepsilon_{t-1}^2\ind{\varepsilon_{t-1}<0}+\beta\sigma_{t-1}^2$ & indicator on negative shocks in the variance (leverage effect, power 2) & scored \\37NAGARCH (Engle--Ng) {\scriptsize[\texttt{nagarch11}]} & 1993 & \citet{engle1993measuring} & $\sigma_t^2=\omega+\alpha(\varepsilon_{t-1}-\theta\sigma_{t-1})^2+\beta\sigma_{t-1}^2$ & nonlinear asymmetric GARCH: news-impact curve shifted by $\theta\sigma_{t-1}$ (scale-free); sign/size bias tests & scored \\38VGARCH (Engle--Ng) & 1993 & \citet{engle1993measuring} & $\sigma_t^2=\omega+\alpha(z_{t-1}+\theta)^2+\beta\sigma_{t-1}^2$, $z_{t-1}=\varepsilon_{t-1}/\sigma_{t-1}$ & news impact in standardized-shock units, independent of the variance level & out of scope \\39TGARCH (Zakoïan) {\scriptsize[\texttt{tarch11}]} & 1994 & \citet{zakoian1994threshold} & $\sigma_t=\omega+\alpha^{+}\varepsilon_{t-1}^{+}-\alpha^{-}\varepsilon_{t-1}^{-}+\beta\sigma_{t-1}$, $\varepsilon^{+}=\max(\varepsilon,0)$, $\varepsilon^{-}=\min(\varepsilon,0)$ & threshold on the conditional standard deviation (power 1) with separate slopes for positive and negative shocks & scored \\40QGARCH (Sentana) {\scriptsize[\texttt{qgarch11}]} & 1995 & \citet{sentana1995quadratic} & $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\psi\varepsilon_{t-1}+\beta\sigma_{t-1}^2$ & general quadratic form in lagged shocks; linear term gives a parabola-shifting asymmetry & scored \\41\midrule42\multicolumn{6}{l}{\textit{Power / absolute-value / Box--Cox}}\\43\addlinespace[1pt]44AVGARCH / Taylor--Schwert {\scriptsize[\texttt{avgarch11}]} & 1986 & \citet{taylor2008modelling} & $\sigma_t=\omega+\alpha|\varepsilon_{t-1}|+\beta\sigma_{t-1}$ & dynamics in the conditional standard deviation (absolute-value GARCH) & scored \\45NARCH / NGARCH (Higgins--Bera) & 1992 & \citet{higgins1992class} & $\sigma_t^{\delta}=\omega+\sum_{i}\alpha_i|\varepsilon_{t-i}|^{\delta}+\sum_{j}\beta_j\sigma_{t-j}^{\delta}$ (power-GARCH form \newline Higgins--Bera's original is the Box--Cox ARCH case) & estimated power transformation of the ARCH recursion (nonlinear ARCH) & out of scope \\46APARCH {\scriptsize[\texttt{aparch11}]} & 1993 & \citet{ding1993long} & $\sigma_t^{\delta}=\omega+\sum_{i}\alpha_i(|\varepsilon_{t-i}|-\gamma_i\varepsilon_{t-i})^{\delta}+\sum_{j}\beta_j\sigma_{t-j}^{\delta}$ & estimated power $\delta$ and asymmetry $\gamma$; nests GARCH, AVGARCH, GJR, TGARCH (Zakoïan), NGARCH & scored \\47Family GARCH (Hentschel) & 1995 & \citet{hentschel1995all} & $\dfrac{\sigma_t^{\lambda}-1}{\lambda}=\omega+\alpha\sigma_{t-1}^{\lambda}f^{\nu}(z_{t-1})+\beta\dfrac{\sigma_{t-1}^{\lambda}-1}{\lambda}$, $f(z)=|z-b|-c(z-b)$ & Box--Cox transform of $\sigma_t$ with a shifted and rotated absolute-value news-impact function: one nest for the asymmetric family & out of scope \\48Augmented GARCH (Duan) & 1997 & \citet{duan1997augmented} & $\phi_t=\omega+\xi_{1,t-1}\phi_{t-1}+\xi_{2,t-1}$, $\sigma_t^2=|\lambda\phi_t-\lambda+1|^{1/\lambda}$ ($\lambda\ne0$), $\xi_{1,t}=\alpha_1+\alpha_2|z_t-c|^{\delta}+\alpha_3\max(0,c-z_t)^{\delta}$, $\xi_{2,t}=\alpha_4\dfrac{|z_t-c|^{\delta}-1}{\delta}+\alpha_5\dfrac{\max(0,c-z_t)^{\delta}-1}{\delta}$ ((1,1) case) & Box--Cox transform of the variance driven by Box--Cox functions of the shock; nests GARCH, EGARCH, GJR, TGARCH, APARCH, NAGARCH, VGARCH and the Hentschel family; diffusion limit & out of scope \\49\midrule50\multicolumn{6}{l}{\textit{Fractional persistence}}\\51\addlinespace[1pt]52FIEGARCH {\scriptsize[\texttt{fiegarch11}]} & 1996 & \citet{bollerslev1996modeling} & $\log\sigma_t^2=\omega+\phi(L)^{-1}(1-L)^{-d}\,[1+\psi(L)]\,g(z_{t-1})$, $g(z)=\theta z+\gamma(|z|-\E|z|)$ & fractional integration in the log variance with EGARCH asymmetry & extension \\53FIGARCH {\scriptsize[\texttt{figarch11}]} & 1996 & \citet{baillie1996fractionally} & $\sigma_t^2=\omega+\big[1-\beta L-(1-\phi L)(1-L)^{d}\big]\varepsilon_t^2+\beta\sigma_{t-1}^2$ (ARCH($\infty$) weights truncated in practice) & fractional differencing: hyperbolic decay of shocks to the variance between GARCH ($d=0$) and IGARCH ($d=1$) & scored \\54FIAPARCH {\scriptsize[\texttt{fiaparch11}]} & 1998 & \citet{tse1998conditional} & $\sigma_t^{\delta}=\omega+\big\{1-[1-\beta(L)]^{-1}\phi(L)(1-L)^{d}\big\}(|\varepsilon_t|-\gamma\varepsilon_t)^{\delta}$ & fractional persistence combined with the APARCH power and asymmetry & extension \\55HYGARCH {\scriptsize[\texttt{hygarch11}]} & 2004 & \citet{davidson2004moment} & $\sigma_t^2=\omega+\big\{1-[1-\beta(L)]^{-1}\phi(L)\big[1+\tau\big((1-L)^{d}-1\big)\big]\big\}\varepsilon_t^2$ & amplitude parameter $\tau$ interpolating GARCH ($\tau=0$) and FIGARCH ($\tau=1$): hyperbolic GARCH & extension \\56\midrule57\multicolumn{6}{l}{\textit{Component and multiplicative}}\\58\addlinespace[1pt]59Long-memory ARCH by aggregation (Ding--Granger) & 1996 & \citet{ding1996modeling} & $\sigma_t^2=\sum_{k=1}^{N}w_k\sigma_{k,t}^2$, $\sigma_{k,t}^2=(1-\alpha_k)\varepsilon_{t-1}^2+\alpha_k\sigma_{k,t-1}^2$, $\sum_k w_k=1$ & aggregation of EWMA-type components with heterogeneous decay rates produces hyperbolic autocorrelations of $\varepsilon_t^2$ & out of scope \\60Asymmetric component GARCH & 1999 & \citet{engle1999long} & $\sigma_t^2-q_t=\alpha(\varepsilon_{t-1}^2-q_{t-1})+\gamma(\varepsilon_{t-1}^2-q_{t-1})\ind{\varepsilon_{t-1}<0}+\beta(\sigma_{t-1}^2-q_{t-1})$, $q_t$ as in CGARCH & leverage (GJR-type indicator) in the transitory component & out of scope \\61Component GARCH (Engle--Lee) {\scriptsize[\texttt{cgarch11}]} & 1999 & \citet{engle1999long} & $\sigma_t^2=q_t+\alpha(\varepsilon_{t-1}^2-q_{t-1})+\beta(\sigma_{t-1}^2-q_{t-1})$, $q_t=\omega+\rho q_{t-1}+\phi(\varepsilon_{t-1}^2-\sigma_{t-1}^2)$ & permanent (slowly mean-reverting) and transitory variance components & scored \\62Spline-GARCH & 2008 & \citet{engle2008spline} & $\sigma_t^2=\tau_t g_t$, $g_t=(1-\alpha-\beta)+\alpha\dfrac{\varepsilon_{t-1}^2}{\tau_{t-1}}+\beta g_{t-1}$, $\tau_t=c\exp\!\big(w_0t+\sum_{k=1}^{K}w_k\,((t-t_{k-1})_+)^2\big)$ & multiplicative low-frequency component as a deterministic exponential quadratic spline of time & out of scope \\63Multiplicative component GARCH (intraday) & 2011 & \citet{engle2011forecasting} & $\sigma_{t,i}^2=h_t\,s_i\,q_{t,i}$, $q_{t,i}=\omega+\alpha\dfrac{\varepsilon_{t,i-1}^2}{h_t s_{i-1}}+\beta q_{t,i-1}$ (day $t$, intraday bin $i$) & daily component $h_t$ $\times$ diurnal pattern $s_i$ $\times$ intraday GARCH $q_{t,i}$ & out of scope \\64GARCH-MIDAS {\scriptsize[\texttt{garch\_midas}]} & 2013 & \citet{engle2013stock} & $\sigma_t^2=\tau_t g_t$, $g_t=(1-\alpha-\beta)+\alpha\dfrac{\varepsilon_{t-1}^2}{\tau_t}+\beta g_{t-1}$, $\log\tau_t=m+\theta\sum_{k=1}^{K}\varphi_k(w)X_{t-k}$ (MIDAS weights on a low-frequency series $X$) & long-run component driven by lagged low-frequency realized variance or macroeconomic variables through a MIDAS polynomial & extension \\65\midrule66\multicolumn{6}{l}{\textit{Regime switching and smooth transition}}\\67\addlinespace[1pt]68MS-ARCH (Cai) & 1994 & \citet{cai1994markov} & $\sigma_t^2=\omega_{S_t}+\sum_{i=1}^{q}\alpha_i\varepsilon_{t-i}^2$, $S_t\in\{1,2\}$ Markov (switching in the intercept) & regime-switching ARCH intercept (switching in the level of the variance) & out of scope \\69SWARCH (Hamilton--Susmel) & 1994 & \citet{hamilton1994autoregressive} & $\varepsilon_t=\sqrt{g_{S_t}}\,u_t$, $u_t=\sigma_t z_t$, $\sigma_t^2=\omega+\sum_{i=1}^{q}\alpha_i u_{t-i}^2$ ($+\xi\,u_{t-1}^2\ind{u_{t-1}<0}$), $S_t$ a $K$-state Markov chain & Markov-switching scale factor on an ARCH process (ARCH only, to avoid path dependence) & out of scope \\70MS-GARCH (Gray, collapsing) & 1996 & \citet{gray1996modeling} & $\sigma_t^2=\omega_{S_t}+\alpha_{S_t}\varepsilon_{t-1}^2+\beta_{S_t}\tilde\sigma_{t-1}^2$, $\tilde\sigma_{t-1}^2=\sum_k p_{k,t-1}(\mu_k^2+\sigma_{k,t-1}^2)-\big(\sum_k p_{k,t-1}\mu_k\big)^2$ & first tractable MS-GARCH: the lagged variance is collapsed across regimes with the regime probabilities (removes path dependence) & out of scope \\71Smooth-transition ARCH (Hagerud) & 1997 & \citet{hagerud1997smooth} & $\sigma_t^2=\omega+\sum_{i=1}^{q}\big[\alpha_i+\gamma_iF(\varepsilon_{t-i})\big]\varepsilon_{t-i}^2$, $F$ logistic (LSTARCH) or exponential (ESTARCH) & smooth logistic/exponential transition between two ARCH regimes indexed by the lagged shock & out of scope \\72Smooth-transition GARCH (González-Rivera) {\scriptsize[\texttt{stgarch}]} & 1998 & \citet{gonzalezrivera1998smooth} & $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\beta\sigma_{t-1}^2+(\delta_0+\delta_1\varepsilon_{t-1}^2)\,F(\varepsilon_{t-1})$, $F(x)=[1+\exp(-\theta x)]^{-1}$ & smooth (logistic) version of the GJR indicator: the speed $\theta$ of the transition is estimated; GJR is the limit $\theta\to\infty$ & extension \\73MS-GARCH (Klaassen) & 2002 & \citet{klaassen2002improving} & $\sigma_t^2=\omega_{S_t}+\alpha_{S_t}\varepsilon_{t-1}^2+\beta_{S_t}\,\E\!\big[\sigma_{S_{t-1},t-1}^2\,\big|\,\F_{t-1},S_t\big]$ & collapses the lagged variance conditional on the current regime; analytic multi-step forecasts & out of scope \\74MS-GARCH (Haas--Mittnik--Paolella) {\scriptsize[\texttt{msgarch2}]} & 2004 & \citet{haas2004new} & $\sigma_{k,t}^2=\omega_k+\alpha_k\varepsilon_{t-1}^2+\beta_k\sigma_{k,t-1}^2$ for each regime $k$ \newline $r_t\,|\,S_t=k\sim D(\mu_k,\sigma_{k,t}^2)$ \newline $S_t$ Markov with $P=(p_{ij})$ & $K$ parallel GARCH recursions driven by the common shock; analytic likelihood (Hamilton filter), stationarity conditions and forecasts & scored \\75Path-dependent MS-GARCH & 2010 & \citet{bauwens2010theory} & $\sigma_t^2=\omega_{S_t}+\alpha_{S_t}\varepsilon_{t-1}^2+\beta_{S_t}\sigma_{t-1}^2$, with $\sigma_{t-1}^2$ depending on the whole regime path $\{S_{t-1},S_{t-2},\dots\}$ & the literal regime-switching GARCH; likelihood integrates over $K^{t}$ paths (Bayesian MCMC, later EM/particle methods) & out of scope \\76\midrule77\multicolumn{6}{l}{\textit{Score-driven (GAS / DCS)}}\\78\addlinespace[1pt]79Beta-$t$-EGARCH (DCS) & 2008 & \citet{harvey2008beta} & $\log\sigma_t^2=\omega+\phi\log\sigma_{t-1}^2+\kappa u_{t-1}$, $u_t=\dfrac{(\nu+1)\varepsilon_t^2}{\nu\sigma_t^2+\varepsilon_t^2}-1$ ($u_t+1$ Beta-distributed) & log variance driven by the bounded Student-$t$ score; closed-form moments and multi-step forecasts & extension \\80GAS / score-driven volatility {\scriptsize[\texttt{gas\_t11}]} & 2013 & \citet{creal2013generalized} & $f_t=\omega+\alpha s_{t-1}+\beta f_{t-1}$, $s_t=S_t\,\partial\log p(r_t\,|\,f_t)/\partial f_t$ \newline Student-$t$, $f_t=\sigma_t^2$: $s_t=\dfrac{\nu+3}{\nu}\Big[\dfrac{(\nu+1)\varepsilon_t^2}{(\nu-2)+\varepsilon_t^2/f_t}-f_t\Big]$ & the update is the scaled score of the conditional density: GARCH for the normal, outlier-robust (Beta-$t$-GARCH) for the Student-$t$ & scored \\81\midrule82\multicolumn{6}{l}{\textit{Realized-measure models and HAR}}\\83\addlinespace[1pt]84GARCH-X (realized regressor) {\scriptsize[\texttt{garchx\_rv11}]} & 2002 & \citet{engle2002new} & $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\gamma x_{t-1}+\beta\sigma_{t-1}^2$ & lagged realized measure enters the GARCH variance additively & scored \\85MEM (multiplicative error model) & 2002 & \citet{engle2002new} & $x_t=\mu_t\eta_t$, $\mu_t=\omega+\alpha x_{t-1}+\beta\mu_{t-1}$, $\eta_t\ge0$ i.i.d., $\E\eta_t=1$ & GARCH-type dynamics for a positive-valued series (realized variance, range, duration); ACD is the duration case & out of scope \\86HAR-RV {\scriptsize[\texttt{har\_rv}]} & 2009 & \citet{corsi2009simple} & $x_{t+1}=\beta_0+\beta_d x_t+\beta_w\bar x_t^{(5)}+\beta_m\bar x_t^{(22)}+e_{t+1}$, $\bar x_t^{(n)}=n^{-1}\sum_{k=0}^{n-1}x_{t-k}$ & heterogeneous autoregression of realized variance on daily, weekly and monthly averages (approximate long memory by OLS) & scored \\87HEAVY {\scriptsize[\texttt{heavy11}]} & 2010 & \citet{shephard2010realising} & $\sigma_t^2=\omega+\alpha x_{t-1}+\beta\sigma_{t-1}^2$ \newline $\mu_t=\E[x_t\,|\,\F_{t-1}]=\omega_R+\alpha_Rx_{t-1}+\beta_R\mu_{t-1}$ & the realized measure replaces the squared return in the return-variance equation; a second MEM equation forecasts the measure & scored \\88Realized GARCH {\scriptsize[\texttt{realgarch11}]} & 2012 & \citet{hansen2012realized} & $\log\sigma_t^2=\omega+\beta\log\sigma_{t-1}^2+\gamma\log x_{t-1}$ \newline $\log x_t=\xi+\varphi\log\sigma_t^2+\tau_1z_t+\tau_2(z_t^2-1)+u_t$ & joint model of returns and the realized measure with a measurement equation and a leverage function $\tau(z)$ & scored \\89HARQ (and HAR-CJ, HAR-RS) & 2016 & \citet{bollerslev2016exploiting} & $x_{t+1}=\beta_0+(\beta_d+\beta_q\,RQ_t^{1/2})x_t+\beta_w\bar x_t^{(5)}+\beta_m\bar x_t^{(22)}+e_{t+1}$ ($RQ_t$ = realized quarticity) & measurement-error-adjusted persistence; HAR-CJ splits continuous and jump variation, HAR-RS uses realized semivariances & out of scope \\90Realized EGARCH {\scriptsize[\texttt{realegarch11}]} & 2016 & \citet{hansen2016exponential} & $\log\sigma_t^2=\omega+\beta\log\sigma_{t-1}^2+\tau(z_{t-1})+\gamma u_{t-1}$ \newline $\log x_t=\xi+\varphi\log\sigma_t^2+\delta(z_t)+u_t$ & EGARCH-type leverage from the return shock plus the measurement innovation $u_t$ in the variance; several realized measures at once & extension \\91\midrule92\multicolumn{6}{l}{\textit{Multivariate}}\\93\addlinespace[1pt]94VEC-GARCH (Bollerslev--Engle--Wooldridge) & 1988 & \citet{bollerslev1988capital} & $\mathrm{vech}(H_t)=C+\sum_{i}A_i\,\mathrm{vech}(\varepsilon_{t-i}\varepsilon_{t-i}')+\sum_{j}B_j\,\mathrm{vech}(H_{t-j})$ (diagonal $A_i,B_j$ in the original) & first multivariate GARCH: every covariance is a linear function of lagged cross-products and covariances & out of scope \\95CCC-GARCH {\scriptsize[\texttt{ccc\_garch11}]} & 1990 & \citet{bollerslev1990modelling} & $H_t=D_tRD_t$, $D_t=\mathrm{diag}(\sigma_{1,t},\dots,\sigma_{N,t})$, $R$ constant & univariate GARCH margins with a constant correlation matrix & scored \\96Factor-ARCH (Engle--Ng--Rothschild) & 1990 & \citet{engle1990asset} & $H_t=\Omega+\sum_{k=1}^{K}\beta_k\beta_k'\,\sigma_{k,t}^2$, $\sigma_{k,t}^2$ a univariate GARCH of the factor portfolio $\phi_k'r_t$ & time-varying covariances through a few GARCH factors (reduced rank); positive definite if $\Omega$ is & out of scope \\97BEKK (full / diagonal) & 1995 & \citet{engle1995multivariate} & $H_t=CC'+\sum_{i}A_i'\varepsilon_{t-i}\varepsilon_{t-i}'A_i+\sum_{j}B_j'H_{t-j}B_j$ & quadratic form guaranteeing positive definiteness; nests the diagonal VEC and factor-ARCH & out of scope \\98Scalar BEKK (covariance targeting) {\scriptsize[\texttt{scalar\_bekk11}]} & 1995 & \citet{engle1995multivariate} & $H_t=(1-a-b)\bar\Sigma+a\,\varepsilon_{t-1}\varepsilon_{t-1}'+b\,H_{t-1}$ & two dynamic parameters; EWMA covariance is the case $a+b=1$, $\bar\Sigma$-term dropped & scored \\99DCC {\scriptsize[\texttt{dcc11}]} & 2002 & \citet{engle2002dynamic} & $H_t=D_tR_tD_t$ \newline $Q_t=(1-a-b)\bar Q+a\,z_{t-1}z_{t-1}'+b\,Q_{t-1}$ \newline $R_t=\mathrm{diag}(Q_t)^{-1/2}Q_t\,\mathrm{diag}(Q_t)^{-1/2}$ & scalar GARCH-type dynamics for the correlation of the standardized residuals; two-step estimation & scored \\100GO-GARCH & 2002 & \citet{vanderweide2002gogarch} & $\varepsilon_t=Zf_t$, $H_t=ZV_tZ'$, $V_t=\mathrm{diag}(\sigma_{1,t}^2,\dots,\sigma_{N,t}^2)$ with univariate GARCH for each factor \newline $Z=P\Lambda^{1/2}U$, $U$ orthogonal & generalized orthogonal factors (an estimated rotation of the principal components) with independent univariate GARCH & out of scope \\101Asymmetric DCC {\scriptsize[\texttt{adcc11}]} & 2006 & \citet{cappiello2006asymmetric} & $Q_t=(1-a-b)\bar Q-g\bar N+a\,z_{t-1}z_{t-1}'+g\,n_{t-1}n_{t-1}'+b\,Q_{t-1}$, $n_t=z_t\odot\ind{z_t<0}$ & correlations respond more to joint negative shocks (GJR logic in the correlation recursion) & scored \\102DECO (dynamic equicorrelation) & 2012 & \citet{engle2012dynamic} & $R_t=(1-\rho_t)I_N+\rho_t\iota\iota'$, $\rho_t=\dfrac{1}{N(N-1)}\big(\iota'R_t^{\mathrm{DCC}}\iota-N\big)$ & all pairwise correlations equal at each date; closed-form inverse and determinant for large $N$ & out of scope \\103Multivariate HEAVY & 2012 & \citet{noureldin2012multivariate} & $H_t=C_H+a_H\,RC_{t-1}+b_H\,H_{t-1}$ \newline $M_t=\E[RC_t\,|\,\F_{t-1}]=C_M+a_M\,RC_{t-1}+b_M\,M_{t-1}$ (scalar form \newline $RC_t$ = realized covariance) & realized covariance drives both the return covariance and its own (Wishart-type) measurement equation & out of scope \\104Corrected DCC (cDCC) & 2013 & \citet{aielli2013dynamic} & $Q_t=(1-a-b)\bar S+a\,Q_{t-1}^{*1/2}z_{t-1}z_{t-1}'Q_{t-1}^{*1/2}+b\,Q_{t-1}$, $Q_t^{*}=\mathrm{diag}(Q_t)$ & rescales the outer product so that $\bar S$ is the unconditional mean of $Q_t$: consistent two-step estimation & out of scope \\105DCC-NL (nonlinear shrinkage) & 2019 & \citet{engle2019large} & DCC/cDCC recursion with $\bar Q$ replaced by the nonlinear-shrinkage estimator of the unconditional correlation matrix and composite-likelihood estimation of $(a,b)$ & large-$N$ DCC: nonlinear shrinkage of the targeting matrix and composite-likelihood estimation & out of scope \\106Multivariate Realized GARCH & 2026 & \citet{archakov2026multivariate} & $v_t=\omega+\beta v_{t-1}+\gamma u_{t-1}$ with $v_t=(\log h_{1,t},\dots,\log h_{N,t},\mathrm{vecl}[\log C_t]')'$ and realized-measure equation $x_t=\xi+\varphi v_t+\tau(z_t)+u_t$ (schematic \newline see reference) & realized-GARCH structure on the log variances and the matrix-logarithm of the correlation matrix (unconstrained parametrization) & out of scope \\107\midrule108\multicolumn{6}{l}{\textit{Stochastic volatility (parameter-driven)}}\\109\addlinespace[1pt]110Stochastic volatility (log-SV, Taylor) {\scriptsize[\texttt{sv}]} & 1986 & \citet{taylor2008modelling} & $\varepsilon_t=\sigma_tz_t$, $\log\sigma_t^2=\omega+\phi\log\sigma_{t-1}^2+\sigma_\eta\eta_t$, $\eta_t\sim N(0,1)$ independent of $z_t$ & latent (parameter-driven) log variance with its own innovation; the variance is not $\F_{t-1}$-measurable & unsup\-ported \\111Multivariate SV, QML (Harvey--Ruiz--Shephard) & 1994 & \citet{harvey1994multivariate} & $\log\varepsilon_{i,t}^2=\log\sigma_{i,t}^2+\log z_{i,t}^2$, $\log\sigma_t^2=\omega+\Phi\log\sigma_{t-1}^2+\eta_t$ (linear state space \newline Kalman-filter QML) & multivariate SV with common factors; quasi-maximum likelihood on $\log\varepsilon^2$ & out of scope \\112SV by MCMC (Kim--Shephard--Chib) & 1998 & \citet{kim1998stochastic} & log-SV as above \newline $\log z_t^2$ approximated by a seven-component normal mixture \newline MCMC with an offset mixture and particle-filter likelihood for model comparison with ARCH & exact-likelihood (Bayesian) inference for SV and a formal comparison with GARCH & unsup\-ported \\113\midrule114\multicolumn{6}{l}{\textit{Other relatives (jumps, periodicity)}}\\115\addlinespace[1pt]116Periodic GARCH & 1996 & \citet{bollerslev1996periodic} & $\sigma_t^2=\omega_{s(t)}+\alpha_{s(t)}\varepsilon_{t-1}^2+\beta_{s(t)}\sigma_{t-1}^2$, $s(t)\in\{1,\dots,S\}$ the stage of a known periodic cycle & GARCH coefficients that vary deterministically with the calendar (day of the week, intraday season) & out of scope \\117GARCH-jump (ARJI, Maheu--McCurdy) & 2004 & \citet{maheu2004news} & $r_t=\mu+\sigma_tz_t+\sum_{k=1}^{n_t}Y_{t,k}$, $n_t\,|\,\F_{t-1}\sim\mathrm{Poisson}(\lambda_t)$, $\lambda_t=\lambda_0+\rho\lambda_{t-1}+\gamma\xi_{t-1}$, $\sigma_t^2$ GARCH/GJR on $\varepsilon_{t-1}=r_{t-1}-\mu$ & autoregressive conditional jump intensity on top of a GARCH diffusion component; total variance $\sigma_t^2+\lambda_t(\mu_J^2+\sigma_J^2)$ & out of scope \\118\end{longtable}}119