1\section{Related literature}\label{sec:literature}23\paragraph{From ARCH to the GARCH family.}4Conditional heteroskedasticity entered econometrics with \citet{engle1982autoregressive}, whose ARCH($q$) model lets the5conditional variance of a return be a linear function of $q$ lagged squared innovations. \citet{bollerslev1986generalized}6added lagged conditional variances, producing the GARCH($p,q$) recursion whose (1,1) member has since become the7reference specification of the field. The decade that followed produced most of the architectures benchmarked here:8the integrated GARCH of \citet{engle1986modelling}, the in-mean specification of \citet{engle1987estimating}, the9exponential GARCH of \citet{nelson1991conditional}, the threshold models of \citet{glosten1993relation} and10\citet{zakoian1994threshold}---which differ in whether the asymmetry acts on the variance or on the standard deviation---the11power ARCH of \citet{ding1993long} that nests the absolute-value models of \citet{taylor2008modelling} and12\citet{schwert1989why}, the nonlinear asymmetric GARCH of \citet{engle1993measuring} (building on \citealp{higgins1992class}),13the quadratic GARCH of \citet{sentana1995quadratic}, and the fractionally integrated GARCH of14\citet{baillie1996fractionally}, later extended to the exponential and power families by \citet{bollerslev1996modeling} and15\citet{tse1998conditional}, and generalized by \citet{davidson2004moment}. \citet{engle1999long} decomposed the conditional16variance into permanent and transitory components; Markov-switching variants were proposed by \citet{hamilton1994autoregressive},17\citet{cai1994markov}, \citet{gray1996modeling} and \citet{klaassen2002improving}, and given a tractable form with18regime-specific recursions by \citet{haas2004new}, whose specification is the one estimated by the \texttt{MSGARCH}19package of \citet{ardia2019markov}. Smooth-transition \citep{gonzalezrivera1998smooth,hagerud1997smooth} and mixed-frequency20\citep{engle2013stock} extensions complete the taxonomy of Section~\ref{sec:taxonomy}. Surveys by21\citet{bollerslev1992arch}, \citet{bollerslev1994arch}, \citet{terasvirta2009introduction}, \citet{bollerslev2010glossary} and22the monograph of \citet{francq2019garch} document the nomenclature inconsistencies that the model cards of this paper23resolve explicitly.2425\paragraph{Innovation laws and estimation.}26Fat-tailed and skewed standardized innovations were introduced by \citet{bollerslev1987conditionally} (Student-$t$),27\citet{nelson1991conditional} (generalized error distribution) and \citet{hansen1994autoregressive}28(skewed Student-$t$; see also \citealp{fernandez1998bayesian}). Quasi-maximum-likelihood theory for GARCH is due to29\citet{bollerslev1992quasi}, \citet{lee1994asymptotic}, \citet{lumsdaine1996consistency}, \citet{jensen2004asymptotic},30\citet{francq2004maximum} and \citet{straumann2006quasi}; strict and weak stationarity conditions to31\citet{nelson1990stationarity} and \citet{bougerol1992stationarity}; moment conditions to \citet{he1999properties} and32\citet{ling2002stationarity}; positivity constraints to \citet{nelson1992inequality} and, for FIGARCH, to33\citet{conrad2006inequality}. The score-driven models of \citet{creal2013generalized} and \citet{harvey2013dynamic}34provide a non-GARCH observation-driven alternative in which the variance is updated by the scaled score of the35conditional density; stochastic-volatility models \citep{taylor1994modeling,kim1998stochastic,jacquier1994bayesian}36are the parameter-driven counterpart.3738\paragraph{Realized measures and realized-measure models.}39High-frequency data turned the latent variance into something that can be measured with error rather than only40forecast: \citet{andersen1998answering} showed that realized variance makes GARCH forecasts look far better than41squared returns suggest, and \citet{andersen2001distribution,andersen2003modeling} and \citet{barndorff2002econometric}42established the asymptotics of realized variance. Microstructure noise \citep{hansen2006realized,zhang2005tale,43aitsahalia2005how}, jumps \citep{barndorff2004power,andersen2007roughing}, kernels \citep{barndorff2008designing} and44semivariances \citep{barndorff2010measuring} refined the measurement; \citet{liu2015does} found that five-minute45realized variance is hard to beat across 400 estimators and 31 assets, and \citet{hansen2005realized} proposed the46whole-day scaling we use as a sensitivity proxy. On the modelling side, the HAR model of \citet{corsi2009simple}47(with the extensions of \citealp{andersen2007roughing,patton2015good,bollerslev2016exploiting}), the GARCH-X of48\citet{engle2002new}, the multiplicative error model of \citet{engle2006multiple}, the HEAVY model of49\citet{shephard2010realising} and the Realized GARCH of \citet{hansen2012realized} (exponential form in50\citealp{hansen2016exponential}; multivariate HEAVY in \citealp{noureldin2012multivariate}) constitute the realized tier51of this benchmark. Multivariate conditional covariance models---CCC \citep{bollerslev1990modelling}, BEKK52\citep{engle1995multivariate}, DCC \citep{engle2002dynamic,engle2001theoretical,aielli2013dynamic}, asymmetric DCC53\citep{cappiello2006asymmetric}, GO-GARCH \citep{vanderweide2002gogarch}; surveys in \citealp{bauwens2006multivariate,54silvennoinen2009multivariate}---are evaluated with the consistent matrix losses of \citet{laurent2013loss} and55\citet{patton2009evaluating}.5657\paragraph{Forecast evaluation and inference.}58The evaluation toolkit is standard but its assembly matters. \citet{patton2011volatility} characterizes the loss59functions that rank variance forecasts consistently when only a noisy proxy of the latent variance is observed: the60MSE and QLIKE families, of which QLIKE is the most robust to proxy noise. Pairwise comparisons use61\citet{diebold1995comparing} with the small-sample adjustment of \citet{harvey1997testing} and HAC variances62\citep{newey1987simple,newey1994automatic}; see \citet{diebold2015comparing} and \citet{giacomini2006tests} for the63conditional-ability interpretation that justifies rolling-window comparisons of possibly misspecified models64\citep{patton2020comparing}. Multiple comparisons are handled with the reality check of \citet{white2000reality}, the65superior predictive ability test of \citet{hansen2005test} and the model confidence set of66\citet{hansen2011model}, all implemented with the stationary bootstrap of \citet{politis1994stationary}. Tail-risk67forecasts are backtested with \citet{kupiec1995techniques}, \citet{christoffersen1998evaluating},68\citet{engle2004caviar} and the joint VaR--ES scoring functions of \citet{fissler2016higher} and69\citet{patton2019dynamic}; \citet{nolde2017elicitability} and \citet{bayer2022regression} discuss comparative versus70absolute backtesting.7172\paragraph{Large-scale comparisons.}73The closest antecedent is \citet{hansen2005forecast}, who compared 330 ARCH-type specifications on two series and found74nothing that beat GARCH(1,1) on exchange rates while asymmetric models dominated on IBM. \citet{marcucci2005forecasting},75\citet{awartani2005predicting}, \citet{brownlees2011practical}, \citet{kourtis2016international} and76\citet{ardia2018forecasting} extended the comparison to regime switching, asymmetry, multiple horizons and hundreds of77assets, but each within a single information tier and without accounting for estimation failures or compute. The recent78realized-volatility benchmarks \citep{lyocsa2021stock,christensen2023machine,zhang2024volatility,branco2024forecasting,79rahimikia2024machine,brini2026forecasting} compare HAR-type and machine-learning forecasters on realized variance at80horizons of 1, 5 and 22 days with QLIKE and the MCS, but treat the GARCH family as, at most, a single baseline. Our search81(cut-off 2026-11-09, Appendix~\ref{app:provenance}) found no peer-reviewed revisit of the ``does anything beat GARCH(1,1)''82question on a cross-asset universe with a pre-registered design; the contribution of this paper is therefore the83combination---a definition-faithful GARCH taxonomy, realized and multivariate tiers scored separately, failure and compute84accounting, and a frozen protocol---rather than any single element, several of which (asymmetry on equities, the value of85intraday information, regime dependence of rankings) are established results that we can only quantify more systematically.86