1\section{Experimental design, estimation, forecasting and inference}\label{sec:design}23\paragraph{Pre-registration.} The protocol---universe, periods, model registry, estimation scheme, losses, proxies,4bootstrap settings, regime definitions and hypotheses---was written after the data audit and a validation-period pilot and5frozen in the repository (git tag \texttt{prereg-frozen}) before any loss on the test period was computed. Everything not in6the protocol is labelled exploratory. The protocol lists six hypotheses: H1, asymmetric models beat GARCH(1,1) at $h=1$ on7equities; H2, non-Gaussian innovations leave variance-forecast rankings materially unchanged but improve 1\% VaR coverage;8H3, realized-measure models beat the best daily model at $h=1$ with an advantage that shrinks with the horizon; H4,9long-memory, component and regime models help, if at all, only at $h=22$; H5, complex models fail more often and cost more,10which reduces their advantage on the failure-adjusted sample; H6, rankings differ across trailing-volatility terciles. Each11is tied to the primary loss, proxy and sample, with ``materially'' defined as a pooled loss ratio outside $1\pm0.02$.1213\paragraph{Information tiers.} Tier~A models use daily returns only. Tier~B models additionally use the lagged primary14realized proxy $y_{t-1}$, observed at the close of day $t-1$; they are compared with the Tier~A models on the same origins but15always with the information-set difference stated, and the realized random walk, HAR and log-HAR provide strong16realized-only baselines. Tier~C models forecast one-step covariance matrices of two ETF baskets (SPY, QQQ, IWM, DIA; and17SPY, TLT, GLD, USO, EEM) and are scored on their own.1819\paragraph{Rolling scheme.} For every instrument and specification, parameters are estimated on a rolling window of the20last 1{,}000 trading days ending at the origin $t$ and re-estimated every five trading days; between re-estimations the21conditional variance is filtered daily with the last estimated parameters, so that the forecast at every origin uses22$\F_t$ and nothing later. Expanding windows, a 2{,}000-day window and a 22-day re-estimation cadence are robustness23variants. At each origin the model produces per-step variance forecasts $\sigma^2_{t+k|t}$, $k=1,\dots,22$, by its own exact24conditional-expectation recursion where one exists (GARCH, GJR with $\kappa$, NAGARCH, QGARCH, CGARCH, IGARCH, GAS,25HEAVY, GARCH-X, FIGARCH, MS-GARCH) or by simulation with 2{,}000 paths and a fixed seed otherwise (EGARCH, APARCH, TGARCH,26AVGARCH, Realized GARCH); HAR forecasts are direct projections. The cumulative forecast $\sum_{k=1}^h\sigma^2_{t+k|t}$ is27matched to the cumulative proxy $\sum_{k=1}^hy_{t+k}$ for $h\in\{1,5,22\}$, so that forecasts and targets share units28(percent$^2$) and horizon. The validation period 2012-01-03 to 2015-12-31 served for design checks only; all confirmatory29results use the test period 2016-01-04 to 2026-10-30, which instruments enter when their window allows.3031\paragraph{Estimation and failures.} Models in the arch package \citep{sheppard2025arch} are estimated by its32quasi-maximum-likelihood routines; the custom models maximize their exact likelihoods with L-BFGS-B on smooth33reparameterizations that enforce the admissibility domain (persistence as a logistic share, positivity by exponentials), three34starting values and a Nelder--Mead polish, with the initial variance set by the same exponentially weighted backcast as the35arch package so that likelihoods are comparable. A re-estimation that raises or fails to converge is logged as a failure;36the specification then keeps its last successful parameter vector (flagged) if one exists and produces no forecast otherwise.37A converged estimate outside the admissibility region (for instance $\alpha+\beta\ge1$) is kept and flagged. No forecast is38ever replaced by another model's. We record, per re-estimation, wall time, iterations, restarts and status, and per job the39CPU time and peak memory, which feed the complexity--accuracy--robustness analysis of Section~\ref{sec:frontier}.4041\paragraph{Verification.} Every implementation passes unit tests of the recursion, likelihood, admissibility, multi-step42forecasts and absence of look-ahead (perturbing post-origin data leaves all forecasts of a job unchanged), seeded Monte Carlo43recovery tests, and a cross-check against \texttt{rugarch} 1.5-6 \citep{galanos2011rugarch} on a shared synthetic fixture:44the custom GARCH(1,1) reproduces the arch recursion to $10^{-9}$ and its analytic forecasts to machine precision, and every45scored architecture available in both packages attains a log-likelihood within 0.3 of the \texttt{rugarch} optimum (FIGARCH:46arch is 1.47 below, a documented truncation difference). Appendix~\ref{app:verification} reports the details.4748\paragraph{Losses and samples.} The primary loss is QLIKE,49$\QLIKE(y,h)=y/h-\log(y/h)-1$ for $y,h>0$, computed in the zero-safe form $\log h+y/h-(\log y+1)$ which coincides with QLIKE50on the support and remains finite when $y=0$ \citep{patton2011volatility}; forecasts are floored at $10^{-8}$ before any51logarithm. MSE, MAE and the mean squared error on the standard-deviation scale are secondary losses, the latter reported only52as a sensitivity because it is not robust to proxy noise. For each instrument and horizon we report three samples: the53\emph{pairwise} sample of origins on which both the specification and the GARCH(1,1)-normal benchmark have valid forecasts54and the target is valid (used for loss ratios and Diebold--Mariano tests); the \emph{common-success} sample on which every55specification of the tier with at least 90\% coverage has a valid forecast (used for the model confidence set and the SPA56test); and each model's \emph{own} sample with its coverage rate (the operational view).5758\paragraph{Inference.} Pairwise tests are Diebold--Mariano statistics with Newey--West variances (Bartlett kernel, bandwidth59$h-1+\lfloor4(n/100)^{2/9}\rfloor$) and the Harvey--Leybourne--Newbold correction. Within each instrument, horizon and tier we60compute the 90\% model confidence set of \citet{hansen2011model} with the range statistic and a stationary bootstrap of61expected block length 22 days and 2{,}000 replications, and the SPA test of \citet{hansen2005test} with GARCH(1,1) as the62benchmark. Across instruments we report, for each specification and horizon, the equally weighted mean of instrument-level63loss ratios together with a stationary block bootstrap over dates that resamples all instruments jointly---so that64cross-sectional dependence is preserved---and the number of instruments on which the specification significantly beats or65loses to the benchmark after a Holm adjustment across the 42 instruments. Statistical significance is reported next to the66economic threshold of the protocol: a pooled ratio within $1\pm0.02$ is ``no material difference'' whatever its $p$-value.6768\paragraph{Tail risk and covariance.} One-day 1\% and 5\% VaR and expected shortfall are formed from each model's own69innovation law (empirical standardized residual quantiles of the estimation window for distribution-free models) and70backtested with the unconditional coverage test of \citet{kupiec1995techniques}, the conditional coverage test of71\citet{christoffersen1998evaluating}, the dynamic quantile test of \citet{engle2004caviar} with four lags and the joint72VaR--ES scoring function FZ0 of \citet{fissler2016higher} in the parameterization of \citet{patton2019dynamic}. Covariance73forecasts are scored against the five-minute realized covariance plus the overnight outer product with the Frobenius loss, the74multivariate QLIKE of \citet{laurent2013loss} and the QLIKE of the equally weighted portfolio variance, after verifying75positive definiteness.76