spb/wp46_uqo

1\section{The complexity--accuracy--robustness frontier}\label{sec:frontier}23\paragraph{Failures and cost (H5).}4Table~\ref{tab:accounting} and Figures~\ref{fig:pareto} and~\ref{fig:failures} price the accuracy gains of5Sections~\ref{sec:daily} and~\ref{sec:realized} in three currencies: the share of re-estimations that fail or end at an6inadmissible optimum, the share of forecasts that are explosive, and the computing time of one re-estimation.7Outright estimation failures are rare for every pre-registered specification after the multi-start and likelihood-floor8policy (below 0.5\% of re-estimations, except Realized EGARCH in the extension set); the common failure of the daily tier9is \emph{inadmissibility}---a converged estimate outside the stationarity region---and it concentrates in the fat-tailed10GARCH family on FX and index ETFs: GARCH(1,1)-$t$ returns $\alpha+\beta\ge1$ on 7\% of re-estimations, GJR-$t$ on 4\%,11the component model on 3\%, the Gaussian versions on 1--2\%, and the models that parameterize persistence differently12(EGARCH, APARCH, FIGARCH, NAGARCH, QGARCH, MS-GARCH) on well under 1\%. Inadmissible estimates are kept and flagged;13their forecasts are not catastrophic---integrated GARCH is a legitimate forecaster---and the failure-adjusted ratios of14Appendix~\ref{app:extra} differ from the common-sample ratios by less than one percent for every specification except15unconstrained EGARCH and the levels HAR, whose explosive forecasts (1.4\% and 0.3\% of origins) are the entire gap between16their trimmed and untrimmed ratios. H5 is supported in a form we did not anticipate: the operational risk of a complex17model is not that it fails to estimate, it is that it estimates perfectly well and then explodes out of sample.1819\paragraph{Compute.}20Computing cost is modest for everything with an analytic recursion and large for everything else. One re-estimation of21GARCH(1,1) with Student-$t$ innovations takes 50~milliseconds on an Apple M-series core with the multi-start policy;22GJR, NAGARCH, QGARCH and the component model 25--70~ms; MS-GARCH 180~ms; APARCH 260~ms; FIGARCH 1.5~s, thirty times23GARCH(1,1), because its truncated ARCH($\infty$) representation is re-evaluated at every likelihood call; the realized24models 50--80~ms; HAR 1~ms. Simulation-based multi-step forecasts add 2--15~ms per origin. The whole pre-registered25benchmark---3{,}276 instrument $\times$ specification jobs, 2.4 million re-estimations, 12 million forecast origins---ran in26about 90 CPU-hours on six Apple-silicon nodes, so that cost is never prohibitive at daily frequency; it becomes a design27variable only at scale (thousands of instruments, intraday re-estimation) or for the fractional and switching models.2829\paragraph{The frontier.}30Figure~\ref{fig:pareto} places every specification by its pooled one-day ratio and its cost. Within the daily tier the31Pareto frontier is short: GARCH(1,1) with Student-$t$ or GED innovations (ratio 0.99, 50~ms), the component model32(0.98, 40~ms), and FIGARCH-GED (0.97, 1.5~s). Everything else is dominated: the asymmetric models cost as much as the33component model for no pooled gain, MS-GARCH costs four times GARCH(1,1) for nothing, and the baselines are cheap but34poor. The realized tier moves the frontier by an order of magnitude more than any daily refinement---log-HAR (0.86,351~ms), GARCH-X and HEAVY (0.86, 80~ms)---at the price of one day of intraday data, which is a data cost rather than a36compute cost. Against the failure rate (Figure~\ref{fig:pareto}, right panel) the frontier is the same with one addition:37Gaussian GARCH(1,1), which is never inadmissible, is the robust anchor, and the fat-tailed variants trade a few percent38of accuracy on calm instruments for a 4--7\% rate of boundary estimates.3940\paragraph{Practical implications.}41For a practitioner forecasting daily variance from daily data, the evidence favours GARCH(1,1) with a fat-tailed law,42re-estimated regularly, with the component or FIGARCH model as the only upgrades worth their cost and only on exchange43rates and at monthly horizons; asymmetric models earn their keep on equities and nowhere else; EGARCH should not be run44without sign constraints; regime switching is not worth its cost for variance point forecasts. For anyone with access to45intraday data the decision is not which GARCH to use but whether to use a realized measure at all: a logarithmic HAR46regression is the cheapest, most robust and most accurate one-day forecaster in the study, and a realized GARCH or47HEAVY model adds a density and a coherent multi-step recursion for a small cost. For tail risk the law matters more than48the architecture, and the skewed Student-$t$ is the only law that calibrates both the 1\% and the 5\% quantiles.4950\begin{table}[tp]\centering\scriptsize51\caption{Failure and compute accounting, test period, all instruments. Failed = re-estimations that raised, did not converge,52or were rejected by the likelihood floor; inadmissible = converged outside the admissibility region (kept and flagged);53coverage = share of origins with a valid forecast; median refit = wall time of one re-estimation (Apple M-series core, one54thread); iterations = optimizer iterations summed over the multi-start sequence.}\label{tab:accounting}55\adjustbox{max width=\textwidth}{\input{tables/tab_accounting}}56\end{table}5758\begin{figure}[tp]\centering59\includegraphics[width=\textwidth]{figures/fig_pareto_time.pdf}\\[4pt]60\includegraphics[width=\textwidth]{figures/fig_pareto_fail.pdf}61\caption{Complexity--accuracy frontiers, $h=1$. Top: pooled QLIKE ratio to GARCH(1,1)-normal against the median wall time of62one re-estimation (log scale). Bottom: against the share of failed or inadmissible re-estimations. Dashed line: Pareto frontier;63shaded band: $\pm2\%$ materiality band; specifications above the trimmed axis limit are drawn at the top edge and named in the64text.}\label{fig:pareto}65\end{figure}6667\begin{figure}[tp]\centering68\includegraphics[width=\textwidth]{figures/fig_failures.pdf}69\caption{Failure diagnostics, test period, all instruments, representative specifications with a non-zero share. Left: share of70re-estimations that failed (blue) or ended at an inadmissible optimum (orange, hatched). Right: share of forecast origins with an71explosive forecast (cumulative forecast above 100 times the trailing 22-day proxy mean), a different denominator.}\label{fig:failures}72\end{figure}73