spb/wp46_uqo

1\section{The daily-information comparison (Tier A)}\label{sec:daily}23\paragraph{Pooled evidence.}4Table~\ref{tab:pooledA} reports, for one representative specification per architecture (Student-$t$ innovations where5available), the equally weighted mean over the \NInstruments{} instruments of the instrument-level QLIKE ratio to6GARCH(1,1)-normal, its 90\% joint block-bootstrap confidence interval, and the number of instruments on which the7specification significantly beats or loses to the benchmark after a Holm adjustment; Figure~\ref{fig:heatA} shows the8instrument-level ratios. The headline is parsimonious: on the pre-registered loss, proxy and test period, \emph{no daily9architecture beats GARCH(1,1) by a material margin at one day}. The best Tier~A specification is10\BestSpecTierAHOne{} with a pooled ratio of \BestRatioTierAHOne{}; FIGARCH with Student-$t$ innovations11(\RatioFigarchOneOneTHOne, CI [\CiLoFigarchOneOneTHOne, \CiHiFigarchOneOneTHOne]) and the component model12(\RatioCgarchOneOneTHOne, [\CiLoCgarchOneOneTHOne, \CiHiCgarchOneOneTHOne]) are the only families whose intervals exclude13one by more than a percentage point, and even they sit inside or at the edge of the $\pm2\%$ band that the protocol14defines as no material difference. Changing the innovation law of GARCH(1,1) itself moves the ratio by about one percent15(Student-$t$ \RatioGarchOneOneTHOne, skew-$t$ \RatioGarchOneOneSkewtHOne, GED \RatioGarchOneOneGedHOne), consistent with16H2: the law matters for the tails (Section~\ref{sec:tail}), not for the variance point forecast. The order grid adds17nothing (GARCH(2,1) \RatioGarchTwoOneTHOne, GARCH(1,2) \RatioGarchOneTwoTHOne), the in-mean term adds nothing18(\RatioGarchmOneOneTHOne), and integrated GARCH is marginally worse (\RatioIgarchOneOneTHOne). The classical baselines19lose clearly: ARCH(5) \RatioArchFiveTHOne, the fixed-$\lambda$ EWMA \RatioEwmaZeroNineFourNormalHOne, the 22-day20historical variance \RatioHistvarTwoTwoEmpiricalHOne{} and the constant variance \RatioConstvarNormalHOne; RiskMetrics~2006,21whose multi-horizon kernel is in effect a long-memory filter without estimation, is close to GARCH(1,1)22(\RatioRmTwoZeroZeroSixNormalHOne) and beats it on a handful of instruments.2324\paragraph{Asymmetry (H1).}25The asymmetric family does not beat GARCH(1,1) on average across assets at one day: GJR \RatioGjrOneOneTHOne,26Zakoïan's TGARCH \RatioTarchOneOneTHOne, APARCH \RatioAparchOneOneTHOne, NAGARCH \RatioNagarchOneOneTHOne{} and QGARCH27\RatioQgarchOneOneTHOne{} have intervals that straddle one or exclude it by less than two percent. The pre-registered28hypothesis H1 was stated for equities, and there the picture differs: Section~\ref{sec:heterogeneity} shows ratios of290.93--0.97 for the asymmetric models on stocks and ETFs, significant on a minority of instruments, against ratios at or30above one on futures and FX. The 1\%-trimmed ratios in Table~\ref{tab:pooledA} are systematically a little lower than the31means for the asymmetric models (GJR 0.97 versus 1.00), which says that their rare large errors are larger than32GARCH(1,1)'s---asymmetric news impact buys a better typical forecast at the price of a fatter error tail.33EGARCH is the extreme case and deserves its own paragraph.3435\paragraph{EGARCH and the geometry of failure.}36Under the pre-registered accounting, EGARCH has the worst mean QLIKE of the whole study, by six orders of magnitude37(Table~\ref{tab:pooledA}, mean ratio \RatioEgarchOneOneTHOne). This is not an estimation failure in the usual sense: the38fits converge, their in-sample likelihoods are at or above GARCH(1,1)'s, and the multi-start procedure of39Appendix~\ref{app:verification} finds the same optima. The parameters are the problem. On about a third of the40instruments---bond and sector ETFs, FX pairs, several stocks---the unconstrained arch parameterization returns41$\alpha<0$ with $\beta$ within 0.003 of one. A negative $\alpha$ means that a small standardized shock \emph{raises}42log-variance; once the variance is large, every shock is small in standardized units, so the log-variance keeps rising43with no information from the data: EURUSD in February~2014 goes from 163 to 927 percent$^2$ in four calm days. The44share of such explosive forecasts is only 1.4\% of origins, but QLIKE penalizes a forecast that is a thousand times too45large by $\log(1000)\approx7$ nats, and the mean never recovers. Nelson's model has no built-in protection because its46news-impact curve is unbounded in $|z|$; the Beta-$t$-EGARCH of the extension section bounds the score and does not47explode, and the constrained twin with $\alpha\ge0$, $\beta\le0.995$ (Section~\ref{sec:extensions}) removes the pathology48at a small cost in fit. We report the median and 1\%-trimmed ratios next to the means throughout, but we do not replace49the pre-registered statistic: an operational user who ran unconstrained EGARCH with weekly re-estimation would have50suffered exactly these forecasts.5152\paragraph{Horizons (H4).}53At five and twenty-two days the ranking tightens around the models with slowly mean-reverting dynamics54(Figure~\ref{fig:horizon}). At $h=22$ the absolute-value models lead the daily tier---TGARCH \RatioTarchOneOneTHTwoTwo{}55and AVGARCH \RatioAvgarchOneOneTHTwoTwo---followed by QGARCH \RatioQgarchOneOneTHTwoTwo, FIGARCH-GED56\RatioFigarchOneOneGedHTwoTwo{} and APARCH \RatioAparchOneOneTHTwoTwo, all with intervals below one; GARCH(1,1)-$t$57itself gains two percent on its Gaussian version (\RatioGarchOneOneTHTwoTwo). The component model, which helps at one58day, does not at one month (\RatioCgarchOneOneTHTwoTwo), integrated GARCH loses (\RatioIgarchOneOneTHTwoTwo), and59the Markov-switching model is never materially different from GARCH(1,1) at any horizon60(\RatioMsgarchTwoTHOne{} and \RatioMsgarchTwoTHTwoTwo), in line with the single-series evidence of61\citet{marcucci2005forecasting}. H4 is therefore supported in its first half (long-memory and persistence-rich models62gain only at the long horizon) and rejected in its second (regime switching does not gain at any horizon).6364\paragraph{Model confidence sets and the limits of the toolkit.}65With \NSpecs{} specifications and heavy-tailed loss differentials, the 90\% model confidence set over all Tier~A66specifications is large: it keeps 59--69 of about 70 specifications on most instruments at every horizon67(Figure~\ref{fig:mcsA}), and GARCH(1,1) is excluded on none. This is a statement about power, not about equivalence: the68range statistic is driven by the worst pair, and with dozens of nearly identical specifications the bootstrap69distribution of the range is wide. The exploratory set restricted to one specification per architecture70(Figure~\ref{fig:mcsrepA}) is sharper---the historical-variance and constant-variance baselines, ARCH(5), EWMA and71unconstrained EGARCH are eliminated on most instruments---but GARCH(1,1) survives in every instrument's set. The SPA test72of \citet{hansen2005test} with GARCH(1,1) as the benchmark rejects the null that no Tier~A specification beats it on a73minority of instruments at one day (details in Appendix~\ref{app:extra}); where it rejects, the beating specification is74usually FIGARCH or the component model, with gains of two to four percent.7576\begin{table}[tp]\centering\scriptsize77\caption{Tier A, pooled QLIKE ratios to GARCH(1,1)-normal (test period 2016--2026, pairwise samples, primary proxy). One78representative specification per architecture (Student-$t$ where available). ``ratio'' = equally weighted mean of79instrument-level ratios; 90\% CI = joint stationary block bootstrap over dates (all instruments resampled together); W/L/N =80instruments on which the specification significantly beats / loses to the benchmark at 10\% after a Holm adjustment across the81N instruments. Stars: bootstrap $p$ for H$_0$: pooled relative difference $=0$ (*** 1\%, ** 5\%, * 10\%). Full table with all82innovation laws, trimmed and median ratios and explosive-forecast shares in Appendix~\ref{app:extra}.}\label{tab:pooledA}83\adjustbox{max width=\textwidth}{\input{tables/tab_pooled_tierA_rep}}84\end{table}8586\begin{figure}[tp]\centering87\includegraphics[width=\textwidth]{figures/fig_heatmap_h1_tierA.pdf}88\caption{Instrument-level QLIKE ratios to GARCH(1,1)-normal, Tier A, $h=1$. Rows: representative specifications in family blocks;89columns: instruments in asset-class blocks. Colour: log ratio on a diverging scale (blue = better than GARCH(1,1), red = worse), flat90grey inside the $\pm2\%$ materiality band, clipped at [0.80, 1.25]; filled dot = Diebold--Mariano rejection at 5\%, open dot at9110\% (HAC, HLN); ``$\times$'' = ratio above 2 or below 1/2 (degenerate, clipped); hatched = not available. In grayscale the two92poles are both dark: the sign is in the tables of Appendix~\ref{app:extra}.}\label{fig:heatA}93\end{figure}9495\begin{figure}[tp]\centering96\includegraphics[width=0.75\textwidth]{figures/fig_horizon.pdf}97\caption{Pooled QLIKE ratio to GARCH(1,1)-normal by horizon for the three best Tier A and three best Tier B specifications at98$h=1$, with 90\% joint block-bootstrap bands. Tier B models use the lagged realized measure (a larger information set).}\label{fig:horizon}99\end{figure}100101\begin{figure}[tp]\centering102\includegraphics[width=\textwidth]{figures/fig_mcsrep_h1_tierA.pdf}103\caption{90\% model confidence set (Hansen, Lunde and Nason 2011; range statistic, stationary bootstrap, block 22, 2{,}000104replications) per instrument, Tier A, $h=1$, one specification per architecture (exploratory restriction; the pre-registered105set over all specifications is in Appendix~\ref{app:extra}). Rows: specifications; columns: instruments. Shading: MCS $p$-value (sequential scale, 0.10 threshold marked on the bar); dot = in the set.}\label{fig:mcsrepA}106\end{figure}107