\section{Cross-market, regime and horizon heterogeneity}\label{sec:heterogeneity}

\paragraph{Asset classes.}
Figure~\ref{fig:class} and Table~\ref{tab:regimes} break the pooled ratios by asset class. Three patterns are robust across
specifications. First, asymmetry is an equity phenomenon: on stocks the asymmetric models reach 0.93--0.97 of
GARCH(1,1)'s loss at one day (TGARCH 0.93, APARCH 0.94, GJR 0.97) and 0.88--0.90 at one month, on ETFs 0.96--0.97, but on
futures and FX they are at or above one---the leverage effect that the asymmetric news-impact curve captures is absent
from commodity, rate and currency returns, and the extra parameter costs forecast variance. Second, fat-tailed and
long-memory dynamics are an FX phenomenon: for the six exchange rates, GARCH(1,1) with Student-$t$ innovations alone
improves on its Gaussian version by eight percent, and FIGARCH and the component model by eleven and ten percent, the
largest daily-tier gains of the study; for the nine futures and fifteen ETFs these models are within two percent of
GARCH(1,1). USDCHF is the sharpest illustration of the second pattern: its 1{,}000-day windows contain the Swiss franc shock of
15~January~2015 until 2019, the Gaussian quasi-likelihood lets that one day dominate the estimates, and every fat-tailed or
long-memory specification beats GARCH(1,1)-normal by 20--50\% on that instrument alone (the instrument-level ratios of
Figure~\ref{fig:heatA}). Third, intraday information is worth most where the session is longest: the realized models gain 20--23\% on
futures and FX (sessions of 23--24 hours, no overnight gap), 15--17\% on stocks, 7--10\% on ETFs, and nothing on the two
cryptocurrencies, whose one-minute aggregate is too noisy to help even in 2022--2026---there the realized models are
worse than GARCH(1,1) and the daily models with fat tails (FIGARCH, MS-GARCH) are the best available. The two
cryptocurrencies are also the only instruments on which GARCH(1,1) with Student-$t$ innovations is worse than the
Gaussian version, a reminder that a two-asset class is not a sample.

\paragraph{Regimes (H6).}
Table~\ref{tab:regimes} splits the test period by the pre-registered regimes. The volatility terciles, defined from each
instrument's trailing 22-day realized variance at the origin with thresholds fixed on the validation period, barely move
the daily-tier ratios: GJR goes from 1.00 in the low tercile to 1.01 in the high tercile, FIGARCH from 0.96 to 0.96, the
component model from 0.97 to 0.99. H6 predicted that asymmetric and realized models gain most in high-volatility
states; the asymmetric models do the opposite (TGARCH 0.99 low versus 1.03 high), and only the realized-GARCH-type models
gain more in the high tercile (HEAVY 0.89 low versus 0.85 high; Realized GARCH 0.90 versus 0.88). The COVID episode is
the exception that illustrates the mechanism: in the 90 trading days from 2020-02-20, HEAVY beats GARCH(1,1) by 29\%
and Realized GARCH by 15\%, because the realized measure reports the March~2020 variance explosion a day before the
squared return has accumulated it, while the asymmetric models are 6--9\% \emph{worse} than GARCH(1,1), because their
leverage term over-reacts to the first negative days. In the 2022 rate shock, a year of elevated but not explosive
volatility, every ratio is within two percent of its full-sample value. The sign of the previous day's return is
immaterial. H6 is therefore rejected as stated and replaced by a narrower finding: rankings are stable across trailing-
volatility states; what reorders them is the speed of a variance jump, which only intraday information measures in
time.

\paragraph{Proxies and losses.}
Appendix~\ref{app:extra} repeats the pooled comparison with the six secondary proxies and the three secondary losses.
Replacing the subsampled five-minute realized variance by its plain, one-minute, bipower or median versions changes no
ranking and moves ratios by less than one percent; the Hansen--Lunde whole-day scaling moves them by two percent in the
direction of the realized models (the scaled proxy is closer to the quantity they fit); the squared daily return, the
noisiest proxy, compresses every ratio toward one---the realized models' gain falls from 14\% to 6\%---exactly as
\citet{patton2011volatility} predicts for a proxy with ten times the noise variance. MSE in variance units is dominated
by a few extreme days and reverses several rankings (the historical variance looks competitive); the standard-deviation
loss, which is not robust to proxy noise, favours the smoothest forecasters. None of these variations changes the two
headline statements: nothing material within the daily tier, a stable 12--15\% from one day of intraday data.

\begin{figure}[tp]\centering
\includegraphics[width=\textwidth]{figures/fig_class_h1.pdf}
\caption{Pooled QLIKE ratio to GARCH(1,1)-normal by asset class, $h=1$, representative specifications: bars = equal-weight mean of
instrument-level ratios, whiskers = 90\% joint block-bootstrap interval; values beyond the displayed range are drawn as arrowheads;
grey band = $\pm2\%$ materiality band.}\label{fig:class}
\end{figure}

\begin{table}[tp]\centering\scriptsize
\caption{Pooled QLIKE ratios to GARCH(1,1)-normal by pre-registered regime, $h=1$, test period (equal-weight means of
instrument-level pairwise ratios; number of instruments in parentheses). Terciles of the trailing 22-day realized variance
at the origin (thresholds from the validation period); sign of the previous day's return; calendar episodes fixed in the
protocol (the 2025 tariff episode has too few origins per instrument and is reported in Appendix~\ref{app:extra}).}\label{tab:regimes}
\adjustbox{max width=\textwidth}{\input{tables/tab_regimes_h1}}
\end{table}

\begin{figure}[tp]\centering
\includegraphics[width=\textwidth]{figures/fig_regimes_h1.pdf}
\caption{QLIKE ratio to GARCH(1,1)-normal by pre-registered regime, $h=1$, for eight representative specifications: line = median
across instruments, band = interquartile range (display statistics; Table~\ref{tab:regimes} reports the equal-weight means);
grey band = $\pm2\%$. The 2025 tariff episode has fewer than 30 origins per instrument and is omitted.}\label{fig:regimes}
\end{figure}