\section{The realized-measure comparison (Tier B)}\label{sec:realized}

Tier~B models observe, in addition to daily returns, the previous day's full-day realized variance. This is a larger
information set and the comparison with Tier~A is reported as such: the question is how much one day of intraday data is
worth, not which model is best. The answer at one day is unambiguous (Table~\ref{tab:pooledB}, Figure~\ref{fig:heatB}):
every realized-measure model except the levels HAR beats GARCH(1,1) by 12 to 15 percent on the pooled ratio---log-HAR
\RatioLogharRvEmpiricalHOne{} [\CiLoLogharRvEmpiricalHOne, \CiHiLogharRvEmpiricalHOne], GARCH-X \RatioGarchxRvOneOneTHOne,
HEAVY \RatioHeavyOneOneTHOne, Realized GARCH \RatioRealgarchOneOneNormalHOne---with significant wins on 25 to 31 of the
42 instruments and losses on at most two. H3 is supported, and the gain is roughly four times the largest gain available
within the daily tier. The four realized models are statistically indistinguishable from each other at one day; what
separates them is the \emph{form} in which they use the measure. The levels HAR of \citet{corsi2009simple}, estimated by
least squares on raw variances, is the one realized model that fails: its mean ratio is astronomical
(\RatioHarRvEmpiricalHOne) because direct projections estimated on windows that contain a variance spike extrapolate that
spike, while its trimmed ratio is close to one; its logarithmic version is the best model of the study at one day. The
realized random walk---tomorrow's variance equals today's measure---is a poor forecaster (trimmed ratio
\RatioRmRwEmpiricalHOne{} after trimming, far worse before), which is the clearest statement that smoothing, not the
measure itself, carries the value of intraday data.

\paragraph{Horizons.}
The advantage of intraday information shrinks with the horizon, as H3 predicted, but not uniformly across models
(Figure~\ref{fig:horizon}). Log-HAR, whose direct monthly projection is estimated on the monthly target, keeps a seven
percent gain at $h=22$ (\RatioLogharRvEmpiricalHTwoTwo) and Realized GARCH with Gaussian innovations five percent
(\RatioRealgarchOneOneNormalHTwoTwo); HEAVY and GARCH-X, whose multi-step forecasts iterate a linear recursion in which
the realized measure reverts to its own mean, lose their edge entirely by one month (\RatioHeavyOneOneTHTwoTwo{} and
\RatioGarchxRvOneOneTHTwoTwo) and are beaten by plain GARCH(1,1) on stocks. Realized GARCH with Student-$t$ innovations
is the second cautionary tale of the study: at one day it is as good as its Gaussian version (\RatioRealgarchOneOneTHOne),
but its multi-step forecasts, which must be simulated because the model is log-linear, inherit the tail of a $t$
distribution whose degrees of freedom are often estimated at the lower bound; one path in ten thousand with $z^2$ in the
hundreds dominates the mean of $\exp(\log\sigma^2)$, and the pooled ratio at $h=22$ is \RatioRealgarchOneOneTHTwoTwo{}
with ten percent of origins flagged as explosive. The model is not wrong; the conditional expectation of a log-normal
with a heavy-tailed driver is simply not what a forecaster wants to report, and a user of Realized GARCH should either use
Gaussian innovations for multi-step variance forecasts or forecast the median.

\paragraph{The matched-information view.}
The appropriate daily competitor of a realized model is not GARCH(1,1) but the best daily model, and GARCH-X is the natural
bridge: it is GARCH(1,1) with one extra regressor. Its ratio to GARCH(1,1) at one day, \RatioGarchxRvOneOneTHOne, is
therefore the cleanest estimate of what the realized measure adds to the squared-return news term, and it is the same as
HEAVY's, in which the squared return is dropped altogether: once the realized measure is available, the squared return
contributes nothing. By asset class the value of the measure is largest for futures and FX (ratios of 0.77--0.80), where
the overnight component is negligible and the measure covers the whole day, intermediate for stocks (0.85), smaller for ETFs
(0.93), whose overnight return is a third of daily variance and is not measured intraday, and absent for the two
cryptocurrencies, whose one-minute data are noisy even after 2022 (Section~\ref{sec:heterogeneity}).

\begin{table}[tp]\centering\scriptsize
\caption{Tier B, pooled QLIKE ratios to GARCH(1,1)-normal (test period, pairwise samples, primary proxy). Models use the lagged
full-day realized variance in addition to daily returns (larger information set than Tier A). Columns as in Table~\ref{tab:pooledA}.}\label{tab:pooledB}
\adjustbox{max width=\textwidth}{\input{tables/tab_pooled_tierB}}
\end{table}

\begin{figure}[tp]\centering
\includegraphics[width=\textwidth]{figures/fig_heatmap_h1_tierB.pdf}
\caption{Instrument-level QLIKE ratios to GARCH(1,1)-normal, Tier B, $h=1$ (with GARCH(1,1)-normal as the reference column).
Conventions as in Figure~\ref{fig:heatA}.}\label{fig:heatB}
\end{figure}