\section{Related literature}
\label{sec:lit}

\paragraph{Realized measures and their software.} The realized variance of \citet{AndersenBollerslevDieboldLabys2003} and \citet{BarndorffNielsenShephard2002}, bipower variation \citep{BarndorffNielsenShephard2004,BarndorffNielsenShephard2006}, realized semivariance \citep{BarndorffNielsenKinnebrockShephard2010}, realized quarticity and jump proxies \citep{AndersenBollerslevDiebold2007,HuangTauchen2005} are the building blocks of Tier~A. Their finite-sample constants and sampling conventions vary across implementations; \citet{LiuPattonSheppard2015} document how much the choice of sampling scheme matters. Noise-robust estimators---two-scale \citep{ZhangMyklandAitSahalia2005}, realized kernels \citep{BarndorffNielsenHansenLundeShephard2008}, pre-averaging \citep{JacodLiMyklandPodolskijVetter2009}---require tick or quote data and are outside the admissible inputs of this paper, which uses 1-minute OHLCV bars only. The reference implementation for our comparisons is the \texttt{highfrequency} package \citep{BoudtKleenSjorup2022}.

\paragraph{HAR and GARCH.} The HAR model of \citet{Corsi2009} and its HARQ extension \citep{BollerslevPattonQuaedvlieg2016}, the signed-jump variants of \citet{PattonSheppard2015}, and the GARCH family \citep{Engle1982,Bollerslev1986,Nelson1991,GlostenJagannathanRunkle1993} with quasi-maximum likelihood inference \citep{BollerslevWooldridge1992,FiorentiniCalzolariPanattoni1996} are the forecasting and conditional-variance workhorses we implement. We compare with \texttt{rugarch} \citep{Ghalanos2024rugarch} and \texttt{arch} \citep{Sheppard2025arch}. Forecast losses follow \citet{Patton2011}; the horse race of \citet{HansenLunde2005} and the comparison framework of \citet{DieboldMariano2002,HansenLundeNason2011} motivate the loss-based evaluation, although model selection is not the object of this paper. Realized GARCH \citep{HansenHuangShek2012}, HEAVY \citep{NoureldinShephardSheppard2012}, DCC \citep{Engle2002} and GARCH-MIDAS \citep{EngleGhyselsSohn2013} are listed in the registry as not implemented.

\paragraph{Inference.} Heteroskedasticity-robust \citep{White1980}, HAC \citep{NeweyWest1987,NeweyWest1994} and cluster-robust \citep{CameronGelbachMiller2011} covariances are implemented with the \texttt{statsmodels} conventions. Dependent-data resampling uses the stationary bootstrap \citep{PolitisRomano1994}; the wild bootstrap for realized volatility \citep{GoncalvesMeddahi2009} is a natural extension we do not pursue. Connectedness measures follow \citet{DieboldYilmaz2012,DieboldYilmaz2014} with the generalized FEVD of \citet{KoopPesaranPotter1996} and \citet{PesaranShin1998}; covariance shrinkage follows \citet{LedoitWolf2004}.

\paragraph{Numerical computing and GPUs in economics.} Compensated summation goes back to \citet{Kahan1965}, \citet{Dekker1971} and \citet{Neumaier1974}; \citet{Higham1993} gives the error analysis we rely on for the double-float accumulators. \citet{AldrichFernandezVillaverdeGallantRubioRamirez2011} introduced GPU computing to dynamic equilibrium models; the econometric literature on GPUs has concentrated on Monte Carlo and Bayesian computation rather than on the estimators studied here. Our systems dependencies are Eigen \citep{Eigen2025} with Apple Accelerate BLAS/LAPACK \citep{Blackford2002}, Apache Arrow and Parquet \citep{Arrow2025}, nanobind \citep{Nanobind2025}, and the Metal Shading Language \citep{AppleMetal2026}. The comparison engines for the pipeline experiment are pandas, Polars and DuckDB on top of NumPy \citep{Harris2020} and SciPy \citep{Virtanen2020}.