1\section{Estimator definitions and conventions}2\label{sec:est}34The method registry (\texttt{docs/methods/registry.md} in the repository) records, for each method, the equation, the inputs, the reference implementation and its options, the test tolerance and the validation status. We reproduce the Tier~A definitions here because conventions---not formulas---are where implementations diverge.56\paragraph{Returns and sessions.} With bar closes $p_{t,0},\dots,p_{t,M_t}$ in session $t$ (regular hours 09:30--16:00 ET inclusive of the 16:00 closing print), intraday log returns are $r_{t,j} = \log(p_{t,j}/p_{t,j-1})$, $j = 1,\dots,M_t$; a normal session has $M_t = 390$ and an early close $M_t = 210$ when no minute is missing. Zero-volume minutes are absent at the source and are left absent.78\paragraph{Realized measures.} For one session with $M$ returns,9\begin{align}10\RV &= \sum_{j=1}^{M} r_j^2, \qquad11\BPV = \mu_1^{-2}\,\frac{M}{M-1}\sum_{j=2}^{M} |r_j|\,|r_{j-1}|, \quad \mu_1 = \sqrt{2/\pi}, \label{eq:rv}\\12\RSV^{-} &= \sum_{j} r_j^2\,\I\{r_j<0\}, \qquad \RSV^{+} = \sum_{j} r_j^2\,\I\{r_j>0\}, \qquad13\RQ = \frac{M}{3}\sum_{j=1}^{M} r_j^4, \label{eq:rq}\\14J &= \max(\RV-\BPV, 0), \qquad15\mathrm{TriPV} = \mu_{4/3}^{-3}\,\frac{M}{M-2}\sum_{j=3}^{M} |r_j r_{j-1} r_{j-2}|^{4/3}, \label{eq:tripv}16\end{align}17with $\mu_p = \E|Z|^p$ for $Z\sim N(0,1)$. The finite-sample factors are options. Two conventions of \texttt{highfrequency} differ: \texttt{rBPCov} with \texttt{makeReturns = FALSE} applies no $M/(M-1)$ factor, and \texttt{rQuar} uses $(M+1)/3$, the number of prices, instead of $M/3$; the reference tests compare after conversion. Realized covariance on synchronised returns is $\sum_j r_j r_j'$ without demeaning.1819\paragraph{Regression and inference.} OLS, WLS and ridge are solved by column-pivoting Householder QR (SVD when rank-deficient; ridge by augmentation). Covariances follow \texttt{statsmodels}: classical $s^2(X'X)^{-1}$; HC0--HC3 with the leverage corrections; HAC with Bartlett weights and $L$ lags (default $\lfloor 4(n/100)^{2/9}\rfloor$, no small-sample correction); one-way cluster with $\frac{G}{G-1}\frac{n-1}{n-k}$. $p$-values use $t(n-k)$. Rolling OLS refits each window.2021\paragraph{HAR family.} With daily $\RV_t$, lags $(1,5,22)$ and horizon $h$,22\begin{equation}23y_{t+h} = \beta_0 + \beta_d \RV_t + \beta_w \overline{\RV}^{(5)}_t + \beta_m \overline{\RV}^{(22)}_t \;[+\,\beta_q(\sqrt{\RQ_t}-\sqrt{\overline{\RQ}})\RV_t] + \varepsilon_{t+h}, \qquad \overline{\RV}^{(k)}_t = \tfrac1k\sum_{j=0}^{k-1}\RV_{t-j},24\label{eq:har}25\end{equation}26where $y_{t+h}$ is the average of $\RV_{t+1},\dots,\RV_{t+h}$ (the single-day target is an option) and the bracketed HARQ term follows \citet{BollerslevPattonQuaedvlieg2016} in the demeaned form used by \texttt{highfrequency}; log-HAR applies the same regression to $\log\RV$ and back-transforms with $\exp$ (the smearing factor is reported, not applied). Overlapping targets for $h>1$ induce MA$(h-1)$ errors, so the HAC lag is at least $h-1$. Rolling estimation is strictly causal: the model fitted on observations up to $t$ forecasts $y_{t+h}$. Forecast losses are MSE and QLIKE $= y/\hat y - \log(y/\hat y) - 1$ \citep{Patton2011}.2728\paragraph{GARCH family.} With $\varepsilon_t = r_t - \mu$,29\begin{align}30\text{GARCH}(1,1):\;& \sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2, \qquad31\text{GJR}:\; \sigma_t^2 = \omega + (\alpha + \gamma\I\{\varepsilon_{t-1}<0\})\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2, \label{eq:garch}\\32\text{EGARCH}:\;& \log\sigma_t^2 = \omega + \alpha(|z_{t-1}| - \E|z|) + \gamma z_{t-1} + \beta\log\sigma_{t-1}^2, \quad z = \varepsilon/\sigma. \label{eq:egarch}33\end{align}34Errors are Gaussian or standardised Student-$t$ with $\nu\in(2.05, 500)$ (the bounds of \texttt{arch}); $\sigma_0^2$ is the sample variance of $\varepsilon$ (as \texttt{rugarch}; \texttt{arch} uses a backcast, which explains the $5\times10^{-4}$ log-likelihood tolerance against it). Estimation is quasi-maximum likelihood by BFGS on transformed parameters (log for $\omega$, logistic for persistence and the $\alpha/\beta$ share, $\tanh$ for the EGARCH $\beta$), with central-difference gradients validated across step sizes; up to two alternative starting points are tried on non-convergence, and the number tried is reported. Standard errors are the Bollerslev--Wooldridge sandwich $H^{-1}\,\mathrm{OPG}\,H^{-1}$ by default (Hessian and OPG are options). Forecasts are analytical for GARCH/GJR and by seeded simulation beyond one step for EGARCH. The \texttt{rugarch} EGARCH maps its \texttt{alpha1} to our $\gamma$ and its \texttt{gamma1} to our $\alpha$.3536\paragraph{Bootstrap.} The stationary bootstrap of \citet{PolitisRomano1994} draws blocks of geometric length (mean $b$, default $n^{1/3}$) starting at uniform indices with circular wrap; the circular block bootstrap uses fixed length; an iid option exists for comparison only. Replica $r$ uses the stream $(\text{seed}, r)$ of a counter-seeded xoshiro256++ generator, so replicas are independent of the thread that computes them.3738\paragraph{Tier B.} VAR$(p)$ with intercept by multivariate least squares, generalized FEVD \citep{KoopPesaranPotter1996,PesaranShin1998} normalised row-wise, Diebold--Yilmaz total, from, to and net connectedness \citep{DieboldYilmaz2012,DieboldYilmaz2014} with rolling windows, and Ledoit--Wolf shrinkage toward the scaled identity \citep{LedoitWolf2004}. Realized GARCH, HEAVY, DCC, GARCH-MIDAS, state-space and change-point methods, pre-averaging, realized kernels and order-flow methods are \emph{not} implemented; the registry says so.39