% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/pmm3_utils.R
\name{pmm3_variance_matrices}
\alias{pmm3_variance_matrices}
\title{Calculate theoretical variance matrices for OLS and PMM3}
\usage{
pmm3_variance_matrices(X, m2, m4, m6)
}
\arguments{
\item{X}{Design matrix with column of ones for the intercept.}

\item{m2, m4, m6}{Second, fourth, and sixth central moments computed
from the initial OLS residuals.}
}
\value{
A list with components \code{ols} (OLS covariance matrix),
\code{pmm3} (PMM3 covariance matrix), and the scalar efficiency
diagnostics \code{gamma4}, \code{gamma6}, \code{g3}.
}
\description{
Returns the asymptotic covariance matrices
\eqn{V_{\mathrm{OLS}} = m_2 (X^\top X)^{-1}} and
\eqn{V_{\mathrm{PMM3}} = g_3\, m_2 (X^\top X)^{-1}} where
\eqn{g_3 = 1 - \gamma_4^2 / (6 + 9\gamma_4 + \gamma_6)}.
}
\details{
The PMM3 formula is the symmetric-platykurtic specialisation of
Kunchenko's polynomial-maximisation framework (s = 3); see
\code{notes/pmm3_vcov_derivation.md} for the derivation and
Zabolotnii et al. (2018, 2022, 2023) for the general method.
Under Gaussian errors \eqn{g_3 = 1} and the matrix coincides with the
OLS covariance.
}
