High Dimensional Covariance Matrix Estimation Using a Factor Model

摘要

High dimensionality comparable to sample size is common in many statisticalproblems. We examine covariance matrix estimation in the asymptotic frameworkthat the dimensionality $p$ tends to $infty$ as the sample size $n$ increases.Motivated by the Arbitrage Pricing Theory in finance, a multi-factor model isemployed to reduce dimensionality and to estimate the covariance matrix. Thefactors are observable and the number of factors $K$ is allowed to grow with$p$. We investigate impact of $p$ and $K$ on the performance of the model-basedcovariance matrix estimator. Under mild assumptions, we have establishedconvergence rates and asymptotic normality of the model-based estimator. Itsperformance is compared with that of the sample covariance matrix. We identifysituations under which the factor approach increases performance substantiallyor marginally. The impacts of covariance matrix estimation on portfolioallocation and risk management are studied. The asymptotic results aresupported by a thorough simulation study.