Large Portfolios' Risks and High-Dimensional Factor Models

Abstract

This dissertation explores two important topics on high-dimensional factor models. We first consider the problem of estimating and assessing the risk of a large portfolio. In financial econometrics literature, the risk is often estimated by a substitution of a good estimator of the volatility matrix. However, the accuracy of such a risk estimator for large portfolios is largely unknown, and a simple inequality in the previous literature gives an infeasible and crude upper bound for the estimation error. In the first half of this dissertation, we study factor-based risk estimators under a large amount of assets and introduce a high-confidence level upper bound (H-CLUB) to assess the accuracy of the risk estimation. The H-CLUB is constructed based on the confidence interval of risk estimators with either known or unknown factors. We derive the limiting distribution of the estimated risks in high dimensionality. We find that when the dimensionality is larger than the sample size, the factor-based risk estimators have the same asymptotic variances no matter factors are known or not and are smaller than that of the sample covariance-based estimators. Our numerical results demonstrate that the proposed upper bounds outperform the traditional crude bounds and provide insightful assessment of the estimation of the portfolio risks.

Penalized estimation principle is fundamental to high-dimensional problems. In the literature, it has been extensively and successfully applied to various models, but only with structural parameters. In the second part, we apply the penalized estimation principle to a linear regression model with both finite-dimensional structural parameters and high-dimensional sparse incidental parameters. For the estimated structural parameters, we derive their consistency and asymptotic distributions, which reveals an oracle property. However, the penalized estimator for the incidental parameters possesses only partial selection consistency but not consistency, which shows a partial consistency phenomenon. In estimation, we consider an alternative two-step penalized estimator, which is more efficient compared with the one-step procedure and is more suitable for constructing confidence regions. Furthermore, we extend methods and results to the case where the dimension of the structural parameters diverges. Data-driven penalty regularization parameters are also provided.