Estimation of a Function of a Large Covariance Matrix Using Classical and Bayesian Methods

Abstract

In this dissertation, we consider the problem of estimating a high dimensional co-

variance matrix in the presence of small sample size. The proposed Bayesian solution

is general and can be applied to dierent functions of the covariance matrix in a wide

range of scientic applications, though we narrowly focus on a specic application of

allocation of assets in a portfolio where the function is vector-valued with components

which sum to unity. While often there exists a high dimension of time series data, in

practice only a shorter length is tenable, to avoid violating the critical assumption of

equal covariance matrix of investment returns over the period.

Using Monte Carlo simulations and real data analysis, we show that for small

sample size, allocation estimates based on the sample covariance matrix can perform

poorly in terms of the traditional measures used to evaluate an allocation for portfolio

analysis. When the sample size is less than the dimension of the covariance matrix,

we encounter diculty computing the allocation estimates because of singularity of

the sample covariance matrix. We evaluate a few classical estimators. Among them,

the allocation estimator based on the well-known POET estimator is developed using

a factor model. While our simulation and data analysis illustrate the good behavior

of POET for large sample size (consistent with the asymptotic theory), our study

indicates that it does not perform well in small samples when compared to our pro-

posed Bayesian estimator. A constrained Bayes estimator of the allocation vector is

proposed that is the best in terms of the posterior risk under a given prior among

all estimators that satisfy the constraint. In this sense, it is better than all classi-

cal plug-in estimators, including POET and the proposed Bayesian estimator. We

compare the proposed Bayesian method with the constrained Bayes using the tradi-

tional evaluation measures used in portfolio analysis and nd that they show similar

behavior. In addition to point estimation, the proposed Bayesian approach yields

a straightforward measure of uncertainty of the estimate and allows construction of

credible intervals for a wide range of parameters.