Corrections for Trading Frictions in Multivariate Returns

Abstract

When observed stock returns are obtained from trades subject to friction, it is known that an individual stock's beta and covariance are measured with error. Univariate models of additive error adjustment are available and are often applied simultaneously to more than one stock. Unfortunately, these multivariate adjustments produce nonpositive definite covariance and correlation matrices, unless the return sample sizes are very large. To prevent this, restrictions on the adjustment matrix are developed and a correction is proposed, which dominates the uncorrected estimator. The estimators are illustrated with asset opportunity set estimates where daily returns have trading frictions. A NUMBER OF UNIVARIATE nonsynchronous trading adjustments are available for beta, but no multivariate adjustment has been specifically designed for a general covariance matrix. For example, the univariate trading friction adjustment of Cohen, Hawawini, Maier, Schwartz, and Whitcomb (CHMSW) (1983) is a generalization of Scholes and Williams (1977). The CHMSW procedure adjusts beta to account for price-adjustment delays exceeding one period, where a portion of the true return is reflected in lagged returns. The procedure is extended and applied in Shanken (1987) to estimation of the daily multivariate covariance matrix of stocks' returns. Observed covariances significantly understated true covariances by a factor of two, suggesting that a daily covariance adjustment is necessary. Because of a large number of studies which employ multivariate daily stock returns, it is therefore important that the adjustment results in a good estimator of the covariance matrix. While no evidence is available regarding other properties, it is shown in CHMSW that their univariate adjusted estimator has the desirable statistical properties of consistency and unbiasedness. Unfortunately, when the adjusted covariance estimator is applied in finite samples, the resulting adjusted covariance matrix may not be a covariance matrix and the implied adjusted correlation matrix, independently, may not be a correlation matrix because the adjustment does not restrict the covariance and correlation matrices to be positive definite.1 When these matrices are not positive definite, functions of these matrices can produce very unreliable inference results. This paper provides empirical examples of the violations of the positive definite