Bias in Estimating Multivariate and Univariate Di¤usions

Multivariate continuous time models are now widely used in economics and …nance. Empirical applications typically rely on some process of discretization so that the system may be estimated with discrete data. This paper introduces a framework for discretizing linear multivariate continuous time systems that includes the commonly used Euler and trapezoidal approximations as special cases and leads to a general class of estimators for the mean reversion matrix. Asymptotic distributions and bias formulae are obtained for estimates of the mean reversion parameter. Explicit expressions are given for the discretization bias and its relationship to estimation bias in both multivariate and in univariate settings. In the univariate context, we compare the performance of the two approximation methods relative to exact maximum likelihood (ML) in terms of bias and variance for the Vasicek process. The bias and the variance of the Euler method are found to be smaller than the trapezoidal method, which are in turn smaller than those of exact ML. Simulations suggest that when the mean reversion is slow the approximation methods work better than ML, the bias formulae are accurate, and for scalar models the estimates obtained from the two approximate methods have smaller bias and variance than exact ML. For the square root process, the Euler method outperforms the Nowman method in terms of both bias and variance. Simulation evidence indicates that the Euler method has smaller bias and variance than exact ML, Nowman’s method and the Milstein method. Keywords: Bias; Di¤usion, Euler approximation; Trapezoidal approximation; Milstein approximation JEL classi…cation: C15, G12 1 Introduction Continuous time models, which are speci…ed in terms of stochastic di¤erential equations, have found wide applications in economics and …nance. Empirical interest in systems of this type has grown particularly rapidly in recent years with the availability of high frequency Xiaohu Wang and Jun Yu (author for correspondence), School of Economics and Sim Kee Boon Institute for Financial Economics, Singapore Management University, 90 Stamford Road, Singapore 178903. Email: [email protected] and [email protected]. Peter Phillips, Cowles Foundation for Research in Economics, Yale University, Box 208281, Yale Station, New Haven, Connecticut 06520-8281. Email: [email protected].