COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1778 Bias in Estimating Multivariate and Univariate Diffusions Xiaohu Wang, Peter C. B. Phillips and Jun Wu January 2011 Multivariate continuous time models are now widely used in economics
and finance. 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. |
