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Bias in Estimating Multivariate and Univariate Diffusions

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Abstract

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.

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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1778.

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Length: 37 pages
Date of creation: Jan 2011
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Publication status: Published in Journal of Econometrics (2011), 161(2): 228-245
Handle: RePEc:cwl:cwldpp:1778

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Bias; Diffusion; Euler approximation; Trapezoidal approximation; Milstein approximation;

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Cited by:
  1. Peter C.B. Phillips, 2013. "Unit Roots in Life -- A Graduate Student Story," Cowles Foundation Discussion Papers 1913, Cowles Foundation for Research in Economics, Yale University.

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