THE FINITE-SAMPLE EFFECTS OF VAR DIMENSIONS ON MLE BIAS, MLE VARIANCE AND MINIMUM MSE ESTIMATORS: PURELY NONSTATIONARY CASE

Steve LAWFORD (ECARES, Université Libre de Bruxelles)
Michalis P. STAMATOGIANNIS (University of York)

Vector autoregressions (VAR's) continue to be an important tool in time series analysis. However, little is currently known about the finite-sample behaviour of parameter estimators, and especially in nonstationary frameworks - important progress on some problems has been made only very recently ([1]). In particular, the form and extent of small sample bias have not been fully investigated. We address this issue here, by investigating maximum likelihood estimators (MLE's) in the context of a purely nonstationary first-order VAR. Using Monte Carlo and numerical optimization techniques, we perform a comprehensive simulation study and derive a series of parsimonious response surfaces for MLE bias, in terms of sample size and VAR dimension, given correct and over-parameterization of the model; we substantially improve upon one of the main results of [1]. We motivate the response surface technique as a means of improving approximate theoretical results in frameworks where derivation of exact results (although desirable) would be very difficult, or intractable, as is frequently the case in even the simplest settings. We also extend univariate results by Abadir and Hadri ([2],[3]) to the multivariate framework. We study non-zero initial values, and conjecture that the curious univariate bias nonmonotonicity noticed by [2] disappears in the multivariate case. Lastly, we examine MLE variance and the correction factors required for the MLE to attain minimum mean squared error (MSE).