Order determination for multivariate autoregressive processes using resampling methods

Let X(1),..., X(n) be observations from a multivariate AR(p) model with unknown order p. A resampling procedure is proposed for estimating the order p. The classical criteria, such as AIC and BIC, estimate the order p as the minimizer of the function delta(k) = ln (\<(Sigma)over cap>(k)\) + C(n)k where n is the sample size, k is the order of the fitted model, <(Sigma)over cap>(2)(k) is an estimate of the white noise covariance matrix, and C-n is a sequence of specified constants (for AIC, C-n = 2m(2)/n, for Hannan and Quinn's modification of BIC, C-n = 2m(2)(ln ln n)/n, where m is the dimension of the data vector). A resampling scheme is proposed to estimate an improved penalty factor C-n. Conditional on the data, this procedure produces a consistent estimate of p. Simulation results support the effectiveness of this procedure when compared with some of the traditional order selection criteria. Comments are also made on the use of Yule-Walker as opposed to conditional least squares estimations for order selection. (C) 1996 Academic Press, Inc.