Order Determination for Multivariate Autoregressive Processes Using Resampling Methods

LetX1, ..., Xnbe observations from a multivariate AR(p) model with unknown orderp. A resampling procedure is proposed for estimating the orderp. The classical criteria, such as AIC and BIC, estimate the orderpas the minimizer of the function[formula]wherenis the sample size,kis the order of the fitted model, [Sigma]2kis an estimate of the white noise covariance matrix, andCnis a sequence of specified constants (for AIC,Cn=2m2/n, for Hannan and Quinn's modification of BIC,Cn=2m2(ln ln n)/n, wheremis the dimension of the data vector). A resampling scheme is proposed to estimate an improved penalty factorCn. Conditional on the data, this procedure produces a consistent estimate ofp. 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.