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, Σ2kis an estimate of the white noise covariance matrix, andCnis a sequence of specified constants (for AIC,Cn=2m2/n, for Hannan and Quinnʹs 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.