Alex Trindade, Colorado State University

Burg-type Algorithms for Subset Multivariate Autoregressive Processes

We present an algorithm that extends Burg's original idea for autoregressive modeling of a time series, to modeling based on a subset of lags. The algorithm is derived in the general multivariate setting for an arbitrary set of lags, univariate results being obtained as special cases. The key step in the algorithm involves the minimization of a truncated scalar sum in the forward and backward prediction error residual vectors, with respect to the highest lag vector autoregressive (VAR) coefficient matrix. Using matrix calculus and appealing to some convexity results, we show that this scalar function of a matrix has a unique global minimum, and give its value in vec form. We present subset versions of the Durbin-Levinson-Whittle algorithm for solving the YuleWalker equations, the popular Vieira-Morf algorithm, and the less known Nuttall-Strand algorithm, via similar Burg-type recursions involving forward and backward prediction error residual vectors. These four VAR modeling methods are then seen to differ only in the way the reflection coefficients (the highest lag coefficient matrices) axe calculated. Although their finite sample performances are quite different, we show that the asymptotic distributions of the Yule-Walker and Burg subset VAR estimators are identical (and equal to that of LeastSquares). Since these four algorithms axe frequently used as quick and easy VAR estimation methods, we present a simulation study of the relative performance of each one in terms of the size of its Gaussian likelihood. We find that when the roots of the AR polynomial are fax from the unit circle, Yule-Walker gives higher likelihoods about 80% of the time. With all roots of the AR polynomial on the real axis and close to unity, Burg, and especially Vieira-Morf, perform better. Between these two extremes we see a gradual shift in dominance, depending on the magnitude of the imaginary component, and proximity to the unit circle. On the whole, Burg and Vieira-Morf perform uniformly better than the others for all configurations of roots, since they average higher likelihoods with smaller variability across a large number of realizations.