## 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.