Weighted least squares approximate restricted likelihood estimation for vector autoregressive processes

We derive a weighted least squares approximate restricted likelihood estimator for a k-dimensional pth-order autoregressive model with intercept. Exact likelihood optimization of this model is generally infeasible due to the parameter space, which is complicated and high-dimensional, involving pk2 parameters. The weighted least squares estimator has significantly reduced bias and mean squared error than the ordinary least squares estimator for both stationary and nonstationary processes. Furthermore, at the unit root, the limiting distribution of the weighted least squares approximate restricted likelihood estimator is shown to be the zero-intercept Dickey–Fuller distribution, unlike the ordinary least squares with intercept estimator that has a different distribution with significantly higher bias.