Weighted Least Squares Approximate Restricted Likelihood Estimation for Vector Autoregressive Processes

We derive a weighted least squares approximate restricted likelihood estimator for a kdimensional pth order autoregressive model with intercept, for which exact likelihood optimization is generally infeasible due to the parameter space which is complicated and highdimensional, 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 non-stationary processes. Furthermore, at the unit root, the limiting distribution of the weighted least squares approximate restricted likelihood estimator is shown to be the zerointercept Dickey–Fuller distribution, unlike the ordinary least squares with intercept estimator which has a different distribution with significantly higher bias.