Testing linear regression model with AR(1) errors against a first-order dynamic linear regression model with white noise errors: A point optimal testing approach

Abstract

We know very little about the performance of point optimal (PO) and approximate point optimal (APO) tests in the presence of unavoidable nuisance parameters. Because marginal likelihood based tests are said to perform well in the presence of unavoidable nuisance parameters, this paper compares the performance of marginal likelihood based APO tests and classical tests using a testing problem which has been largely overlooked by econometric practitioners, namely testing for a static linear regression model with AR(1) errors against a dynamic linear regression model with white noise errors. It is well known that the classical tests are specifically designed for nested testing, they are applied to test for the significance of the dynamic coefficient of a dynamic linear regression model with AR(1) errors.A testing procedure is proposed, where the size and power comparisons used are based on near-exact non-similar critical values of tests obtained using the simulated annealing (SA) algorithm, as the near-exact non-similar critical values control the sizes of the tests well overall.Among marginal likelihood based classical tests, the likelihood ratio (LR) test and Lagrange multiplier (LM) test seem to perform well under the alternative hypothesis, particularly when the dynamic parameter is large and the sample size is reasonably big. The Wald (W) test is the worst performer overall. This concurs with previous observations that the W test performs poorly in small samples. Compared to the classical approach, APO tests appear to have good power properties, particularly in the neighborhood of the chosen parameter point under the alternative hypothesis. This finding may advance the use of PO and APO tests.