Assessing and improving the performance of nearly efficient unit root tests in small samples

The development of unit root tests continues unabated, with many recent contributions using techniques such as generalized least squares (GLS) detrending and recursive detrending to improve the power of the test. In this article, the relation between the seemingly disparate tests is demonstrated by algebraically nesting all of them as ratios of quadratic forms in normal variables. By doing so, and using the exact sampling distribution of the ratio, it is straightforward to compute, examine, and compare the test’ critical values and power functions. It is shown that use of GLS detrending parameters other than those recommended in the literature can lead to substantial power improvements. The open and important question regarding the nature of the first observation is addressed. Tests with high power are proposed irrespective of the distribution of the initial observation, which should be of great use in practical applications.