The Calculation of Some Limiting Distributions Arising in Near-Integrated Models with GLS Detrending

Many unit root test statistics are nowadays constructed using detrended data, with the method of GLS detrending being popular in the setting of a near-integrated model. This paper determines the properties of some associated limiting distributions when the GLS detrending is based on a linear time trend. A fundamental result for the moment generating function of two key functionals of the relevant stochastic process is provided and used to compute probability density functions and cumulative distribution functions, as well as means and variances, of the limiting distributions of some statistics of interest. Some further applications, including a comparison of limiting power functions and the consideration of a more complicated statistic, are also provided.