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The Calculation of Some Limiting Distributions Arising in Near‐Integrated Models with GLS Detrending
Author(s) -
Chambers Marcus J.
Publication year - 2015
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/jtsa.12123
Subject(s) - mathematics , unit root , statistic , asymptotic distribution , statistics , test statistic , moment (physics) , statistical hypothesis testing , physics , classical mechanics , estimator
Many unit root test statistics are based on detrended data, with the method of generalized least squares (GLS) detrending being popular in the setting of a near‐integrated model. This article 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. The exact moments and percentiles of some of these distributions are compared with those obtained by simulations, and it is found that, even with a large number of replications and a large sample size, the errors resulting from the simulation methods are not negligible. Some further applications, including a comparison of limiting power functions of different unit root test statistics and the consideration of a more complicated statistic, are also provided.