Approximation to the limiting distribution of t- and F-statistics in testing for seasonal unit roots

Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215-2 38), Beaulieu and Miron (1993, Journal of Econometrics 55, 305-328), Ghysel s, Lee, and Noh ( 1994, Journal of Econometrics 62, 415- 442), Smith and Ta ylor (1998, Journal of Econometrics 85, 269-288; 1999, Journal of Time Seri es Analysis 20, 453-476; 1999, Discussion paper 99-15 in economics, Univers ity of Birmingham), and Taylor (1998, Journal of Time Series Analysis 19, 3 49-368) have developed a method of testing for seasonal unit roots of zero and nonzero frequencies. They propose to use t- and F-statistics as criteri a that are obtained from an auxiliary regression and find their limiting di stributions as the number of observations becomes large. Their limiting dis tributions are expressed by means of Brownian motions. In this paper the mo ment generating functions associated with the limiting distributions are de rived, and it is shown, as in Nabeya (2000, Econometric Theory 16, 200-230) , that the limiting distribution of t is well approximated by a distributio n given in Gram-Charlier series. The limiting distribution of F is also wel l approximated by another type of distribution.