Semiparametric tests for seasonal unit roots based on a semiparametric feasible GLSE

Semiparametric extensions of the seasonal unit root tests for the model of Dickey et al. (1984, J. Amer. Statist. Assoc. 79, 355) are proposed. Development of semiparametric extensions based on the ordinary least-squares estimator (OLSE) is impossible for the regression of Dickey et al. (1984) since the limiting null distribution of the OLSE-based test statistic is entangled with nuisance parameters under the usual normalization. This is in contrast with the successful development of the OLSE-based semiparametric unit root tests of Phillips (Econometrica 55 (1987) 277). To overcome the difficulty, we propose tests based on a feasible generalized least-squares estimator (GLSE), instead of the OLSE, and the spectral decomposition of the generalized sum of products of the regressor variables. The key advantage of the proposed method is that one can construct the feasible GLSE and hence tests of the seasonal unit root without specifying a parametric model for the error process.