An extension of Gaussian reduced rank estimation of Ahn and Reinsel (Journal of Econometrics, Vol. 62, pp. 317-350, 1994) to seasonal periods other than four is presented. Simple adjustments for estimation that are necessary because of complex-valued seasonal unit roots are presented in detail and the asymptotic distribution of the estimators that takes the same form as that in Ahn and Reinsel (1994) is derived. Tests for contemporaneous cointegration and common polynomial cointegrating vectors (PCIVs) for different seasonal unit roots are presented. Finite sample properties are briefly examined through a small Monte Carlo simulation study and a numerical example is presented to illustrate the methods.
The maximum eigenvalue (ME) test for seasonal cointegrating ranks is presented using the approach of Cubadda [Oxford ]. The asymptotic distributions of the ME test statistics are obtained for several cases that depend on the nature of deterministic terms. Monte Carlo experiments are conducted to evaluate the relative performances of the proposed ME test and the trace test, and we illustrate these tests using a monthly time series.
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