PERIODIC STATIONARITY CONDITIONS FOR PERIODIC AUTOREGRESSIVE MOVING AVERAGE PROCESSES AS EIGENVALUE PROBLEMS

The determination of periodic stationarity conditions for periodic aut oregressive moving average (PARMA) processes is a prerequisite to thei r analysis. Means of obtaining these conditions in analytically simple forms are sought. It is shown that periodic stationarity conditions f or univariate and multivariate PARMA processes can always be reduced t o eigenvalue problems, which are computationally and analytically easi er to deal with. Two different lumpings of the periodic process are co nsidered along this line. The first is the common w-span lumping over all w periods. The second lumping considered is the p-span lumping of the pth order periodic autoregressive process over p periods, which is based on a recently introduced lumping technique. It is shown that p- span lumping may yield the periodic stationarity conditions in an anal ytically simpler form as compared to w-span lumping when p < w.