NAG Library Routine Document G13AEF

where rrs xt is the result of applying non-seasonal differencing of order d and seasonal differencing of seasonality s and order D to the series xt, as outlined in the description of G13AAF. The differenced series is then of length N 1⁄4 n d0, where d0 1⁄4 dþ D s ð Þ is the generalized order of differencing. The scalar c is the expected value of the differenced series, and the series w1; w2; . . . ; wN follows a zero-mean stationary autoregressive moving average (ARMA) model defined by a pair of recurrence equations. These express wt in terms of an uncorrelated series at, via an intermediate series et. The first equation describes the seasonal structure: wt 1⁄4 1wt s þ 2wt 2 s þ þ Pwt P s þ et 1et s 2et 2 s Qet Q s.