A sufficient condition for the invertibility of univariate periodic movingaverage models has been given by Cipra and Ghysels and Hall. We show that this condition is not a necessary one, and provide a necessary and sufficient condition for the general rn-variate, d-periodical moving-average MA(q) case.Much attention has been given recently to periodical autoregressive movingaverage (ARMA) processes, mainly because of their applications as an alternative to traditional seasonal models, in econometrics (see, for example, Lutkepohl, 1991a,b, or Ghysels andHall, 1992a) as well as in hydrology and environmental studies (McLeod, 1993).Though the identification and estimation of periodic ARMA models has been studied in some detail (see, for example, Vecchia, 1985;Anderson and Vecchia, 1993), the available results remain somewhat incomplete, in the moving-average (MA) case, due to the lack of an appropriate characterization of invertibility. Without such a property, periodic ARMA processes, just as their non-periodic counterparts, remain unidentifiable. Moreover, non-invertible MA models, whether periodical or not, produce inefficient forecasts; see Hallin (1986).Let L denote, as usual, the lag operator, and consider the rn-dimensional linear difference operator of order q , @,(L) = 8 , ; o + 8,;IL + * * . + 8,;,L4 t E Z(1) where eti0, . . ., @, ; , are m x m matrices such that \8,;ol # 0, t E Z. If, for some integer d > 1,( 2 ) the operator Q),(L) is called periodical. If, moreover, d is the smallest integer such that (2) holds, d is called the period, and we say that 8 , ( L ) is d-periodical.Denote by { E , , t E Z} a second-order, m-dimensional white noise process: E(E,E;) = 6(s; t ) Z 0143-9782/94/03 263-268 JOURNAL OF TIME SERIES ANALYSIS Vol. 15, No. 3 0