ABSTRACT
There is an intuitive equivalence between multivariate stationarity and periodic correlation. First, suppose that { X 11 } is PC with period T and consider the T variate series {Y 11 } obtained by setting Y1111 = X( 11 _ 1 l T+,, for 1 :::; v:::; T ( Yii denotes the jth component of Yi). It follows from Eq. (2.1) that {Y 11 } is T-variate stationary. Conversely, for a T-variate stationary series {Y 11 }, define for each n the "year" m and "season" v via m = l (n - 1) / T J and v = n - m T; here lx J denotes the greatest integer less than or equal to x. Then a PC univariate series { X 11 } with period T is obtained by setting X11 = Y111 ,;·
Whereas the equivalence between periodic correlation and multivariate stationarity is a useful mathematical tool, analysis of PC series through multivariate time series techniques is seldom practical due to the large dimensionality frequently encountered in applications. For example, in a monthly series, T = 12 and a first order periodic autoregressive model with general monthly means has 36 parameters (see Sec. 4).
