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By Hay J.L., Pettitt A.N.

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X(n - m 1) have been removed by regression. Proof. We establish the formula by showing that + I ( t - R,~) = (1 - R;-~)(I - p2(m 1 , . . , m - 1)). In turn this follows from a classical formula of regression theory (Cramtr, 1946, p. 319) if we show that p(m 1, . ,m - 1) is the partial correlation as stated in Theorem 6. Let 2(n) be the regression of z ( n ) on z ( n - l), . . , z(n - m 1). > Let 2(n - m) be the regression of z ( n - m ) on the same variables. (n) on z ( n - m) - 2(n - m) to the mean square of z ( n ) - 2(n).

MuZtzple Time Series E. J. HANNAN Copyright Q 1970, by John Wiky 8 Sons Inc CHAPTER I1 The Spectral Theory of Vector Processes 1. INTRODUCTION This chapter is concerned with the Fourier analysis of time series. These Fourier methods are intimately linked with the notion of stationarity so that this kind of stochastic process will be of central interest, although departures from stationarity will also be considered. It is best, perhaps, to begin by explaining why Fourier methods play such an essential part in the theory.

I) Integral Operators We consider first the operation which replaces z(t) by y(t> =S_mmA(s) 4 t - s) ds, where A ( s ) is such that h ( l ) =b ( f ) e " " dt exists in the mean-square sense defined above. (I and M ( s ) is a matrix of functions of bounded variation in every finite interval and for which h(l)= ( m J-m e'"M(ds) exists in the sense of mean-square convergence with respect to F defined above. Theorem 6. The random process y ( t ) is well defined as a limit in mean square of integrals over finite intervals, as the intervals increase to (- m, a),when and only when h ( l ) exists as a limit in mean square (F).