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7.5 Best Linear Predictors of Second-Order Random Vectors 245
immediately from the properties of the prediction operator (Section 2.5) that
Pn(Y) µ + A1(Xn − µn) +···+An(X1 − µ1) (7.5.2)
for some matrices A1,...,An, and that
Y − Pn(Y) ⊥ Xn+1−i, i 1,...,n, (7.5.3)
where we say that two m-dimensional random vectors X and Y are orthogonal (written
X ⊥ Y)ifE(XY ′ ) is a matrix of zeros. The vector of best predictors (7.5.1) is uniquely
determined by (7.5.2) and (7.5.3), although it is possible that there may be more than
one possible choice for A1,...,An.
As a special case of the above, if {Xt} is a zero-mean time series, the best linear
predictor ˆXn+1 of Xn+1 in terms of X1,...,Xn is obtained on replacing Y by Xn+1 in
(7.5.1). Thus
ˆXn+1
Hence, we can write
0, if n 0,
Pn(Xn+1), if n ≥ 1.
ˆXn+1 n1Xn +···+nnX1, n 1, 2,..., (7.5.4)
where, from (7.5.3), the coefficients nj ,j 1,...,n, are such that
E
i.e.,
ˆXn+1X ′
n+1−i
E Xn+1X ′
n+1−i , i 1,...,n, (7.5.5)
n
nj K(n + 1 − j,n + 1 − i) K(n + 1,n+ 1 − i), i 1,...,n.
j1
In the case where {Xt} is stationary with K(i,j) Ɣ(i − j), the prediction equations
simplify to the m-dimensional analogues of (2.5.7), i.e.,
n
nj Ɣ(i − j) Ɣ(i), i 1,...,n. (7.5.6)
j1
Provided that the covariance matrix of the nm components of X1,...,Xn is nonsingular
for every n ≥ 1, the coefficients {nj } can be determined recursively using
a multivariate version of the Durbin–Levinson algorithm given by Whittle (1963)
(for details see TSTM, Proposition 11.4.1). Whittle’s recursions also determine the
covariance matrices of the one-step prediction errors, namely, V0 Ɣ(0) and, for
n ≥ 1,
Vn E(Xn+1 − ˆXn+1)(Xn+1 − ˆXn+1) ′
Ɣ(0) − n1Ɣ(−1) −···−nnƔ(−n). (7.5.7)