Introduction to the Mathematical and Statistical Foundations of Econometrics
A.5. Proof of the Wold Decomposition
Let Xt be a zero-mean covariance stationary process and E[X2] = a2. Then the Xt’s are members of the Hilbert space U0 defined in Section 7.A.2. Let S—TO be the subspace spanned by Xt_j, j > 1, and let Xt be the projection of Xt on S——1,. Then Ut = Xt — Xt is orthogonal to all Xt _ j, j > 1, that is, E[UtXt —j] = 0 for j > 1. Because Ut —j є S—TO for j > 1, the Ut’s are also orthogonal to each other: E[UtUt—j] = 0 for j > 1.
Note that, in general, Xt takes the form Xt = fit, jXt—j, where the
coefficients et, j are such that WYt ||2 = E [Yt2] < to. However, because Xt is covariance stationary the coefficients fit, j do not depend on the time index t, for they are the solutions of the normal equations
TO
Y(m) = E[XtXt—m] = J2 вjE[Xt—jX— m]
j=1
TO
= Y! вj Y (j—m), m = 1, 2, 3,....
j=1
Thus, the projections Xt = J2TO=1 ejXt—j are covariance stationary and so are the Ut’s because
a2 = ||Xt ||2 = || Ut + Xt ||2 = || Ut ||2+||Xt ||2 + 2{Ut, Xt)
= II Ut ||2 +||Xt ||2 = E [Uj] + E[X?];
thus, E [U2] = a2 < a2.
Next, let Zt, m = Y."!= 1 ajUt-j, where aj = {Xt, Ut - j > = E[XU-j]. Then
 |
 |
 |
2
= E [X?] - ajE[XtUt-j]
j=1
m m [ ] m + E E ai ajE [U Uj ] = E [X2] - E a?2 > 0
i=1 j=1 j=1
for all m > 1; hence, 2^= a1? < to. The latter implies that J]°=m a1? — 0 for m —— to, and thus for fixed t, Zt, m is a Cauchy sequence in S--,, and Xt - Ztm is a Cauchy sequence in S-to. Consequently, Zt =Yj=1 ajUt-j є S--, and Wt = Xt - Ej=i aj Ut-j є S-to exist.
As to the latter, it follows easily from (7.8) that Wt є S-7 for every m; hence,
Wt є П S-to. (7.67)
-TO<t<TO
Consequently, E[Ut +m Wt] = 0 for all integers t and m. Moreover, it follows from (7.67) that the projection of Wt on any S1- is Wt itself; hence, Wt is perfectly predictable from any set {Xt - j, j > 1} of past values of Xt as well as from any set {Wt-j, j > 1} of past values of Wt.
