The Effects of Filtering on One-Sided and Two-Sided Projections

Consider a jointly covariance stationary, linearly indeterministic process \((y_t,x_t)\). Consider the projections

(335)\[y_t = \sum_{j=-\infty}^{\infty} b_j x_{t-j} + \epsilon_t, \qquad E\epsilon_t x_{t-j} = 0 \quad \text{for all integer } j\]

(336)\[y_t = \sum_{j=0}^{\infty} h_j x_{t-j} + v_t, \qquad Ev_t x_{t-j} = 0 \quad \text{for all integer } j \geq 0\]

Assume that \(x_t\) has Wold representation \(x_t=d(L)\eta_t\), where \(E\eta_t^2 = 1\). We have seen that the generating functions \(b(L)\) and \(h(L)\) are given by the formulas

(337)\[b(z) = \frac{g_{yx}(z)}{d(z^{-1})d(z)}\]

(338)\[h(z) = \left[\frac{g_{yx}(z)}{d(z^{-1})}\right]_+\frac{1}{d(z)}\]

where \(x_t=d(L)\eta_t\) is a Wold representation for \(x_t\). Now consider the filtered series

\[ y_t^f=f(L)y_t, \qquad x_t^f=f(L)x_t \]

where

\[ f(L) = \sum_{j=-\infty}^\infty f_jL^j, \qquad \sum_{j=-\infty}^\infty f_j^2<+\infty. \]

We have that the covariance generating functions \(g_{y^fx^f}(z),g_{x^f}(z),g_{y^f}(z)\) are given by

(339)\[\begin{split}\begin{aligned} g_{y^f}(z) &= f(z)f(z^{-1})g_{y}(z) \\ g_{x^f}(z) &= f(z)f(z^{-1})g_{x}(z) \\ g_{y^fx^f}(z) &= f(z)f(z^{-1})g_{yx}(z) \end{aligned}\end{split}\]

Using these formulas, we consider the projections analogous to (335) and (336) with the filtered series, \(y_t^f,x_t^f\):

(340)\[y_t^f = \sum_{j=-\infty}^{\infty} b^f_j x^f_{t-j} + \tilde{\epsilon}_t, \qquad E\tilde{\epsilon}_t x^f_{t-j} = 0 \quad \text{for all integer } j\]

(341)\[y_t^f = \sum_{j=0}^{\infty} h^f_j x^f_{t-j} + \tilde{v}_t, \qquad E\tilde{v}_t x^f_{t-j} = 0 \quad \text{for all integer } j \geq 0\]

Let \(b^f(L) =\sum_{j=-\infty}^{\infty}b^f_jL^j\), \(h^f(L) =\sum_{j=0}^{\infty}h^f_jL^j\). To implement the filtering formula (338), we require a Wold representation for the filtered series \(x_t^f\). We can obtain such a representation

\[ x_t^f=c(L)a_t \]

where \(c(z)\) has no zeros inside the unit circle and \(Ea_t^2=1\); \(c(z)\) satisfies

(342)\[c(z)c(z^{-1})=g_{x^f}(z) = f(z)f(z^{-1})d(z)d(z^{-1}).\]

However, in general, \(f(z)d(z) \neq c(z)\). Now applying the two-sided filtering formula (337) gives

(343)\[b^f(z)=\frac{f(z)f(z^{-1})g_{yx}(z)}{f(z)f(z^{-1})g_{x}(z)}\]

or

\[ b^f(z)=\frac{g_{yx}(z)}{d(z)d(z^{-1})} = b(z) \]

Applying the one-sided filtering formula (338) gives

(344)\[h^f(z)=\left[\frac{f(z)f(z^{-1})g_{yx}(z)}{c(z^{-1})}\right]_+\frac{1}{c(z)}.\]

Comparison of (337) and (343) shows that filtering both series by a common filter leaves unaltered the two-sided infinite projection of \(y\) on \(x\). Comparison of (338) and (344) shows that in general, filtering \(y\) and \(x\) with a common filter alters the projection of \(y\) on the one-sided infinite past of \(x\). However, there is an important special case in which the one-sided projection is unaltered by filtering both \(y\) and \(x\). This special case occurs when the one-sided projection equals the two-sided projection, which we know to be unaltered by filtering. Sims's theorem establishes that this case obtains when the \(y\) process fails to Granger cause \(x\). We thus have established the following proposition:

If \(y\) fails to Granger cause \(x\), filtering fails to alter the projection of \(y\) on \(x\).

A converse of this proposition also holds:

Consider the one-sided projection of \(y_t\) on \([x_t,x_{t-1},\ldots]\),

(345)\[y_t=\sum_{j=0}^\infty h_jx_{t-j} + v_t = h(L)x_t + v_t\]

where \(Ev_tx_{t-j}=0\) for \(j\geq0\). Let \(F\) be the space of one-sided, square summable sequences with elements \(f=\{f_j\}_{j=0}^\infty\), \(\sum_{j=0}^\infty f_j^2 < +\infty\). Assume that for every \(f \in F\), the one-sided projection of \(f(L)y_t\) on \([f(L)x_t,f(L)x_{t-1},\ldots]\) can be represented as

(346)\[f(L)y_t=h(L)[f(L)x_t] + f(L)v_t\]

where \(E[f(L)v_t]\cdot[f(L)x_{t-k}] = 0\) for \(k\geq0\), and where \(h(L)\) is the same operator \(h(L)\) that appears in (345). Then \(\{y_t\}\) fails to Granger-cause \(\{x_t\}\).

To prove the proposition, we begin by evaluating \(E[f(L)v_t]\cdot[f(L)x_{t-k}]\), which is assumed to be zero for all \(k\geq0\) and all \(f\in F\). For \(k=0\), we obtain

\[ E(f_0v_t + f_1v_{t-1} + f_2v_{t-2} + \cdots)(f_0x_t + f_1x_{t-1}+ \cdots) = \sum_{h=0}^\infty f_h \sum_{k=h+1}^\infty f_kEx_{t-h}v_{t-k}. \]

Evidently, unless \(Ex_t v_{t-k}=0\) for all \(k\) (that is, \(k>0\) as well as \(k\leq 0\)), \(f\) can be chosen to make the above expression unequal to zero. Therefore, the condition that the one-sided projection of \(y_t^f \equiv f(L)y_t\) on \(x_t^f \equiv f(L)x_t\) have the same generating function \(h(L)\) for all \(f \in F\) implies that \(v_t\) is orthogonal to all future \(x_t\)’s \(\{x_{t+k}, k\geq 0\}\), as well as current and past \(x_t\)’s. Therefore, the one-sided projection of \(\{y_t\}\) on current and past \(x_t\) equals the two-sided projection of \(\{y_t\}\) on past, present, and future \(\{x_t\}\). It follows from Sims’s theorem 2 (The Relationship Between Wiener-Granger Causality and Econometric Exogeneity) that \(\{y_t\}\) fails to Granger-cause \(\{x_t\}\).

These propositions provide the foundation of the result in econometrics that generalized least squares estimates of projection equations like (336) are consistent if and only if \(x_t\) fails to be Granger-caused by \(y_t\). Under standard regularity conditions, generalized least squares estimates of \(h(z)\) converge to \(h^f(z)\) given by (344), where \(f(z)\) is a filter that is one-sided in nonnegative powers of \(z\) and that whitens \(v_t\).