The Cross Covariogram

The Cross Covariogram#

Suppose we have two wide-sense stationary stochastic processes \(y_t\) and \(x_t\). The processes are said to be jointly wide-sense stationary if the cross covariance \(E(y_t - Ey_t)(x_{t-k} - Ex_{t-k})\) depends only on \(k\) and not on \(t\). The cross covariogram is this list of covariances as a function of \(k\):

\[ c_{yx}(k) = E(y_t - Ey_t)(x_{t-k} - Ex_{t-k}). \]

Now suppose \(y_t\) and \(x_t\) can be expressed as (possibly two-sided) distributed lags of a single white-noise process \(\epsilon_t\):

\[ y_t = B(L)\epsilon_t, \qquad x_t = D(L)\epsilon_t, \]

where

\[ B(L) = \sum_{j=-\infty}^{\infty} b_j L^j, \quad D(L) = \sum_{j=-\infty}^{\infty} d_j L^j, \quad \sum_{j=-\infty}^{\infty} b_j^2 < \infty, \quad \sum_{j=-\infty}^{\infty} d_j^2 < \infty. \]

Since \(E\epsilon_t = 0\),

\[ c_{yx}(k) = Ey_t x_{t-k} = \sigma_\epsilon^2 \sum_{j=-\infty}^{\infty} b_j d_{j-k}. \]

The cross-covariance generating function \(g_{yx}(z)\) is defined by

\[ g_{yx}(z) = \sum_{k=-\infty}^{\infty} c_{yx}(k)\, z^k. \]

In the present case, we have

\[ g_{yx}(z) = \sigma_\epsilon^2 \sum_{k=-\infty}^{\infty} \sum_{j=-\infty}^{\infty} b_j d_{j-k}\, z^k. \]

Letting \(h = j-k\) so that \(k = j-h\), we have

\[ g_{yx}(z) = \sigma_\epsilon^2 \sum_{j=-\infty}^{\infty}\sum_{h=-\infty}^{\infty} b_j d_h\, z^{j-h} = \sigma_\epsilon^2 \sum_{j=-\infty}^{\infty} b_j z^j \sum_{h=-\infty}^{\infty} d_h z^{-h}. \]

The last equation gives

(170)#\[g_{yx}(z) = \sigma_\epsilon^2\, B(z)\, D(z^{-1}).\]

This is a counterpart to (159) and includes it as a special case.

A More General Bivariate System#

Suppose instead we have the system

(171)#\[y_t = A(L)\epsilon_t + B(L)u_t, \qquad x_t = C(L)\epsilon_t + D(L)u_t,\]

where \(\epsilon_t\) and \(u_t\) are two mutually uncorrelated white-noise processes with variances \(\sigma_\epsilon^2\) and \(\sigma_u^2\), and \(Eu_t\epsilon_{t-k} = 0\) for all \(k\). By calculations analogous to those above, the cross-covariance generating function is

(172)#\[g_{yx}(z) = \sigma_\epsilon^2\, A(z)\, C(z^{-1}) + \sigma_u^2\, B(z)\, D(z^{-1}).\]

The representation (171) is in fact very general—it includes all jointly wide-sense stationary, indeterministic bivariate processes.[1]

Symmetry Relations#

We also define the cross-covariance going the other way:

\[ c_{xy}(k) = E(x_t - Ex_t)(y_{t-k} - Ey_{t-k}) = c_{yx}(-k). \]

Correspondingly, we have

\[ g_{yx}(z) = \sum_{k=-\infty}^{\infty} c_{yx}(k)\, z^k = \sum_{h=-\infty}^{\infty} c_{xy}(h)\, z^{-h}, \]

where the second equality follows by substituting \(k = -h\) and using \(c_{xy}(h) = c_{yx}(-h)\).

The particular system (171) implies the symmetric counterpart:

\[ g_{xy}(z) = \sigma_\epsilon^2\, A(z^{-1})\, C(z) + \sigma_u^2\, B(z^{-1})\, D(z). \]