Vector Stochastic Difference Equations#
Let \(x_t\) be an \((n \times 1)\)-vector wide-sense stationary stochastic process that is governed by the matrix difference equation
where \(\epsilon_t\) is now an \(n \times 1\) vector of white noises with means of zero and contemporaneous covariance matrix \(E \epsilon_t \epsilon_t' = V\), an \(n \times n\) matrix. We assume \(E \epsilon_t \epsilon_{t-s}' = 0_{n \times n}\) for all \(s \neq 0\). In (292), \(C(L)\) is an \(n \times n\) matrix of (finite order) polynomials in the lag operator \(L\):
where each \(C_{i j}(L)\) is a finite order polynomial in the lag operator.
We assume that the matrix \(C(L)\) has an inverse under convolution \(C(L)^{-1} \equiv B(L)\); \(C(L)^{-1}\) is defined as the matrix that satisfies
where \(I_{n \times n}\) is the \(n \times n\) identity matrix. If it exists, \(C(L)^{-1}\) can be found as follows. Evaluate the matrix \(z\) transform \(C(z)\) at \(z = e^{-i\omega}\) to get \(C(e^{-i\omega})\). Then invert \(C(e^{-i\omega})\), frequency by frequency, to get \(C(e^{-i\omega})^{-1}\). Finally, the matrix coefficients \(C(L)^{-1} = B(L) = \sum_{j=0}^\infty B_j L^j\), \(B_j\) being an \(n \times n\) matrix, can be found from the inversion formula
where by integrating a matrix we mean to denote element-by-element integration.
A solution of (292) is found by premultiplying by \(B(L)\) to obtain
The vector stochastic difference equation \(C(L)x_t = \epsilon_t\) is said to be an autoregressive representation for the vector process \(x_t\). The solution \(x_t = B(L)\epsilon_t\) is said to be a vector moving average representation for the process \(x_t\). The cross-spectral density matrix of the \(n\times 1\) \(x_t\) process (which has the cross spectrum between the \(i\)th and \(j\)th components of \(x\) in the \((i,j)\)th position) is given by
where the prime denotes transposition. Formula (294) is analogous to the univariate equation (159), and can be derived by comparable methods.
Equation (294) is a very compact formula for calculating the cross spectra of the \(n\times 1\) \(x_t\) process as a function of the fundamental parameters, the covariance matrix \(V\) and the coefficients in \(C(L)\) (or \(B(L)\)). Equation (292) is quite a general representation and is flexible enough to incorporate exogenous variables and serially correlated noises.
In Equation (292) a variable \(x_{it}\) is said to be exogenous if \(C_{ij}(L) = 0\) for all \(j\) not equal to \(i\). This means that the row of Equation (292) corresponding to \(x_{it}\) becomes \(C_{ii}(L)x_{it}\) so that \(x_{it}\) is governed by only its own past interacting with the random shock \(\epsilon_{it}\). In this sense the evolution of \(x_{it}\) is not affected by interactions with other variables in \(x_t\). This is not to say however that \(x_{it}\) is uncorrelated with other components of \(x_t\), since \(\epsilon_{it}\) can be correlated contemporaneously with other \(\epsilon\)’s (i.e., \(V\) need not be diagonal). The definition of exogeneity given here turns out to be precisely the one used by econometricians in a time series context (see The Relationship Between Wiener-Granger Causality and Econometric Exogeneity).
Serially correlated errors can be incorporated by suitably redefining the errors as components of \(x_t\) and then modeling them as exogenous processes that affect but are not affected by other components of \(x_t\).