SSM Procedure

Multivariate ARMA

You can specify a state vector that follows a multivariate autoregressive, moving average (VARMA) model by using the STATE statement option TYPE=VARMA. The autoregressive and moving average orders can be either 0 or 1 (0 less-than-or-equal-to p less-than-or-equal-to 1 and 0 less-than-or-equal-to q less-than-or-equal-to 1 )—that is, only VAR(1), MA(1), and VARMA(1,1) models can be specified. The notation and the state space form of the VARMA model described here is taken from Reinsel (1997), which is a good reference for VARMA modeling.

A dim-dimensional vector process zeta zeta Subscript t follows a zero-mean, autoregressive order p, moving average order q (VARMA(p, q)) model if it satisfies the following matrix difference equation:

zeta zeta Subscript t Baseline equals sigma-summation Underscript i equals 1 Overscript p Endscripts normal upper Phi normal upper Phi Subscript i Baseline zeta zeta Subscript t minus i Baseline plus epsilon epsilon Subscript t Baseline minus sigma-summation Underscript j equals 1 Overscript q Endscripts normal upper Theta normal upper Theta Subscript j Baseline epsilon epsilon Subscript t minus j

Here normal upper Phi normal upper Phi Subscript i and normal upper Theta normal upper Theta Subscript j are dim-dimensional square matrices and epsilon epsilon Subscript t is a dim-dimensional, Gaussian, white noise sequence with covariance matrix normal upper Sigma normal upper Sigma. If autoregressive order p is 0, the term that involves normal upper Phi normal upper Phi Subscript i is absent; similarly, if the moving average order q is 0, the term that involves normal upper Theta normal upper Theta Subscript j is absent. Since AR and MA orders can be at most 1, the subscripts of normal upper Phi normal upper Phi Subscript i and normal upper Theta normal upper Theta Subscript j can be ignored in this discussion—when applicable, an AR coefficient matrix is denoted by normal upper Phi normal upper Phi and an MA coefficient matrix is denoted by normal upper Theta normal upper Theta. The unknown elements of normal upper Phi normal upper Phi, normal upper Theta normal upper Theta, and normal upper Sigma normal upper Sigma constitute the parameter vector that is associated with a VARMA state. The process zeta zeta Subscript t defined by the VARMA difference equation is stationary and invertible (Reinsel 1997) if and only if the eigenvalues of normal upper Phi normal upper Phi and normal upper Theta normal upper Theta are strictly less than 1 in magnitude. By default, the SSM procedure imposes these stationarity and invertibility restrictions on normal upper Phi normal upper Phi and normal upper Theta normal upper Theta. However, you can specify normal upper Phi normal upper Phi to be an identity matrix, in which case the resulting process is nonstationary.

A VARMA model can be cast into a state space form. The state space form used by the SSM procedure is described in Reinsel (1997, pp. 52–53). The system matrices for the supported VARMA models are as follows:

  • The VAR(1) form is the simplest. In this case, the underlying state alpha alpha Subscript t is the same as the VAR(1) process zeta zeta Subscript t. Therefore, bold upper T equals normal upper Phi normal upper Phi and bold upper Q Subscript bold t Baseline equals normal upper Sigma normal upper Sigma.

  • Taking normal upper Phi normal upper Phi equal to the zero matrix if p equals 0, the VARMA(1,1) and MA(1) cases can be treated together. In this case, the underlying state alpha alpha Subscript t is 2*dim dimensional and the desired VARMA process zeta zeta Subscript t corresponds to its first dim elements. Let normal upper Psi normal upper Psi equals normal upper Phi normal upper Phi minus normal upper Theta normal upper Theta. Then, in the blocked form,

    bold upper T equals Start 2 By 2 Matrix 1st Row 1st Column bold 0 2nd Column bold upper I Subscript d i m Baseline 2nd Row 1st Column bold 0 2nd Column normal upper Phi normal upper Phi EndMatrix and bold upper Q Subscript t Baseline equals bold upper Q equals Start 2 By 2 Matrix 1st Row 1st Column normal upper Sigma normal upper Sigma 2nd Column normal upper Sigma normal upper Sigma normal upper Psi normal upper Psi Superscript Super Superscript prime Superscript Baseline 2nd Row 1st Column normal upper Psi normal upper Psi normal upper Sigma normal upper Sigma 2nd Column normal upper Psi normal upper Psi normal upper Sigma normal upper Sigma normal upper Psi normal upper Psi Superscript prime EndMatrix

Unless normal upper Phi normal upper Phi is restricted to be identity, the underlying state alpha alpha Subscript t is stationary and the covariance of the initial condition is computed by

v e c left-parenthesis bold upper Q 1 right-parenthesis equals left-parenthesis bold upper I minus bold upper T circled-times bold upper T right-parenthesis Superscript negative 1 Baseline v e c left-parenthesis bold upper Q right-parenthesis

where circled-times denotes the Kronecker product and the v e c operation on a matrix creates a vector formed by vertically stacking the rows of that matrix. If normal upper Phi normal upper Phi is restricted to be identity, the initial condition is fully diffuse.

Last updated: June 19, 2025