Stationarity and Invertibility

For stationarity and invertibility of the VARMA process, the roots of and are outside the unit circle.

Parameter Estimation

Under the assumption of normality of the with mean vector zero and nonsingular covariance matrix , consider the conditional (approximate) log-likelihood function of a VARMA(,) model with mean zero.

Define and with and ; define and . Then

where and .

Then, the conditional (approximate) log-likelihood function can be written as follows (Reinsel 1997):

where , and is such that .

For the exact log-likelihood function of a VARMA(,) model, the Kalman filtering method is used transforming the VARMA process into the state-space form (Reinsel 1997).

The state-space form of the VARMA(,) model consists of a state equation

and an observation equation

where for

and

The Kalman filtering approach is used for evaluation of the likelihood function. The updating equation is

with

and the prediction equation is

with for .

The log-likelihood function can be expressed as

where and are determined recursively from the Kalman filter procedure. To construct the likelihood function from Kalman filtering, you obtain , , and .

Define the vector

where and .

The log-likelihood equations are solved by iterative numerical procedures such as the quasi-Newton optimization. The starting values for the AR and MA parameters are obtained from the least squares estimates.

Asymptotic Distribution of the Parameter Estimates

Under the assumptions of stationarity and invertibility for the VARMA model and the assumption that is a white noise process, is a consistent estimator for and converges in distribution to the multivariate normal as , where is the asymptotic information matrix of .

Asymptotic Distributions of Impulse Response Functions

Defining the vector

the asymptotic distribution of the impulse response function for a VARMA() model is

where is the covariance matrix of the parameter estimates and

where is a matrix with the second following after block matrices; is a matrix; is a matrix,

where

    is a zero matrix, and

An Example of a VARMA(1,1) Model

Consider a VARMA(1,1) model with mean zero

where is the white noise process with a mean zero vector and the positive-definite covariance matrix .

The following IML procedure statements simulate a bivariate vector time series from this model to provide test data for the VARMAX procedure:

   proc iml;
      sig = {1.0  0.5, 0.5 1.25};
      phi = {1.2 -0.5, 0.6 0.3};
      theta = {0.5 -0.2, 0.1 0.3};
      /* to simulate the vector time series */
      call varmasim(y,phi,theta) sigma=sig n=100 seed=34657;
      cn = {'y1' 'y2'};
      create simul3 from y[colname=cn];
      append from y;
   run;

The following statements fit a VARMA(1,1) model to the simulated data. You specify the order of the autoregressive model with the P= option and the order of moving-average model with the Q= option. You specify the quasi-Newton optimization in the NLOPTIONS statement as an optimization method.

   proc varmax data=simul3;
      nloptions tech=qn;
      model y1 y2 / p=1 q=1 noint print=(estimates);
   run;

Figure 30.46 shows the initial values of parameters. The initial values were estimated using the least squares method.