The ARMACOV subroutine computes an autocovariance sequence for an autoregressive moving average (ARMA) model. The input arguments to the ARMACOV subroutine are as follows:

phi

refers to a matrix that contains the autoregressive parameters. The first element is assumed to have the value 1.

theta

refers to a matrix that contains the moving average parameters. The first element is assumed to have the value 1.

num

refers to a scalar that contains , the number of autocovariances to be computed, which must be a positive number.

The ARMACOV subroutine returns the following values:

auto

specifies a variable to contain the returned matrix that contains the autocovariances of the specified ARMA model, assuming unit variance for the innovation sequence.

cross

specifies a variable to contain the returned matrix that contains the covariances of the moving-average term with lagged values of the process.

convol

specifies a variable to contain the returned matrix that contains the autocovariance sequence of the moving-average term.

The ARMACOV subroutine computes the autocovariance sequence that corresponds to a given autoregressive moving-average (ARMA) time series model. An arbitrary number of terms in the sequence can be requested. Two related covariance sequences are also returned.

The model notation for the ARMACOV and ARMALIK subroutines is the same. The ARMA model is denoted

with . The notation is the same as that of Box and Jenkins (1976) except that the model parameters are opposite in sign. The innovations satisfy and if = 0, and are zero otherwise. The formula for the th element of the convol argument is

for . The formula for the th element of the cross argument is

for , where is the th impulse response value. The sequence, if desired, can be computed with the RATIO function. It can be shown that is the same as , which is used by Box and Jenkins (1976) in their formulation of the autocovariances. The th autocovariance, denoted and returned as the element of the auto argument , is defined implicitly for by

where is the th element of the cross argument. See Box and Jenkins (1976) or McLeod (1975) for more information.

Consider the model

To compute the autocovariance function at lags zero through four for this model, use the following statements:

/* an ARMA(1,1) model */
phi   = {1 -0.5};
theta = {1 0.8};
call armacov(auto, cross, convol, phi, theta, 5);
print auto, cross convol;

The result is show in Figure 23.41.