The VARMALIK subroutine computes the log-likelihood function for a VARMA() model.

The input arguments to the VARMALIK subroutine are as follows:

series

specifies an matrix that contains the vector time series (assuming mean zero), where is the number of observations and is the number of variables.

phi

specifies a matrix that contains the autoregressive coefficient matrices, where is the number of the elements in the subset of the AR order. You must specify either phi or theta.

theta

specifies a matrix that contains the moving average coefficient matrices, where is the number of the elements in the subset of the MA order. You must specify either phi or theta.

sigma

specifies a covariance matrix of the innovation series. If you do not specify sigma, an identity matrix is used.

p

specifies the subset of the AR order. See the VARMACOV subroutine.

q

specifies the subset of the MA order. See the VARMACOV subroutine.

opt

specifies the method of computing the log-likelihood function:

opt=0

requests the multivariate innovations algorithm. This algorithm requires that the time series is stationary and does not contain missing observations.

opt=1

requests the conditional log-likelihood function. This algorithm requires that the number of the observations in the time series must be greater than pq and that the series does not contain missing observations.

opt=2

requests the Kalman filtering algorithm. This is the default and is used if the required conditions in opt=0 and opt=1 are not satisfied.

The VARMALIK subroutine returns the following value:

lnl

is a matrix that contains the log-likelihood function, the sum of log determinant of the innovation variance, and the weighted sum of squares of residuals. The log-likelihood function is computed as (the sum of last two terms).

The options opt=0 and opt=2 are equivalent for stationary time series without missing values. Setting opt=0 is useful for a small number of the observations and a high order of p and q; opt=1 is useful for a high order of P and q; opt=2 is useful for a low order of p and q, or for missing values in the observations.

Consider the following bivariate () VARMA(1,1) model:

To compute the log-likelihood function of this model, you can use the following statements:

  phi  = { 1.2 -0.5, 0.6 0.3 };
  theta= {-0.6  0.3, 0.3 0.6 };
  sigma= { 1.0  0.5, 0.5 1.25};
  call varmasim(yt, phi, theta) sigma=sigma;
  call varmalik(lnl, yt, phi, theta, sigma);