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 p
q 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:
![\[ \mb {y}_ t = \Phi \mb {y}_{t-1} + \bm {\epsilon }_ t - \Theta \bm {\epsilon }_{t-1} \]](images/imlug_langref1585.png){kind=small}
![\[ \Phi =\left[\begin{matrix} 1.2 & -0.5 \cr 0.6 & 0.3 \cr \end{matrix}\right] ~ ~ \Theta =\left[\begin{matrix} -0.6 & 0.3 \cr 0.3 & 0.6 \cr \end{matrix}\right] ~ ~ \Sigma =\left[\begin{matrix} 1.0 & 0.5 \cr 0.5 & 1.25 \cr \end{matrix}\right] \]](images/imlug_langref1589.png){kind=small}
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 seed=123;
call varmalik(lnl, yt, phi, theta, sigma);
labl = {"LogLik", "SumLogDet", "SSE"};
print lnl[rowname=labl];