VARMAX Procedure

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

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

Define and with and ; define and . Then

where and .

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

where and is such that . You can specify METHOD=CML in the MODEL statement to apply conditional maximum likelihood estimation.

For the exact log-likelihood function of a VARMA model, the VARMA model is transformed into the equivalent state space form and then the Kalman filtering method is applied.

The state space form of the zero-mean VARMA(p,q) model consists of a state equation

and an observation equation

where

and

where and for .

The Kalman filtering approach is used to evaluate the likelihood function. The updating equation is

where

The prediction equation is

where for .

The log-likelihood function can be expressed as

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

When you specify METHOD=ML in the MODEL statement, the exact log likelihood is evaluated and used in the maximum likelihood estimation.

Define the vector as

where and . All elements of are estimated through the preceding (conditional) maximum likelihood method. The estimates of , and , are output in the ParameterEstimates ODS table. The estimates of the covariance matrix () are output in the CovarianceParameterEstimates ODS table. If you specify the OUTEST=, OUTCOV, PRINT=(COVB), or PRINT=(CORRB) option, you can see all elements of , including the covariance matrix , in the parameter estimates, covariance of parameter estimates, or correlation of parameter estimates. You can also apply the BOUND, INITIAL, RESTRICT, and TEST statements to any elements of , including the covariance matrix . For more information, see the syntax of the corresponding statement.

The (conditional) log-likelihood equations are solved by iterative numerical methods such as quasi-Newton optimization. The starting values for the AR and MA parameters are obtained from the least squares estimates. Although the small-sample properties of CML estimates might not be as good as the ML estimates, the CML method is much faster than the ML method. Depending on the sample size and number of parameters to be estimated, the CML method can be hundreds or even thousands of times faster than the ML method. In the following example code, the CML method is about 100 times faster than the ML method, with very similar estimation and forecast results:

proc iml;
   phi = (0.9 * I(4)) // (-0.7* I(4));
   theta = 0.8 * I(4);
   sig = I(4);
   /* to simulate the vector time series */
   call varmasim(y,phi,theta) sigma=sig n=400 seed=2;

   cn = {'y1' 'y2' 'y3' 'y4'};
   create simul6 from y[colname=cn];
   append from y;
   close;
quit;

proc varmax data=simul6;
   model y1 y2 y3 y4 / noint p=2 q=1 method=cml;
   nloptions pall maxit=5000 tech=qn;
   output out=ocml back=12 lead=24;
run;

proc varmax data=simul6;
   model y1 y2 y3 y4 / noint p=2 q=1 method=ml;
   nloptions pall maxit=5000 tech=qn;
   output out=oml back=12 lead=24;
run;

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 V is the asymptotic information matrix of .

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 p block matrices; is a matrix; is a matrix,

where

    is a zero matrix, and

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 by using the P= option and specify the order of moving-average model by using 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 66 shows the initial values of parameters. The initial values were estimated by using the least squares method.

Figure 67 shows the default option settings for the quasi-Newton optimization technique.

Figure 68 shows the iteration history of parameter estimates.

Figure 69 shows the final parameter estimates.

Figure 70 shows the AR coefficient matrix in terms of lag 1, the MA coefficient matrix in terms of lag 1, the parameter estimates, and their significance, which is one indication of how well the model fits the data.

The fitted VARMA(1,1) model with estimated standard errors in parentheses is given as

and

A general VARMAX() process is written as

or

where and . The vector consists of all possible deterministic terms, namely constant, seasonal dummies, linear trend, quadratic trend, and exogenous variables. The vector , where ; ; , are seasonal dummies and is based on the NSEASON= option; ; A is the parameter matrix corresponding to and for .

The state space form of the VARMAX(p,q,s) model consists of a state equation

and an observation equation

where

and

where , for , and u is the dimension of .

Kalman filtering is used to evaluate the likelihood function. The updating equation is

where

The prediction equation is

where for .

The log-likelihood function can be expressed as

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

In the preceding state space form of a VARMAX model, the exogenous variables are treated as determined terms, which implies that the values of the exogenous variables must be provided to forecast the out-of-sample dependent variables. If you do not have the future values of the exogenous variables, either you predict the exogenous variables in a separate model, or you express both the exogenous variables and the dependent variables in one combined model and predict them together (Reinsel 1997).

The dimension of the state space vector of the Kalman filtering method for the VARMAX(p,q,s) model might be large, so it might take a lot of time and memory for computing.

Two examples of VARMAX modeling follow:

model y1 y2 = x1 / q=1;
nloptions tech=qn;

model y1 y2 = x1 / p=1 q=1 xlag=1 nocurrentx;
nloptions tech=qn;