proc varmax: conditional maximum likelihood estimation

[tugasakhir]

I am curently using method=cml in proc varmax. Referring to Reinsel’s "Elements of Multivariate Time Series Analysis," I would like to clarify how SAS handles the initial observations for the conditional likelihood.

For a model with AR order p, does SAS treat the first p observations (y_1, y_2,.., y_p) as fixed values, effectively starting the estimation and the calculation of residuals from t = p + 1?

[SASCom1]

Yes your understanding of method = cml for VAR(p) model estimation is correct. The first p observations are treated as fixed values(for initializing lags), and the estimation and calculation of residuals effectively start from t = p+1.

I hope this helps.

[tugasakhir]

thankyou so much. but there is another question i want to ask. How do I get the initial values of the constant parameters, AR, MA, and their covariances? I read in the documentation that it is done using least squares, but I don't understand the steps taken to get the initial parameter value initialization. Can someone explain it to me? for example, if I want to estimate varma(1,1) with 3 variables then I need to have initial values for the parameters c1,c2,c3 (constanta), AR parameters such as phi11,phi12,phi13,...,phi33 and the MA parameters such as theta11,theta12,..,theta33, and the last, the covariance.

[SASCom1]

For VARX(p,s) model, the initial parameter values can be obtained directly by fitting the VARX model using least squares method;

For VARMAX(p,q,s) model, this can be done in steps:

1. fit a higher autoregressive order VARX model using CML method, (the autoregressive order can be determined using information criteria), and gets the residuals. Compute the lagged periods residuals up to order q.

2. fit regression of Y_t's on constant, X_t's, Y_t-1,.., Y_t-p, -(e_t-1),.., -(e_t-q) using least squares method, where Y_t-1, ..., Y_t-p, are the lagged periods dependent variables, e_t-1, ..., e_t-q are the lagged periods residuals from the VARX model in step 1. The resulting parameter estimates from the least squares regression are the initial parameter values for the Constants, X_t coefficients, AR coefficients, and MA coefficients for the CML method. The resulting covariance of innovation matrix from the least squares regression will be the initial values for the innovation covariance matrix parameters for the CML method.

I hope this helps.

[tugasakhir]

Thank you for your explanation. I think I'm still confused about one point.

In PROC VARMAX, when I use

proc varmax data=work.data;
model x y z /
minic lagmax=12 printall;
nloptions pall maxit=5000 tech=newrap;
output out=out_forecast lead=24;
run;

the MINIC procedure automatically selects the final model (for example, VARMA(1,1)).

However, your explanation mentions first fitting a higher autoregressive order VARX model to obtain the initial values.

Could you please explain more about what "higher autoregressive order" refers to in this context?

Is this higher-order VARX model selected automatically by PROC VARMAX as part of its internal algorithm, or is it something that the user should specify? Also, when you mention that "the autoregressive order can be determined using information criteria," does this refer to the final MINIC-selected order (e.g., AR order = 1 for VARMA(1,1)), or to a different higher-order VARX model used only for initialization?

[SASCom1]

In step 1, a 'higher order VARX model' is fit to get 'residuals' to be used in the second step least squares regression. To fit this VARX model the procedure needs to determine the autoregressive order to use(it is about this first step(in order to get initial parameter values) VARX model's AR order to use, not about the AR order in the VARMAX model you are trying to estimate). The autoregressive order *for this first step VARX model* is internally determined in the procedure algorithm using information criteria.

I hope this helps.