The VARMAX Procedure

Given a multivariate time series, the VARMAX procedure estimates the model parameters and generates forecasts that are associated with vector autoregressive moving average processes with exogenous regressors (VARMAX) models. Often, economic or financial variables are not only contemporaneously correlated with each other, but also correlated with each other’s past values. You can use the VARMAX procedure to model these types of time relationships. In many economic and financial applications, the variables of interest (dependent, response, or endogenous variables) are influenced by variables external to the system under consideration (independent, input, predictor, regressor, or exogenous variables). The VARMAX procedure enables you to model the dynamic relationships both among the dependent variables and between the dependent and independent variables.

A VARMAX model is defined in terms of the orders of the autoregressive or moving average processes (or both). When you use the VARMAX procedure, these orders can be specified by options or they can be automatically determined according to the information criteria. The VARMAX procedure supports the following information criteria: Akaike’s information criterion (AIC), the corrected AIC (AICC), the Hannan-Quinn criterion (HQC), the final prediction error (FPE), and the Schwarz Bayesian criterion (SBC), which is also known as the Bayesian information criterion (BIC). For the definitions and usages of the information criteria, see the section The Minimum Information Criterion (MINIC) Method for more details.

If you do not want to use automatic order selection, the VARMAX procedure provides the following autoregressive order identification aids: partial cross-correlations, partial autoregressive coefficients, partial canonical correlations, and Yule-Walker estimates.

For situations where the stationarity of the time series is in question, the VARMAX procedure provides the following tests to aid in determining the presence of unit roots and cointegration: Dickey-Fuller tests, the Stock-Watson common trends test for the possibility of cointegration among nonstationary vector processes of integrated order one, and Johansen cointegration tests for nonstationary vector processes of integrated order one and order two.

For stationary vector times series or nonstationary series that are made stationary by appropriate differencing or cointegration, the VARMAX procedure provides the vector autoregressive and moving average (VARMA) model and the vector error correction model (VECM). The vector error correction model can be in both autoregressive (AR) and autoregressive and moving average (ARMA) forms.

To cope with the problem of high dimensionality in the parameters of the VAR model and the VECM, the VARMAX procedure provides both the Bayesian vector autoregressive (BVAR) model and the Bayesian vector error correction model (BVECM). Bayesian models are used when prior information about the model parameters is available.

The VARMAX procedure also allows independent (exogenous) variables and their distributed lags to influence dependent (endogenous) variables in various models. These models are identified by an X suffix added to the original model name; for example, VARMAX, VECMX, BVARX, and BVECMX.

Correlations in the second moments of the vector time series might exist; this is called conditional heteroscedasticity. The VARMAX procedure supports three forms of multivariate generalized autoregressive conditional heteroscedasticity (GARCH) models to model the conditional heteroscedasticity: the Baba-Engle-Kroner-Kraft (BEKK) GARCH model, the constant conditional correlation (CCC) GARCH model, and the dynamic conditional correlation (DCC) GARCH model. For CCC and DCC GARCH models, five subforms of univariate GARCH models are supported: the GARCH model, the exponential GARCH (EGARCH) model, the quadratic GARCH (QGARCH) model, the threshold GARCH (TGARCH) model, and the power GARCH (PGARCH) model.

You can use the VARMAX-GARCH model or the VEC-ARMAX-GARCH model to simultaneously model both the first and second moments of the time series.

Forecasting is one of the main objectives of multivariate time series analysis. After successfully fitting the VARMAX, BVARX, VECMX, BVECMX, and multivariate GARCH models, the VARMAX procedure computes predicted values and conditional heteroscedasticity based on the parameter estimates and the past values of the vector time series. Out-of-sample multistep-ahead forecasts are also supported.

The following model parameter estimation methods are supported:

When you use the ML or CML method, you can start your optimization with the default or with different initial parameter values.

The VARMAX procedure supports the estimation of the restricted model when you impose linear constraints on the parameters of interest. The VARMAX procedure also supports various hypothesis tests of long-run effects and adjustment coefficients by using the likelihood ratio test that is based on Johansen cointegration analysis. It also supports the likelihood ratio test of weak exogeneity for each variable. In fact, because the VARMAX procedure outputs log-likelihood values for all models, you can always use the likelihood ratio test to check any linear hypothesis on parameters that are estimated in the models by estimating the restricted and unrestricted models separately. The VARMAX procedure also supports another alternative test, the Wald test.

After fitting the model parameters, the VARMAX procedure uses the following tests to provide model checks and residual analysis: Durbin-Watson (DW) statistics, the F test for autoregressive conditional heteroscedastic (ARCH) disturbance, the F test for AR disturbance, the Jarque-Bera normality test, and the Portmanteau test.

The VARMAX procedure supports several modeling features, including seasonal deterministic terms, linear and quadratic time trends, subset models, multiple regression with distributed lags, the dead-start model (which does not have present values of the exogenous variables), and so on.

The VARMAX procedure provides a Granger causality test to determine the Granger-causal relationships between two distinct groups of variables. It also provides the following: the infinite order AR representation, the impulse response function (also called infinite order MA representation), the decomposition of the predicted error covariances, roots of the characteristic functions for both the AR and MA parts to evaluate the proximity of the roots to the unit circle, and contemporaneous relationships among the components of the vector time series.