System Estimation Methods

EViews will estimate the parameters of a system of equations using:

• Ordinary least squares.

• Equation weighted regression.

• Seemingly unrelated regression (SUR).

• System two-state least squares.

• Weighted two-stage least squares.

• Three-stage least squares.

• Full information maximum likelihood (FIML) (with both unrestricted and restricted covariance matrices).

• Generalized method of moments (GMM).

• Autoregressive Conditional Heteroskedasticity (ARCH).

The equations in the system may be linear or nonlinear, and may contain autoregressive error terms.

In the remainder of this section, we describe each technique at a general level. Users who are interested in the technical details are referred to the “Technical Discussion”.

Ordinary Least Squares

This technique minimizes the sum-of-squared residuals for each equation, accounting for any cross-equation restrictions on the parameters of the system. If there are no such restrictions, this method is identical to estimating each equation using single-equation ordinary least squares.

Cross-Equation Weighting

This method accounts for cross-equation heteroskedasticity by minimizing the weighted sum-of-squared residuals. The equation weights are the inverses of the estimated equation variances, and are derived from unweighted estimation of the parameters of the system. This method yields identical results to unweighted single-equation least squares if there are no cross-equation restrictions.

Seemingly Unrelated Regression

The seemingly unrelated regression (SUR) method, also known as the multivariate regression, or Zellner's method, estimates the parameters of the system, accounting for heteroskedasticity and contemporaneous correlation in the errors across equations. The estimates of the cross-equation covariance matrix are based upon parameter estimates of the unweighted system.

Note that EViews estimates a more general form of SUR than is typically described in the literature, since it allows for cross-equation restrictions on parameters.

Two-Stage Least Squares

The system two-stage least squares (STSLS) estimator is the system version of the single equation two-stage least squares estimator described above. STSLS is an appropriate technique when some of the right-hand side variables are correlated with the error terms, and there is neither heteroskedasticity, nor contemporaneous correlation in the residuals. EViews estimates STSLS by applying TSLS equation by equation to the unweighted system, enforcing any cross-equation parameter restrictions. If there are no cross-equation restrictions, the results will be identical to unweighted single-equation TSLS.

Weighted Two-Stage Least Squares

The weighted two-stage least squares (WTSLS) estimator is the two-stage version of the weighted least squares estimator. WTSLS is an appropriate technique when some of the right-hand side variables are correlated with the error terms, and there is heteroskedasticity, but no contemporaneous correlation in the residuals.

EViews first applies STSLS to the unweighted system. The results from this estimation are used to form the equation weights, based upon the estimated equation variances. If there are no cross-equation restrictions, these first-stage results will be identical to unweighted single-equation TSLS.

Three-Stage Least Squares

Three-stage least squares (3SLS) is the two-stage least squares version of the SUR method. It is an appropriate technique when right-hand side variables are correlated with the error terms, and there is both heteroskedasticity, and contemporaneous correlation in the residuals.

EViews applies TSLS to the unweighted system, enforcing any cross-equation parameter restrictions. These estimates are used to form an estimate of the full cross-equation covariance matrix which, in turn, is used to transform the equations to eliminate the cross-equation correlation. TSLS is applied to the transformed model.

Full Information Maximum Likelihood (FIML)

Full Information Maximum Likelihood (FIML) estimates the likelihood function under the assumption that the contemporaneous errors have a joint normal distribution. Provided that the likelihood function is specified correctly, FIML is fully efficient.

Generalized Method of Moments (GMM)

The GMM estimator belongs to a class of estimators known as M-estimators that are defined by minimizing some criterion function. GMM is a robust estimator in that it does not require information of the exact distribution of the disturbances.

GMM estimation is based upon the assumption that the disturbances in the equations are uncorrelated with a set of instrumental variables. The GMM estimator selects parameter estimates so that the correlations between the instruments and disturbances are as close to zero as possible, as defined by a criterion function. By choosing the weighting matrix in the criterion function appropriately, GMM can be made robust to heteroskedasticity and/or autocorrelation of unknown form.

Many standard estimators, including all of the system estimators provided in EViews, can be set up as special cases of GMM. For example, the ordinary least squares estimator can be viewed as a GMM estimator, based upon the conditions that each of the right-hand side variables is uncorrelated with the residual.

Autogressive Conditional Heteroskedasticity (ARCH)

The System ARCH estimator is the multivariate version of ARCH estimator. System ARCH is an appropriate technique when one wants to model the variance and covariance of the error terms, generally in an autoregressive form. System ARCH allows you to choose from the most popular multivariate ARCH specifications: Constant Conditional Correlation, the Diagonal VECH, and (indirectly) the Diagonal BEKK.