676—Chapter 37. Panel Estimation

The next portion of output provides additional test detail, showing the coefficient estimates from both the random and fixed effects estimators, along with the variance of the difference and associated p-values for the hypothesis that there is no difference. Note that in some cases, the estimated variances can be negative so that the probabilities cannot be computed.

Cross-section random effects test comparisons:

The bottom portion of the output contains the results from the corresponding fixed effects estimation:

Cross-section random effects test equation:

Dependent Variable: I

Method: Panel Least Squares

Date: 08/18/09 Time: 12:51

Sample: 1935 1954

Periods included: 20

Cross-sections included: 10

Total panel (balanced) observations: 200

Effects Specification

Cross-section fixed (dummy variables)

In some cases, EViews will automatically drop non-varying variables in order to construct the test statistic. These dropped variables will be indicated in this latter estimation output.

Estimation Background

The basic class of models that can be estimated using panel techniques may be written as:

678—Chapter 37. Panel Estimation

with respect to b for a suitably chosen p ¥ p weighting matrix H .

In the simple linear case where f(Xit, b) = Xit¢b , we may write the coefficient estimator in closed form as,

The basics of GMM estimation involve: (1) specifying the instruments Z , (2) choosing the weighting matrix H , and (3) determining an estimator for L .

It is worth pointing out that the summations here are taken over individuals; we may equivalently write the expressions in terms of summations taken over periods. This symmetry will prove useful in describing some of the GMM specifications that EViews supports.

A wide range of specifications may be viewed as specific cases in the GMM framework. For example, the simple 2SLS estimator, using ordinary estimates of the coefficient covariance, specifies:

Estimation Background—679

Standard errors that are robust to conditional or unconditional heteroskedasticity and contemporaneous correlation may be computed by substituting a new expression for L ,

so that we have a White cross-section robust coefficient covariance estimator. Additional robust covariance methods are described in detail in “Robust Coefficient Covariances” on page 611.

In addition, EViews supports a variety of weighting matrix choices. All of the choices available for covariance calculation are also available for weight calculations in the standard panel GMM setting: 2SLS, White cross-section, White period, White diagonal, Cross-section SUR (3SLS), Cross-section weights, Period SUR, Period weights. An additional differenced error weighting matrix may be employed when estimating a dynamic panel data specification using GMM.

The formulae for these weights are follow immediately from the choices given in “Robust Coefficient Covariances” on page 611. For example, the Cross-section SUR (3SLS) weighting matrix is computed as:

ˆ

where QM is an estimator of the contemporaneous covariance matrix. Similarly, the White period weights are given by:

These latter GMM weights are associated with specifications that have arbitrary serial correlation and time-varying variances in the disturbances.

680—Chapter 37. Panel Estimation

GLS Specifications

EViews allows you to estimate a GMM specification on GLS transformed data. Note that the moment conditions are modified to reflect the GLS weighting:

j = 1

First-differencing this specification eliminates the individual effect and produces an equation of the form:

j = 1

which may be estimated using GMM techniques.

Efficient GMM estimation of this equation will typically employ a different number of instruments for each period, with the period-specific instruments corresponding to the different numbers of lagged dependent and predetermined variables available at a given period. Thus, along with any strictly exogenous variables, one may use period-specific sets of instruments corresponding to lagged values of the dependent and other predetermined variables.

Consider, for example, the motivation behind the use of the lagged values of the dependent variable as instruments in Equation (37.19). If the innovations in the original equation are i.i.d., then in t = 3 , the first period available for analysis of the specification, it is obvious

Similar sets of instruments may be formed for each predetermined variables.