662—Chapter 21. Discrete and Limited Dependent Variable Models

Negative Binomial

If we maximize the negative binomial log likelihood, given above, for fixed η2 , we obtain the QMLE of the conditional mean parameters β . This QML estimator is consistent even if the conditional distribution of y is not negative binomial, provided that mi is correctly specified.

EViews sets η2 = 1 by default, which is a special case known as the geometric distribution. You may specify any other (positive) value by changing the number in the Fixed variance parameter field box. For the negative binomial QMLE, EViews by default reports the robust QMLE standard errors.

Views of Count Models

EViews provides a full complement of views of count models. You can examine the estimation output, compute frequencies for the dependent variable, view the covariance matrix, or perform coefficient tests. Additionally, you can select View/Actual, Fitted, Residual…

and pick from a number of views describing the ordinary residuals oi i− ( i ˆ ) , e = y m x , β

or you can examine the correlogram and histogram of these residuals. For the most part, all of these views are self-explanatory.

Note, however, that the LR test statistics presented in the summary statistics at the bottom of the equation output, or as computed under the View/Coefficient Tests/Redundant Variables - Likelihood Ratio… have a known asymptotic distribution only if the conditional distribution is correctly specified. Under the weaker GLM assumption that the true variance is proportional to the nominal variance, we can form a quasi-likelihood ratio,

tistic has an asymptotic χ2 distribution under the assumption that the mean is correctly specified and that the variances follow the GLM structure. EViews does not compute the

dardized residuals. We provide an example of the use of the QLR test statistic below.

If the GLM assumption does not hold, then there is no usable QLR test statistic with a known distribution; see Wooldridge (1996).

Procedures for Count Models

Most of the procedures are self-explanatory. Some details are required for the forecasting and residual creation procedures.

664—Chapter 21. Discrete and Limited Dependent Variable Models

Dependent Variable: NUMB

Method: ML/QML - Poisson Count

Date: 09/14/97 Time: 10:58

Sample: 1 103

Included observations: 103

Convergence achieved after 4 iterations

Covariance matrix computed using second derivatives

Cameron and Trivedi (1990) propose a regression based test of the Poisson restriction

v( xi, β) = m( xi, β) . To carry out the test, first estimate the Poisson model and obtain the fitted values of the dependent variable. Click Forecast and provide a name for the forecasted dependent variable, say NUMB_F. The test is based on an auxiliary regression of

The t-statistic of the coefficient is highly significant, leading us to reject the Poisson restriction. Moreover, the estimated coefficient is significantly positive, indicating overdispersion in the residuals.

666—Chapter 21. Discrete and Limited Dependent Variable Models

Dependent Variable: NUMB

Method: QML - Negative Binomial Count

Date: 10/11/97 Time: 23:53

Sample: 1 103

Included observations: 103

QML parameter used in estimation: 0.22124

Convergence achieved after 3 iterations

GLM Robust Standard Errors & Covariance

Variance factor estimate = 2.465660162

Covariance matrix computed using second derivatives

The header indicates that the estimated GLM variance factor is 2.4, suggesting that the negative binomial ML would not have been an appropriate specification. Nevertheless, the negative binomial QML should be consistent, and under the GLM assumption, the standard errors should be consistently estimated. It is worth noting that the coefficients on IP and FEB, which were strongly statistically significant in the Poisson specification, are no longer significantly different from zero at conventional significance levels.

Quasi-likelihood Ratio Statistic

As described by Wooldridge (1996), specification testing using likelihood ratio statistics requires some care when based upon QML models. We illustrate here the differences between a standard LR test for significant coefficients and the corresponding QLR statistic.

From the results above, we know that the overall likelihood ratio statistic for the Poisson model is 16.85, with a corresponding p-value of 0.0002. This statistic is valid under the assumption that m(xi, β) is specified correctly and that the mean-variance equality holds.

We can decisively reject the latter hypothesis, suggesting that we should derive the QML estimator with consistently estimated covariance matrix under the GLM variance assumption. While EViews does not automatically adjust the LR statistic to reflect the QML assumption, it is easy enough to compute the adjustment by hand. Following Wooldridge, we construct the QLR statistic by dividing the original LR statistic by the estimated GLM variance factor.