Technical Details

320—Chapter 27. Generalized Linear Models

for yi = 0, 1, 2, º The dispersion is restricted to be 1 and prior weighting is not permitted.

• Binomial Proportion

322—Chapter 27. Generalized Linear Models

EViews does not restrict the link choices associated with a given distributional family. Thus, it is possible for you to choose a link function that returns invalid mean values for the specified distribution at some parameter values, in which case your likelihood evaluation and estimation will fail.

One important role of the inverse link function is to map the real number domain of the linear index into the range of the dependent variable. Consequently the choice of link function is often governed in part by the desire to enforce range restrictions on the fitted mean. For example, the mean of a binomial proportions or negative binomial model must be between 0 and 1, while the Poisson and Gamma distributions require a positive mean value. Accordingly, the use of a Logit, Probit, Log-Log, Complementary Log-Log, Power Odds Ratio, or Box-Cox Odds Ratio is common with a binomial distribution, while the Log, Power, and Box-Cox families are generally viewed as more appropriate for Poisson or Gamma distribution data.

EViews will default to use the canonical link for a given distribution. The canonical link is the function that equates the canonical parameter v of the exponential family distribution and the linear predictor h = g(m) = v(m). The canonical links for relevant distributions are given by:

Technical Details—323

The negative binomial canonical link is not supported in EViews so the log link is used as the default choice in this case. We note that while the canonical link offers computational and conceptual convenience, it is not necessarily the best choice for a given problem.

Quasi-likelihoods

Wedderburn (1974) proposed the method of maximum quasi-likelihood for estimating regression parameters when one has knowledge of a mean-variance relationship for the response, but is unwilling or unable to commit to a valid fully specified distribution function.

if it exists, should behave very much like a log-likelihood contribution. We may use to the individual contributions Qi to define the quasi-log-likelihood, and the scaled and unscaled quasi-deviance functions

We may obtain estimates of the coefficients by treating the quasi-likelihood q(m, y, f, w) as though it were a conventional likelihood and maximizing it respect to b . As with conventional GLM likelihoods, the quasi-ML estimate of b does not depend on the value of the dispersion parameter f . The dispersion parameter is conventionally estimated using the Pearson x2 statistic, but if the mean-variance assumption corresponds to a valid exponential family distribution, one may also employ the deviance statistic.

For some mean-variance specifications, the quasi-likelihood function corresponds to an ordinary likelihood in the linear exponential family, and the method of maximum quasi-like-

324—Chapter 27. Generalized Linear Models

lihood is equivalent to ordinary maximum likelihood. For other specifications, there is no corresponding likelihood function. In both cases, the distributional properties of the maximum quasi-likelihood estimator will be analogous to those obtained from maximizing a valid likelihood (McCullagh 1983).

We emphasize the fact that quasi-likelihoods offer flexibility in the mean-variance specification, allowing for variance assumptions that extend beyond those implied by exponential family distribution functions. One important example occurs when we modify the variance function for a Poisson, Binomial Proportion, or Negative Binomial distribution to allow a free dispersion parameter.

Furthermore, since the quasi-likelihood framework only requires specification of the mean and variance, it may be used to relax distributional restrictions on the form of the response data. For example, while we are unable to evaluate the Poisson likelihood for non-integer data, there are no such problems for the corresponding quasi-likelihood based on meanvariance equality.

A list of common quasi-likelihood mean-variance assumptions is provided below, along with names for the corresponding exponential family distribution:

Note that the power-mean mr , exponential mean exp(m), and squared binomial proportion m2(1 – m)2 variance assumptions do not correspond to exponential family distributions.

Estimation

Estimation of GLM models may be divided into the estimation of three basic components: the b coefficients, the coefficient covariance matrix S , and the dispersion parameter f .

326—Chapter 27. Generalized Linear Models

employs the Fisher Information (negative of the expected Hessian matrix) as the update weighting matrix in place of the negative of the observed Hessian matrix used in standard Newton-Raphson, or the outer-product of the gradients (OPG) used in BHHH.

In the GLM context, the IRLS-Fisher Scoring coefficient updates have a particularly simple form that may be implemented using weighted least squares, where the weights are known functions of the fitted mean that are updated at each iteration. For this reason, IRLS is particularly attractive in cases where one does not have access to custom software for estimating GLMs. Moreover, in cases where one’s preference is for an observed-Hessian Newton method, the least squares nature of the IRLS updates make the latter well-suited to refining starting values prior to employing one of the other methods.

Coefficient Covariance Estimation

You may choose from a variety of estimators for , the covariance matrix of ˆ . In describ-

S b

ing the various approaches, it will be useful to have expressions at hand for the expected Hessian (I ), the observed Hessian (H ), and the outer-product of the gradients (J ) for GLM

Given correct specification of the likelihood, asymptotically consistent estimators for the S may be obtained by taking the inverse of one of these estimators of the information matrix. In practice, one typically matches the covariance matrix estimator with the method of esti-

328—Chapter 27. Generalized Linear Models

Estimation,” on page 326. The resulting covariance matrix yields what are sometimes termed GLM standard errors.

Bear in mind that estimating ˆ given a fixed dispersion distribution violates the assump- f

tions of the likelihood so that standard ML theory does not apply. This approach is, however, consistent with a quasi-likelihood estimation framework (Wedderburn 1974), under which the coefficient estimator and covariance calculations are theoretically justified (see “Quasi-likelihoods,” beginning on page 323). We also caution that overdispersion may be evidence of more serious problems with your specification. You should take care to evaluate the appropriateness of your model.

Computational Details

The following provides additional details for the computation of results:

Residuals

There are several different types of residuals that are computed for a GLM specification:

• The ordinary or response residuals are defined as

The ordinary residuals are simply the deviations from the mean in the original scale of the responses.

The weighted residuals divide the ordinary response variables by the square root of the unscaled variance. For models with fixed dispersion, the resulting residuals should have unit variance. For models with free dispersion, the weighted residuals may be used to form an estimator of f .

• The standardized or scaled Pearson residuals) are computed as

The standardized residuals are constructed to have approximately unit variance.

The scores of the GLM specification are obtained by multiplying the explanatory variables by the generalized residuals (Equation (27.18)). Not surprisingly, the generalized residuals may be used in the construction of LM hypothesis tests.

Technical Details—329

Sum of Squared Residuals

EViews reports two different sums-of-squared residuals: a basic sum of squared residuals, SSR = Âeˆoi2 , and the Pearson SSR, SSRP = Âeˆ2pi .

Dividing the Pearson SSR by (N – k) produces the Pearson x2 statistic which may be used as an estimator of f , (“Dispersion Estimation” on page 327) and, in some cases, as a measure of goodness-of-fit.

Log-likelihood and Information Criteria

EViews always computes GLM log-likelihoods using the full specification of the density function: scale factors, inessential constants, and all. The likelihood functions are listed in “Distribution,” beginning on page 319.

If your dispersion specification calls for a fixed value for f , the fixed value will be used to compute the likelihood. If the distribution and dispersion specification call for f to be estimated, fˆ will be used in the evaluation of the likelihood. If the specified distribution calls for a fixed value for f but you have asked EViews to estimate the dispersion, or if the specified value is not consistent with a valid likelihood, the log-likelihood will not be computed.

The AIC, SIC, and Hannan-Quinn information criteria are computed using the log-likelihood value and the usual definitions (Appendix D. “Information Criteria,” on page 771).

It is worth mentioning that computed GLM likelihood value for the normal family will differ slightly from the likelihood reported by the corresponding LS estimator. The GLM likelihood follows convention in using a degree-of-freedom corrected estimator for the dispersion while the LS likelihood uses the uncorrected ML estimator of the residual variance. Accordingly, you should take care not compare likelihood functions estimated using the two methods.

Deviance and Quasi-likelihood

EViews reports the unscaled deviance D(m, y, w) or quasi-deviance. The quasi-deviance and quasi-likelihood will be reported if the evaluation of the likelihood function is invalid. You may divide the reported deviance by (N – k) to obtain an estimator of the dispersion, or use the deviance to construct likelihood ratio or F-tests.

In addition, you may divide the deviance by the dispersion to obtain the scaled deviance. In some cases, the scaled deviance may be used as a measure of goodness-of-fit.

Restricted Deviance and LR Statistic

The restricted deviance and restricted quasi-likelihood reported on the main page are the values for the constant only model.