Applying the GLM Variance Assumption to Overcome the Scale-Dependence of the Negative Binomial QGPML Estimator

Abstract

Recently, various studies have used the Poisson Pseudo-Maximal Likehood (PML) to estimate gravity specifications of trade flows and non-count data models more generally. Some papers also report results based on the Negative Binomial Quasi-Generalised Pseudo-Maximum Likelihood (NB QGPML) estimator, which encompasses the Poisson assumption as a special case. This note shows that the NB QGPML estimators that have been used so far are unappealing when applied to a continuous dependent variable which unit choice is arbitrary, because estimates artificially depend on that choice. A new NB QGPML estimator is introduced to overcome this shortcoming.

Keywords:

• Gamma PML
• Negative binomial estimator
• Pseudo-maximum likelihood methods
• Poisson regression

JEL Classification:

• C13
• C21
• F10

ACKNOWLEDGMENT

We are indebted to two anonymous referees for valuable comments and to Thierry Mayer who provided data as well as useful suggestions at an early stage. We also would like to greatly thank Joao Santos Silva, Thierry Magnac, Pierre-Philippe Combes, and Lionel Fontagné, as well as participants of the GREQAM PhD Students lunch seminar, the 2009 RIEF doctoral meetings, the 2009 EEA Congress, the 2009 AFSE Congress and the 2009 ASSET Congress.

Notes

Because E(Log y) ≠ Log E(y), the expected value of the logarithm of trade flows depends on higher moments, including the variance. Since the conditional variance of the residuals is likely to depend on explanatory variables, estimators using the log specification might be biased. Also, as highlighted by one referee, estimating models such as gravity equations directly in levels avoids the problem of re-transforming logged outcomes back into their original scales when one wishes to make predictions of the outcome measures. Manning and Mullahy (2001) argue that in the presence of heteroskedasticity in the log-linear specification, the exponentiated log-scale prediction might provide a biased estimate of the conditional expectation.

Cameron and Trivedi (1986) refer to the Negbin II parameterization of the negative binomial specification in that case.

We follow here the first specification of Santos Silva and Tenreyro (2006). It is indifferent to specify the model as y _i  = exp(X _i β) + u _i with E(u _i  | X _i ) = 0.

Another way to see this is as follows. The NB PML assumption is . Under the condition that is independent from λ (except ), this becomes: . Independence of this estimator (with robust standard errors) with respect to λ implies that .

In Stata, one can use the glm command with family(nbinomial a) where , which does not allow a to be negative. Some studies with a continuous dependent variable might have used the full NB likelihood estimator nbreg in Stata. While this is clearly inappropriate in such a case - we are grateful to one referee for highlighting this -, the NB ML estimator is also clearly scale-dependent because when y is a NB variable, is not due to the Γ function.

Estimating β and α jointly would require using the full NB log-likelihood (Wooldridge, 1999, p. 378), which is not suitable for non count data.

PPML point estimates are scale-invariant as discussed above, and estimated robust standard errors are also scale invariant, see e.g., Santos Silva and Tenreyro (2006). As will be obvious, the geometric mean used is the first step for NB QGPML _HMR is also scale-dependent, which adds a second source of scale-dependence in that case.

The Stata code for both NB QGPML and NB QGPML is available at https://sites. google.com/site/clementbosquet/. Using the consistent PPML estimate in the first step, is estimated from Eq. (14) or (16), respectively.

Standard error between parentheses. ^a , ^b , ^c Significant at the 1%, 5%, and 10% level, respectively. PML =Pseudo-Maximum Likelihood, QGPML =Quasi-Generalised PML, PPML =Poisson PML, NB =Negative Binomial, GPML =gamma PML.

Subsection 2.2 has proved that our NB QGPML ^GLM estimators are scale-invariant. Of course, we checked that this theoretical property is confirmed by the empirical analysis!

PML =Pseudo-Maximum Likelihood, QGPML =Quasi-Generalized PML, PPML =Poisson PML, NB =Negative Binomial QGPML, GPML =gamma PML.

(*) These columns indicate the percentage of occurrences for which the estimated values are positive.

Otherwise, it is very often the case that the estimate of the conditional variance becomes negative for at least a few observations, which prevents the second-step estimation.

As suggested by referees, we tested different settings such as α = 2 and/or a gamma random variable for η (instead of log normal) and obtained similar results.

Focusing on nonzero flows is sufficient for illustration purposes. Including zero flows or focusing on other years unsurprisingly leads to the same conclusion as the proof in Section 2 is general.

http://www.cepii.fr/anglaisgraph/bdd/distances.htm, Centre d'Etudes Prospectives et d'Informations Internationales.

Standard error between brackets. ^a , ^b , ^c Significant at the 1%, 5%, and 10% level, respectively. PML =Pseudo-Maximum Likelihood, QGPML =Quasi-Generalised PML, PPML =Poisson PML, NB =Negative Binomial GQPML, GPML =gamma PML; USD =United States Dollars, Tr. = Trillions, B. = Billions, M. = Millions, Th. = Thousands. Fixed effects are importer and exporter country fixed effects.