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From |
abasu@medicine.bsd.uchicago.edu |

To |
statalist@hsphsun2.harvard.edu |

Subject |
st: Interpretation of coefficients in the Fixed Effects Negative Binomial Model |

Date |
Mon, 7 Mar 2005 10:55:17 -0600 |

Hi, Last week I posted a question regarding a fixed effects negative binomial model and asked about the interpretation of coefficients on covariates that do not vary over time. Nick Cox referred me to the FAQ which clearly points out why and how these coefficients are estimated in the xtreg context. However, this explanation does not carry over to the nonlinear negative binomial models. Fortunately, Paulo Guimaraes referred me to a very interesting article(Paul Allison and Richard Waterman. 2002. "Fixed Effects Negative Binomial Regression Models" Sociological Methodology 32:247-265) that explained why coefficients on time-invariant covariate can be estimated in a fixed effects negative binomial model. I have tried to explain this below along with Paulo's input. Hope this will be useful to Stata users. Also, we would like to hear any comments that people may have. The Fixed effects negative binomial (FENB) version that Stata implements follows from the analysis of Hausman, Hall and Griliches (Econometrica, 1984) where the overdispersion (shape) parameter in the NB distribution is modeled as a function of covariates while the individual level fixed effect is modeled through the scale parameter that is assumed to be fixed across time for a specific individual. Specifically, Let E(Y(it)|X(i), Z(it)) = theta(i)*lambda(it) and V(Y(i)|X(i), Z(it)) = (1+ theta(i))* E(Y(it)|X(i), Z(it)) Where theta(i) is the scale parameter, and Lambda(it) is the shape parameter X(i) are the time invariant covariates and Z(it) are the time variant covariates. Therefore, ln{E(Y(it)|X(i))} = ln(theta(i)) + ln(lambda(it)) = ln(theta(i)) + [alpha + beta*X(i) + gamma*(Z(it))] The conditional likelihood for this model lets the theta(i) parameter drop out thereby overcoming the incidental parameter problems with fixed effects. However, the lambda parameter or any part of it is not eliminated in the conditional likelihood. Therefore, parameters such as the common intercept (alpha), those on time invariant covariates (betas) and the time variant covariates (gammas) are all estimated in the FENB model, since the covariates are assumed to affect lambda and not theta. All the coefficients can still be interpreted in the conventional way as the effect of covariates on the mean in a log-link model. However, the model implements specific assumptions about how these covariates and the fixed effects affect the variance parameter. Anirban ________________________________ Anirban Basu PhD Section of General Internal Medicine Department of Medicine University of Chicago 5841 S. Maryland Ave, MC-2007 AMD B201 Chicago IL 60637 Tel: +1 773 834 1796 Fax: +1 773 834 2238 NOTICE OF CONFIDENTIALITY - This material is intended for the use of the individual or entity to which it is addressed, and may contain information that is privileged, confidential and exempt from disclosure under applicable laws. If the reader of this material is not the intended recipient, you are hereby notified that any dissemination, distribution or copying of this communication is strictly prohibited. Please notify the sender of the error and destroy the Email you received. * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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