Negative Binomial Fixed Effects Regression with Time-invariant Variables and Hausman Test

[DY Kim]

Hello,

Given that the dependent variable is count data and overdispersed, I used negative binomial regression using the command of xtnbreg.

I have both time-invariant and -varying independent variables. I had to use time-invariant variables due to lack of monthly information over time. They were drawn from the census survey 5-year estimates.

I knew that unlike regular fixed effects models, negative binomial fixed effects models allow for estimating non-zero coefficient estimates for those time invariant variables. I just wonder whether I can rely on those estimates of time-invariant variables.

Also, I want to know whether I can conduct a Hausman test to choose between fixed and random effects models when time-invariant independent variables are included in the models.

Any advice would be greatly appreciated. Please let me know if my explanation is not clear or you need additional information. Thank you.

[Joao Santos Silva]

Dear DY Kim,

I would not trust the NB results and note that you cannot really know whether the data is really conditionally overdispersed. I would stick to Poisson regression; see here.

Best wishes,

Joao

[DY Kim]

Dear DY Kim,

I would not trust the NB results and note that you cannot really know whether the data is really conditionally overdispersed. I would stick to Poisson regression; see here.

Best wishes,

Joao

Dear Joao,

Thank you for the helpful link. You suggested using xtpoisson with FE, as opposed to xtnbreg with FX. If I use FE xtpoisson, I will lose all time-invariant variables and also all groups with all zero outcomes. For my research, it is important to keep time-invariant variables in the models. I want to keep them in my models.

Can I use xtpoisson with RE instead? I just wonder whether xtpoisson with RE fully is also robust to overdispersion (My data has numerous zeros), like fixed effects xtpoisson. I found that xtpoisson with RE allows for keeping all time-invariant variables.

I also wonder whether xtnbreg with RE can be an alternative to xtnbreg with FE or xtpoisson with FE.

Any suggestion would be greatly appreciated.

[Jeff Wooldridge]

DY: That the time-invariant variables are kept in the fixed effects NegBin is not a good thing. It's a very peculiar thing and shows FENB is not a true "fixed effects" estimator. As Joao said, you can't really trust the estimator.

What I recommend is combing Poisson FE, which is robust to everything, with a correlated random effects version of Poisson regression. I would estimate the CRE model using a pooled Poisson regression (not RE Poisson, which is not robust to overdispersion or serial correlation). Cluster your standard errors for serial correlation and misspecification of the Poisson distribution. Then you can keep time-constant variables in the equation. Is your panel balanced?

Also, dropping groups with all zero outcomes is exactly what FE Poisson should do because there is no information in estimating the coefficients if all outcomes are zero. This happens with FENB, too.

By the way, in my experience, studies where interest is in the time-constant variables are not very compelling because it's very unlikely your recovering a causal relationship. If you're purely after a descriptive analysis then it's fine.

[DY Kim]

What I recommend is combing Poisson FE, which is robust to everything, with a correlated random effects version of Poisson regression. I would estimate the CRE model using a pooled Poisson regression (not RE Poisson, which is not robust to overdispersion or serial correlation). Cluster your standard errors for serial correlation and misspecification of the Poisson distribution. Then you can keep time-constant variables in the equation. Is your panel balanced?

Hello Jeff,

Thank you for sharing your knowledge and experience.

For my study, the unit of analysis is monthly counts per census tract. There are 2,165 census tracts over 51 time points. My data is strongly balanced. There are many zeros, and the data is over-dispersed. The unavailability of monthly data at the census tract led me to rely on time-invariant variables, even though they are constant and limited in finding a causality.

While examining the effect of the pandemic on crime, I included time-invariant variables as controls. I also wanted to examine interactions between the pandemic and time-invariant variable. Is it statistically okay to examine interaction effects of the pandemic with time-invariant variables?

You suggested that I combine Poisson FE with a correlated random effects version of Poisson regression. I am not familiar with the correlated random effects models associated with a pooled Poisson regression. Would you please provide Stata codes or any reference information to execute your suggestion?

In addition, you suggested that I cluster standard errors for serial correlation and misspecification of the Poisson distribution. To execute your suggestion, should I just add the option of cluster(ct) to my regression, in which ct specifies the variable that defines the group/cluster (census tract in my data).

Any advice would be greatly appreciated.