Heteroskedasticity removed through fixed effect estimation?

Question

I have a large panel data set. Examination of a pooled OLS regression with Breusch Pagan showed heteroskedasticity with all model specifications. I consequently chose to use panel-corrected standard error parameter estimates (PCSE, after Beck and Katz, 1996).

Nonetheless, I decided to test the robustness of my model against one with (country) fixed effects. Now, at least in Stata, the Hausman test doesn't work with robust standard errors. Since it looks at the coefficients, not the standard deviation, though, I can use the FE with uncorrected standard errors without problems for the Hausman test.

My question is, though, whether there is any reason to assume that the FE model's error term might not be heteroskedastic. That is, is it necessary to perform another Breusch Pagan test on the FE model, even though I know from the Pooled OLS that there is heteroskedasticity? I'd rather just run the Hausman test on the uncorrected FE and the PCSE model, then report the corrected FE together with the PCSE model in my appendix. The reason being that Stata is a little sturdy when it comes to postestimation tests after xtreg, i.e. the FE estimations.

Answer 1

My question is, though, whether there is any reason to assume that the FE model's error term might not be heteroskedastic.

My interpretation is that what you really want to know is whether heteroscedasticity in the pooled OLS regression implies heteroscedasticity in the FE regression. To that the answer is no. In other words, you cannot test on the pooled OLS regression and conclude that the result also holds for the FE regression.

The model underlying the FE-estimator in its simplest form can be written as $$y_{i,t}=x_{i,t}\beta +\alpha_i + u_{i,t},$$ where we now for simplicity assume $u_{i,t}$ is iid. If you fit a model given by $$y_{i,t}=x_{i,t}\beta +e_{i,t}$$ using pooled OLS and data is generated by the fixed effects model, you have in effect set $e_{i,t} = \alpha_i + u_{i,t}$. Decompose the variance of the error term in the pooled OLS model to get: $$\operatorname{Var}(e_{i,t})=\operatorname{Cov}(\alpha_i + u_{i,t},\alpha_i + u_{i,t})=\operatorname{Var}(\alpha_i)+\operatorname{Var}(u_{i,t})+2\operatorname{Cov}(\alpha_i,u_{i,t}).$$ From this equation it is quite clear that while $u_{i,t}$ is of constant variance (it is even iid), $e_{i,t}$ can very well have non-constant variance. Therefore, evidence of an heteroscedastic error term in the pooled OLS regression is in general not evidence of an heteroscedastic error term in the fixed effects regression.