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I have a datset of 25 counties over 11 years, with response variable unemployment ( in %), and 6 explanatory variables (proportion with high school, some economic indicators, etc).

After some tests to choose between FE and RE, I get that a one-way (individual) FE model is preferable.

Checking for heteroskedasticity and autocorrelation I use a Breusch-Pagan test that returns a p-value of 0.034 (therefore heteroskedastic variance for estimates). Checking for auto-correlation for errors I use Breusch-Godfrey/Wooldridge test for serial correlation in panel models, which returns a p-value of 0.0001, which means errors are serially correlated.

As I understand from the literature im reading: There are 2 solutions (listed atleast), FGLS instead of OLS estimation on the transformed data. Cluster-robust SE is the second fix.

I have applied fgls instead of ols on the transformed data, changing the SE slightly. Running the Breusch-Godfrey/Wooldridge on the fgls model returns the same result, which confuses me, wouldnt the test now result in the null being accepted (thus no autocorrelation)? Does applying FGLS to the transformed data also handle the heteroskedasticity problem?

The main use for the model is prediction (to be compared to a model of only time-series aspect). As I understand breaking the heteroskedasticity assumption and auto-correlation assumption means that the model is still unbiased and consistent, but not efficient. Does this mean I have to correct the underlying autocorrelation and heteroskedasticity of the model to properly use it for prediction?

Sorry if the question is very general, the text we're using is not very clear how the assumptions are handled in panel data.

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