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I obtained the following output after running the Hausman test:

1) CASE 1 Hausman Test chisq = 13.943, df = 4, p-value = 0.007478 alternative hypothesis: one model is inconsistent

2) CASE 2 Hausman Test chisq = 0.49157, df = 4, p-value = 0.9743 alternative hypothesis: one model is inconsistent

According to the p-values and for significance <0.05, should I go for the fixed effects in CASE 1 and for the random effects in CASE 2?

Thanks

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Yes. See the following taken from a Princeton slide:

To decide between fixed or random effects you can run a Hausman test where the null hypothesis is that the preferred model is random effects vs. the alternative the fixed effects (see Green, 2008, chapter 9). It basically tests whether the unique errors (ui) are correlated with the regressors, the null hypothesis is they are not. Run a fixed effects model and save the estimates, then run a random model and save the estimates, then perform the test. If the p-value is significant (for example <0.05) then use fixed effects, if not use random effects.

see: https://dss.princeton.edu/training/Panel101R.pdf

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