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So I have a panel data with two time periods. My dependent variable is an index that lies in the range of 0 to 1. I did a Breusch- Pagan test (in stata) to see whether I should use random effect or pooled estimation. In the results the variance for u is 0 and the p value is 1 which means I cant reject the null and hence have to do a pooled regression. My question is that isn't the variance of u (unobserved fixed effects) being 0 a bit odd as I am in a panel data setting? What does this really mean?

Furthermore if I can't reject the null does this mean that I can't use a fixed effects estimation as well?

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In a panel model, you have $$y_{it} = \alpha + \beta x_{it} + u_{i} + \varepsilon_{it},$$ so you have two components in your error.

The BP test's null is that the variance of the random effect is zero: $Var[u_i]=0$. Effectively, this would mean that everybody has the same intercept $\tilde \alpha = \alpha + v$, and you can run a pooled regression. I have never been able to reject this null on non-simulated panel data, and you have found that as well. I might consider using het-robust errors and testing again.

To distinguish between RE and FE, you will want to do a Hausman test. The BP test does not help with this decision.

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