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I am working with panel data, and the Hausman test gave a p-value of 0.0001.

From this, can I infer that there is significant heterogeneity of intercepts across entities of my panel?

Im thinking so because the results indicate the fixed estimator is more appropriate for the model, which assumes intercept heterogeneity. Can there be an indirect implication?

(Also, if this does not imply intercept heterogeneity, is there a test for it?)

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The Hausman test can be used in many settings, but you appear to be referring to the case of panel data in which you are considering either fixed effects or random effects. As a reference point, I'm writing a baseline equation below.

$$y_{it}=\beta_0 +\beta_1 x_{it} +\alpha_i +u_{it}$$

The null hypothesis of the Hausman test is that the random effects estimator is consistent, mathematically, $cov(\alpha_i, x_{it})=0$. (Note that, both fixed effects and random effects estimators make the assumption that $cov(u_{it},x_{it})=0$ ).

You have rejected the null hypothesis, which implies that the random effects estimator is inconsistent, and $cov(\alpha_i, x_{it})\ne 0$.

Rejecting this null hypothesis does not directly inform you regarding the values of the $\alpha_i$ and how varied they are, only that the values are correlated with $x$. However, if $\alpha_i$ were constant for all $i$, then it would definitely be uncorrelated with any variable, notably $x_i$. Thus, it is safe to say that "there is heterogeneity of intercepts".

I think that, if you are interested in the extent of heterogeneity of intercepts, it would be more interesting to implement the dummy-variable fixed effects estimator and compare the coefficient estimates for the dummy variables. You can use an F-test with null hypothesis that they are all the same for example.

Cheers!

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