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Wooldridge's graduate textbook (p.285) says that:

We can formally test the assumptions underlying the consistency of the FE and FD estimators by using a Hausman test. [...] If T = 2, it is easy to test for strict exogeneity. In the equation $\Delta y_i = \Delta x_i \beta + \Delta u_i $, neither $x_{i1}$ nor $x_{i2}$ should be significant as additional explanatory variables in the first-differenced equation.

Why is this? The strict exogeneity condition for FD is in differences, not levels: $E[\Delta x \Delta u]=0$ ...

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I think the idea is a little heuristic of Wooldridge. If the model is specified correctly (strict exogeneity is true), then $x_{1i}$ and $x_{2i}$ should only be predictive of $\Delta y_i$ through $\Delta x_i$. The test is to evaluate if that is the case.

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  • $\begingroup$ Thank you Michael, that makes sense. So it is simply a question of whether the model is dynamically complete. $\endgroup$
    – ABCBAA
    Mar 25 at 14:43

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