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

CC BY-SA 4.0
2024-03-25T14:43:07.713Z
Thank you Michael, that makes sense. So it is simply a question of whether the model is dynamically complete.
ABCBAA
CC BY-SA 4.0
2024-03-25T11:05:05.410Z
2024-03-25T11:05:05.410Z
Michael Gmeiner
<p>Wooldridge's graduate textbook (p.285) says that:</p>
<blockquote>
<p>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 <span class="math-container">$\Delta y_i = \Delta x_i \beta + \Delta u_i $</span>, <strong>neither <span class="math-container">$x_{i1}$</span> nor <span class="math-container">$x_{i2}$</span> should be significant as additional explanatory variables in the first-differenced equation</strong>.</p>
</blockquote>
<p>Why is this? The strict exogeneity condition for FD is in differences, not levels: <span class="math-container">$E[\Delta x \Delta u]=0$</span> ...</p>

<econometrics>
Hausman test from Wooldridge (adding level variables)
ABCBAA
# Hausman test from Wooldridge (adding level variables)

(Asked by: username_0 on 2024-03-25)

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: $E[\Delta x \Delta u]=0$ ...

**Tags:** econometrics

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## 1 answers

### Answer by username_1 on 2024-03-25. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you Michael, that makes sense. So it is simply a question of whether the model is dynamically complete.  

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