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I just studied (again) some statements of the Durbin-Wu-Hausman (DWH) test.

After I thought about the theory behind the DWH test, is it not basically kind of useless since I cannot tell if there's actually a problem regarding endogeneity or not?

For completeness, the hypothesis of the test:

$H_0: \ \ y = X\boldsymbol{\beta} + u, \ \ \ \ u \sim IID\left(0,\sigma^2I_n\right), \ \ E(X^Tu)=0$ $H_1: \ \ y = X\boldsymbol{\beta} + u, \ \ \ \ u \sim IID\left(0,\sigma^2I_n\right), \ \ E(W^Tu)=0$ Where $W$ is the matrix of instruments.

If I'm not mistaken, we need to define some instruments in order to perform the DWH test even if we are not sure about the potential endogeneity of some of our regressors $X$.

Now, in case I observe a significant outcome of the DWH test, that is, I reject $H_0$, two interpretations are possible:

  1. The first is that at least some columns of $X$ in $H_0$ are endogenous such the $H_1$ is correctly specified. This interpretation would the desirable one.

  2. The second involves the regressions $y = X\beta + P_W Y\delta + u$ where $Y$ are the columns of $X$ which are assumed to be endogenous (i.e. a submatrix of $X$) and $P_W$ is the orthogonal projection matrix onto the column space of $W$. If now $\hat{\delta}$ (estimated via OLS) differs significantly from $0$ (tested via F-test) then we would know that the nature of the $H_0$ model is not that columns of $X$ are actually endogenous rather then the linear combinations of the instruments, i.e., $P_WY$ have explanatory power for the dependent variable $y$. This would actually tell us that the instruments which we used are not actually good instruments because there is not supposed to be any explanatory power within the instruments if the regressors $X$ are involved.

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  • $\begingroup$ I don't understand the problem here. In your point 2, you are not testing the endogeneity of $\mathbf{X}$, you are testing for the endogeneity of $\mathbf{Y}$. In which case, the idea of the DWH test is that the coefficients on the projections $\mathbf{P}_{\mathbf{W}}\mathbf{Y}$ will be significantly different from the coefficients on $\mathbf{Y}$, which would not be the case if $\mathbf{Y}$ were exogenous. It is true that in either case you need access to an overidentifying set of instruments. $\endgroup$ – tchakravarty Dec 7 '12 at 19:33
  • $\begingroup$ So since $Y$ is a submatrix of $X$ and if $Y$ is actually exogenous we could just omit $P_WY$ in $y = X\beta + P_WY\delta + u$? Anyway.. In my opinion the use of porper instruments makes this test rather vulnerable to the weak instrument problem if theres actually no overidentifying source. $\endgroup$ – Druss2k Dec 9 '12 at 20:36
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    $\begingroup$ Druss, why would you omit $\mathbf{Y}$? You are interested in its effect on the outcome in the structural equation, that is, you are interested in the parameter vector $\boldsymbol{\delta}$. $\endgroup$ – tchakravarty Dec 9 '12 at 20:40
  • $\begingroup$ Not $Y$ but $P_WY$. If $Y$ is actually exogenous $\delta$ should not differ significantly from zero? $\endgroup$ – Druss2k Dec 9 '12 at 20:45
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    $\begingroup$ Ok, I understand the setup now. Let me think about what you are saying. $\endgroup$ – tchakravarty Dec 9 '12 at 20:52

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