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Suppose that we run the simple linear regression $Y = \alpha + \beta X + \epsilon$. I want to test whether the independent variable $X$ is exogenous. If the correlation between the independent variable $X$ and the residual of linear regression $\epsilon$ is almost zero, i.e. $cor(X, \epsilon) \approx 0$, can I then conclude that this simple test suggests that the independent variable $X$ is exogenous?

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    $\begingroup$ the issue is that in practice you don't observe $\epsilon$ (only $\hat{\epsilon}$) so you can never compute $\mbox{cor}(X,\epsilon)$ $\endgroup$ – user603 Dec 27 '13 at 23:45
  • $\begingroup$ As said above, $\epsilon$ is unobservable so you need to rely on your own knowledge and use the common sense to decide about exogeneity. $\endgroup$ – Stat Dec 27 '13 at 23:48
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    $\begingroup$ The correlation between regressors and residuals (not errors) in a linear regression model estimated by least squares is always zero. You cannot test exogeneity (conditional uncorrelatedness) without instrumental variables. $\endgroup$ – tchakravarty Dec 27 '13 at 23:49
  • $\begingroup$ Thank you for your comments, as putting them all together answers my question. Is anyone willing to organize these together into an answer which I can accept? $\endgroup$ – I Like to Code Dec 28 '13 at 1:59
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The linear regression model is $$ \boldsymbol{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon} $$ together with the conditional uncorrelatedness assumption $\mathbb{E}( \mathbf{X}\boldsymbol{\varepsilon}) = \boldsymbol{0}$.

If estimation of the parameters $\boldsymbol{\beta}$ proceeds by least squares, then the first order conditions (normal equations) are $$ \begin{align} \mathbf{X}'\left(\boldsymbol{Y} - \mathbf{X}\hat{\boldsymbol{\beta}} \right) &= \boldsymbol{0} \\ \mathbf{X}'\hat{\boldsymbol{\varepsilon}} &= \boldsymbol{0} \end{align} $$

where the last line indicates that the correlation between the residuals and the regressors is always zero for a linear regression model estimated by least squares. Thus this cannot form the basis of a test for the unconditional uncorrelatedness assumption in the linear regression model.

In order to test the exogeneity assumption, you will need access to instrumental variables. See the Durbin-Wu-Hausman test.

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