# If $cor(X,\epsilon) \approx 0$ in linear regression, can we conclude $X$ is exogenous?

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?

• the issue is that in practice you don't observe $\epsilon$ (only $\hat{\epsilon}$) so you can never compute $\mbox{cor}(X,\epsilon)$ – user603 Dec 27 '13 at 23:45
• As said above, $\epsilon$ is unobservable so you need to rely on your own knowledge and use the common sense to decide about exogeneity. – Stat Dec 27 '13 at 23:48
• 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. – tchakravarty Dec 27 '13 at 23:49
• 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? – I Like to Code Dec 28 '13 at 1:59

## 1 Answer

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.