I'm reading a paper in which is used the following endogeneity test:
- First of all, we have the initial linear model: $$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + e$$ $x_3$ is the endogenous regressor and $z$ is the instrument.
- We regress the endogenous regressor on the instrument and the exogenous regressors: $$x_3 = b_0 + b_1x_1 +b_2x_2 + b_3z + e$$
- We recover the residual $u$ of the linear regression of the previous point. Then we estimate the following linear model: $$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \rho u + e$$
The paper says that this is an endogeneity test: if the estimated coefficients in step 3 are very similar to those in step 1 then regressor $x_3$ was not endogenous.
Could anyone explain me the intuition behind this test?