Suppose we have the following simple regression model (time series framework)"
$$y_1=\beta_0+\beta_1 y_2+\beta_2 z_1 +\beta_3 z_2 +u,$$
where $z_1$ and $z_2$ are exogenous and $y_2$ is either exogeneous or endougenous (this is what we want to check). In order to determine whether or not $y_2$ is endogenous, we can apply endogeneity test (Hausman test), which follows the following procedure:
Estimate the reduced form for $y_2$, i.e. estiamte the following equation:
$$y_2=\alpha_0+\alpha_1z_1+\alpha_2z_2+\alpha_3z_3+\alpha_4z_4+\nu,$$ where $z_3$ and $z_4$ are instruments. Since each $z_j$ is uncorrelated with $u$, $y_2$ is uncorrelated with $u$ if and only if $v$ is uncorrelated with $u$; this is what we want to test. The easiest way to test this is to include $v$ as an additional regressor in structural eqaution and to do a $t$ test, i.e. estimate the model
$$y_1=\beta_0+\beta_1 y_2+\beta_2 z_1 +\beta_3 z_2 +\gamma_1\widehat\nu+error.$$ I don't understad this part. We want to determine whether $u$ and $\nu$ are correlated, but how we unleash it by including $\widehat \nu$ at the structural equation? Doing so, we esimate the impact of $\widehat \nu$ on $y_1$, rather than on $u$. Please explain the intuition.
P.S. In my understanding in order to determine whether $u$ and $\nu$ are correlated, we can apply the following steps:
- Estimate $\widehat u$ from the structural equation,
- Estimate $\widehat \nu$ from the reduced eqaution,
- Regress $\widehat u$ on $\widehat \nu$.