Intuition behind the Endogeneity Test (the Hausman Test)

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 $$\widehat \nu$$.