Hausman Test

Definition

Consider a regression model $y_1=a_0+a_1{y}_2+a_2{x}_1+u_1$ We want to test whether $y_2$ and $u_1$ are correlated or not. i.e. $y_2$ is exogenous. The null hypothesis for the test is defined as $H_0:\mathrm{Cov}(y_1,u_1)=0$. Then, the test Statistic is defined as $({\hat{\alpha }}_{OLS}-{\hat{\alpha }}_{IV}{)}^{⊺}[\mathrm{Var}({\hat{\alpha }}_{OLS}-{\hat{\alpha }}_{IV}){]}^{-1}({\hat{\alpha }}_{OLS}-{\hat{\alpha }}_{IV})\stackrel{a}{\sim }{\chi }^2(k)$ where $k$ is the degrees of freedom, which is equal to the number of parameters