Hausman test for endogenous variables

Summary

Setup

• The linear regression model with random sampling,

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+ε_i, i=1,…,n\]

• You suspect some explanatory variables are endogenous
• You have at least one instrument for each variable you suspect is endogenous, both IV and OLS is available.

Hausman test

• The null hypothesis is that all your explanatory variables are exogenous.
• Under the null hypothesis OLS and IV will both be consistent (only OLS will be efficient).
• If the null hypothesis is false, then OLS will be inconsistent while IV will be consistent.
• The Hausman test is based on the difference between the OLS and the IV estimates. If the difference is significantly different from zero, the null hypothesis is rejected.