0

Suppose we have a linear IV model with: $$ y_i=\beta x_i + \epsilon_i$$ $$x_i=\gamma z_i + u_i $$with $E(x_i \epsilon_i)\neq 0$ and $E(z_i \epsilon_i)=0$.

Then we can estimate the residuals in the reduced form $\hat{u}_i$ and run the regression: $$ y_i=\beta x_i + \alpha \hat{u}_i + v_i$$

which yields a consistent estimator for $\beta$ (equivalent to 2SLS).

We can run the test for $H_0: \alpha = 0$ using a standard t-test.

Would this be a valid test to check the validity of the instrument (i.e. $H_0: E(z_i \epsilon_i)=0$)? I've read that it's not possible to test for instrument validity but I don't understand why this test is not acceptable. What is this testing for?

Lorenzo
  • 63
  • 5
  • 1
    What you describe is a Wu-Hausman test. Have a look at https://stats.stackexchange.com/a/495476/264076 and the link in that post. If you can’t reject alpha=0 it suggests that X is exogenous, assuming the instrument is valid. – Jonathan Jan 09 '21 at 18:31
  • so is this a test for endogeneity of $x_i$ and not for instrument validity? – Lorenzo Jan 14 '21 at 16:55
  • Exactly, but remember that it is conditional on that the instrument is valid. – Jonathan Jan 14 '21 at 20:46

0 Answers0