Why even use a Hausman test? Isn't the logic behind it circular?

[Carl Isaac]

Suppose I want to estimate the following:

y = a + B₁x₁ + B₂x₂ + e

I highly suspect that x₁ is endogenous, therefore I use z₁ to instrument for it. If I want to test whether z₁ itself is endogenous, I have to have at least one other exogenous instrument, z₂, to run a Hausman test. The question I have is whether this is ever useful for actually testing whether an instrument meets the exclusion restriction?

For instance, if it is not totally clear that my other exogenous instrument, z₂, is truly exogenous, then won't this cause the test results to be meaningless? It seems like the test is circular unless you already have a good instrument to use, in which case, the test only seems useful for testing whether or not the model actually needs to be 2SLS as opposed to OLS.

[Jeff Wooldridge]

Here's my take on it. If you have z1 and z2 and the Hausman test rejects, you have strong evidence that at least one of these is endogenous. If you feel very good about exogeneity of z1 then it becomes a test of exogeneity of z2. For example, maybe z1 is effectively randomized but not very strong. You have some doubts about z2, but you'd like to add it if possible to improve efficiency. Then I think the test can be useful.

Problems arise when z1 and z2 are both suspect -- say, mother's education and father's education as IVs for child's education. They are both likely to be endogenous but the Hausman test probably won't pick that up because they're endogenous in a similar way.

A rejection using the Hausman test means you have to rethink your IVs, and possible drop one where the story isn't convincing. A failure to reject need not be good news because the test can have poor power when z1 and z2 are endogenous in ways that cause similar bias.

[Carl Isaac]

Thanks Jeff! This is useful.

[Sebastian Kripfganz]

Here's my take on it. If you have z1 and z2 and the Hausman test rejects, you have strong evidence that at least one of these is endogenous. If you feel very good about exogeneity of z1 then it becomes a test of exogeneity of z2. For example, maybe z1 is effectively randomized but not very strong. You have some doubts about z2, but you'd like to add it if possible to improve efficiency. Then I think the test can be useful.

A problem is however that the Hausman test may have low power if the exogenous instrument z1 is weak. I have recently looked at this in an Economics Letters paper with Jan Kiviet for the Sargan/Hansen test, but I expect the result to carry over to the Hausman test as well.

[Jeff Wooldridge]

A problem is however that the Hausman test may have low power if the exogenous instrument z1 is weak. I have recently looked at this in an Economics Letters paper with Jan Kiviet for the Sargan/Hansen test, but I expect the result to carry over to the Hausman test as well.

Indeed. Everything hinges on sufficiently strong IVs.