Questions: Test of Overidentification: Hansen J-Test

Question 1

A researcher has one endogenous regressor and one instrument. They attempt to run the Hansen J-test. What happens?

• A. The test runs normally with 1 degree of freedom
• B. The test cannot be computed — there are zero overidentifying restrictions when m = k
• C. The test runs but requires a heteroskedasticity correction
• D. The test is equivalent to a standard t-test on the first-stage coefficient

Answer: (B)

The J-statistic has χ²(m − k) degrees of freedom. When m = k (just-identified), m − k = 0 — there are no surplus instruments to test. Every instrument has been used for identification and none remain as free restrictions to check against. The test is undefined in the just-identified case; overidentification is the prerequisite for running it.

Question 2

A researcher uses quarter-of-birth and distance-to-college as instruments for education and finds the J-test rejects at the 5% level. What can they conclude?

• A. Both instruments violate the exclusion restriction
• B. The first stage is too weak to support IV estimation
• C. At least one instrument correlates with the structural error, but the test cannot identify which one
• D. The two instruments are collinear and cannot be used together

Answer: (C)

Rejection indicates that the instruments imply inconsistent estimates of β — a sign that at least one is correlated with the error term. But the J-test cannot decompose this into 'which instrument is the culprit.' Searching over instrument subsets until the test passes exploits in-sample correlation structure and is a form of specification search, not a solution.

Question 3

Passing the J-test is sufficient evidence that most instruments satisfy the exclusion restriction.

• T. True
• F. False

Answer: False

Passing the J-test is consistent with all instruments being valid, but it does not prove validity. The test only has power to detect deviations that produce inconsistency *between* instruments. If all instruments are biased in the same direction — for example, all correlated with omitted ability in a wage regression — the J-test will pass even though every instrument violates the exclusion restriction. The exclusion restriction remains untestable for the component shared by all instruments.

Question 4

The Hansen J-test detects instrument invalidity by checking whether the 2SLS residuals are correlated with the instruments.

• T. True
• F. False

Answer: True

If all instruments are valid (uncorrelated with the structural error u), they should be uncorrelated with the residuals from a consistent 2SLS estimator. The J-statistic is computed as n × R² from regressing 2SLS residuals on all instruments. If any instrument is invalid — correlated with u — it will also correlate with the residuals, inflating this R² and pushing J above the χ²(m − k) critical value.

Question 5

Why can the J-test only be run in the overidentified case, and what does each additional instrument add to the test?

Model answer: In the just-identified case (m = k), all instruments are used up to estimate the structural coefficients and there are no remaining degrees of freedom to test anything. Each additional instrument beyond the number of endogenous regressors adds one overidentifying restriction — one testable implication of joint instrument validity. A second instrument means you can check whether both instruments imply the same β estimate; if they don't, at least one is invalid. The J-test's power therefore grows with the number of extra instruments.

This is the fundamental tension in IV: you need instruments to achieve identification, but identification exhausts the instruments. Only surplus instruments can be tested. This is why researchers often seek more instruments than strictly necessary — overidentification allows at least a partial check on validity, which the just-identified case forbids entirely.