Two Questions That Require Different Tests
Instrumental-variable analysis raises two related but distinct questions. Is a regressor correlated with the structural error, so that ordinary least squares is inconsistent? And do the instruments satisfy the moment restrictions needed to identify the model? An endogeneity test addresses the first question; an overidentification test addresses the second. A result from one does not answer the other.
The distinction matters in practice. A researcher might find evidence that a regressor is endogenous while the available overidentifying restrictions are not rejected. Alternatively, the restrictions may be rejected even though an endogeneity test does not reject its null. The second pattern is especially difficult to interpret: if the instruments are invalid, the standard endogeneity test that relies on them may not have its claimed reference distribution.
The Model and the Two Null Hypotheses
Consider a linear structural equation with one potentially endogenous regressor \(x_i\), included exogenous regressors \(c_i\), and structural error \(u_i\):
Let \(w_i\) denote excluded instruments, and let \(z_i=(c_i',w_i')'\) collect the included exogenous variables and excluded instruments. The instrumental-variable approach requires relevant instruments and the moment restrictions \(E[z_i u_i]=0\). In particular, the excluded instruments must be uncorrelated with the structural error and excluded from the structural equation except through the endogenous regressor. The regressor is exogenous, in the relevant linear moment sense, if \(E[x_i u_i]=0\); it is endogenous if that moment is nonzero.
An endogeneity test takes exogeneity as its null hypothesis. In a Durbin–Wu–Hausman (DWH) test, the null is that the specified regressor is exogenous, while the alternative is that the OLS and IV approaches differ in a way consistent with endogeneity. The standard comparison requires a correctly specified structural equation, identification, and valid instruments. Under the null, both estimators are consistent, and OLS is typically more efficient; under the alternative, OLS is inconsistent while IV remains consistent if the instruments are valid.
An overidentification test instead takes the instrument restrictions as its null. If there are more independent instrument moment conditions than structural parameters to estimate, the model is overidentified. A GMM J test assesses whether the sample moments can be made sufficiently close to zero at the estimated parameters. Its null is that the overidentifying moment restrictions are jointly compatible with the model. Rejection can reflect an invalid instrument, but it can also reflect a misspecified structural equation or other failure of the maintained moment conditions.
What the Statistics Measure
One form of the DWH test uses a control-function regression. Regress \(x_i\) on the included exogenous variables and excluded instruments, and let \(\hat v_i\) be the fitted residual. Add \(\hat v_i\) to the structural regression. Under the stated linear-model and instrument assumptions, exogeneity corresponds to the coefficient on \(\hat v_i\) being zero. A Wald statistic for that restriction has an asymptotic chi-square distribution with one degree of freedom in this single-regressor case. The test is not a direct test of instrument validity: its interpretation depends on instruments being valid in the first place.
For the J test, write \(\bar g(\hat\theta)\) for the vector of sample moment averages evaluated at the GMM estimate \(\hat\theta\), and let \(W\) be the weighting matrix used to form the test. The statistic is
Under the null and standard large-sample regularity and identification conditions, an appropriately constructed J statistic has an asymptotic chi-square distribution. If there are \(q\) independent moment conditions and \(p\) parameters identified by those conditions, its degrees of freedom are \(q-p\). This is the number of restrictions left over after estimating the parameters: the estimation uses \(p\) degrees of freedom to fit the moments, leaving \(q-p\) overidentifying restrictions to assess. A just-identified model has \(q=p\), so there is no overidentifying restriction to test.
This count is about independent restrictions, not simply the number of variables given the label “instrument.” Collinear or redundant moments do not supply additional independent restrictions. A J-test rejection is joint: it does not identify which particular instrument or moment condition is responsible.
Worked Examples: Different Outcomes, Different Conclusions
Worked Example: Endogeneity Rejected, Overidentifying Restrictions Not Rejected
Suppose a hypothetical study estimates an outcome equation with one potentially endogenous treatment-intensity regressor, \(x_i\), and two excluded instruments. There are two independent excluded-instrument moments for the one endogenous regressor, so the model has \(q-p=2-1=1\) overidentifying restriction. Assume the structural form and other required conditions are plausible, and suppose the instruments are valid.
In the residual-inclusion version of the DWH test, the estimated coefficient on the first-stage residual is \(0.30\), with standard error \(0.10\). The one-restriction Wald statistic is
Compared with a chi-square distribution with one degree of freedom, this gives \(p\approx 0.0027\). At the 5% significance level, the exogeneity null is rejected. Conditional on the validity of the instruments and the other maintained assumptions, the result is evidence against treating this regressor as exogenous.
Suppose the GMM output also reports \(n=200\) and \(\bar g(\hat\theta)'W\bar g(\hat\theta)=0.002\). Then
With one overidentifying restriction, this has a chi-square reference distribution with one degree of freedom and \(p\approx 0.527\). The J test does not reject the joint moment restrictions at the 5% level. It does not prove that the instruments are valid. Together, these results are consistent with evidence of endogeneity and no detected failure of the overidentifying restrictions; they do not establish either conclusion with certainty.
Worked Example: Overidentifying Restrictions Rejected, Endogeneity Test Not Rejected
Now consider hypothetical output for a similar model with one endogenous regressor and two excluded instruments. The reported DWH residual-inclusion coefficient is \(0.05\), with standard error \(0.10\). The nominal one-restriction statistic is
Against a chi-square distribution with one degree of freedom, this gives \(p\approx 0.617\), so the test does not reject exogeneity at the 5% level. That is a failure to find evidence against the exogeneity null, not evidence that the regressor is exogenous.
Suppose the J-test output gives \(n=200\) and \(\bar g(\hat\theta)'W\bar g(\hat\theta)=0.040\). The calculation is
With one degree of freedom, \(p\approx 0.0047\), so the overidentifying restrictions are rejected at the 5% level. This is evidence that the set of moment restrictions is incompatible with the model. It does not show that the regressor is endogenous: a violation could instead arise because an instrument is invalid or because the structural equation is misspecified.
There is an important qualification to the apparent pairing of results. The standard DWH reference distribution assumes valid instruments. Once the J test rejects, that assumption is in doubt, so the DWH non-rejection cannot be relied on as a valid finding of exogeneity. The two calculated outputs illustrate that the tests ask different questions; the J rejection calls the standard interpretation of the DWH comparison into question.
Worked Example: A Just-Identified Model Has No J Test
Suppose there is one endogenous regressor and exactly one excluded instrument. For these excluded-instrument moments, \(q=1\) and \(p=1\), so
There is no overidentifying restriction, and a J test of the excluded-instrument moments is unavailable. A DWH test may still be computed if its requirements, including a valid instrument, are met. For example, a reported DWH statistic of \(4.00\) has a one-degree-of-freedom chi-square \(p\)-value of about \(0.0455\), which rejects exogeneity at the 5% level. This result is conditional on the instrument being valid, but this just-identified setup cannot use a J test to check that validity. The absence of a J test is not evidence that the instrument is valid.
How to Interpret the Pair of Results
A useful interpretation starts with the assumptions rather than a mechanical decision rule. First ask whether the model is overidentified; if it is not, an overidentification test is not available. If it is, a J test can assess the joint restrictions, subject to correct specification and the relevant GMM conditions. Separately, ask whether the DWH comparison is justified, particularly whether the instruments can be treated as valid. A non-rejection from either test is not confirmation of its null.
A J-test rejection does not say which moment fails, and it does not distinguish instrument invalidity from structural misspecification. Investigators may have to use substantive knowledge, design information, or further analyses to diagnose the source. Conversely, a J-test non-rejection does not certify every instrument: the test may have limited power, especially in finite samples.
The tests can therefore be thought of as addressing different links in an instrumental-variable argument. The endogeneity test asks whether ordinary least squares is compromised by the regressor’s relation to the structural error. The overidentification test asks whether the extra instrument restrictions fit with the estimated model. Neither supplies a general-purpose guarantee of causal identification.
Common Mistakes
- “A non-significant J test proves the instruments are valid.” It does not. It means the test did not detect a joint violation at the chosen significance level; power may be limited.
- “A significant J test proves the regressor is endogenous.” It does not. The rejection concerns the joint moment restrictions and could arise from an invalid instrument or model misspecification.
- “A non-significant DWH test proves exogeneity.” It does not. A non-rejection may reflect limited power, and the usual test also relies on valid instruments.
- “The J test identifies the bad instrument.” A standard overidentification test is a joint test. Rejection alone does not identify which restriction is responsible.
- “Every IV model has a J test.” A just-identified model has no overidentifying restrictions. With \(q=p\), the usual J-test degrees of freedom are zero.
Check Your Understanding
For each question, distinguish the test’s null from what its result can establish.
- In a model with one endogenous regressor and three independent excluded-instrument moments, how many overidentifying restrictions are there?
- A DWH test rejects exogeneity, but the J test does not reject. What conditional conclusion is justified, and what is not proved?
- A J test rejects while the reported DWH test does not. Why might the DWH non-rejection be difficult to interpret?
- Can a just-identified model use a J test to assess its instrument restrictions? Explain using the degrees-of-freedom count.
- Does a J-test rejection identify which instrument is invalid? What other explanation for rejection should be considered?