Why Test the Rank Condition?
Instrumental-variable estimation requires more than a list of plausible instruments. The excluded instruments must contain enough independent predictive information about the endogenous regressors. If that information is insufficient, the model is underidentified: the data cannot separately determine the coefficients of interest through the instrument-based variation.
An underidentification test asks whether the population relationship between excluded instruments and endogenous regressors has deficient rank. It is a test of an exact rank condition, not a general certificate that the instruments are strong enough for reliable estimation. That distinction matters especially when instruments are only weakly related to the endogenous regressors.
The rank condition is about independent directions of prediction, not simply the number of instruments. For example, two excluded instruments do not identify two endogenous regressors if their relationships with those regressors are proportional in the population. Included exogenous controls are conditioned on because the relevant question is whether the excluded instruments contribute identifying variation beyond those controls.
The Kleibergen–Paap rk LM Test
The Kleibergen–Paap rk LM test is a rank-based underidentification test that allows for heteroskedasticity. Its null is rank deficiency, and its alternative is full column rank. Under the null and standard large-sample regularity conditions, its statistic is compared with a chi-square reference distribution. With \(q\) excluded instruments and \(k\) endogenous regressors, the reference degrees of freedom are \(q-k+1\), provided \(q \geq k\).
The statistic is a score-type test of whether the observed instrument–regressor relationships contain evidence against deficient rank. In general, it is not obtained by treating each first-stage coefficient as an unrelated restriction and applying an ordinary joint test. The rank null is a matrix-rank hypothesis, and the Kleibergen–Paap procedure accounts for that structure. With multiple endogenous regressors, use the reported rk LM statistic and its rank-test reference distribution rather than substituting an arbitrary first-stage \(F\) statistic.
For one endogenous regressor, the rank condition reduces to a simple first-stage question: are all \(q\) excluded-instrument coefficients zero, or is at least one nonzero? In this special case, a heteroskedasticity-robust score test gives a useful explicit formulation. Let \(z_i\) be the vector of excluded instruments residualized on the included controls, and let \(v_i\) be the residual from the restricted first-stage regression of the endogenous regressor on those controls alone. Define the scaled score \(S\) and its robust covariance estimate \(V\) as follows.
Here the restricted model imposes the null that the excluded instruments have no partial predictive relationship with the endogenous regressor. The covariance estimate allows the conditional variance of \(v_i\) to differ across observations. Provided \(V\) is nonsingular and the observations meet the usual independence and finite-moment conditions, the null distribution is asymptotically chi-square with \(q\) degrees of freedom. This single-regressor score formulation illustrates the logic of an underidentification test; with multiple endogenous regressors, the Kleibergen–Paap rk LM test addresses the rank hypothesis itself.
Why the Score Statistic Has a Chi-Square Reference
The single-regressor formulation makes the test logic transparent. Under \(H_0\), the excluded instruments have no partial predictive relationship with the endogenous regressor, so the population score has mean zero. With independent observations and appropriate moment conditions, the central limit theorem gives an approximately normal distribution for the scaled sample score. If its covariance is estimated consistently by \(V\), then the score standardized by that covariance has an asymptotic standard normal distribution in \(q\) dimensions.
The quadratic form \(S'V^{-1}S\) measures the squared length of that standardized score. The sum of squares of \(q\) independent standard normal components has a chi-square distribution with \(q\) degrees of freedom. Thus, under the null,
This argument explains the score-test reference distribution in the single-regressor case. For a general rank restriction, the null involves a matrix whose rank is below \(k\), rather than a regular collection of \(qk\) independent zero restrictions. The Kleibergen–Paap rk LM statistic is constructed for that rank problem; its chi-square reference degrees of freedom are \(q-k+1\), not \(qk\).
Worked Examples
Worked Example: Robust Score Test With One Endogenous Regressor
Suppose a hypothetical analysis has one endogenous regressor and two excluded instruments, after including and accounting for the exogenous controls. A robust score calculation gives \(S=(1.20,1.60)'\) and \(V=I_2\), where \(I_2\) is the two-by-two identity matrix. Test underidentification at \(\alpha=0.05\).
With one endogenous regressor, the null is that both excluded-instrument coefficients are zero. The score statistic is
There are \(q=2\) excluded instruments, so the reference distribution is \(\chi^2_2\). The p-value is \(P(\chi^2_2 \geq 4.00)=0.1353\). Since \(0.1353>0.05\), this test does not reject the null of underidentification at the 5% level. The result is not proof that the instruments have no predictive relationship; it says the data do not provide sufficient evidence against the rank-deficiency null at this threshold.
Worked Example: Rank Test With Multiple Endogenous Regressors
Consider a hypothetical model with \(q=4\) excluded instruments and \(k=2\) endogenous regressors. A heteroskedasticity-robust Kleibergen–Paap rk LM test reports a statistic of 9.20. Test underidentification at the 5% level.
The null is \(H_0:\operatorname{rank}(\Pi)<2\), and the alternative is \(H_a:\operatorname{rank}(\Pi)=2\). The reference degrees of freedom are
For a chi-square variable with 3 degrees of freedom, the upper-tail probability at 9.20 is approximately 0.0267. Because \(0.0267<0.05\), reject the underidentification null at the 5% level. The test provides evidence that the instrument–regressor relationship has full column rank in this model. It does not establish that the relationships are sufficiently strong for conventional 2SLS inference to perform well.
Worked Example: Rejection Does Not Establish Strong Identification
Suppose a hypothetical study has \(q=3\) excluded instruments and \(k=2\) endogenous regressors. Its robust rk LM statistic is 20.0. The reference degrees of freedom are \(3-2+1=2\), and the p-value is
The test strongly rejects rank deficiency at conventional significance levels. Now suppose separate first-stage strength diagnostics indicate that the instrument-related variation is small relative to residual variation. These results can coexist: the rank test asks whether there is evidence of any full-rank relationship, while strength diagnostics ask whether the identifying relationships are large enough to support reliable conventional inference. In a sufficiently large sample, even a small but nonzero relationship may yield a very small underidentification-test p-value.
Therefore, the correct conclusion is that the data reject deficient rank, but the strength diagnostics still raise concern about weak identification. A small rank-test p-value alone is not a reason to proceed as if weak-instrument concerns have been resolved.
What the Test Does—and Does Not—Tell You
Underidentification and weak identification are related but distinct. Underidentification concerns a failure of the rank condition: the population relationship does not provide enough independent directions to identify all endogenous-regressor coefficients. Weak identification concerns relationships that may have full rank but are too small, relative to sampling uncertainty, for standard large-sample approximations to be dependable.
This difference also explains why a significance test is not a strength threshold. The test's p-value depends on both the underlying relationships and the amount of information in the sample. As the sample grows, a small nonzero relationship can become statistically distinguishable from exact rank deficiency without becoming practically strong for estimation or inference.
A rank test also does not test whether instruments satisfy the substantive assumptions needed for causal interpretation. In particular, rejecting underidentification does not establish that the instruments are independent of unmeasured causes of the outcome or that they affect the outcome only through the endogenous regressors. Those are validity questions, not rank questions.
| Question | What it concerns | What rejection does not establish |
|---|---|---|
| Is the model underidentified? | Whether the instrument–regressor coefficient matrix has full column rank | That the identifying relationships are strong |
| Are instruments weak? | The magnitude and sampling precision of identifying relationships | That the instruments are valid |
| Are instruments valid? | Substantive assumptions supporting causal interpretation | That the rank condition holds or that instruments are strong |
Common Mistakes
- “The rank test rejects, so the instruments are strong.” Rejection is evidence against exact rank deficiency, not evidence that the relationships are large enough for reliable conventional inference. Examine strength diagnostics as a separate question.
- “There are at least as many instruments as endogenous regressors, so identification is guaranteed.” The order condition \(q \geq k\) is necessary, but the rank condition also requires independent predictive directions. Count alone cannot establish it.
- “A non-significant result proves underidentification.” Failure to reject may reflect limited information. It is not proof that the population rank is deficient.
- “Rejecting the null validates the instruments.” The test concerns relevance and rank. It does not verify the assumptions required for causal interpretation.
- “Any first-stage joint test is the multiple-regressor rank test.” With multiple endogenous regressors, the relevant null is deficient matrix rank. Use the rank-based statistic and its stated reference distribution, rather than treating all first-stage coefficients as unrelated restrictions.
Check Your Understanding
Answer each question by distinguishing the rank hypothesis from claims about strength or validity.
- A model has three excluded instruments and two endogenous regressors. What are the underidentification null and the alternative, and how many degrees of freedom does the rk LM reference distribution use?
- In the one-endogenous-regressor case, a robust score statistic is 6.00 with two excluded instruments. Write its null reference distribution and explain what a small p-value would indicate.
- Why can a very small rk LM p-value occur even when separate diagnostics suggest weak instruments?
- A test does not reject underidentification at the 5% level. Does that prove the population rank is deficient? Explain.
- What does rejection of the underidentification null establish, and what does it not establish about causal validity?