When clustered data make weak instruments harder to trust

Three problems that interact

Instrumental-variable methods help researchers study effects when an explanatory variable is entangled with unobserved influences on the outcome. An instrument supplies variation related to that explanatory variable but, under the model’s assumptions, unrelated to the outcome’s unobserved determinants.

Three difficulties can arise together. Instruments may be weak, meaning they provide little information about the explanatory variable. Having many instruments can undermine conventional statistical approximations. And observations may be clustered: errors can be related within groups rather than independent across every recorded observation.

Ligtenberg’s preprint, Inference in clustered IV models with many and weak instruments, addresses this combination. Its contribution concerns testing proposed effect sizes and constructing confidence sets, rather than providing a new point estimator.

Clustering changes the information available

The paper first studies identification strength: how much information the instruments provide about the effect being estimated. Its intuition is that observations sharing common disturbances contain less independent information than their raw count suggests. With strong within-group dependence, comparing the number of instruments with the number of clusters can therefore be more revealing than comparing it with the number of observations.

A theoretical result formalizes this concern in a particular Gaussian model with known error covariance and one endogenous explanatory variable. The relevant measure of instrument strength, called the concentration parameter, must account for within-cluster dependence. If this adjusted signal measure fails to grow sufficiently quickly relative to the number of instruments, uniformly reliable detection of a fixed alternative cannot approach certainty.

The conclusion is not that every correlation pattern necessarily weakens identification. The paper’s examples show that the effect depends on how the instrument signal aligns with the dependence structure. Clustering can reduce the concentration parameter and can make it grow more slowly as the sample expands.

Remove within-cluster contributions, not just individual ones

The proposed methods extend two existing procedures: the jackknife Anderson–Rubin test and the jackknife score test. Both are designed to handle many instruments without requiring those instruments to be strong.

For independent observations, jackknifing removes particular self-products that would otherwise give the test statistic a nonzero expected value under the hypothesis being tested. With clustered data, that correction is insufficient: products involving different observations in the same cluster can also have nonzero expectations.

The cluster jackknife removes all these within-cluster contributions, retaining cross-cluster terms. Under the paper’s assumptions, appropriately standardized statistics jointly approach a normal distribution as the numbers of clusters and instruments increase.

The paper also develops cross-fitted variance estimators. These use fitted residuals calculated while leaving out relevant clusters, reducing a source of inflated variance estimates that can make tests less able to reject false hypotheses.

Simulations show benefits and trade-offs

The main simulation design uses 800 observations in 100 clusters. Size comparisons use 10,000 simulated datasets and a 5% significance threshold. Here, correct size means rejecting a true null hypothesis about 5% of the time.

Tests that ignore clustering reject too frequently in these experiments. The cluster jackknife score test stays close to the intended rejection rate across the instrument counts examined. The cluster jackknife Anderson–Rubin test with cross-fitted variance slightly overrejects; its version without cross-fitting performs closer to 5%.

Power experiments—testing the ability to reject false hypotheses—favor the cluster jackknife procedures over their conventional cluster-robust counterparts in the many-instrument settings studied. Neither new test dominates universally: the score version performs better in a design with particularly strong differences in error variability. A dominant cluster can also make the jackknife Anderson–Rubin test reject too often. These are simulation findings, not universal performance guarantees.

A historical application changes some conclusions

The empirical illustration revisits Dube and Harish’s study of queenly reign and warfare. It contains 3,586 polity-year observations from 18 European polities, grouped into 176 reign clusters. Instruments use the previous monarch’s family composition; expanded specifications add potentially weak interactions.

For the full specification, the conventional cluster Anderson–Rubin 95% confidence set runs from −0.021 to 0.547, including zero. The cluster jackknife Anderson–Rubin set is 0.021 to 0.570, while the cluster jackknife score set is 0.038 to 0.531. Both exclude zero and support a positive effect within the model. These coefficients represent changes in war probability, not percentage changes.

What robustness does not establish

The theory assumes independence between clusters, valid instruments, constant effects, moment conditions and restrictions preventing excessive cluster dominance. Cross-fitted variance estimation requires stronger conditions than the simpler variance estimators. The paper assumes the chosen clustering level is adequate; it does not determine that level.

The historical results remain conditional on the identification assumptions. Better-calibrated tests do not establish instrument validity or justify general claims about female leadership. The preprint’s narrower lesson is that clustering, instrument weakness and instrument count must be considered together when assessing statistical evidence.