Cluster-Robust First-Stage Strength

Why Clustered First-Stage Inference Matters

A first-stage regression asks whether the excluded instruments help predict the endogenous regressor after accounting for included covariates. When observations within a cluster may share shocks—for example, patients within a clinic or workers within a firm—their errors need not be independent. Treating every observation as independent can then give a misleading estimate of uncertainty about instrument relevance.

The earlier material on the First-Stage F Statistic and Heteroskedasticity-Robust First-Stage Tests establishes the ordinary and heteroskedasticity-robust approaches. Here the key change is that the variance estimate allows arbitrary dependence within each cluster, while requiring independent clusters. This changes the uncertainty calculation, not the first-stage regression itself.

Definition: Let $i$ index observations and $g=1,\ldots,G$ index clusters, with $N$ total observations. For one endogenous regressor $D_{ig}$, let $W_{ig}$ be a vector of included covariates and $Z_{ig}$ a vector of $q$ excluded instruments. The first-stage linear regression is $$ D_{ig}=W_{ig}'\gamma+Z_{ig}'\pi+u_{ig}. $$ The joint first-stage relevance null is $H_0:\pi=0$, meaning that all $q$ excluded-instrument coefficients are zero conditional on $W_{ig}$.

A rejection of this null is evidence that the instruments are jointly associated with the endogenous regressor, conditional on the included covariates, under the specified model. It is not a test of the exclusion restriction or instrument exogeneity. Nor does rejection by itself establish that the instruments are strong enough for reliable conventional instrumental-variables inference.

The Cluster-Robust Variance and Joint Test

Assume the first-stage conditional mean is correctly specified, the relevant regressors have adequate variation, and the $G$ clusters are independent of one another. Within a cluster, the errors may be heteroskedastic and arbitrarily correlated. Inference also relies on there being enough independent clusters for the chosen approximation; a very large $N$ does not compensate automatically for very small $G$.

Write $X$ for the design matrix containing all included covariates and excluded instruments, and $\hat u_g$ for the vector of first-stage OLS residuals in cluster $g$. A common uncorrected cluster-robust (CR0) covariance estimator is

\widehat{\operatorname{Var}}_{\mathrm{cl}}(\hat\beta) =(X'X)^{-1} \left[\sum_{g=1}^{G}X_g'\hat u_g\hat u_g'X_g\right] (X'X)^{-1},

where $\hat\beta=(\hat\gamma',\hat\pi')'$, and $X_g$ contains the rows of $X$ for cluster $g$. The middle term is the cluster “meat”: it forms a score contribution for each cluster and then sums the outer products of those contributions. Thus, dependence among observations within a cluster is retained rather than treated as if each observation supplied an independent score.

Many implementations multiply this estimator by a finite-sample correction, such as $c=\frac{G}{G-1}\frac{N-1}{N-k}$, where $k$ is the number of estimated regression coefficients. Other corrections and reference degrees of freedom are also used. Therefore, a reported cluster-robust test should identify the correction and reference distribution rather than relying on the label “clustered” alone.

Let $R$ select the $q$ entries of $\pi$ from $\beta$. The cluster-robust Wald statistic for $H_0:R\beta=0$ is

W=(R\hat\beta)'[R\widehat{\operatorname{Var}}_{\mathrm{cl}}(\hat\beta)R']^{-1}(R\hat\beta).

Under a large-number-of-independent-clusters approximation, $W$ is compared with a chi-square distribution with $q$ degrees of freedom. Equivalently, $F_{\mathrm{cl}}=W/q$ can be compared with an $F$ reference distribution. A commonly used finite-sample convention is $F(q,G-1)$, but software may use a different correction or denominator degrees of freedom. The convention must be stated when interpreting a p-value.

Why the Variance Uses Cluster-Level Contributions

The structure of the estimator follows from the way regression uncertainty accumulates. For OLS, the coefficient error depends on the sum of regressor-residual products. If errors were independent across observations, each observation could contribute separately to the variance. With within-cluster dependence, the relevant independent units are instead the clusters: observations in the same cluster can move together.

Form a score for each cluster.
For cluster $g$, the OLS score contribution is $X_g'\hat u_g$, which adds the regressor-residual products over that cluster’s observations.

Aggregate uncertainty across clusters.
Take the outer product of each cluster score and sum across $g$. This preserves the within-cluster cross-products that would be lost if observations were incorrectly treated as independent.

Scale to obtain coefficient uncertainty.
Premultiply and postmultiply the summed score outer products by $(X'X)^{-1}$, producing the cluster-robust covariance estimate used in the joint Wald test.

For intuition, suppose two observations in a cluster have similar residuals and similar instrument values. Their score contributions may reinforce each other. An observation-level variance calculation omits the resulting cross-product; the cluster-level outer product includes it. Dependence can increase or decrease a particular coefficient variance, however, so clustering does not mechanically inflate every standard error.

For a single excluded instrument, the joint test reduces to a one-coefficient test: $W=(\hat\pi/SE_{\mathrm{cl}}(\hat\pi))^2$. For multiple instruments, the test uses the full covariance matrix of their estimated coefficients. Testing each coefficient separately is not a substitute for testing the joint null, because the coefficients may be correlated.

Worked Example: A Single Instrument and Cluster Score Contributions

Suppose a hypothetical first-stage regression has one excluded instrument, and the instrument has been residualized with respect to the included covariates. Let $A=\sum_{ig}\tilde Z_{ig}^{\,2}=100$, and suppose the six cluster-level score contributions $S_g=\sum_{i\in g}\tilde Z_{ig}\hat u_{ig}$ are $2,-2,1,-1,0,0$. Use the uncorrected CR0 variance and the finite-sample reference $F(1,G-1)$. The estimated instrument coefficient is $\hat\pi=0.20$ units of $D$ per instrument unit.

For this one-coefficient regression, the cluster-robust variance is $A^{-2}\sum_g S_g^2$. Substitution gives

\widehat{\operatorname{Var}}_{\mathrm{cl}}(\hat\pi) =\frac{2^2+(-2)^2+1^2+(-1)^2+0^2+0^2}{100^2} =\frac{10}{10000}=0.001, \qquad SE_{\mathrm{cl}}(\hat\pi)=\sqrt{0.001}=0.03162.

The coefficient test statistic is $t=0.20/0.03162=6.325$, so $W=t^2=40.00$, and $F_{\mathrm{cl}}=40.00$ because $q=1$. With $G=6$, the specified reference is $F(1,5)$, giving a p-value of approximately $0.0015$. Under this model and reference convention, the data provide evidence against the null that the instrument coefficient is zero. The calculation supports conditional relevance; it does not establish instrument validity or settle whether the instrument is sufficiently strong for all IV inference.

Joint Tests, Cluster Counts, and Interpretation

Worked Example: Testing Two Instruments Jointly

In a hypothetical first stage with two excluded instruments, suppose $\hat\pi=(0.18,0.12)'$, and the estimated cluster-robust covariance matrix for these coefficients is

\widehat{\operatorname{Var}}_{\mathrm{cl}}(\hat\pi) = \begin{pmatrix} 0.010 & 0.006\\ 0.006 & 0.009 \end{pmatrix}.

There are $G=20$ independent clusters. Use the finite-sample reference $F(2,19)$. The determinant of the covariance matrix is $0.010(0.009)-0.006^2=0.000054$, so its inverse is

\begin{pmatrix} 0.010 & 0.006\\ 0.006 & 0.009 \end{pmatrix}^{-1} = \begin{pmatrix} 166.667 & -111.111\\ -111.111 & 185.185 \end{pmatrix}.

The joint Wald statistic is

W = \begin{pmatrix}0.18&0.12\end{pmatrix} \begin{pmatrix} 166.667 & -111.111\\ -111.111 & 185.185 \end{pmatrix} \begin{pmatrix}0.18\\0.12\end{pmatrix} =3.267, \qquad F_{\mathrm{cl}}=\frac{3.267}{2}=1.633.

The $F(2,19)$ p-value is approximately $0.220$. At $\alpha=0.05$, this test does not reject the joint null that both coefficients are zero. That conclusion is about the joint test under the stated clustered-inference convention; it does not prove the instruments have no predictive relationship. The covariance matters: the test combines both coefficient estimates and their estimated dependence rather than treating the two individual tests as independent.

Worked Example: Few Clusters Change the Reference Comparison

Suppose a hypothetical first-stage coefficient is $0.30$, with a cluster-robust standard error of $0.10$, and the analysis has $G=5$ independent clusters. For one excluded instrument, $W=(0.30/0.10)^2=9.00$, so $F_{\mathrm{cl}}=9.00$. With the specified finite-sample convention, the reference distribution is $F(1,4)$, for which the two-sided coefficient-test p-value is approximately $0.0399$.

The large-cluster chi-square comparison would instead compare $W=9.00$ with $\chi^2_1$, giving a p-value of approximately $0.0027$. These are different approximations, and with only five clusters the large-$G$ approximation is especially questionable. Under the stated $F(1,4)$ convention, the test rejects at the 5% level, but the result should be reported with the small cluster count made explicit. A reference distribution cannot create information that the design does not contain.

Diagnostics, Alternatives, and Limits

Before interpreting a cluster-robust first-stage test, check whether the clustering matches the dependence structure and the study design. If treatment or instrument assignment occurs at the cluster level, the number of independent clusters carrying assignment variation is particularly important. A large number of observations within a few clusters can make the data look extensive without providing many independent sources of instrument variation.

Check the number and size of clusters. Very few clusters can make conventional cluster-robust approximations unreliable. Unequal cluster sizes or a small number of dominant clusters may also make asymptotic approximations poor.
Check where the instrument varies. If the instrument is constant within clusters, its identifying variation may come mainly from differences across clusters. A high observation count does not replace a sufficient number of independent clusters.
Inspect the test specification. The null must include all excluded-instrument coefficients intended for the joint relevance test, conditional on the included covariates. Verify that the reported covariance estimator is clustered at the intended level.
Compare uncertainty estimates thoughtfully. A conventional first-stage F test assumes independent, homoskedastic errors; a heteroskedasticity-robust test allows heteroskedasticity but not arbitrary within-cluster dependence. A cluster-robust test addresses within-cluster dependence under independent-cluster assumptions. Differences across results can reveal sensitivity, but the model and design—not whichever statistic is largest—determine the appropriate analysis.

The cluster-robust Wald test is a test of the first-stage coefficient restrictions, not a universal weak-instrument diagnostic with a single critical value that applies across designs. Its null distribution is an approximation for the relevance test. In particular, a significant result should not be read as a guarantee that conventional IV confidence intervals or tests have reliable coverage when instruments are weak. Conversely, a non-significant result may reflect imprecision, including the limited independent information in a small number of clusters.

The clustered variance estimator also does not solve misspecification, invalid instruments, or a lack of meaningful instrument variation. If clusters are not independent—for example, if dependence persists across clusters—the stated cluster-robust justification does not apply without an approach suited to that dependence. Small-sample corrections can change estimated uncertainty and reference degrees of freedom, but they are not a substitute for an adequate design or enough independent clusters.

Key takeaway: Cluster-robust first-stage testing aggregates score contributions by cluster and tests the excluded-instrument coefficients jointly. The number of independent clusters, within-cluster dependence, covariance correction, and reference distribution all matter for the reported uncertainty and p-value.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A first stage has three excluded instruments. What is the joint null, and how many restrictions does the test impose?
Why can a cluster-robust standard error differ from an observation-level heteroskedasticity-robust standard error?
A test has $W=12$, $q=3$, and $G=16$. Under the stated convention $F(q,G-1)$, what statistic and reference distribution are used?
Does rejection of the cluster-robust first-stage null establish that the instruments satisfy the exclusion restriction? Explain.

Solutions. (1) The null is that all three excluded-instrument coefficients equal zero conditional on included covariates; it imposes $q=3$ restrictions. (2) The cluster estimator includes cross-products among score contributions from observations in the same cluster, while the observation-level estimator treats those contributions as independent. (3) $F_{\mathrm{cl}}=W/q=12/3=4.00$, compared with $F(3,15)$. (4) No. Rejection is evidence of conditional first-stage relevance under the model and inference assumptions; it does not test the exclusion restriction or instrument exogeneity.