Worked Problems in Clustered and HAC Inference

Choosing Inference for the Dependence Structure

Regression inference depends not only on the fitted equation but also on how observations may be dependent. If errors can move together within a school, firm, or other group, treating every observation as independent can understate uncertainty. If observations are ordered over time, serial dependence can create the same problem. The practical task is to identify the relevant independent sampling units, estimate uncertainty at the appropriate level, and use a reference distribution suited to the number and arrangement of those units.

The examples below compare four settings: unequal-sized independent clusters, few clusters, two-way dependence, and serially correlated errors. In every case, the regression coefficient is an ordinary least-squares estimate; what changes is the covariance estimator and the justification for the reference distribution.

Unequal Cluster Sizes: Use Cluster Scores

Suppose observations are partitioned into \(G\) clusters. A one-way cluster-robust covariance estimator allows arbitrary dependence among observations in the same cluster, while requiring clusters to be independent of one another. In a standard sandwich form, let \(X_g\) be the regressors for cluster \(g\), and let \(\hat{u}_g\) be its vector of fitted residuals. Then the uncorrected estimator is

The quantity \(X_g'\hat{u}_g\) is the cluster’s score contribution. Its outer product retains the within-cluster covariation that an observation-level heteroskedasticity-robust estimator would omit. Unequal cluster sizes require no separate averaging step: the regression design and residuals determine each cluster’s contribution.

A common finite-sample version multiplies this estimate by \( \frac{G}{G-1}\frac{n-1}{n-k} \), when these denominators are defined. This CR1 scaling is a convention, not a general solution to having very few clusters. In particular, it does not by itself make a normal reference distribution reliable when \(G\) is small.

Few Clusters: Adjust the Covariance and the Reference Distribution

With few independent clusters, cluster-robust standard errors can be downward biased, and normal or large-\(G\) approximations can be inaccurate. A careful analysis should examine cluster count, cluster leverage, and whether a small number of clusters dominate the coefficient of interest. CR2 adjusts cluster residual contributions for leverage under a working model with independent, homoskedastic errors across observations. That working model is used to calibrate the adjustment; the target inference can still allow cluster-level dependence.

CR2 does not supply a universal degrees-of-freedom correction. For a coefficient or contrast, Satterthwaite degrees of freedom approximate the sampling variation of the adjusted variance estimate. When the estimated variance can be represented approximately as a weighted sum of independent squared standard normal variables with nonnegative weights \( \lambda_1,\ldots,\lambda_m \), the corresponding degrees of freedom are

The resulting statistic is compared with a \(t\) distribution using these approximate degrees of freedom. A small \(df_{\mathrm{Satt}}\) reflects that the information for the particular contrast is concentrated in relatively few independent contributions—even if the total number of clusters is somewhat larger.

Two-Way Dependence: Inclusion and Exclusion

Some regressions have two overlapping sources of dependence. For example, observations may be related within both firms and years. If there are many independent clusters in each dimension, the two-way cluster-robust covariance estimator combines the one-way estimates by inclusion-exclusion:

Here \(A\) and \(B\) identify the two clustering dimensions, and \(A\cap B\) identifies their intersections (such as firm-year cells). The intersection term is subtracted because dependence shared by both dimensions has otherwise been counted twice. Each component is formed from cluster-summed regression scores; finite-sample scaling can be applied to components according to the chosen implementation. This estimator relies on having a sufficiently informative number of clusters in each dimension. A large number of intersections cannot compensate for having only a handful of clusters in one dimension.

Serial Dependence: HAC Inference

For observations ordered over time, heteroskedasticity-and-autocorrelation-consistent (HAC) covariance estimation allows a specified range of serial dependence while relying on a time-series asymptotic approximation. Let \(x_t\) be the row of regressors at time \(t\), \(\hat{u}_t\) its fitted residual, and \(\hat{g}_t=x_t'\hat{u}_t\) the score vector. A kernel HAC covariance estimate has the form

Here \(T\) is the number of time observations, \(L\) is the bandwidth, and \(w_\ell\) is the kernel weight at lag \(\ell\). For the Bartlett kernel, \(w_\ell=1-\ell/(L+1)\). The choice of kernel and bandwidth determines how much estimated serial covariance contributes. Valid large-sample inference requires appropriate time-series regularity, such as weak dependence and a bandwidth that grows suitably slowly with the sample size. HAC inference usually uses an asymptotic normal reference distribution; it is not a general exact small-sample \(t\) procedure.

A Practical Selection Sequence

The same regression output can support different inferences depending on the design. Before reporting a standard error or test, make the dependence structure explicit and check which asymptotic information is available.

Common Mistake: Treating a Correction as a Cure

A frequent error is to apply a finite-sample multiplier to a cluster-robust covariance estimate and then assume that the resulting standard error makes a normal test reliable with very few clusters. The multiplier adjusts scale; it does not create additional independent information or guarantee a suitable reference distribution. Few-cluster inference requires attention to leverage and degrees of freedom, such as CR2 with Satterthwaite inference when appropriate. Likewise, multiway clustering requires enough clusters in each dimension, and HAC methods require a defensible time-series approximation and bandwidth. A method should be chosen from the dependence structure and effective number of independent units—not from the largest available observation count.