A 🐴 race: The Wild Cluster Bootstrap vs Satterthwaite-corrected Sandwich Estimators when the number of Clusters is small

Figure 1: Toulose-Lautrec, Le Jockey, 1899

Simulation 1: Small number of clusters G & balanced cluster sizes

To initiate the horse race, you simply have to run the sim_balanced_clusters() function, though I want to note that on my laptop, this takes around 2h while using multiple cores. By default, the wild cluster bootstrap will run with \(B = 9999\) bootstrap iterations throughout all simulations.

library(clusteredErrorsSims)
set.seed(1234)
sim_balanced_clusters(n = 1000, n_sims = 1000, rho = 0.7, workers = 4)

Here are the results for the first simulation:

Figure 2: Simulation results. N = 1000, rho = 0.7, balanced cluster sizes

There are three takeaways from figure 2:

• As expected, inference with non-robust standard errors is severely biased.

• For less than 50 clusters, the coverage rate for the CRVE based confidence intervals is always lower than 95%: inference based on uncorrected CRVEs underestimate the variability of the parameter of interest, \(\beta_1\).

• The wild cluster bootstrap and cluster robust variance estimator with Satterthwaite correction perform astonishingly well for 3 or more clusters. For a number of cluster with of \(3 \leq G \leq 10\), the Satterthwaite correction seems to perform slightly better than the wild cluster bootstrap.

Simulation 2: Wildly different cluster sizes

Instead of simulating balanced cluster sizes, I now follow MacKinnon & Webb and simulate group sizes that mimic the relative size of the US states (minus Washington DC) for \(G=50\) and \(G = 100\) clusters. The dgp is unchanged, but in parallel to MacKinnon & Webb’s work, I set the number of observations \(N\) to \(2.000\). I also increase the number of Monte Carlo simulations to n_sim = 10.000 and repeat the analysis for a range of intra-cluster correlations \(\rho\).

Figure 3: Simulation results. N = 2000, 50 and 100 clusters, wildly different cluster sizes

Both for \(G = 50\) and \(G = 100\), the wild cluster bootstrap and Satterthwaite corrected errors perform equally well and achieve close to 95% coverage for all intra-cluster correlations \(\rho\). The unadjusted CRVEs instead only achieve coverage rates of around 93 and 94%.

You can reproduce Figure 2 by running

wildly_different_sim(n = 2000, n_sims = 5000, workers = 4)

Once again, note that this function will run for a very long time.

Treatment Effects

The last simulation investigates a topic that has received considerable attention: regression models where the treatment occurs at the cluster level, but only few clusters are treated (e.g. “How much should we trust DiD estimates?“)? For the sake of simplicity, I do not simulate a”full” DiD model with 2-way fixed effects and potential error correlations across time but restrict myself to replacing \(X_{ig}\) in the model above by a treatment assignment dummy \(D_{ig}\) – the simulation hence mirrors a cluster randomized experiment. (Loosely) following MacKinnon & Webb once again, I then simulate \(N=2000\) observations with intra-cluster correlation \(\rho = 0.5\) and vary the number of clusters that are treated. The simulations are repeated for different proportions of treated clusters \(P \in \{1/50, …, 1\}\), where the clusters are a) of equal size, b) US-state sized and sorted in increasing order and c) US-state sized and sorted in decreasing order.

treatment_effect_sim(n = 2000, n_sims = 10000, workers = 4)

Figure 4: Treatment Effect Simulations, N = 2000, 50 clusters, treatment effects. The x axis denotes the share of treated clusters, either in increasing or decreasing cluster size.

Once again, both the wild cluster bootstrap and Satterthwaite correction perform well for a wide range of treated clusters. For more extreme shares of treated clusters, the wild cluster bootstrap’s coverage rate approaches 1, while the Satterthwaite corrected CRVE’s coverage rate is as low as the regular CRVE’s, whose coverage rate starts to deviate from the desired coverage level of 95% already at less extreme shares of treated clusters than both bootstrap and Satterthwaite correction.