Asymptotic theory and wild bootstrap inference with clustered errors

Abstract

We study asymptotic inference based on cluster-robust variance estimators for regression models with clustered errors, focusing on the wild cluster bootstrap and the ordinary wild bootstrap. We state conditions under which both asymptotic and bootstrap tests and confidence intervals will be asymptotically valid. These conditions put limits on the rates at which the cluster sizes can increase as the number of clusters tends to infinity. To include power in the analysis, we allow the data to be generated under sequences of local alternatives. Under a somewhat stronger set of conditions, we also derive formal Edgeworth expansions for the asymptotic and bootstrap test statistics. Simulation experiments illustrate the theoretical results, and the Edgeworth expansions explain the overrejection of the asymptotic test and shed light on the choice of auxiliary distribution for the wild bootstrap.

Asymptotic theory for clustered samples

“…(11)- (12) is weaker than (5)-(8) when λ n diverges to infinity (which occurs when X n converges at a rate slower than n −1/2 ). Since the sequence λ n is unknown in an application it is difficult to interpret the assumptions (11)- (12). Hence we prefer the assumptions (5)- (8).…”

Improved inference on the rank of a matrix

“…Assumption holds for and some centered Gaussian matrix , by the Lindeberg–Feller‐type central limit theorem. Following Cameron, Gelbach, and Miller (), we may construct where may be a multinomial vector over G categories with probabilities (corresponding to the pairs cluster bootstrap) or other weights (such as those leading to the cluster wild bootstrap); see also Djogbenou, MacKinnon, and Nielsen ().…”

“…where (W 1 W G ) may be a multinomial vector over G categories with probabilities (1/G /1G) (corresponding to the pairs cluster bootstrap) or other weights (such as those leading to the cluster wild bootstrap); see also Djogbenou, MacKinnon, and Nielsen (2018).…”

Inference with Dependent Data in Accounting and Finance Applications

“…Note that, because the are generated independently, this bootstrap produces bootstrap data that are independent across groups. This modification of the wild bootstrap is called a cluster wild bootstrap; see Cameron, Gelbach, and Miller [] and Djogbenou, MacKinnon, and Nielsen []. Because it holds the observed values of the regressors fixed, the wild bootstrap is well suited to unbalanced firm‐level panels, where it implicitly holds fixed the timing of the observations for each firm.…”

“…We will use OLS point estimates for . However, we note that the estimate of β could be constrained to impose the null hypothesis, which could improve performance of the bootstrap, as noted by Djogbenou, MacKinnon, and Nielsen [].…”