The Wild Bootstrap with a 'Small' Number of 'Large' Clusters

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. AbstractThis paper studies the properties of the wild bootstrap-based test proposed in Cameron et al. (2008) in settings with clustered data. Cameron et al. (2008) provide simulations that suggest this test works well even in settings with as few as five clusters, but existing theoretical analyses of its properties all rely on an asymptotic framework in which the number of clusters is "large." In contrast to these analyses, we employ an asymptotic framework in which the number of clusters is "small," but the number of observations per cluster is "large." In this framework, we provide conditions under which the limiting rejection probability of an un-Studentized version of the test does not exceed the nominal level. Importantly, these conditions require, among other things, certain homogeneity restrictions on the distribution of covariates. We further establish that the limiting rejection probability of a Studentized version of the test does not exceed the nominal level by more than an amount that decreases exponentially with the number of clusters. We study the relevance of our theoretical results for finite samples via a simulation study.

Inference with Dependent Data in Accounting and Finance Applications

“…Even if one excludes consideration of LN_AGE , only one scheme controls size to be at most 10% for a 5% level test: eight‐year time block clustering with fixed effects for year and eight‐year time block by firm. Given the evidence for substantial across‐group heterogeneity for most grouping structures discussed in section 4.1, this performance is in line with results in Mackinnon and Webb [] and Canay, Santos, and Shaikh [] that suggest the cluster wild bootstrap performs poorly with heterogeneous groups.…”

“…It is possible for cluster wild bootstraps to work well when a homogeneity condition holds across groups. In recent work, Canay, Santos, and Shaikh [] show that the cluster wild bootstrap delivers valid inference with a small number of clusters using a strong homogeneity condition and particular choice of weights. Although the homogeneity condition of Canay, Santos, and Shaikh [] is weaker than that used in obtaining the few clusters HAC approximation in , this version of the cluster wild bootstrap will also struggle with across‐group heterogeneity in firm‐level panels where complex dependence motivates the use of a few, large clusters.…”

The Propagation of Regional Shocks in Housing Markets: Evidence from Oil Price Shocks in Canada

“…28 Because the number of clusters is small, the fixed effect cannot be estimated consistently, invalidating conventional bootstrap confidence intervals and standard errors. Asymptotically valid p -values may be constructed by bootstrapping the model under the null hypothesis of zero coefficients (Canay, Santos and Shaikh 2018). Yet another concern is that the increased global demand for crude oil and other industrial commodities in the 2000s coincided with the increased availability of inexpensive consumer goods imports from emerging Asia, representing a favorable terms-of-trade shock, as Canadian import prices dropped.…”

Public Works Programmes and Cooperation for the Common Good: Evidence from Malawi

“…The reported standard errors are heteroscedasticity-robust but not clustered or bootstrapped because it is not clear whether any alternative would be more accurate. 9 With just three catchments and a moderately high, and widely varying, number of observations per catchment, clustering standard errors at the catchment level is not recommended (Cameron and Miller 2014 ; Canay et al 2017 , 2019 ; MacKinnon and Webb 2018 ; Roodman et al 2019 ). 10…”