We consider statistical inference for regression when data are grouped into clusters, with regression model errors independent across clusters but correlated within clusters. Examples include data on individuals with clustering on village or region or other category such as industry, and state-year differences-in-differences studies with clustering on state. In such settings default standard errors can greatly overstate estimator precision. Instead, if the number of clusters is large, statistical inference after OLS should be based on cluster-robust standard errors. We outline the basic method as well as many complications that can arise in practice. These include cluster-specific fixed effects, few clusters, multi-way clustering, and estimators other than OLS.

Researchers have increasingly realized the need to account for within-group dependence in estimating standard errors of regression parameter estimates. The usual solution is to calculate cluster-robust standard errors that permit heteroskedasticity and within-cluster error correlation, but presume that the number of clusters is large. Standard asymptotic tests can over-reject, however, with few (5-30) clusters. We investigate inference using cluster bootstrap-t procedures that provide asymptotic refinement. These procedures are evaluated using Monte Carlos, including the example of Bertrand, Duflo and Mullainathan (2004). Rejection rates of ten percent using standard methods can be reduced to the nominal size of five percent using our methods.

In this paper we propose a new variance estimator for OLS as well as for nonlinear estimators such as logit, probit and GMM, that provcides cluster-robust inference when there is two-way or multi-way clustering that is non-nested. The variance estimator extends the standard cluster-robust variance estimator or sandwich estimator for one-way clustering (e.g. Liang and Zeger (1986), Arellano (1987)) and relies on similar relatively weak distributional assumptions. Our method is easily implemented in statistical packages, such as Stata and SAS, that already offer cluster-robust standard errors when there is one-way clustering. The method is demonstrated by a Monte Carlo analysis for a two-way random effects model; a Monte Carlo analysis of a placebo law that extends the state-year effects example of Bertrand et al. (2004) to two dimensions; and by application to two studies in the empirical public/labor literature where two-way clustering is present.

Researchers have increasingly realized the need to account for within-group dependence in estimating standard errors of regression parameter estimates. The usual solution is to calculate cluster-robust standard errors that permit heteroskedasticity and within-cluster error correlation, but presume that the number of clusters is large. Standard asymptotic tests can over-reject, however, with few (five to thirty) clusters. We investigate inference using cluster bootstrap-t procedures that provide asymptotic refinement. These procedures are evaluated using Monte Carlos, including the example of Bertrand, Duflo, and Mullainathan (2004). Rejection rates of 10% using standard methods can be reduced to the nominal size of 5% using our methods. Copyright by the President and Fellows of Harvard College and the Massachusetts Institute of Technology.

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. This paper exploits a new source of variation in Head Start funding to identify the programs effects on health and schooling. In 1965 the Office of Economic Opportunity (OEO) provided technical assistance to the 300 poorest counties in the U.S. to develop Head Start funding proposals. The result was a large and lasting discontinuity in Head Start funding rates at the OEO cutoff for grant-writing assistance, but no discontinuity in other forms of federal social spending. We find evidence of a large negative discontinuity at the OEO cutoff in mortality rates for children ages 5-9 from causes that could be affected by Head Start, but not for other mortality causes or birth cohorts that should not be affected by the program. We also find suggestive evidence for a positive effect of Head Start on educational attainment in both the 1990 Census, concentrated among those cohorts born late enough to have been exposed to the program, and among respondents in the National Education Longitudinal Study of 1988. Terms of use: Documents in Does Head Start Improve Children's Life Chances?Evidence from a Regression Discontinuity Design AbstractThis paper exploits a new source of variation in Head Start funding to identify the program s effects on health and schooling. In 1965 the Office of Economic Opportunity (OEO) provided technical assistance to the 300 poorest counties in the U.S. to develop Head Start funding proposals. The result was a large and lasting discontinuity in Head Start funding rates at the OEO cutoff for grant-writing assistance, but no discontinuity in other forms of federal social spending. We find evidence of a large negative discontinuity at the OEO cutoff in mortality rates for children ages 5-9 from causes that could be affected by Head Start, but not for other mortality causes or birth cohorts that should not be affected by the program. We also find suggestive evidence for a positive effect of Head Start on educational attainment in both the 1990 Census, concentrated among those cohorts born late enough to have been exposed to the program, and among respondents in the National Education Longitudinal Study of 1988.