Inference with Few Heterogeneous Clusters∗

Suppose estimating a model on each of a small number of potentially heterogeneous clusters yields approximately independent, unbiased and Gaussian parameter estimators. We make two contributions in this set-up. First, we show how to compare a ∗The authors would like to thank three anonymous referees and seminar and conference participants at various universities and conferences for helpful comments. Ibragimov gratefully acknowledges partial support via NSF grant SES-0820124 and grants from the GDN-SEE and CIS Research Competition, the Russian Ministry of Education and Science (Innopolis University) and the Russian Government Program of Competitive Growth of Kazan Federal University (Higher Institute of Information Technologies and Information Systems), and Müller gratefully acknowledges support by the NSF via grant SES-0518036. We are indebted to Nail Bakirov (1952-2010) and Daniyar Mushtari (1945-2013) for inspiring discussions, comments and attention to our work. We are also thankful to Aprajit Mahajan for sharing data and useful discussions, and to Chenchuan (Mark) Li for outstanding research assistance. scalar parameter of interest between treatment and control units using a two-sample t-statistic, extending previous results for the one-sample t-statistic. Second, we develop a test for the appropriate level of clustering, which tests the null hypothesis that clustered standard errors from a much finer partition are correct. We illustrate the approach by revisiting empirical studies involving clustered, time series and spatially correlated data. JEL classification: C12, C14, C32