Abstracts
Abstract
Confidence intervals based on cluster-robust covariance matrices can be constructed in many ways. In addition to conventional intervals obtained by inverting Wald (t) tests, the paper studies intervals obtained by inverting LM tests, studentized bootstrap intervals based on the wild cluster bootstrap, and restricted bootstrap intervals obtained by inverting bootstrap Wald and LM tests. It also studies the choice of an auxiliary distribution for the wild bootstrap, a modified covariance matrix based on transforming the residuals that was proposed some years ago, and new wild bootstrap procedures based on the same idea. Some procedures perform extraordinarily well even with the number of clusters is small.
Appendices
Bibliography
- Angrist, J. D., and J.-S. Pischke (2009). Mostly Harmless Econometrics. Princeton: Princeton University Press.10.1162/003355304772839588 Google Scholar Search this bibliographic reference on Google Scholar
- Bell, R. M., and D. F. McCaffey (2002). “Bias Reduction in Standard Errors for Linear Regression with Multi-stage Samples.’’ Survey Methodology, 28: 169-181.10.1016/j.jeconom.2011.01.007 Google Scholar Search this bibliographic reference on Google Scholar
- Bertrand, M., E. Duflo, and S. Mullainathan (2004). “How much Should we Trust Differences in Differences Estimates?’’ Quarterly Journal of Economics, 119: 249-275.10.1162/003355304772839588 Google Scholar Search this bibliographic reference on Google Scholar
- Bester, C. A., T. G. Conley, and C. B. Hansen (2011). “Inference with Dependent Data Using Cluster Covariance Estimators.’’ Journal of Econometrics, 165: 137-151.10.1016/j.jeconom.2011.01.007 Google Scholar Search this bibliographic reference on Google Scholar
- Breusch, T. S. (1979). “Confict among Criteria for Testing Hypotheses: Extensions and Comments.’’ Econometrica, 47: 203-207.10.2307/1912356 Google Scholar Search this bibliographic reference on Google Scholar
- Cameron, A. C., J. B. Gelbach, and D. L. Miller (2008). ‘‘Bootstrap-based Improvements for Inference with Clustered Errors.’’ Review of Economics and Statistics, 90: 414-427.10.1162/rest.90.3.414 Google Scholar Search this bibliographic reference on Google Scholar
- Cameron, A. C., and D. L. Miller (2015). ‘‘A Practitioner’s Guide to Clusterrobust Inference.’’ Journal of Human Resources, 50: forthcoming.10.1017/S0266466699153040 Google Scholar Search this bibliographic reference on Google Scholar
- Carter, A. V., K. T. Schnepel, and D. G. Steigerwald (2013). ‘‘Asymptotic Behavior of a t Test Robust to Cluster Heterogeneity.’’ Technical report, University of California, Santa Barbara.10.1080/07474930008800459 Google Scholar Search this bibliographic reference on Google Scholar
- Davidson, J., A. Monticini, and D. Peel (2007). ‘‘Implementing the Wild Bootstrap Using a Two-point Distribution.’’ Economics Letters, 96: 309-315.10.1016/j.econlet.2007.01.020 Google Scholar Search this bibliographic reference on Google Scholar
- Davidson, R., and E. Flachaire (2008). ‘‘The Wild Bootstrap, Tamed at Last.’’ Journal of Econometrics, 146: 162-169.10.1016/j.jeconom.2008.08.003 Google Scholar Search this bibliographic reference on Google Scholar
- Davidson, R., and J. G. MacKinnon (1993). Estimation and Inference in Econometrics. New York, NY: Oxford University Press.10.1162/rest.89.2.221 Google Scholar Search this bibliographic reference on Google Scholar
- Davidson, R., and J. G. MacKinnon (1999). ‘‘The Size Distortion of Bootstrap Tests.’’ Econometric Theory, 15: 361-376.10.1017/S0266466699153040 Google Scholar Search this bibliographic reference on Google Scholar
- Davidson, R., and J. G. MacKinnon (2000). ‘‘Bootstrap Tests: How many Bootstraps?’’ Econometric Reviews, 19: 55-68.10.1080/07474930008800459 Google Scholar Search this bibliographic reference on Google Scholar
- Davidson, R. and J. G. MacKinnon (2004). Econometric Theory and Methods. New York, NY: Oxford University Press.10.1016/0304-4076(85)90158-7 Google Scholar Search this bibliographic reference on Google Scholar
- Davidson, R. and J. G. MacKinnon (2014). ‘‘Bootstrap Confidence Sets with Weak Instruments.’’ Econometric Reviews, 33: 651-675.10.1080/07474938.2013.825177 Google Scholar Search this bibliographic reference on Google Scholar
- Davison, A. C., and D. V. Hinkley (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge University Press.10.1017/CBO9780511802843 Google Scholar Search this bibliographic reference on Google Scholar
- Donald, S. G., and K. Lang (2007). ‘‘Inference with Difference-in-differences and Other Panel Data.’’ The Review of Economics and Statistics, 89: 221-233.10.1162/rest.89.2.221 Google Scholar Search this bibliographic reference on Google Scholar
- Imbens, G. W., and M. Kolesar (2012). ‘‘Robust Standard Errors in Small Samples: Some Practical Advice.’’ Working Paper 18478, National Bureau of Economic Research.Google Scholar Search this bibliographic reference on Google Scholar
- Liang, K.-Y., and S. L. Zeger (1986). ‘‘Longitudinal Data Analysis Using Generalized Linear Models.’’ Biometrika, 73: 13-22.10.1093/biomet/73.1.13 Google Scholar Search this bibliographic reference on Google Scholar
- Liu, R. Y. (1988). ‘‘Bootstrap Procedures Under Some non-I.I.D. Models.’’ Annals of Statistics, 16: 1696-1708.10.1214/aos/1176351062 Google Scholar Search this bibliographic reference on Google Scholar
- MacKinnon, J. G. (2012). ‘‘Thirty Years of Heteroskedasticity-robust Inference.’’ in X. Chen and N. R. Swanson (Eds) Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis, New York: Springer, 2012, pp. 437-461. Google Scholar Search this bibliographic reference on Google Scholar
- MacKinnon, J. G., and M. D. Webb (2014). ‘‘Wild Bootstrap Inference for Wildly Different Cluster Sizes.’’ Queen’s University, Department of Economics, Queen’s University, QED Working Paper 1314 (revised).Google Scholar Search this bibliographic reference on Google Scholar
- MacKinnon, J. G., and H. White (1985). ‘‘Some Heteroskedasticity Consistent Covariance Matrix Estimators with Improved Finite Sample Properties.’’ Journal of Econometrics, 29: 305-325.10.1016/0304-4076(85)90158-7 Google Scholar Search this bibliographic reference on Google Scholar
- Mammen, E. (1993). ‘‘Bootstrap and Wild Bootstrap for High Dimensional Linear Models.’’ Annals of Statistics 21: 255-285.10.1214/aos/1176349025 Google Scholar Search this bibliographic reference on Google Scholar
- Webb, M. D. (2013). ‘‘Reworking Wild Bootstrap Based Inference for Clustered Errors.’’ Department of Economics, Queen’s University, QED Working Paper 1315.Google Scholar Search this bibliographic reference on Google Scholar
- Wu, C. F. J. (1986). ‘‘Jackknife, Bootstrap and other Resampling Methods in Regression Analysis.’’ Annals of Statistics, 14: 1261-1295.10.1214/aos/1176350142 Google Scholar Search this bibliographic reference on Google Scholar
