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.
- 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.
- Bertrand, M., E. Duflo, and S. Mullainathan (2004). “How much Should we Trust Differences in Differences Estimates?’’ Quarterly Journal of Economics, 119: 249-275.
- 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.
- Breusch, T. S. (1979). “Confict among Criteria for Testing Hypotheses: Extensions and Comments.’’ Econometrica, 47: 203-207.
- 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.
- Cameron, A. C., and D. L. Miller (2015). ‘‘A Practitioner’s Guide to Clusterrobust Inference.’’ Journal of Human Resources, 50: forthcoming.
- 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.
- Davidson, J., A. Monticini, and D. Peel (2007). ‘‘Implementing the Wild Bootstrap Using a Two-point Distribution.’’ Economics Letters, 96: 309-315.
- Davidson, R., and E. Flachaire (2008). ‘‘The Wild Bootstrap, Tamed at Last.’’ Journal of Econometrics, 146: 162-169.
- Davidson, R., and J. G. MacKinnon (1993). Estimation and Inference in Econometrics. New York, NY: Oxford University Press.
- Davidson, R., and J. G. MacKinnon (1999). ‘‘The Size Distortion of Bootstrap Tests.’’ Econometric Theory, 15: 361-376.
- Davidson, R., and J. G. MacKinnon (2000). ‘‘Bootstrap Tests: How many Bootstraps?’’ Econometric Reviews, 19: 55-68.
- Davidson, R. and J. G. MacKinnon (2004). Econometric Theory and Methods. New York, NY: Oxford University Press.
- Davidson, R. and J. G. MacKinnon (2014). ‘‘Bootstrap Confidence Sets with Weak Instruments.’’ Econometric Reviews, 33: 651-675.
- Davison, A. C., and D. V. Hinkley (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge University Press.
- 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.
- Imbens, G. W., and M. Kolesar (2012). ‘‘Robust Standard Errors in Small Samples: Some Practical Advice.’’ Working Paper 18478, National Bureau of Economic Research.
- Liang, K.-Y., and S. L. Zeger (1986). ‘‘Longitudinal Data Analysis Using Generalized Linear Models.’’ Biometrika, 73: 13-22.
- Liu, R. Y. (1988). ‘‘Bootstrap Procedures Under Some non-I.I.D. Models.’’ Annals of Statistics, 16: 1696-1708.
- 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.
- 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).
- MacKinnon, J. G., and H. White (1985). ‘‘Some Heteroskedasticity Consistent Covariance Matrix Estimators with Improved Finite Sample Properties.’’ Journal of Econometrics, 29: 305-325.
- Mammen, E. (1993). ‘‘Bootstrap and Wild Bootstrap for High Dimensional Linear Models.’’ Annals of Statistics 21: 255-285.
- Webb, M. D. (2013). ‘‘Reworking Wild Bootstrap Based Inference for Clustered Errors.’’ Department of Economics, Queen’s University, QED Working Paper 1315.
- Wu, C. F. J. (1986). ‘‘Jackknife, Bootstrap and other Resampling Methods in Regression Analysis.’’ Annals of Statistics, 14: 1261-1295.