Wild Bootstrap for Instrumental Variables Regression with Weak Instruments and Few Clusters

Wang, Wenjie (2021): Wild Bootstrap for Instrumental Variables Regression with Weak Instruments and Few Clusters.

Abstract

Under a framework with a small number of clusters but large numbers of observations per cluster for instrumental variable (IV) regression, we show that an unstudentized wild bootstrap test based on IV estimators such as the two-stage least squares estimator is valid as long as the instruments are strong for at least one cluster. This is different from alternative methods proposed in the literature for inference with a small number of clusters, whose validity would require that the instruments be strong for all clusters. Moreover, for the leading case in empirical applications with a single instrument, the unstudentized wild bootstrap test generated by our procedure is fully robust to weak instrument in the sense that its limiting null rejection probability is no greater than the nominal level even if all clusters are ``weak". However, such robustness is not shared by its studentized version; the wild bootstrap test that is based on the t-test statistic can have serious size distortion in this case. Furthermore, in the general case with multiple instruments, we show that an unstudentized version of bootstrap Anderson-Rubin (AR) test is fully robust to weak instruments, and is superior with regard to both size and power properties to alternative asymptotic and bootstrap AR tests that employ cluster-robust variance estimators. By contrast, we �find that bootstrapping other weak-instrument-robust tests such as the Lagrange multiplier test and the conditional quasi-likelihood ratio test, no matter studentized or unstudentized, does not guarantee correct limiting null rejection probability when all clusters are ``weak".

• Item Type: MPRA Paper
• Language: English
• Keywords: Weak Instrument, Wild Bootstrap, Clustered Data, Randomization Test
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C26 - Instrumental Variables (IV) Estimation

References

Abadie, A., J. Gu, and S. Shen (2019): ``Instrumental Variable Estimation with First Stage Heterogeneity," Discussion paper, Working paper, MIT.

Anderson, T. W., and H. Rubin (1949): ``Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations," Annals of Mathematical Statistics, 20(1), 46-63.

Andrews, D. W., and P. Guggenberger (2019): ``Identification-and singularity-robust inference for moment condition models," Quantitative Economics, 10(4), 1703-1746.

Andrews, I. (2016): ``Conditional linear combination tests for weakly identified models," Econometrica, 84(6), 2155-2182.

Andrews, I. (2018): ``Valid two-step identification-robust confidence sets for GMM," Review of Economics and Statistics, 100(2), 337-348.

Andrews, I., and A. Mikusheva (2016): ``Conditional inference with a functional nuisance parameter," Econometrica, 84(4), 1571-1612.

Andrews, I., J. H. Stock, and L. Sun (2019): ``Weak instruments in instrumental variables regression: Theory and practice," Annual Review of Economics, 11, 727-753.

Angrist, J., G. Imbens, and A. Krueger (1999): ``Jackknife Instrumental Variables Estimates," Journal of Applied Econometrics, 14(1), 57-67.

Bekker, P. (1994): ``Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, 62(3), 657-681.

Beran, R. (1988): ``Prepivoting test statistics: a bootstrap view of asymptotic refinements," Journal of the American Statistical Association, 83(403), 687-697.

Bester, C. A., T. G. Conley, and C. B. Hansen (2011): ``Inference with dependent data using cluster covariance estimators," Journal of Econometrics, 165(2), 137-151.

Brodeur, A., N. Cook, and A. Heyes (2020): ``Methods Matter: P-Hacking and Publication Bias in Causal Analysis in Economics," American Economic Review.

Cameron, A. C., J. B. Gelbach, and D. L. Miller (2008): ``Bootstrap-based improvements for inference with clustered errors," The Review of Economics and Statistics, 90(3), 414-427.

Cameron, A. C., and D. L. Miller (2015): ``A practitioner's guide to cluster-robust inference," Journal of human resources, 50(2), 317-372.

Canay, I. A., J. P. Romano, and A. M. Shaikh (2017): ``Randomization tests under an approximate symmetry assumption," Econometrica, 85(3), 1013-1030.

Canay, I. A., A. Santos, and A. M. Shaikh (2020): ``The wild bootstrap with a "small" number of" large" clusters," Review of Economics and Statistics, Forthcoming.

Chao, J. C., and N. R. Swanson (2005): ``Consistent Estimation with a Large Number of Weak Instruments," Econometrica, 73(5), 1673-1692.

Chao, J. C., N. R. Swanson, J. A. Hausman, W. K. Newey, and T. Woutersen (2012): ``Asymptotic Distribution Of JIVE In A Heteroskedastic IV Regression With Many Instruments," Econometric Theory, 28(1), 42-86.

Chernozhukov, V., and C. Hansen (2008): ``Instrumental variable quantile regression: A robust inference approach," Journal of Econometrics, 142(1), 379-398.

Chernozhukov, V., and C. Hansen (2008): ``The reduced form: A simple approach to inference with weak instruments," Economics Letters, 100(1), 68-71.

Davidson, R., and J. G. MacKinnon (2008): ``Bootstrap inference in a linear equation estimated by instrumental variables," The Econometrics Journal, 11(3), 443-477.

Davidson, R., and J. G. MacKinnon (2010): ``Wild bootstrap tests for IV regression," Journal of Business & Economic Statistics, 28(1), 128-144.

Davidson, R., and J. G. MacKinnon (2014): ``Bootstrap confidence sets with weak instruments," Econometric Reviews, 33(5-6), 651-675.

Djogbenou, A. A., J. G. MacKinnon, and M. Ø. Nielsen (2019): ``Asymptotic theory and wild bootstrap inference with clustered errors," Journal of Econometrics, 212(2), 393-412.

Finlay, K., and L. M. Magnusson (2009): ``Implementing weak-instrument robust tests for a general class of instrumental-variables models," The Stata Journal, 9(3), 398-421.

Finlay, K., and L. M. Magnusson (2019): ``Two applications of wild bootstrap methods to improve inference in cluster-IV models," Journal of Applied Econometrics, 34(6), 911-933.

Guggenberger, P., F. Kleibergen, and S. Mavroeidis (2019): ``A more powerful subvector Anderson Rubin test in linear instrumental variables regression," Quantitative Economics, 10(2), 487-526.

Guggenberger, P., F. Kleibergen, S. Mavroeidis, and L. Chen (2012): ``On the asymptotic sizes of subset Anderson-Rubin and Lagrange multiplier tests in linear instrumental variables regression," Econometrica, 80(6), 2649-2666.

Hagemann, A. (2019): ``Permutation inference with a finite number of heterogeneous clusters," arXiv preprint arXiv:1907.01049.

Hall, P. (1992): ``The bootstrap and Edgeworth expansion," in The bootstrap and Edgeworth expansion, ed. by P. Hall. Springer-Verlag New York, Inc, New York.

Hansen, B. E., and S. Lee (2019): ``Asymptotic theory for clustered samples," Journal of Econometrics, 210(2), 268-290.

Hausman, J. A., W. K. Newey, T. Woutersen, J. C. Chao, and N. R. Swanson (2012): ``Instrumental variable estimation with heteroskedasticity and many instruments," Quantitative Economics, 3(2), 211-255.

Horowitz, J. L. (2001): ``The bootstrap," in Handbook of Econometrics, ed. by J. Heckman, and E. E. Leamer. Elsevier Science, Amsterdam, The Netherlands.

Ibragimov, R., and U. K. Müller (2010): ``t-Statistic based correlation and heterogeneity robust inference," Journal of Business & Economic Statistics, 28(4), 453-468.

Ibragimov, R., and U. K. Müller (2016): ``Inference with few heterogeneous clusters," Review of Economics and Statistics, 98(1), 83-96.

Imbens, G. W., and P. R. Rosenbaum (2005): ``Robust, accurate confidence intervals with a weak instrument: quarter of birth and education," Journal of the Royal Statistical Society: Series A (Statistics in Society), 168(1), 109-126.

Kaffo, M., and W. Wang (2017): ``On bootstrap validity for specification testing with many weak instruments," Economics Letters, 157, 107-111.

Kleibergen, F. (2002): ``Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, 70(5), 1781-1803.

Kleibergen, F. (2005): ``Testing Parameters in GMM Without Assuming That They are Identified," Econometrica, 73, 1103-1124.

MacKinnon, J. G., M. Ø. Nielsen, and M. D. Webb (2019): ``Wild bootstrap and asymptotic inference with multiway clustering," Journal of Business & Economic Statistics, pp. 1-15.

MacKinnon, J. G., and M. D. Webb (2017): ``Wild bootstrap inference for wildly different cluster sizes," Journal of Applied Econometrics, 32(2), 233-254.

Moreira, H., and M. J. Moreira (2019): ``Optimal two-sided tests for instrumental variables regression with heteroskedastic and autocorrelated errors," Journal of Econometrics, 213(2), 398-433.

Moreira, M. J. (2003): ``A Conditional Likelihood Ratio Test for Structural Models," Econometrica, 71(4), 1027-1048.

Moreira, M. J., J. Porter, and G. Suarez (2009): ``Bootstrap Validity for the Score Test when Instruments may be Weak," Journal of Econometrics, 149(1), 52-64.

Muralidharan, K., P. Niehaus, and S. Sukhtankar (2016): ``Building state capacity: Evidence from biometric smart cards in India," American Economic Review, 106(10), 2895-2929.

Nagar, A. L. (1959): ``The Bias and Moment Matrix of the Generalized k-Class Estimators of the Parameters in Simultaneous Equations," Econometrica, 27, 575-595.

Newey, W. K., and F. Windmeijer (2009): ``Generalized method of moments with many weak moment conditions," Econometrica, 77(3), 687-719.

Olea, J. L. M., and C. Pflueger (2013): ``A robust test for weak instruments," Journal of Business & Economic Statistics, 31(3), 358-369.

Phillips, G., and C. Hale (1977): ``The Bias of Instrumental Variable Estimators of Simultaneous Equation Systems," International Economic Review, 18(1), 219-228.

Roodman, D., M. Ø. Nielsen, J. G. MacKinnon, and M. D. Webb (2019): ``Fast and wild: Bootstrap inference in Stata using boottest," The Stata Journal, 19(1), 4-60.

Rosenbaum, P. R. (1996): ``Identification of causal effects using instrumental variables: Comment," Journal of the American Statistical Association, 91(434), 465-468.

Rothenberg, T. (1984): ``Approximating the Distributions of Econometric Estimators and Test Statistics." Ch. 14 in: Handbook of Econometrics, vol 2, ed. Z. Griliches and M. Intriligator.

Staiger, D., and J. H. Stock (1997): ``Instrumental Variables Regression with Weak Instruments," Econometrica, 65(3), 557-586.

Wang, W. (2020): ``On the inconsistency of nonparametric bootstraps for the subvector Anderson-Rubin test," Economics Letters, 109157.

Wang, W., and F. Doko Tchatoka (2018): ``On bootstrap inconsistency and Bonferroni-based size-correction for the subset Anderson-Rubin test under conditional homoskedasticity," Journal of Econometrics, 207(1), 188-211.

Wang, W., and M. Kaffo (2016): ``Bootstrap inference for instrumental variable models with many weak instruments," Journal of Econometrics, 192(1), 231-268.

Young, A. (2020): ``Consistency without inference: Instrumental variables in practical application," Discussion paper, London School of Economics.