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ALTERNATIVE ASYMPTOTICS AND THE PARTIALLY LINEAR MODEL WITH MANY REGRESSORS

Published online by Cambridge University Press:  19 October 2016

Matias D. Cattaneo ,
Michael Jansson  and
Whitney K. Newey
Matias D. Cattaneo
Affiliation:
University of Michigan
Michael Jansson
Affiliation:
University of California Berkeley and CREATES
Whitney K. Newey*
Affiliation:
Massachusetts Institute of Technology
*
*Address correspondence to Whitney K. Newey, Department of Economics, MIT, E52-424, Cambridge, MA 02139, USA; e-mail: [email protected].
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Abstract

Many empirical studies estimate the structural effect of some variable on an outcome of interest while allowing for many covariates. We present inference methods that account for many covariates. The methods are based on asymptotics where the number of covariates grows as fast as the sample size. We find a limiting normal distribution with variance that is larger than the standard one. We also find that with homoskedasticity this larger variance can be accounted for by using degrees-of-freedom-adjusted standard errors. We link this asymptotic theory to previous results for many instruments and for small bandwidth(s) distributional approximations.

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 2 , April 2018 , pp. 277 - 301
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

The authors are grateful for comments from Xiaohong Chen, Victor Chernozhukov, Alfonso Flores–Lagunes, Lutz Kilian, seminar participants at Bristol, Brown, Cambridge, Exeter, Indiana, LSE, Michigan, MSU, NYU, Princeton, Rutgers, Stanford, UCL, UCLA, UCSD, UC-Irvine, USC, Warwick and Yale, and conference participants at the 2010 Joint Statistical Meetings and the 2010 LACEA Impact Evaluation Network Conference. We also thank the editor, Peter Phillips, the co-editor and two reviewers for their comments. The first author gratefully acknowledges financial support from the National Science Foundation (SES 1122994 and SES 1459931). The second author gratefully acknowledges financial support from the National Science Foundation (SES 1124174 and SES 1459967) and the research support of CREATES (funded by the Danish National Research Foundation under Grant No. DNRF78). The third author gratefully acknowledges financial support from the National Science Foundation (SES 1132399).

References

REFERENCES

Alvarez, J. & Arellano, M. (2003) The time series and cross-section asymptotics of dynamic panel data estimators. Econometrica 71(4), 1121–1159.Google Scholar
Angrist, J., Imbens, G.W., & Krueger, A. (1999) Jackknife instrumental variables estimation. Journal of Applied Econometrics 14(1), 57–67.Google Scholar
Aradillas–Lopéz, A., Honoré, B.E., & Powell, J.L. (2007) Pairwise difference estimation with nonparametric control variables. International Economic Review 48, 1119–1158.CrossRefGoogle Scholar
Atchadé, Y.F. & Cattaneo, M.D. (2014) A martingale decomposition for quadratic forms of Markov chains (with applications). Stochastic Processes and their Applications 124(1), 646–677.CrossRefGoogle Scholar
Bekker, P.A. (1994) Alternative approximations to the distributions of instrumental variables estimators. Econometrica 62, 657–681.Google Scholar
Belloni, A., Chernozhukov, V., & Hansen, C. (2014) Inference on treatment effects after selection among high-dimensional controls. Review of Economic Studies 81(2), 608–650.CrossRefGoogle Scholar
Cattaneo, M.D., Crump, R.K., & Jansson, M. (2010) Robust data-driven inference for density-weighted average derivatives. Journal of the American Statistical Association 105(491), 1070–1083.Google Scholar
Cattaneo, M.D., Crump, R.K., & Jansson, M. (2014a) Bootstrapping density-weighted average derivatives. Econometric Theory 30(6), 1135–1164.Google Scholar
Cattaneo, M.D., Crump, R.K., & Jansson, M. (2014b) Small bandwidth asymptotics for density-weighted average derivatives. Econometric Theory 30(1), 176–200.Google Scholar
Cattaneo, M.D. & Jansson, M. (2015) Bootstrapping Kernel-Based Semiparametric Estimators. Working paper, University of Michigan.Google Scholar
Cattaneo, M.D., Jansson, M., & Newey, W.K. (2015) Treatment Effects With Many Covariates and Heteroskedasticity. Working paper, University of Michigan.CrossRefGoogle Scholar
Chao, J.C., Swanson, N.R., Hausman, J.A., Newey, W.K., & Woutersen, T. (2012) Asymptotic distribution of JIVE in a heteroskedastic IV regression with many instruments. Econometric Theory 28(1), 42–86.CrossRefGoogle Scholar
Chen, X. (2007) Large sample sieve estimation of semi-nonparametric models. In Heckman, J., & Leamer, E. (eds.), Handbook of Econometrics, vol. 6, pp. 5550–5632. Elsevier Science B.V.Google Scholar
de Jong, P. (1987) A central limit theorem for generalized quadratic forms. Probability Theory and Related Fields 75, 261–277.Google Scholar
Donald, S.G. & Newey, W.K. (1994) Series estimation of semilinear models. Journal of Multivariate Analysis 50(1), 30–40.Google Scholar
El Karoui, N., Bean, D., Bickel, P.J., Lim, C., & Yu, B. (2013) On robust regression with high-dimensional predictors. Proceedings of the National Academy of Sciences 110(36), 14557–14562.Google Scholar
Escanciano, J.C. & Jacho-Chavez, D. (2012) $\sqrt n $ -Uniformly consistent density estimation in nonparametric regression. Journal of Econometrics 167(1), 305–316.Google Scholar
Giné, E. & Nickl, R. (2008) A simple adaptive estimator of the integrated square of a density. Bernoulli 14(1), 47–61.CrossRefGoogle Scholar
Hahn, J. & Newey, W.K. (2004) Jackknife and analytical bias reduction for nonlinear panel data models. Econometrica 72(4), 1295–1319.Google Scholar
Hansen, C., Hausman, J., & Newey, W.K. (2008) Estimation with many instrumental variables. Journal of Business and Economic Statistics 26(4), 398–422.Google Scholar
Heckman, J.J. & Vytlacil, E.J. (2007) Econometric evaluation of social programs, Part I: Causal models, structural models and econometric policy evaluation. In Heckman, J., & Leamer, E. (eds.), Handbook of Econometrics, vol. 6, pp. 4780–4874. Elsevier Science B.V.Google Scholar
Huber, P.J. (1973) Robust regression: Asymptotics, conjectures, and Monte Carlo. Annals of Stastistics 1(5), 799–821.Google Scholar
Imbens, G.W. & Wooldridge, J.M. (2009) Recent developments in the econometrics of program evaluation. Journal of Economic Literature 47(1), 5–86.CrossRefGoogle Scholar
Kunitomo, N. (1980) Asymptotic expansions of the distributions of estimators in a linear functional relationship and simultaneous equations. Journal of the American Statistical Association 75(371), 693–700.Google Scholar
Li, Q. & Racine, S. (2007) Nonparametric Econometrics. Princeton University Press.Google Scholar
Linton, O. (1995) Second order approximation in the partialy linear regression model. Econometrica 63(5), 1079–1112.Google Scholar
Morimune, K. (1983) Approximate distributions of k-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica 51(3), 821–841.Google Scholar
Newey, W.K. (2009) Two-step series estimation of sample selection models. Econometrics Journal 12(1), S217–S229.Google Scholar
Newey, W.K., Hsieh, F., & Robins, J.M. (2004) Twicing kernels and a small bias property of semiparametric estimators. Econometrica 72(1), 947–962.Google Scholar
Powell, J.L., Stock, J.H., & Stoker, T.M. (1989) Semiparametric estimation of index coefficients. Econometrica 57(6), 1403–1430.Google Scholar
Schick, A. & Wefelmeyer, W. (2013) Uniform convergence of convolution estimators for the response density in nonparametric regression. Bernoulli 19(5B), 2250–2276.Google Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.Google Scholar