Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T18:36:26.308Z Has data issue: false hasContentIssue false
  • English
  • Français

INFERENCE IN INSTRUMENTAL VARIABLE MODELS WITH HETEROSKEDASTICITY AND MANY INSTRUMENTS

Published online by Cambridge University Press:  26 March 2020

Federico Crudu ,
Giovanni Mellace  and
Zsolt Sándor
Federico Crudu
Affiliation:
Università di Siena and CRENoS
Giovanni Mellace*
Affiliation:
University of Southern Denmark
Zsolt Sándor
Affiliation:
Sapientia Hungarian University of Transylvania
*
Address correspondence to Giovanni Mellace, Department of Business and Economics, Campusvej 55, 5230Odense M, Denmark; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proposes novel inference procedures for instrumental variable models in the presence of many, potentially weak instruments that are robust to the presence of heteroskedasticity. First, we provide an Anderson–Rubin-type test for the entire parameter vector that is valid under assumptions weaker than previously proposed Anderson–Rubin-type tests. Second, we consider the case of testing a subset of parameters under the assumption that a consistent estimator for the parameters not under test exists. We show that under the null, the proposed statistics have Gaussian limiting distributions and derive alternative chi-square approximations. An extensive simulation study shows the competitive finite sample properties in terms of size and power of our procedures. Finally, we provide an empirical application using college proximity instruments to estimate the returns to education.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 2 , April 2021 , pp. 281 - 310
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We thank two anonymous referees, the Co-Editor Patrik Guggenberger, and the Editor Peter Phillips for valuable comments and suggestions that greatly improved the paper. We are also grateful to Stanislav Anatolyev, Samuele Centorrino, and Neil Davies for valuable help. F.C. thanks financial support from the Chilean Government through CONICYT’s grant FONDECYT Iniciacion n. 11140433. Z.S. thanks financial support from grant PN-II-ID-PCE-2012-4-0066 of the Romanian Ministry of National Education, CNCS-UEFISCDI.

References

REFERENCES

Anatolyev, S. (2018) Many instruments and/or regressors: A friendly guide. Journal of Economic Surveys 33, 689–726.CrossRefGoogle Scholar
Anatolyev, S. & Gospodinov, N. (2011) Specification testing in models with many instruments. Econometric Theory 27, 427–441.CrossRefGoogle Scholar
Anatolyev, S. & Yaskov, P. (2017) Asymptotics of diagonal elements of projection matrices under many instruments/regressors. Econometric Theory 33, 717–738.CrossRefGoogle Scholar
Anderson, T.W. & Rubin, H. (1949) Estimators of the parameters of a single equation in a complete set of stochastic equations. The Annals of Mathematical Statistics 21, 570–582.CrossRefGoogle Scholar
Andrews, D.W.K., Marmer, V. & Yu, Z. (2019) On optimal inference in the linear IV model. Quantitative Economics 10, 457–485.CrossRefGoogle Scholar
Andrews, D.W.K., Moreira, M.J. & Stock, J.H. (2006) Optimal two-sided invariant similar tests for instrumental variable regression. Econometrica 73, 715–752.CrossRefGoogle Scholar
Andrews, D.W.K. & Stock, J.H. (2007) Testing with many weak instruments. Journal of Econometrics 138, 24–46.CrossRefGoogle Scholar
Angrist, J.D. & Pischke, J.S. (2008) Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton University Press.CrossRefGoogle Scholar
Bekker, P.A. (1994) Alternative approximations to the distributions of instrumental variable estimators. Econometrica 54, 657–682.CrossRefGoogle Scholar
Bekker, P.A. & Crudu, F. (2015) Jackknife instrumental variable estimation with heteroskedasticity. Journal of Econometrics 185, 332–342.CrossRefGoogle Scholar
Bekker, P.A. & Van der Ploeg, J. (2005) Instrumental variable estimation based on grouped data. Statistica Neerlandica 59, 239–267.CrossRefGoogle Scholar
Bun, M., Farbmacher, H. & Poldermans, R. (2019) Finite Sample Properties of the Anderson and Rubin (1949) Test. Working Paper.Google Scholar
Card, D. (1995) Using geographic variation in college proximity to estimate the return to schooling. In Christofides, L., Grant, E. & Swidinsky, R. (eds.), Aspects of Labor Market Behaviour: Essays in Honour of John Vanderkamp, pp. 201–222. University of Toronto Press.Google Scholar
Chao, J.C., Hausman, J.A., Newey, W.K., Swanson, N.R. & Woutersen, T. (2014) Testing overidentifying restrictions with many instruments and heteroskedasticity. Journal of Econometrics 178, 15–21.CrossRefGoogle Scholar
Chao, J.C. & Swanson, N.R. (2005) Consistent estimation with a large number of weak instruments. Econometrica 73, 1673–1692.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, 42–86.CrossRefGoogle Scholar
Davidson, R. & MacKinnon, J.G. (1998) Graphical methods for investigating the size and power of hypothesis tests. The Manchester School 66, 1–26.CrossRefGoogle Scholar
Donald, S.G., Imbens, G.W. & Newey, W.K. (2003) Empirical likelihood estimation and consistent tests with conditional moment restrictions. Journal of Econometrics 117, 55–93.CrossRefGoogle Scholar
Guggenberger, P., Kleibergen, F. & Mavroeidis, S. (2019) A more powerful subvector Anderson Rubin test in linear instrumental variables regression. Quantitative Economics 10, 487–526.CrossRefGoogle Scholar
Guggenberger, P., Kleibergen, F., Mavroeidis, S. & Chen, L. (2012) On the asymptotic sizes of subset Anderson–Rubin and Lagrange multiplier tests in linear instrumental variables regression. Econometrica 80, 2649–2666.Google Scholar
Guggenberger, P. & Smith, R.J. (2005) Generalized empirical likelihood estimators and tests under partial, weak, and strong identification. Econometric Theory 21, 667–709.CrossRefGoogle Scholar
Hausman, J.A., Newey, W.K., Woutersen, T., Chao, J.C. & Swanson, N.R. (2012) Instrumental variable estimation with heteroskedasticity and many instruments. Quantitative Economics 3, 211–255.CrossRefGoogle Scholar
Imbens, G.W. & Rubin, D. (1997) Estimating outcome distributions for compliers in instrumental variables models. Review of Economic Studies 64, 555–574.CrossRefGoogle Scholar
Jiang, Y., Kang, H. & Small, D. (2016) ivmodel: Statistical Inference and Sensitivity Analysis for Instrumental Variables Model. Available at https://CRAN.R-project.org/package=ivmodel, r package version 1.2.Google Scholar
Kang, H., Zhang, A., Cai, T.T. & Small, D.S. (2016) Instrumental variables estimation with some invalid instruments and its application to Mendelian randomization. Journal of the American Statistical Association 111, 132–144.CrossRefGoogle Scholar
Kleibergen, F. (2002) Pivotal statistics for testing structural parameters in instrumental variables regression. Econometrica 70, 1781–1803.CrossRefGoogle Scholar
Kleibergen, F. (2004) Testing subsets of structural parameters in the instrumental variables. The Review of Economics and Statistics 86, 418–423.CrossRefGoogle Scholar
Kleibergen, F. (2005) Testing parameters in GMM without assuming they are identified. Econometrica 73, 1103–1123.CrossRefGoogle Scholar
Lee, Y. & Okui, R. (2012) Hahn–Hausman test as a specification test. Journal of Econometrics 167, 133–139.CrossRefGoogle Scholar
Moreira, M.J. (2009) Tests with correct size when instruments can be arbitrarily weak. Journal of Econometrics 152, 131–140.CrossRefGoogle Scholar
Newey, W.K. & Windmeijer, F. (2009) Generalized method of moments with many weak moment conditions. Econometrica 77, 687–719.Google Scholar
Phillips, P.C. & Gao, W.Y. (2017) Structural inference from reduced forms with many instruments. Journal of Econometrics 199, 96–116.CrossRefGoogle Scholar
Staiger, D. & Stock, J.H. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557–586.CrossRefGoogle Scholar
Stock, J.H., Wright, J.H. & Yogo, M. (2002) A survey of weak instruments and weak identification in generalized method of moments. Journal of Business and Economic Statistics 20, 518–529.CrossRefGoogle Scholar
Von Hinke, S., Davey Smith, G., Lawlor, D.A., Propper, C. & Windmeijer, F. (2016) Genetic markers as instrumental variables. Journal of Health Economics 45, 131–148.CrossRefGoogle ScholarPubMed
Wang, J. & Zivot, E. (1998) Inference on a structural parameter in instrumental variables regression with weak instruments. Econometrica 66, 1389–1404.CrossRefGoogle Scholar
Windmeijer, F., Farbmacher, H., Davies, N. & Davey Smith, G. (2017) On the Use of the Lasso for Instrumental Variables Estimation with Some Invalid Instruments. Bristol Economics Discussion Papers, Department of Economics, University of Bristol, UK.Google Scholar
Zivot, E., Startz, R. & Nelson, C.R. (1998) Valid confidence intervals and inference in the presence of weak instruments. International Economic Review 39, 1119–1144.CrossRefGoogle Scholar
Supplementary material: File

Crudu et al. Supplementary Materials

Crudu et al. Supplementary Materials

Download Crudu et al. Supplementary Materials(File)
File 29.7 KB