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A POWERFUL SUBVECTOR ANDERSON–RUBIN TEST IN LINEAR INSTRUMENTAL VARIABLES REGRESSION WITH CONDITIONAL HETEROSKEDASTICITY

Published online by Cambridge University Press:  14 April 2023

Patrik Guggenberger ,
Frank Kleibergen  and
Sophocles Mavroeidis
Patrik Guggenberger
Affiliation:
Pennsylvania State University
Frank Kleibergen*
Affiliation:
University of Amsterdam
Sophocles Mavroeidis
Affiliation:
University of Oxford
*
Address correspondence to Frank Kleibergen, Department of Quantitative Economics, University of Amsterdam, Amsterdam, Netherlands; e-mail: [email protected].
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Abstract

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We introduce a new test for a two-sided hypothesis involving a subset of the structural parameter vector in the linear instrumental variables (IVs) model. Guggenberger, Kleibergen, and Mavroeidis (2019, Quantitative Economics, 10, 487–526; hereafter GKM19) introduce a subvector Anderson–Rubin (AR) test with data-dependent critical values that has asymptotic size equal to nominal size for a parameter space that allows for arbitrary strength or weakness of the IVs and has uniformly nonsmaller power than the projected AR test studied in Guggenberger et al. (2012, Econometrica, 80(6), 2649–2666). However, GKM19 imposes the restrictive assumption of conditional homoskedasticity (CHOM). The main contribution here is to robustify the procedure in GKM19 to arbitrary forms of conditional heteroskedasticity. We first adapt the method in GKM19 to a setup where a certain covariance matrix has an approximate Kronecker product (AKP) structure which nests CHOM. The new test equals this adaptation when the data are consistent with AKP structure as decided by a model selection procedure. Otherwise, the test equals the AR/AR test in Andrews (2017, Identification-Robust Subvector Inference, Cowles Foundation Discussion Papers 3005, Yale University) that is fully robust to conditional heteroskedasticity but less powerful than the adapted method. We show theoretically that the new test has asymptotic size bounded by the nominal size and document improved power relative to the AR/AR test in a wide array of Monte Carlo simulations when the covariance matrix is not too far from AKP.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 5 , October 2024 , pp. 957 - 1002
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Footnotes

We would like to thank the Editor (Peter Phillips), the Co-Editor (Michael Jansson), and two referees for very helpful comments. Guggenberger thanks the European University Institute for its hospitality while parts of this paper were drafted. Mavroeidis gratefully acknowledges the research support of the European Research Council via Consolidator grant number 647152. We would also like to thank Donald Andrews for detailed comments and for providing his Gauss code of, and explanations about, Andrews (2017) and Lixiong Li for outstanding research assistance for the Monte Carlo study. We would also like to thank seminar participants in Amsterdam, Bologna, Bristol, Florence (EUI), Indiana, Konstanz, Manchester, Mannheim, Paris (PSE), Pompeu Fabra, Regensburg, Rotterdam, Singapore (NUS and SMU), Tilburg, Toulouse, Tübingen, and Zurich, and conference participants at the Institute for Fiscal Studies (London) for helpful comments.

References

REFERENCES

Anderson, T.W. & Rubin, H. (1949) Estimation of the parameters of a single equation in a complete system of stochastic equations. Annals of Mathematical Statistics 20, 46–63.CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59(3), 817–858.CrossRefGoogle Scholar
Andrews, D.W.K. (2017) Identification-Robust Subvector Inference. Cowles Foundation Discussion Papers 3005, Cowles Foundation for Research in Economics, Yale University.Google Scholar
Andrews, D.W.K. & Cheng, X. (2014) GMM estimation and uniform subvector inference with possible identification failure. Econometric Theory 30(2), 287–333.CrossRefGoogle Scholar
Andrews, D.W.K., Cheng, X., & Guggenberger, P. (2020) Generic results for establishing the asymptotic size of confidence sets and tests. Journal of Econometrics 218(2), 496–531.CrossRefGoogle Scholar
Andrews, D.W.K. & Guggenberger, P. (2019) Identification-and singularity-robust inference for moment condition models. Quantitative Economics 10(4), 1703–1746.CrossRefGoogle Scholar
Andrews, D.W.K., Marmer, V., & Yu, Z. (2019) On optimal inference in the linear IV model. Quantitative Economics 10(2), 457–485.CrossRefGoogle Scholar
Andrews, D.W.K., Moreira, M.J., & Stock, J.H. (2006) Optimal two-sided invariant similar tests for instrumental variables regression. Econometrica 74(3), 715–752.CrossRefGoogle Scholar
Andrews, D.W.K. & Soares, G. (2010) Inference for parameters defined by moment inequalities using generalized moment selection. Econometrica 78(1), 119–157.Google Scholar
Andrews, I. & Mikusheva, A. (2016) A geometric approach to nonlinear econometric models. Econometrica 84(3), 1249–1264.CrossRefGoogle Scholar
Bernanke, B. (2004) Essays on the Great Depression. Princeton University Press.Google Scholar
Bugni, F.A., Canay, I.A., & Shi, X. (2017) Inference for subvectors and other functions of partially identified parameters in moment inequality models. Quantitative Economics 8(1), 1–38.CrossRefGoogle Scholar
Chaudhuri, S. & Zivot, E. (2011) A new method of projection-based inference in GMM with weakly identified nuisance parameters. Journal of Econometrics 164(2), 239–251.Google Scholar
Dufour, J.-M. & Taamouti, M. (2005) Projection-based statistical inference in linear structural models with possibly weak instruments. Econometrica 73(4), 1351–1365.CrossRefGoogle Scholar
Gafarov, T. (2019) Inference in high-dimensional set-identified affine models. arXiv:1904.00111 Google Scholar
Golub, G.H. & van Loan, C.F. (1989) Matrix Computations, vol. 3. Johns Hopkins Studies in Mathematical Sciences.Google 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. (2023) A test for Kronecker product structure covariance matrix. Journal of Econometrics 233, 88–112.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(6), 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(4), 667–709.CrossRefGoogle Scholar
Hahn, J. & Kuersteiner, G. (2002) Discontinuities of weak instrument limiting distributions. Economics Letters 75(3), 325–331.CrossRefGoogle Scholar
Han, S. & McCloskey, A. (2019) Estimation and inference with a (nearly) singular Jacobian. Quantitative Economics 10(3), 1019–1068.CrossRefGoogle Scholar
Kaido, H., Molinari, F., & Stoye, J. (2019) Confidence intervals for projections of partially identified parameters. Econometrica 87(4), 1397–1432.CrossRefGoogle Scholar
Kelejian, H.H. (1982) An extension of a standard test for heteroskedasticity to a systems framework. Journal of Econometrics 20(2), 325–333.CrossRefGoogle Scholar
Kleibergen, F. (2021) Efficient size correct subset inference in homoskedastic linear instrumental variables regression. Journal of Econometrics 221(1), 78–96.CrossRefGoogle Scholar
Li, R.-C. (1998) Relative perturbation theory: II. Eigenspace and singular subspace variations. SIAM Journal on Matrix Analysis and Applications 20(2), 471–492.CrossRefGoogle Scholar
McCloskey, A. (2017) Bonferroni-based size-correction for nonstandard testing problems. Journal of Econometrics 200(1), 17–35.CrossRefGoogle Scholar
Muirhead, R.J. (1978) Latent roots and matrix variates: A review of some asymptotic results. Annals of Statistics 6, 5–33.CrossRefGoogle Scholar
Newey, W.K. & West, K.D. (1987) A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3), 703–708.CrossRefGoogle Scholar
Sims, C.A. (1980) Macroeconomics and reality. Econometrica 48(1), 1–48.CrossRefGoogle Scholar
Staiger, D. & Stock, J.H. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557–586.CrossRefGoogle Scholar
Stewart, G. & Sun, J.-G. (1990) Matrix Perturbation Theory. Academic Press, Inc.Google Scholar
Stock, J.H. & Watson, M.W. (2018) Identification and estimation of dynamic causal effects in macroeconomics using external instruments. Economic Journal 128(610), 917–948.CrossRefGoogle Scholar
Stock, J.H. & Wright, J.H. (2000) GMM with weak identification. Econometrica 68(5), 1055–1096.CrossRefGoogle Scholar
van Loan, C.F. & Pitsianis, N. (1993) Approximation with Kronecker products. In M.S. Moonen, G.H. Golub, & B.L.R. De Moor (eds), Linear Algebra for Large Scale and Real-Time Applications, pp. 293–314. Springer.CrossRefGoogle Scholar
Wedin, P.-Å. (1972) Perturbation bounds in connection with singular value decomposition. BIT Numerical Mathematics 12(1), 99–111.CrossRefGoogle Scholar