Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-09T07:01:14.740Z Has data issue: false hasContentIssue false
  • English
  • Français

ON GMM INFERENCE: PARTIAL IDENTIFICATION, IDENTIFICATION STRENGTH, AND NONSTANDARD ASYMPTOTICS

Published online by Cambridge University Press:  18 September 2023

Donald S. Poskitt
Donald S. Poskitt*
Affiliation:
Monash University
*
Address correspondence to Donald S. Poskitt, Department of Econometrics and Business Statistics, Monash University, Clayton, VIC 3800, Australia; e-mail: Donald.Poskitt@monash.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper analyses aspects of generalized method of moments (GMM) inference in moment equality models in settings where standard regularity conditions may break down. Explicit analytic formulations for the asymptotic distributions of estimable functions of the GMM estimator and statistics based on the GMM criterion function are derived under relatively mild assumptions. The moment Jacobian is allowed to be rank deficient, so first order identification may fail, the values of the Jacobian singular values are not constrained, thereby allowing for varying levels of identification strength, the long-run variance of the moment conditions can be singular, and the GMM criterion function weighting matrix may also be chosen sub-optimally. The large-sample properties are derived without imposing a specific structure on the functional form of the moment conditions. Closed-form expressions for the distributions are presented that can be evaluated using standard software without recourse to bootstrap or simulation methods. The practical operation of the results is illustrated via examples involving instrumental variables estimation of a structural equation with endogenous regressors and a common CH features model.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 4 , August 2024 , pp. 875 - 925
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

I am grateful to two anonymous referees for valuable comments which helped to improve the paper. I am indebted to the Co-Editor (Patrik Guggenberger) for insightful and constructive criticism and for helpful suggestions on the content and presentation of the paper. I am also grateful to the Editor (Peter C.B. Phillips) for correcting errors, directing my attention to related extant literature, and editorial assistance. This research has been supported by the Australian Research Council (ARC) Discovery Grant DP120102344.

References

REFERENCES

Ambrisio, L. & Tilli, P. (2004) Topics on Analysis in Metric Spaces . Oxford University Press.Google Scholar
Andrews, D.K.W. (1997) A stopping rule for the computation of generalized method of moments estimators. Econometrica 65(4), 913–931.CrossRefGoogle Scholar
Andrews, D.W.K. & Guggenberger, P. (2017) Asymptotic size of Kleibergen’s LM and conditional LR tests for moment condition models. Econometric Theory 33, 1046–1080.CrossRefGoogle Scholar
Andrews, D.W.K. & Guggenberger, P. (2019) Identification- and singularity-robust inference for moment condition models. Quantitative Economics 10, 1703–1746.CrossRefGoogle Scholar
Andrews, D.W.K. & Stock, J.H. (2007) Inference with weak instruments. In Blundell, R., Newey, W.K., & Persson, T. (eds.), Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, vol. III , pp. 122–173. Cambridge University Press.Google Scholar
Antoine, B. & Renault, E. (2009) Efficient GMM with nearly-weak instruments. Journal of Econometrics 12, S135–S171.CrossRefGoogle Scholar
Bodnar, T., Mazur, S., & Podgorski, K. (2016) Singular inverse Wishart distribution and its application to portfolio theory. Journal of Multivariate Analysis 143, 314–326.CrossRefGoogle Scholar
Caner, M. (2010) Testing, estimation in GMM and CUE with nearly weak identification. Econometric Reviews 29, 330–363.CrossRefGoogle Scholar
Chao, J.C. & Swanson, N.R. (2005) Consistent estimation with a large number of weak instruments. Econometrica 73, 1673–1692.CrossRefGoogle Scholar
Chernozhukov, V., Hong, H., & Tamer, E. (2007) Estimation and confidence regions for parameter sets in econometric models 1. Econometrica 75(5), 1243–1284.CrossRefGoogle Scholar
Choi, I. & Phillips, P.C.B. (1992) Asymptotic and finite sample distribution theory for IV estimators and tests in partially identified models. Journal of Econometrics 51(1–2), 113–150.CrossRefGoogle Scholar
Cragg, J.G. & Donald, S.G. (1993) Testing Identifiability and specification in instrument variable models. Econometric Theory 9, 222–240.CrossRefGoogle Scholar
Dovonon, P. & Renault, E. (2013) Testing for common conditionally heteroskedastic factors. Econometrica 81, 2561–2586.Google Scholar
Engle, R.F. & Kozicki, S. (1993) Testing for common features. Journal of Buisiness and Economic Statistics 11, 369–395.CrossRefGoogle Scholar
Engle, R.F. & Susmel, R. (1993) Common volatility in international equity markets. Journal of Business and Economic Statistics 11, 369–395.CrossRefGoogle 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
Hahn, J. & Hausman, J. (2003) Weak instruments: Diagnosis and cures in empirical econometrics. American Economic Review 93(2), 118–125.CrossRefGoogle Scholar
Hall, A.R. (2005) Generalized Method of Moments . Oxford University Press.Google Scholar
Han, C. & Phillips, P.C.B. (2006) GMM with many moment conditions. Econometrica 74, 147–192.CrossRefGoogle Scholar
Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054.CrossRefGoogle Scholar
Hansen, L.P., Heaton, J., & Yaron, A. (1996) Finite sample properties of some alternative GMM estimators. Journal of Business and Economic Statistics 14, 262–280.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S., & Balakrishnan, N. (1995) Continuous Univariate Distributions , vol. 2, 2nd ed. Wiley.Google Scholar
Kleibergen, F. (2005) Testing parameters in GMM without assuming that they are identified. Econometrica 73(4), 1103–1123.CrossRefGoogle Scholar
Lee, J. & Liao, Z. (2018) On standard inference for GMM with local identification failure of known forms. Econometric Theory 34, 790–814.CrossRefGoogle Scholar
Marsden, J.E. (1974) Elementary Classical Analysis . W.H. Freeman.Google Scholar
Menzel, K. (2014) Consistent estimation with many moment inequalities. Journal of Econometrics 182, 329–350.CrossRefGoogle Scholar
Newey, W.K. (1985) Generalized method of moments specification testing. Journal of Econometrics 29, 229–256.CrossRefGoogle Scholar
Newey, W.K. & Windmeijer, F. (2009) Generalized method of moments with many weak moment conditions. Econometrica 77, 687–719.Google Scholar
Pagan, A. & Robertson, J. (1997) GMM and Its Problems. Discussion paper, Australia National University.Google Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5(2), 181–240.CrossRefGoogle Scholar
Phillips, P.C.B. (2006) A remark on bimodality and weak instrumentation in structural equation models. Econometric Theory 22(5), 947–960.CrossRefGoogle Scholar
Phillips, P.C.B. (2016) Inference in near-singular regression. In González-Rivera, G., Hill, R.C., & Lee, T.-H. (eds.), Essays in Honor of Aman Ullah . Advances in Econometrics, vol. 36, pp. 461–486. Emerald Group Publishing.Google Scholar
Podivinsky, J.M. (1999) Finite sample properties of GMM estimators and tests. In Matyas, L. (ed.), Generlized Method of Moments Estimation , Chap. 5, pp. 128–148. Cambridge University Press.CrossRefGoogle Scholar
Poskitt, D.S. & Skeels, C.L. (2008) Assessing the magnitude of the concentration parameter in a simultaneous equations model. Econometrics Journal 12(1), 26–44.CrossRefGoogle Scholar
Poskitt, D.S. & Skeels, C.L. (2013) Inference in the Presence of Weak Instruments: A Selected Survey . Foundations and Trends in Econometrics, vol. 6.1. Now Publishers Inc. Google Scholar
Puri, M.L., Russell, C.T., & Mathew, T. (1984) Convergence of generalized inverses with applications to asymptotic hypothesis testing. Sankhya: The Indian Journal of Statistics, Series A 46(2), 277–286.Google Scholar
Rao, C.R. & Mitra, S.K. (1971) Generalized Inverse of Matrices and its Applications . Wiley.Google Scholar
Rothenberg, T.J. (1971) Identification in parametric models. Econometrica 39(3), 577–591.CrossRefGoogle Scholar
Sargan, J.D. (1958) The estimation of economic relationships using instrumental variables. Econometrica 26(3), 393–415.CrossRefGoogle Scholar
Srivastava, M. (2003) Singular Wishart and multivariate beta distributions. Annals of Statistics 31, 1537–1560.CrossRefGoogle Scholar
Staiger, D. & Stock, J.H. (1997) Instrumental variables regression with weak instruments. Econometrica 65(3), 557–586.CrossRefGoogle Scholar
Stock, J.H. & Wright, J.H. (2000) GMM with weak identification. Econometrica 68, 1055–1096.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 & Economic Statistics 20(4), 518–529.CrossRefGoogle Scholar
Stock, J.H. & Yogo, M. (2005) Testing for weak instruments in linear IV regression. In Andrews, D.W.K. and Stock, J.H. (eds.), Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg , Chap. 5, pp. 80–108. Cambridge University Press.CrossRefGoogle Scholar
Tauchen, G. (1986) Statistical properties of generalized method-of-moments estimators of structural parameters obtained from financial market data. Journal of Business and Economic Statistics 4, 397–416.CrossRefGoogle Scholar
Windmeijer, F. (2005) A finite sample correction for the variance of linear efficient two-step GMM estimators. Journal of Econometrics 126, 25–51.CrossRefGoogle Scholar
Wright, J.H. (2003) Detecting lack of identification in GMM. Econometric Theory 19, 322–330.CrossRefGoogle Scholar
Supplementary material: PDF

Poskitt supplementary material

Poskitt supplementary material
Download Poskitt supplementary material(PDF)
PDF 798.7 KB