Weak Identification Robust Tests for Subvectors Using Implied Probabilities
Abstract
1. Introduction
2. Implied Probabilities for Hypothesis on Subsets of Parameters in the GMM Framework
2.1. Background
2.2. Assumptions
- (i)
- .
- (ii)
- .
- (iii)
- for .
- (iv)
- .
- (i)
- , ,and .
- (ii)
- where and stand for the smallest and largest eigenvalues, respectively, of .
2.3. Properties of GEL Implied Probabilities
- Assumption : (GEL function)
- is a continuous function such that
- (i)
- is concave on its domain , which is an open interval containing 0.
- (ii)
- is twice continuously differentiable on its domain. Defining for and , let (standardization for convenience).
- (iii)
- There exists a positive constant b such that for each , hold.
- (B)
- ,
- (B)
- .
3. Score Test for Subsets of Parameters Using the Implied Probabilities
3.1. Score Vector and Score Statistic Using the Implied Probabilities
3.2. Refined Projection Score Test Using the Implied Probabilities
- Step 1: Construct a confidence interval for under the restriction of the null hypothesis . is a random subset of the parameter space of and is defined as follows:
- (i)
- The asymptotic size of the test cannot exceed for any choice of and with under a restriction in (8) that .
- (ii)
- If all elements of θ are strongly identified as in Newey and West [6], and , then the test with any given such that is non-empty and is asymptotically equivalent to the infeasible efficient score test that rejects if .
4. Monte Carlo Experiment
4.1. Design
4.2. Results
5. Application to the Impact of Veteran Status on Earnings
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix A.1. Assumptions Involving Weak Identification
- For , let and for , let interior , where is compact.
- where
- (a)
- is such that uniformly for where is bounded and continuous and . For , , uniformly. where, for and , the matrix is bounded and continuous.
- (b)
- is a continuous function and if and only if . For , is bounded and continuous. has full column rank. Here, where for .
- D1.
- uniformly for where by imposing interchangeability of the order of differentiation and integration (and from Assumption ).
- D2.
- where (the partition of , and is conformable to the partition of , the partition of the weakly identified elements of into those from and , respectively)is bounded, continuous, and positive definite. Refining Assumption 2 and with the obvious correspondence of notation between the V’s and the ’s that specify what the estimators are, we also make the following assumptions: uniformly for . is bounded and continuous. uniformly for . (It is worth noting that the matrix where is the l-th element () of .) For the matrices are such that uniformly for .
Appendix A.2. Proofs
- In the proofs, we use the notation
References
- Hall, A. Generalized Method of Moments; Advanced Texts in Econometrics; Oxford University Press: Oxford, UK, 2005. [Google Scholar]
- Staiger, D.; Stock, J. Intrumal Variables Regression with Weak Instruments. Econometrica 1997, 65, 557–586. [Google Scholar]
- Antoine, B.; Renault, E. Efficient GMM with Nearly-Weak Instruments. Econom. J. 2009, 12, S135–S171. [Google Scholar]
- Andrews, D.; Cheng, X. Estimation and Inference with Weak, Semi-Strong, and Strong Identification. Econometrica 2012, 80, 2153–2211. [Google Scholar]
- Dufour, J.M. Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models. Econometrica 1997, 65, 1365–1387. [Google Scholar]
- Newey, W.K.; West, K.D. Hypothesis Testing with Efficient Method of Moments Estimation. Int. Econ. Rev. 1987, 28, 777–787. [Google Scholar]
- Wang, J.; Zivot, E. Inference on Structural Parameters in Instrumental Variables Regression with Weak Instruments. Econometrica 1998, 66, 1389–1404. [Google Scholar]
- Chaudhuri, S.; Renault, E. Shrinkage of Variance for Minimum Distnce Based Tests. Econom. Rev. 2015, 34, 328–351. [Google Scholar]
- Chaudhuri, S.; Renault, E. Score Tests in GMM: Why Use Implied Probabilities. J. Econom. 2020, 219, 260–280. [Google Scholar]
- Newey, W.K.; Smith, R.J. Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. Econometrica 2004, 72, 219–255. [Google Scholar]
- Kitamura, Y.; Stutzer, M. An Information-Theoretic Alternative to Generalized Method of Moments Estimation. Econometrica 1997, 65, 861–874. [Google Scholar]
- Golan, A. Information and entropy economerics—A review and synthesis. Found. Trends Econom. 2006, 2, 1–145. [Google Scholar]
- Guggenberger, P.; Kleibergen, F.; Mavroeidis, S.; Chen, L. On the Asymptotic Sizes of Subset Anderson-Rubin and Lagrange Multiplier Tests in Linear Instrumental Variables Regression. Econometrica 2012, 80, 2649–2666. [Google Scholar]
- Chaudhuri, S.; Zivot, E. A new method of projection-based inference in GMM with weakly identified nuisance parameters. J. Econom. 2011, 164, 239–251. [Google Scholar]
- Smith, R.J. Alternative semi-parametric likelihood approaches to generalized method of moments estimation. Econ. J. 1997, 107, 503–519. [Google Scholar]
- Chaudhuri, S.; Rose, E.; Estimating the Veteran Effect with Endogenous Schooling when Instruments are Potentially Weak. Technical Report, University of North Carolina, Chapel Hill and University of Washington. 2010. Available online: https://saraswata.research.mcgill.ca/sc_er_10.pdf (accessed on 2 February 2025).
- Angrist, J. Lifetime Earnings and the Vietnam Era Draft Lottery: Evidence from Social Security Administrative Records. Am. Econ. Rev. 1990, 80, 313–336. [Google Scholar]
- Card, D. Using Geographical Variation in College Proximity to Estimate the Return to Schooling. Aspects of Labour Market Behavior: Essays in Honor of John Vanderkamp; University of Toronto Press: Toronto, ON, Canada, 1995. [Google Scholar]
- Golan, A.; Judge, G.; Miller, D. Maximum Entropy Econometrics: Robust Estimation with Limited Data; John Wiley & Sons: New York, NY, USA, 1996. [Google Scholar]
- Imbens, G.W.; Spady, R.H.; Johnson, P. Information Theoretic Approaches to Inference in Moment Condition Models. Econometrica 1998, 66, 333–357. [Google Scholar]
- Smith, R.J. Empirical Likelihood Estimation and Inference. In Applications of Differential Geometry to Econometrics; Salmon, M., Marriott, P., Eds.; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Kitamura, Y. Empirical Likelihood Methods in Econometrics: Theory and Practice. In Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress; Blundell, R., Newey, W., Persson, T., Eds.; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Guggenberger, P.; Smith, R. Generalized Empirical Likelihood Estimators and Tests under Partial, Weak and Strong Identification. Econom. Theory 2005, 21, 667–709. [Google Scholar]
- Caner, M. Exponential tilting with weak instruments: Estimation and testing. Oxf. Bull. Econ. Stat. 2010, 72, 307–326. [Google Scholar]
- Andrews, A.; Stock, J.; Sun, L. Weak Instruments in IV Regression: Theory and Practice. Annu. Rev. Econ. 2019, 11, 727–753. [Google Scholar]
- Andrews, D.W.K. Identification-Robust Subvector Inference; Yale University Working Paper; Yale University: New Haven, CT, USA, 2017. [Google Scholar]
- Andrews, I. Valid Two-Step Identification-Robust Confidence Sets for GMM. Rev. Econ. Stat. 2018, 100, 337–348. [Google Scholar]
- Otsu, T. Generalized Empirical Likelihood Inference for Nonlinear and Time Series Models under Weak Identification. Econom. Theory 2006, 22, 513–527. [Google Scholar] [CrossRef]
- Guggenberger, P.; Smith, R. Generalized empirical likelihood tests in time series models with potential identification failure. J. Econom. 2008, 142, 134–161. [Google Scholar] [CrossRef]
- Guggenberger, P.; Kleibergen, F.; Mavroeidis, S. A more powerful subvector Anderson Rubin test in linear instrumental variables regression. Quant. Econ. 2019, 10, 487–526. [Google Scholar] [CrossRef]
- Kleibergen, F. Efficient size correct subset inference in homoskedastic linear instrumental variables regression. J. Econom. 2021, 221, 78–96. [Google Scholar] [CrossRef]
- Kleibergen, F.; Mavroeidis, S.; Guggenberger, P. A powerful subvector Anderson-Rubin test in linear instrumental variables regression with conditional heteroskedasticity. Econom. Theory 2024, 40, 957–1002. [Google Scholar]
- Kleibergen, F. Testing Parameters In GMM Without Assuming That They Are Identified. Econometrica 2005, 73, 1103–1123. [Google Scholar] [CrossRef]
- Schennach, S.M. Point estimation with exponentially tilted empirical likelihood. Ann. Stat. 2007, 35, 634–672. [Google Scholar] [CrossRef]
- Dufour, J.M.; Taamouti, M. Projection-Based Statistical Inference in Linear Structural Models with Possibly Weak Instruments. Econometrica 2005, 73, 1351–1365. [Google Scholar] [CrossRef]
- Dufour, J.M.; Taamouti, M. Further Results on Projection-Based Inference in IV Regressions with Weak, Collinear or Missing Instruments. J. Econom. 2007, 139, 133–153. [Google Scholar] [CrossRef]
- Chaudhuri, S. Projection-Type Score Tests for Subsets of Parameters. Ph.D. Thesis, University of Washington, Seattle, WA, USA, 2008. [Google Scholar]
- Zivot, E.; Chaudhuri, S. Comment: Weak Instrument Robust Tests in GMM and the New Keynesian Phillips Curve by F. Kleibergen and S. Mavroeidis. J. Bus. Econ. Stat. 2009, 27, 328–331. [Google Scholar] [CrossRef]
- Chaudhuri, S.; Richardson, T.; Robins, J.M.; Zivot, E. Split-Sample Score Tests in Linear Instrumental Variables Regression. Econom. Theory 2010, 26, 1820–1837. [Google Scholar] [CrossRef]
- Cox, D.R.; Hinkley, D.V. Theoretical Statistics; Chapman and Hall: London, UK, 1974. [Google Scholar]
- Neyman, J. Optimal Asymptotic Test of Composite Statistical Hypothesis. In Probability and Statistics, the Harald Cramer Volume; Grenander, U., Ed.; Almqvist and Wiksell: Uppsala, Sweden, 1959; pp. 313–334. [Google Scholar]
- Bera, A.K.; Bilias, Y. Rao’s score, Neyman’s C(α) and Silvey’s LM tests: An essay on historical developments and some new results. J. Stat. Plan. Inference 2001, 97, 9–44. [Google Scholar] [CrossRef]
- Kleibergen, F. Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression. Econometrica 2002, 70, 1781–1803. [Google Scholar]
- Stock, J.H.; Wright, J.H. GMM with Weak Identification. Econometrica 2000, 68, 1055–1096. [Google Scholar]
- Sun, L. Implementing valid two-step identification-robust confidence sets for linear instrumental-variables model. Stata J. 2018, 18, 803–825. [Google Scholar] [CrossRef]
- Chaudhuri, S.; Renault, E. Finite-Sample Improvements of Score Tests by the Use of Implied Probabilities from Generalized Empirical Likelihood; Technical report; McGill University: Montreal, QC, Canada, 2011. [Google Scholar]
- Antoine, B.; Bonnal, H.; Renault, E. On the efficient use of the informational content of estimating equations: Implied probabilities and Euclidean empirical likelihood. J. Econom. 2007, 138, 461–487. [Google Scholar]
- Angrist, J. The Draft Lottery and Voluntary Enlistment in the Vietnam Era. J. Am. Stat. Assoc. 1991, 86, 584–595. [Google Scholar]
- Card, D. The Causal Effect of Education on Earnings. Handb. Labor Econ. 1999, 3, 1801–1863. [Google Scholar]
- Card, D. Estimating the Return to Schooling: Progress on Some Persistent Econometric Problems. Econometrica 2001, 69, 1127–1160. [Google Scholar]
Nominal Level | Plug-in and Refined Projection Score Tests with and | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
: EEL | : EL | |||||||||
n | Plug-in | Plug-in | Plug-in | |||||||
100 | −1 | 99.7 | 99.3 | 98.4 | 97.5 | 96.3 | 95.2 | 99.5 | 99.4 | 98.9 |
100 | −0.8 | 98.6 | 96.9 | 93.7 | 92.5 | 89.5 | 87.0 | 97.6 | 96.7 | 95.2 |
100 | −0.6 | 93.4 | 87.7 | 79.2 | 78.4 | 72.3 | 67.3 | 91.2 | 88.7 | 83.5 |
100 | −0.4 | 76.2 | 64.0 | 50.4 | 50.9 | 42.9 | 37.6 | 72.6 | 64.9 | 55.4 |
100 | −0.2 | 42.1 | 28.6 | 18.1 | 20.3 | 15.2 | 12.2 | 39.1 | 29.3 | 21.3 |
100 | 0 | 10.6 | 6.1 | 2.9 | 7.8 | 5.6 | 4.6 | 13.0 | 7.7 | 5.4 |
100 | 0.2 | 6.6 | 6.1 | 6.0 | 21.8 | 18.6 | 17.9 | 26.1 | 20.5 | 20.1 |
100 | 0.4 | 34.4 | 34.4 | 34.4 | 57.4 | 50.8 | 48.5 | 68.4 | 63.8 | 63.6 |
100 | 0.6 | 74.4 | 74.4 | 74.4 | 76.4 | 66.5 | 61.6 | 94.7 | 93.4 | 93.4 |
100 | 0.8 | 92.3 | 92.3 | 92.3 | 62.5 | 49.0 | 43.1 | 99.8 | 99.8 | 99.8 |
100 | 1 | 96.2 | 96.2 | 96.2 | 37.8 | 23.2 | 18.5 | 100.0 | 100.0 | 100.0 |
1000 | −0.3162 | 99.3 | 99.3 | 99.1 | 99.3 | 97.6 | 92.9 | 97.1 | 96.1 | 94.9 |
1000 | −0.253 | 96.9 | 96.3 | 95.7 | 97.6 | 93.4 | 84.7 | 91.6 | 88.1 | 86.1 |
1000 | −0.1897 | 86.9 | 84.9 | 83.1 | 91.2 | 81.5 | 68.4 | 76.4 | 71.1 | 67.1 |
1000 | −0.1265 | 60.5 | 57.3 | 53.8 | 73.1 | 56.9 | 41.4 | 48.1 | 42.3 | 37.5 |
1000 | −0.0632 | 25.6 | 23.2 | 20.6 | 39.5 | 24.7 | 14.8 | 18.9 | 14.9 | 12.4 |
1000 | 0 | 6.2 | 5.6 | 4.8 | 11.9 | 6.1 | 3.1 | 6.8 | 4.8 | 4.0 |
1000 | 0.0632 | 11.4 | 11.4 | 11.3 | 15.8 | 7.4 | 5.8 | 21.3 | 18.8 | 18.8 |
1000 | 0.1265 | 45.1 | 45.1 | 45.1 | 33.2 | 16.0 | 13.0 | 61.1 | 58.2 | 58.2 |
1000 | 0.1897 | 85.6 | 85.6 | 85.6 | 24.8 | 13.3 | 11.1 | 92.0 | 90.8 | 90.8 |
1000 | 0.253 | 98.5 | 98.5 | 98.5 | 8.9 | 5.1 | 4.3 | 99.5 | 99.4 | 99.4 |
1000 | 0.3162 | 100.0 | 100.0 | 100.0 | 1.5 | 1.0 | 0.8 | 100.0 | 100.0 | 100.0 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Carrasco, M.; Chaudhuri, S. Weak Identification Robust Tests for Subvectors Using Implied Probabilities. Entropy 2025, 27, 396. https://doi.org/10.3390/e27040396
Carrasco M, Chaudhuri S. Weak Identification Robust Tests for Subvectors Using Implied Probabilities. Entropy. 2025; 27(4):396. https://doi.org/10.3390/e27040396
Chicago/Turabian StyleCarrasco, Marine, and Saraswata Chaudhuri. 2025. "Weak Identification Robust Tests for Subvectors Using Implied Probabilities" Entropy 27, no. 4: 396. https://doi.org/10.3390/e27040396
APA StyleCarrasco, M., & Chaudhuri, S. (2025). Weak Identification Robust Tests for Subvectors Using Implied Probabilities. Entropy, 27(4), 396. https://doi.org/10.3390/e27040396