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Finite Sample Inference Methods for Simultaneous Equations and Models with Unobserved and Generated Regressors

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  • Jean-Marie Dufour

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  • Joanna Jasiak

Abstract

We propose finite sample tests and confidence sets for models with unobserved and generated regressors as well as various models estimated by instrumental variables methods. The validity of the procedures is unaffected by the presence of identification problems or weak instruments, so no detection of such problems is required. We study two distinct approaches for various models considered by Pagan (1984). The first one is an instrument substitution method which generalizes an approach proposed by Anderson and Rubin (1949) and Fuller (1987) for different (although related) problems, while the second one is based on splitting the sample. The instrument substitution method uses the instruments directly, instead of generating regressors, in order to test hypotheses about the structural parameters of interest and build confidence sets. The second approach relies on generated regressors, which allows a gain in degrees of freedom, and a sample split technique. For inference about general possibly nonlinear transformations of model parameters, projection techniques are proposed. A distributional theory is obtained under the assumptions of Gaussian errors and stricly exogenous regressors. We show that the various tests and confidence sets proposed are (locally) asymptotically valid under much weaker assumptions. The properties of the tests proposed are examined in simulation experiments. In general, they outperform the usual asymptotic inference methods in terms of both reliability and power. Finally, the techniques suggested are applied to a model of Tobin's q and to a model of academic performance. Nous proposons des tests et régions de confiance exacts pour des modèles comportant des variables inobservées ou des régresseurs estimés de même que pour divers modèles estimés par la méthode des variables instrumentales. La validité des procédures proposées n'est pas influencée par la présence de problèmes d'identification ou d'instruments faibles, de sorte que la détection de tels problèmes n'est pas requise pour les appliquer. De façon plus spécifique, nous étudions deux approches différentes pour divers modèles considérés par Pagan (1984). La première est une méthode de substitution d'instruments qui généralise des techniques proposées par Anderson et Rubin (1949) et Fuller (1984) pour des problèmes différents, tandis que la seconde méthode est fondée sur une subdivision de l'échantillon. La méthode de substitution d'instruments utilise directement les instruments disponibles, plutôt que des régresseurs estimés, afin de tester des hypothèses et de construire des régions de confiance sur les paramètres structuraux du modèle. La seconde méthode s'appuie sur des régresseurs estimés, ce qui permet un gain de degrés de liberté, ainsi que sur une technique de subdivision de l'échantillon. Pour faire de l'inférence sur des transformations générales, possiblement non-linéaires, des paramètres du modèle, nous proposons l'utilisation de techniques de projection. Nous fournissons une théorie distributionnelle exacte sous une hypothèse de normalité des perturbations et de régresseurs strictement exogènes. Nous montrons que les tests et régions de confiance ainsi obtenus sont aussi (localement) asymptotiquement valides sous des hypothèses distributionnelles beaucoup plus faibles. Nous étudions les propriétés des tests proposés dans le cadre d'une expérience de simulation. En général, celles-ci sont plus fiables et ont une meilleure puissance que les techniques traditionnelles. Finalement, les techniques proposées sont appliquées à un modèle du q de Tobin et à un modèle de performance scolaire.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 2000s-13.

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Date of creation: 01 Apr 2000
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Handle: RePEc:cir:cirwor:2000s-13

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Keywords: Simultaneous equations; structural model; instrumental variables; weak instruments; generated regressor; Anderson-Rubin method; pivotal function; samplesplit; exact test; confidence region; projection techniques; Tobin's q; academic performance; Équations simultanées; modèle structurel; variables instrumentales; instruments faibles; régresseur estimé; méthode d'Anderson-Rubin; fonction pivotale; subdivision d'échantillon; inférence à distance finie; test exact; région de confiance; techniques de projection; q de Tobin; performance scolaire;

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  1. Oxley, Les & McAleer, Michael, 1993. " Econometric Issues in Macroeconomic Models with Generated Regressors," Journal of Economic Surveys, Wiley Blackwell, vol. 7(1), pages 1-40.
  2. Murphy, Kevin M & Topel, Robert H, 1985. "Estimation and Inference in Two-Step Econometric Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 3(4), pages 370-79, October.
  3. Dufour, Jean-Marie, 1989. "Nonlinear Hypotheses, Inequality Restrictions, and Non-nested Hypotheses: Exact Simultaneous Tests in Linear Regressions," Econometrica, Econometric Society, vol. 57(2), pages 335-55, March.
  4. Andrew B. Abel & Janice C. Eberly, 1995. "A Unified Model of Investment Under Uncertainty," NBER Working Papers 4296, National Bureau of Economic Research, Inc.
  5. Nelson, C. & Startz, R., 1988. "The Distribution Of The Instrumental Variables Estimator And Its T-Ratio When The Instrument Is A Poor One," Working Papers 88-07, University of Washington, Department of Economics.
  6. Robert J. Barro, 1976. "Unanticipated Money Growth and Unemployment in the United States," Working Papers 234, Queen's University, Department of Economics.
  7. Montmarquette, Claude & Mahseredjian, Sophie, 1989. "Could teacher grading practices account for unexplained variation in school achievements?," Economics of Education Review, Elsevier, vol. 8(4), pages 335-343, August.
  8. Hall, Alastair R & Rudebusch, Glenn D & Wilcox, David W, 1996. "Judging Instrument Relevance in Instrumental Variables Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 37(2), pages 283-98, May.
  9. Kiviet, Jan F. & Dufour, Jean-Marie, 1997. "Exact tests in single equation autoregressive distributed lag models," Journal of Econometrics, Elsevier, vol. 80(2), pages 325-353, October.
  10. Maddala, G S & Jeong, Jinook, 1992. "On the Exact Small Sample Distribution of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 60(1), pages 181-83, January.
  11. Abel, Andrew B & Blanchard, Olivier J, 1986. "The Present Value of Profits and Cyclical Movements in Investment," Econometrica, Econometric Society, vol. 54(2), pages 249-73, March.
  12. Dagenais, M.G. & Dufour, J.M., 1987. "Invariance, Nonlinear Models and Asymptotic Tests," Cahiers de recherche 8738, Universite de Montreal, Departement de sciences economiques.
  13. Maddala, G S, 1974. "Some Small Sample Evidence on Tests of Significance in Simultaneous Equations Models," Econometrica, Econometric Society, vol. 42(5), pages 841-51, September.
  14. Dufour, Jean-Marie & Kiviet, Jan F., 1996. "Exact tests for structural change in first-order dynamic models," Journal of Econometrics, Elsevier, vol. 70(1), pages 39-68, January.
  15. Pagan, Adrian, 1984. "Econometric Issues in the Analysis of Regressions with Generated Regressors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(1), pages 221-47, February.
  16. Joshua Angrist & Alan Krueger, 1993. "Split Sample Instrumental Variables," Working Papers 699, Princeton University, Department of Economics, Industrial Relations Section..
  17. Fumio Hayashi, 1981. "Tobin's Marginal q and Average a : A Neoclassical Interpretation," Discussion Papers 457, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  18. Douglas Staiger & James H. Stock, 1994. "Instrumental Variables Regression with Weak Instruments," NBER Technical Working Papers 0151, National Bureau of Economic Research, Inc.
  19. Buse, A, 1992. "The Bias of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 60(1), pages 173-80, January.
  20. Jean-Marie Dufour, 1997. "Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models," Econometrica, Econometric Society, vol. 65(6), pages 1365-1388, November.
  21. Savin, N.E., 1984. "Multiple hypothesis testing," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 14, pages 827-879 Elsevier.
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Cited by:
  1. Benoit Perron, 2003. "Semiparametric Weak-Instrument Regressions with an Application to the Risk-Return Tradeoff," The Review of Economics and Statistics, MIT Press, vol. 85(2), pages 424-443, May.
  2. Dufour, Jean-Marie, 2001. "Logique et tests d’hypothèses," L'Actualité Economique, Société Canadienne de Science Economique, vol. 77(2), pages 171-190, juin.
  3. DUFOUR, Jean-Marie, 2001. "Logique et tests d'hypotheses: reflexions sur les problemes mal poses en econometrie," Cahiers de recherche 2001-15, Universite de Montreal, Departement de sciences economiques.
  4. Debdulal Mallick, 2007. "The Role of Elasticity of Substitution in Economic Growth: A Cross-Country Test of the La Grandville Hypothesis," Economics Series 2007_04, Deakin University, Faculty of Business and Law, School of Accounting, Economics and Finance.

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