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Projection-Based Statistical Inference in Linear Structural Models with Possibly Weak Instruments


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  • Jean-Marie Dufour


  • Mohamed Taamouti


It is well known that standard asymptotic theory is not valid or is extremely unreliable in models with identification problems or weak instruments [Dufour (1997, Econometrica), Staiger and Stock (1997, Econometrica), Wang and Zivot (1998, Econometrica), Stock and Wright (2000, Econometrica), Dufour and Jasiak (2001, International Economic Review)]. One possible way out consists here in using a variant of the Anderson-Rubin (1949, Ann. Math. Stat.) procedure. The latter, however, allows one to build exact tests and confidence sets only for the full vector of the coefficients of the endogenous explanatory variables in a structural equation, which in general does not allow for individual coefficients. This problem may in principle be overcome by using projection techniques [Dufour (1997, Econometrica), Dufour and Jasiak (2001, International Economic Review)]. Artypes are emphasized because they are robust to both weak instruments and instrument exclusion. However, these techniques can be implemented only by using costly numerical techniques. In this paper, we provide a complete analytic solution to the problem of building projection-based confidence sets from Anderson-Rubin-type confidence sets. The latter involves the geometric properties of "quadrics" and can be viewed as an extension of usual confidence intervals and ellipsoids. Only least squares techniques are required for building the confidence intervals. We also study by simulation how "conservative" projection-based confidence sets are. Finally, we illustrate the methods proposed by applying them to three different examples: the relationship between trade and growth in a cross-section of countries, returns to education, and a study of production functions in the U.S. economy. L'une des questions les plus étudiées récemment en économétrie est celle des modèles présentant des problèmes de quasi non-identification ou d'instruments faibles. L'une des conséquences importantes de ce problème est la non validité de la théorie asymptotique standard [Dufour (1997, Econometrica), Staiger et Stock (1997, Econometrica), Wang et Zivot (1998, Econometrica), Stock et Wright (2000, Econometrica), Dufour et Jasiak (2001, International Economic Review)]. Le défi majeur dans ce cas consiste à trouver des méthodes d'inférence robustes à ce problème. Une solution possible consiste à utiliser la statistique d'Anderson-Rubin (1949, Ann. Math. Stat.). Nous mettons l'emphase sur les procédures de type Anderson-Rubin, car celles-ci sont robustes tant à la présence d'instruments faibles et à l'exclusion d'instruments. Cette dernière ne fournit cependant des tests exacts que pour les hypothèses spécifiant le vecteur entier des coefficients des variables endogènes dans un modèle structurel, et de façon correspondante, que des régions de confiance simultanées pour ces coefficients. Elle ne permet pas de tester des hypothèses spécifiant des coefficients individuels ou sur des transformations de ces coefficients. Ce problème peut être résolu en principe par des techniques de projection [Dufour (1997, Econometrica), Dufour et Jasiak (2001, International Economic Review)]. Cependant , ces techniques ne sont pas toujours faciles à appliquer et requièrent en général l'emploi de méthodes numériques. Dans ce texte, nous proposons une solution explicite complète au problème de la construction de régions de confiance par projection basées sur des statistiques de type Anderson-Rubin. Cette solution exploite les propriétés géométriques des "quadriques" et peut s'interpréter comme une extension des intervalles et ellipsoïdes de confiance usuels. Le calcul de ces régions ne requièrent que des techniques de moindres carrés. Nous étudions également par simulation le degré de conservatisme des régions de confiance obtenues par projection. Enfin, nous illustrons les méthodes proposées par trois applications différentes: la relation entre l'ouverture commerciale et la croissance, le rendement de l'éducation et une étude sur les rendement d'échelles dans l'économie américaine.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 2003s-39.

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Date of creation: 01 May 2003
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Handle: RePEc:cir:cirwor:2003s-39

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Keywords: Simultaneous equations; structural model; instrumental variable; weak instrument; confidence interval; testing; projection; simultaneous inference; exact inference; asymptotic theory; équations simultanées ; modèle structurel ; variable instrumentale; instruments faibles; intervalle de confiance ; test ; projection ; inférence simultanée ; inférence exacte; théorie asymptotique;

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  1. Cragg, John G. & Donald, Stephen G., 1993. "Testing Identifiability and Specification in Instrumental Variable Models," Econometric Theory, Cambridge University Press, vol. 9(02), pages 222-240, April.
  2. Douglas A. Irwin & Marko Tervio, 2000. "Does Trade Raise Income? Evidence from the Twentieth Century," NBER Working Papers 7745, National Bureau of Economic Research, Inc.
  3. 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.
  4. James H. Stock & Motohiro Yogo, 2002. "Testing for Weak Instruments in Linear IV Regression," NBER Technical Working Papers 0284, National Bureau of Economic Research, Inc.
  5. Ann Harrison, 1995. "Openness and Growth: A Time-Series, Cross-Country Analysis for Developing Countries," NBER Working Papers 5221, National Bureau of Economic Research, Inc.
  6. Dufour, J.M. & Kiviet, J.F., 1995. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Cahiers de recherche 9547, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  7. Stock, James H & Wright, Jonathan H & Yogo, Motohiro, 2002. "A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(4), pages 518-29, October.
  8. Alastair Hall & Fernanda Peixe, 2003. "A Consistent Method for the Selection of Relevant Instruments," Econometric Reviews, Taylor & Francis Journals, vol. 22(3), pages 269-287.
  9. Davidson, Russell & MacKinnon, James G., 1993. "Estimation and Inference in Econometrics," OUP Catalogue, Oxford University Press, number 9780195060119, September.
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