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

Authors

  • Jean-Marie Dufour,

    1. Departement de Sciences Économiques, Université de Montréal, C.P. 6128 Succursale Centre-ville, Montréal, Québec, Canada H3C 3J7; and Centre Interuniversitaire de Recherche en Analyse des Organisations and Centre Interuniversitaire de Recherche en Économie Quantitative, Université de Montréal; jean.marie.dufour@umontreal.ca
      and
      Centre Interuniversitaire de Recherche en Économic Quantitative, Université de Montréal, C.P. 6128 Succursale Centre-ville, Montréal, Québec, Canada H3C 3J7; and INSEA, Rabat, Morocco; taamouti@insea.ac.ma.
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  • Mohamed Taamouti

    1. Departement de Sciences Économiques, Université de Montréal, C.P. 6128 Succursale Centre-ville, Montréal, Québec, Canada H3C 3J7; and Centre Interuniversitaire de Recherche en Analyse des Organisations and Centre Interuniversitaire de Recherche en Économie Quantitative, Université de Montréal; jean.marie.dufour@umontreal.ca
      and
      Centre Interuniversitaire de Recherche en Économic Quantitative, Université de Montréal, C.P. 6128 Succursale Centre-ville, Montréal, Québec, Canada H3C 3J7; and INSEA, Rabat, Morocco; taamouti@insea.ac.ma.
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    • The authors thank Laurence Broze, John Cragg, Jean-Pierre Florens, Christian Gouriéroux, Joanna Jasiak, Frédéric Jouneau, Lynda Khalaf, Nour Meddahi, Benoît Perron, Tim Vogelsang, Eric Zivot, three anonymous referees, and a co-editor for several useful comments. This work was supported by the Canada Research Chair Program (Chair in Econometrics, Université de Montréal), the Alexander-von-Humboldt Foundation (Germany), the Canadian Network of Centres of Excellence (program on Mathematics of Information Technology and Complex Systems), the Canada Council for the Arts (Killam Fellowship), the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, the Fonds de Recherche sur la Société et la Culture (Québec), and the Fonds de Recherche sur la Nature et les Technologies (Québec). Taamouti was also supported by a fellowship from the Canadian International Development Agency. Dufour holds the Canada Research Chair (Econometrics).


Abstract

It is well known that standard asymptotic theory is not applicable or is very unreliable in models with identification problems or weak instruments. One possible way out consists of using a variant of the Anderson–Rubin ((1949), AR) procedure. The latter 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, but not for individual coefficients. This problem may in principle be overcome by using projection methods (Dufour (1997), Dufour and Jasiak (2001)). At first sight, however, this technique requires the application of costly numerical algorithms. In this paper, we give a general necessary and sufficient condition that allows one to check whether an AR-type confidence set is bounded. Furthermore, we provide an analytic solution to the problem of building projection-based confidence sets from AR-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 needed to build the confidence intervals.

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