Inference in Nonlinear Regression

Inference in Nonlinear Regression - Inference in Nonlinear...

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Inference in Nonlinear Regression when the Strength of Identi f cation is Unknown Timothy Cogley and Richard Startz February 1, 2011 (Preliminary and Incomplete) Abstract We present a Bayesian approach to inference in the presence of weak iden- ti f cation. A number of important weak identi f cation problems can be cast in the form of a nonlinear regression. We suggest an approach to computing Bayesian posteriors for such problems. The f rst contribution of the paper is a solution to what can be a di cult numerical computation task. We then analyze the simplest such weak identi f cation problem, the ratio of coe cients in a linear regression where the denominator is close to zero. This problem is also of signi f cant independent interest. A Monte Carlo analysis shows that the proposed method has good frequentist properties. The second contribution of the paper is to show that the Bayesian approach, when used with weakly informative priors, provides a useful frequentist estimator. An application to the accelerationist Phillips curve is included. 1 Introduction This paper reconsiders the problem of constructing con f dence intervals in a non- linear regression in which the strength of identi f cation is unknown a priori. We study regressions of the form =  (  )+ (1) where is the parameter of interest and and 2 are nuisance parameters. The variable is strictly exogenous, and the innovation is  (0  2 ) . The parameter For comments and suggestions, we are grateful to James Ramsey, Tao Zha, and seminar partic- ipants at U.C. Davis and U.C. Santa Barbara. Department of Economics, New York University, 19 W. 4th St., 6FL, New York, NY 10012. Email: tim.cogley@nyu.edu. Department of Economics University of Washington Box 353330 Seattle, WA 98l95-3330. Email: startz@u.washington.edu 1
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is unidenti f ed when =0 , and it is weakly identi f ed when 0 We assume only weak prior information about  so the strength of identi f cation is unknown a priori. We want a procedure for estimating and constructing con f dence regions that works well regardless of whether is strongly or weakly identi f ed. I.e., the coverage probability should be close to the nominal size, and the intervals should not be excessively long. An example of equation (1) is a Phillips curve a la Staiger et. al. (1997), = ( 1 )+ (2) where is in F ation, is unemployment, and is the natural rate of unemployment, which is assumed to be constant. The parameters of interest are the natural rate and the slope of the Phillips curve  The natural rate is unidenti f ed when and it is weakly identi f ed when 0 ForUSdatacover ingtheper iodo ftheGreat Moderation (1985.Q1-2008.Q4), the OLS estimate of is -0.012 with an asymptotic standard error of 0.1. The point estimate is close enough to zero and its standard error is large enough to raise concerns that
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Inference in Nonlinear Regression - Inference in Nonlinear...

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