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Exponential Tilting with Weak Instruments

Exponential Tilting with Weak Instruments - Exponential...

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Exponential Tilting with Weak Instruments: Estimation and Testing Mehmet Caner North Carolina State University January 2008 Abstract This article analyzes exponential tilting estimator with weak instruments in a nonlinear framework. The limits of these estimators under standard identification assumptions are derived by Imbens, Spady and Johnson (1998) and Kitamura and Stutzer (1997). There are also papers by Guggenberger and Smith (2005), and Otsu (2006) in a similar context to our paper. They derive the limits of generalized empirical likelihood estimators under weak identification. Our paper differs from them in the context of consistency proof. Tests that are robust to the identification problem are also obtained. These are Anderson-Rubin and Kleibergen type of test statistics. The limits are nuisance parameter free and χ 2 distributed. We can also build confidence intervals by inverting these test statistics. We also conduct a simulation study where we compare empirical likelihood and continuous updating based tests with exponential tilting based ones. The designs involve GARCH (1,1) and contaminated structural errors. We find that exponential tilting based Kleibergen test has the best size among these competitors. JEL Classification: C2,C4,C5. Keywords: Lagrange multipliers, weak instruments, information theory. Department of Economics, 4168 Nelson Hall, College of Management, Raleigh, NC 27695. email: [email protected] 0
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1 Introduction In the recent literature Stock and Wright (2000) have shown that GMM’s asymptotic properties change when the instruments are weakly correlated with moment conditions. They also show that the limits are not asymptotically normal and the new limits involve nuisance parameters. The weak instrument asymptotics provides better results in small samples. Inference that is robust to identification is also pursued by Stock and Wright (2000), and they propose an Anderson- Rubin like (1949) test statistic. The limit is χ 2 , with degrees of freedom equal to the number of orthogonality conditions. Kleibergen (2005) also provides an LM-like test statistic which is nuisance parameter free. This statistic has also χ 2 limit with degrees of freedom equal to the number of parameters being tested. This has usually better power properties than the Anderson-Rubin like test when there are many instruments. Confidence intervals are built by inverting these two test statistics. Confidence intervals that are based on LM like statistic of Kleibergen (2005) are never empty, whereas Anderson-Rubin based confidence intervals may be empty when the overidentifying restrictions are invalid. Recently Caner (2007) has developed boundedly pivotal structural change tests in weakly identified models with nonlinear moment restrictions. To improve the small sample properties of GMM, Newey and Smith (2004) take a different direction. In a recent article, they propose Generalized Empirical Likelihood Estimators. These in- clude continuous updating, exponential tilting, and empirical likelihood estimators. They compare higher-order asymptotic properties of these estimators and GMM. They find that the bias-corrected
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