Exponential Tilting with Weak Instruments: Estimation and Testing*
| Author | Mehmet Caner |
| DOI | http://doi.org/10.1111/j.1468-0084.2009.00579.x |
| Date | 01 June 2010 |
| Published date | 01 June 2010 |
307
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 72, 3 (2010) 0305-9049
doi: 10.1111/j.1468-0084.2009.00579.x
Exponential Tilting with Weak Instruments:
Estimation and TestingÅ
Mehmet Caner
Department of Economics, North Carolina State University, 4168 Nelson Hall, Raleigh,
NC 27695, USA (email: mcaner@ncsu.edu)
Abstract
This article analyses exponential tilting estimator with weak instruments in a non-
linear framework. Our paper differs from the previous literature in the context of
consistency proof. Teststhat are robust to the identification problem are also analysed.
These are Anderson–Rubin and Kleibergen types of test statistics. We also conduct
a simulation study wherein we compare empirical likelihood and continuous
updating-based tests with exponential tilting (ET)-based ones. The designs involve
GARCH(1,1) and contaminated structural errors. We find that ET-based Kleibergen
test has the best size among these competitors.
I. Introduction
In the recent literature Stock and Wright (2000) have shown that GMM’sasymptotic
properties change when the instruments 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 anAnderson and Rubin (1949)-like test statistic.
The limit is chi-squared, 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 chi-squared limit with degrees of freedom equal
to the number of parameters being tested. This has usually power properties better
than the Anderson–Rubin-like test when there are many instruments. Confidence
intervals are built by inverting these two test statistics. Confidence intervals that
ÅI thank the Associate Editor,Anders Rahbek, and two anonymous referees for comments.
JEL Classification numbers: C2, C4, C5.
308 Bulletin
are based on LM-like statistic of Kleibergen (2005) are never empty, whereas
Anderson–Rubin-based confidence intervals may be empty when the over-
identifying 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 include continuous updating (CUE), exponential tilting (ET) and
empirical likelihood (EL) estimators. They compare higher order asymptotic
properties of these estimators and GMM. They find that the bias-corrected EL is
asymptotically efficient relative to the other bias-corrected ET, CUE and GMM two-
step estimators. However, as stated in Imbens, Spady and Johnson (1998) ET has also
desirable properties compared with EL. The influence function of ET is less affected
by perturbation in the Lagrange multipliers compared with EL. ET is more robust to
misspecification problems.
In this paper, we analyse ET with weak instruments. Imbens et al. (1998) and
Kitamura and Stutzer (1997) consider the same model with standard identification
conditions. Our paper analyses the case with weak instruments. Weconsider the weak
instrument set-up of Stock and Wright(2000). This is important to applied researchers
as we have to see how the asymptotics of ET may be changing when there is an identi-
fication problem. Weanalyse both consistency and testing issues. We analyse a model
with nonlinear moment restrictions.
We propose two tests that are robust to the identification problem: Anderson–
Rubin and Kleibergen types of test statistics. Weshow that their limits are chi-squared
and nuisance parameter free. Confidence intervals can also be built using these test
statistics. Wealso conduct simulation exercises to analyse the small-sample properties
of these tests. Our simulation involves weak instruments set-up with two different
designs. In the first set-up, structural errors are generalized autoregressive conditional
heteroscedasticity, (GARCH(1,1)) and in the second one there is a contaminated error
distribution. Wecompare the tests with EL and CUE-based ones. Wefind that ET based
Kleibergen test has the best size.
In a related paper, Guggenberger (2003), considers generalized EL estimators
(GEL) in weakly identified linear models. A subsequent paper by Guggenberger
and Smith (2005) generalize Guggenberger (2003) to nonlinear models. Otsu (2006)
also generalizes the findings to time-series models and introduces an alternative
Kleibergen-type test. Guggenberger and Smith (2005) also have time-series exten-
sion. Their research is carried independently of ours.
The differences between Guggenberger and Smith’s(2005) paper and this one can
be related to the contents of the proof methods. To see theoretical differences with
the Guggenberger and Smith (2005) article, our consistency proof, which includes
Lemma A.1 and Theorem 1, is new in the weak identification literature. The proof of
Theorem 1 extends the Wald (1949) and Wolfowitz (1949) consistency approach to
a mixed set-up of both weakly and fully identified parameters. The assumption and
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