A Note on the Regularized Approach to Biased 2SLS Estimation with Weak Instruments
| Author | Winfried Pohlmeier,Namhyun Kim |
| DOI | http://doi.org/10.1111/obes.12144 |
| Published date | 01 December 2016 |
| Date | 01 December 2016 |
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©2016 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 78, 6 (2016) 0305–9049
doi: 10.1111/obes.12144
A Note on the Regularized Approach to Biased 2SLS
Estimation with Weak Instruments*
Namhyun Kim† and Winfried Pohlmeier‡
†University of Exeter Business School, Streatham Court, Rennes Drive, Exeter, UK
(e-mail: n.kim@exeter.ac.uk)
‡Department of Economics, University of Konstanz, D-78457, Konstanz, Germany
(e-mail: winfried.pohlmeier@uni-konstanz.de)
Abstract
The presence of weak instruments is translated into a nearly singular problem in a control
function representation. Therefore, the L2-norm type of regularization is proposed to
implement the 2SLS estimation for addressing the weak instrument problem. The L2-norm
regularization with a regularized parameter O(n) allowsus to obtain the Rothenberg (1984)
type of higher-order approximation of the 2SLS estimator in the weak instrument asymp-
totic framework. The proposed regularized parameter yields the regularized concentration
parameter O(n), which is used as a standardized factor in the higher-order approximation.
We also show that the proposed L2-norm regularization consequently reduces the finite
sample bias. A number of existing estimators that address finite sample bias in the pres-
ence of weak instruments, especially Fuller’s limited information maximum likelihood
estimator, are compared with our proposed estimator in a simple Monte Carlo exercise.
I. Introduction
A substantial number of studies have examined the performance of the two-stage least
squares (2SLS) estimator in the presence of weak instruments over the past few decades.
The presence of weakinstr uments is knownto cause the 2SLS estimator to be biased towards
the ordinary least squares estimator in small samples (see Bound, Jaeger and Baker, 1995;
Staiger and Stock, 1997; and Stock, Wright andYogo, 2002 for details).A number of studies,
such as Hahn and Hausman (2002), Chao and Swanson (2005), and Bun and Windmeijer
(2011) analytically examined the higher-order approximations of the 2SLS estimator and
showed the severe small sample biases. On the other hand, studies such as those of Nelson
and Startz (1990a, b), and Cruz and Moreira (2005) investigated the performance of the
2SLS estimator in small samples by using Monte Carlo simulation exercises.
JEL Classification numbers: C1, C2, C4, C5
*‘Financial support by the German Research Foundation (DFG) through research unit FOR 1882 \Psycho-
economics’ is gratefully acknowledged.The authors also would like to thank Sebastian Bayer for helpful research
assistance. Early versions of the paper have been presented at workshopsand conferences in Rimini, Pisa, Cologne
and Lancaster.All remaining er rors are ours.
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