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MANY INSTRUMENTS ASYMPTOTIC APPROXIMATIONSUNDER NONNORMAL ERROR DISTRIBUTIONS

Published online by Cambridge University Press:  18 August 2009

Martijn van Hasselt
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Abstract

In this paper we derive an alternative asymptoticapproximation to the sampling distribution of thelimited information maximum likelihood estimator anda bias-corrected version of the two-stage leastsquares estimator. The approximation is obtained byallowing the number of instruments and theconcentration parameter to grow at the same rate asthe sample size. More specifically, we allow forpotentially nonnormal error distributions and obtainthe conventional asymptotic distribution and theresults of Bekker (1994,Econometrica 62, 657–681) andBekker and Van der Ploeg (2005, StatisticaNeerlandica 59, 139–267) as specialcases. The results show that when the errordistribution is not normal, in general both theproperties of the instruments and the third andfourth moments of the errors affect the asymptoticvariance. We compare our findings with those in therecent literature on many and weak instruments.

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Type
Brief Report
Information
Econometric Theory , Volume 26 , Issue 2 , April 2010 , pp. 633 - 645
Copyright
Copyright © Cambridge University Press 2009

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Footnotes

I thank Paul Bekker, John Knight, FrankKleibergen, and Tony Lancaster for their supportand suggestions. The co-editor and two anonymousreferees provided valuable comments that greatlyimproved the presentation in this paper. I amresponsible for any remaining errors.

References

REFERENCES

Anderson, T. & Kunitomo, N. (2006) On Asymptotic Properties of the LIML Estimator with Possibly Many Instruments. Working paper, University of Tokyo.Google Scholar
Bekker, P.A. (1994) Alternative approximations to the distributions of instrumental variables estimators. Econometrica 62, 657–681.10.2307/2951662CrossRefGoogle Scholar
Bekker, P.A. & Van der Ploeg, J. (2005) Instrumental variable estimation based on grouped data. Statistica Neerlandica 59, 239–267.10.1111/j.1467-9574.2005.00296.xCrossRefGoogle Scholar
Billingsley, P. (1995) Probability and Measure. Wiley.Google Scholar
Chao, J.C. & Swanson, N.R. (2004) Asymptotic Distribution of JIVE in a Heteroscedastic IV Regression with Many Weak Instruments. Working paper, Rutgers University.Google Scholar
Chao, J.C. & Swanson, N.R. (2005) Consistent estimation with a large number of weak instruments. Econometrica 73, 1673–1692.10.1111/j.1468-0262.2005.00632.xCrossRefGoogle Scholar
Chao, J.C. & Swanson, N.R. (2006) Asymptotic normality of single-equation estimators for the case with a large number of weak instruments. In Corbae, D., Durlauf, S.N., & Hansen, B.E. (eds.), Econometric Theory and Practice: Frontiers of Analysis and Applied Research, pp. 82–124. Cambridge University Press.10.1017/CBO9781139164863.006CrossRefGoogle Scholar
Chao, J.C. & Swanson, N.R. (2007) Alternative approximations of the bias and MSE of the IV estimator under weak identification with an application to bias correction. Journal of Econometrics 137, 515–555.10.1016/j.jeconom.2005.09.002CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.10.1093/0198774036.001.0001CrossRefGoogle Scholar
Hansen, C., Hausman, J.A., & Newey, W.K. (2006) Estimation with Many Instrumental Variables. Working paper, MIT.CrossRefGoogle Scholar
Hausman, J.A., Newey, W.K., Woutersen, T., Chao, J.C., & Swanson, N.R. (2007) Instrumental Variable Estimation with Heteroscedasticity and Many Instruments. Working paper, MIT.CrossRefGoogle Scholar
Morimune, K. (1983) Approximate distributions of k-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica 51, 821–842.10.2307/1912160CrossRefGoogle Scholar
Nagar, A.L. (1959) The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 575–595.10.2307/1909352CrossRefGoogle Scholar
Rohatgi, V.K. (1971) Convergence of weighted sums of independent random variables. Proceedings of the Cambridge Philosophical Society 69, 305–307.10.1017/S0305004100046685CrossRefGoogle Scholar
Stock, J.H. & Yogo, M. (2005) Asymptotic distributions of instrumental variables statistics with many instruments. In Andrews, D.W.K. & Stock, J.H. (eds.), Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, pp. 109–120. Cambridge University Press.10.1017/CBO9780511614491.007CrossRefGoogle Scholar
Van der Ploeg, J. (1997) Instrumental Variable Estimation and Group-Asymptotics. Ph.D. dissertation, Rijksuniversiteit Groningen, The Netherlands.Google Scholar