Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T15:32:36.415Z Has data issue: false hasContentIssue false
  • English
  • Français

MANY INSTRUMENTS ASYMPTOTIC APPROXIMATIONS UNDER NONNORMAL ERROR DISTRIBUTIONS

Published online by Cambridge University Press:  18 August 2009

Martijn van Hasselt
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we derive an alternative asymptotic approximation to the sampling distribution of the limited information maximum likelihood estimator and a bias-corrected version of the two-stage least squares estimator. The approximation is obtained by allowing the number of instruments and the concentration parameter to grow at the same rate as the sample size. More specifically, we allow for potentially nonnormal error distributions and obtain the conventional asymptotic distribution and the results of Bekker (1994, Econometrica 62, 657–681) and Bekker and Van der Ploeg (2005, Statistica Neerlandica 59, 139–267) as special cases. The results show that when the error distribution is not normal, in general both the properties of the instruments and the third and fourth moments of the errors affect the asymptotic variance. We compare our findings with those in the recent literature on many and weak instruments.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

I thank Paul Bekker, John Knight, Frank Kleibergen, and Tony Lancaster for their support and suggestions. The co-editor and two anonymous referees provided valuable comments that greatly improved the presentation in this paper. I am responsible 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