Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-22T02:25:37.606Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTIC DISTRIBUTION OF JIVE IN A HETEROSKEDASTIC IV REGRESSION WITH MANY INSTRUMENTS

Published online by Cambridge University Press:  13 September 2011

John C. Chao ,
Norman R. Swanson ,
Jerry A. Hausman ,
Whitney K. Newey  and
Tiemen Woutersen
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper derives the limiting distributions of alternative jackknife instrumental variables (JIV) estimators and gives formulas for accompanying consistent standard errors in the presence of heteroskedasticity and many instruments. The asymptotic framework includes the many instrument sequence of Bekker (1994, Econometrica 62, 657–681) and the many weak instrument sequence of Chao and Swanson (2005, Econometrica 73, 1673–1691). We show that JIV estimators are asymptotically normal and that standard errors are consistent provided that as n→∞, where Kn and rn denote, respectively, the number of instruments and the concentration parameter. This is in contrast to the asymptotic behavior of such classical instrumental variables estimators as limited information maximum likelihood, bias-corrected two-stage least squares, and two-stage least squares, all of which are inconsistent in the presence of heteroskedasticity, unless Kn/rn→0. We also show that the rate of convergence and the form of the asymptotic covariance matrix of the JIV estimators will in general depend on the strength of the instruments as measured by the relative orders of magnitude of rn and Kn.

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 1 , February 2012 , pp. 42 - 86
Copyright
Copyright © Cambridge University Press 2011

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

Earlier versions of this paper were presented at the NSF/NBER conference on weak and/or many instruments at MIT in 2003 and at the 2004 winter meetings of the Econometric Society in San Diego, where conference participants provided many useful comments and suggestions. Particular thanks are owed to D. Ackerberg, D. Andrews, J. Angrist, M. Caner, M. Carrasco, P. Guggenberger, J. Hahn, G. Imbens, R. Klein, N. Lott, M. Moriera, G.D.A. Phillips, P.C.B. Phillips, J. Stock, J. Wright, two anonymous referees, and a co-editor for helpful comments and suggestions.

References

REFERENCES

Abadir, K.M. & Magnus, J.R. (2005) Matrix Algebra. Cambridge University Press.10.1017/CBO9780511810800CrossRefGoogle Scholar
Ackerberg, D.A. & Devereux, P. (2009) Improved JIVE estimators for overidentified models with and without heteroskedasticity. Review of Economics and Statistics 91, 351–362.10.1162/rest.91.2.351CrossRefGoogle Scholar
Angrist, J.D., Imbens, G.W., & Krueger, A. (1999) Jackknife instrumental variables estimation. Journal of Applied Econometrics 14, 57–67.10.1002/(SICI)1099-1255(199901/02)14:1<57::AID-JAE501>3.0.CO;2-GGoogle Scholar
Bekker, P.A. (1994) Alternative approximations to the distributions of instrumental variable 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, 506–508.10.1111/j.1467-9574.2005.00296.xCrossRefGoogle Scholar
Billingsley, P. (1986) Probability and Measure, 2nd ed. Wiley.Google Scholar
Blomquist, S. and Dahlberg, M. (1999) Small sample properties of LIML and jackknife IV estimators: Experiments with weak instruments. Journal of Applied Econometrics 14, 69–88.10.1002/(SICI)1099-1255(199901/02)14:1<69::AID-JAE521>3.0.CO;2-73.0.CO;2-7>CrossRefGoogle Scholar
Chao, J.C., Hausman, J.A., Newey, W.K., Swanson, N.R., & Woutersen, T. (2010) Testing Overidentifying Restrictions with Many Instruments and Heteroskedasticity. Working paper, MIT.Google Scholar
Chao, J.C. & Swanson, N.R. (2004) Estimation and Testing Using Jackknife IV in Heteroskedastic Regressions with Many Weak Instruments. Working paper, Rutgers University.CrossRefGoogle 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.xGoogle 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.), Frontiers of Analysis and Applied Research: Essays in Honor of Peter C. B. Phillips, pp. 82–124. Cambridge University Press.Google Scholar
Davidson, R. & MacKinnon, J.G. (2006) The case against JIVE. Journal of Applied Econometrics 21, 827–833.10.1002/jae.873CrossRefGoogle Scholar
Donald, S.G. & Newey, W.K. (2001) Choosing the number of instruments. Econometrica 69, 1161–1191.10.1111/1468-0262.00238Google Scholar
Fuller, W.A. (1977) Some properties of a modification of the limited information estimator. Econometrica 45, 939–954.10.2307/1912683CrossRefGoogle Scholar
Han, C. & Phillips, P.C.B. (2006) GMM with many moment conditions. Econometrica 74, 147–192.10.1111/j.1468-0262.2006.00652.xCrossRefGoogle Scholar
Hansen, C., Hausman, J.A., & Newey, W.K. (2008) Estimation with many instrumental variables. Journal of Business & Economic Statistics 26, 398–422.10.1198/073500108000000024Google Scholar
Hausman, J.A., Newey, W.K., Woutersen, T., Chao, J., & Swanson, N.R. (2007) IV Estimation with Heteroskedasticity and Many Instruments. Working paper, MIT.CrossRefGoogle Scholar
Kunitomo, N. (1980) Asymptotic expansions of distributions of estimators in a linear functional relationship and simultaneous equations. Journal of the American Statistical Association 75, 693–700.10.1080/01621459.1980.10477535CrossRefGoogle Scholar
Magnus, J.R. & Neudecker, H. (1988) Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley.Google Scholar
Morimune, K. (1983) Approximate distributions of k-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica 51, 821–841.10.2307/1912160CrossRefGoogle Scholar
Newey, W.K. (1990) Efficient instrumental variable estimation of nonlinear models. Econometrica 58, 809–837.10.2307/2938351Google Scholar
Phillips, G.D.A. & Hale, C. (1977) The bias of instrumental variable estimators of simultaneous equation systems. International Economic Review 18, 219–228.10.2307/2525779Google Scholar
Phillips, P.C.B. (1983) Small sample distribution theory in econometric models of simultaneous equations. In Griliches, Z. & Intriligator, M.D. (eds.), Handbook of Econometrics, vol. 1, pp. 449–516. North-Holland.CrossRefGoogle Scholar
Sawa, T. (1968) The exact sampling distribution of ordinary least squares and two-stage least squares estimators. Journal of the American Statistical Association 64, 923–937.10.1080/01621459.1969.10501024CrossRefGoogle Scholar
Staiger, D. & Stock, J.H. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557–586.10.2307/2171753Google Scholar
Stock, J.H. and Yogo, M. (2005) Asymptotic distributions of instrumental variables statistics with many weak instruments. In Andrews, D.W.K. & Stock, J.H. (eds.), Identification and Inference for Econometric Models: Essays in Honor of Thomas J. Rothenberg, pp. 109–120. Cambridge University Press.10.1017/CBO9780511614491.007Google Scholar