Skip to main content


  • John C. Chao (a1)

This paper considers estimating a panel data simultaneous equations model under both coefficient and covariance matrix restrictions in a scenario where one or the other set of identifying restrictions may be invalid or may hold only weakly. We study the limiting properties of various estimators in an asymptotic framework, which takes both the cross-sectional dimension N and the time dimension T to infinity. In this setting as in the pure cross-sectional setup, the performance of the 2SLS estimator depends on the strength of the identifying conditions imposed on the coefficients of the model, and it fails to be consistent once these conditions break down sufficiently resulting in instruments that are too weakly correlated with the endogenous regressors. On the other hand, the between-group (BG) estimator is consistent and asymptotically normal even when coefficient restrictions fail, but it has the shortcoming that its precision depends only on variations in the cross-sectional dimension; and, hence, it is less efficient and has slower rate of convergence than alternatives, which make better use of the large time dimension. A GMM estimator, which combines the moment conditions of the BG estimator with that of the within-group IV estimator, is more robust to instrument weakness than 2SLS and is more efficient than the BG estimator, but it has a second-order bias even under strong instruments if the assumed covariance restrictions do not hold. To remedy the deficiency of the aforementioned estimators, we propose in this paper two new model averaging estimators, which are weighted averages of the GMM estimator and a bias-corrected GMM estimator. The two proposed estimators have weighting functions that depend on alternative transformations of the Bayesian Information Criterion (BIC), which is employed here to assess the validity of the covariance restrictions. We show that these new estimators have some nice robustness properties against possible failure of either the coefficient restrictions or the covariance restrictions.

Corresponding author
*Address correspondence to John C. Chao, Department of Economics, University of Maryland, College Park,MD 20742. e-mail:
Hide All
Alonso-Borrego, C. & Arellano, M. (1999) Symmetrically normalized instrumental-variable estimation using panel data. Journal of Business and Economic Statistics 17, 3649.
Anderson, T.W. (1971). The Statistical Analysis of Time Series. John Wiley & Sons.
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817858.
Baltagi, B.H. (1981) Simultaneous equations with error components. Journal of Econometrics 17, 189200.
Baltagi, B.H. & Li, Q. (1992) A note on the estimation of simultaneous equations with error components. Econometric Theory 8, 113119.
Buckland, S.T., Burnham, K.P., & Augustin, N.H. (1997) Model selection: An integral part of inference. Biometrics 53, 603618.
Bun, M. & Windmeijer, F. (2007) The weak instrument problem of the system GMM estimator in dynamic panel data models. CEMMP working paper.
Chao, J.C. & Swanson, N.R. (2005) Consistent estimation with a large number of weak instruments. Econometrica 73, 16731692.
Choi, I. & Philllips, P.C.B. (1992) Asymptotic and finite sample distribution theory for IV estimators and tests in partially identified structural equations. Journal of Econometrics 51, 113150.
Han, C. & Phillips, P.C.B. (2006) GMM with many moment conditions. Econometrica 74, 147192.
Hannan, E.J. (1970) Multiple Time Series. John Wiley & Sons.
Hansen, B.E. (2007) Least squares model averaging. Econometrica 75, 11751189.
Hausman, J.A. & Taylor, W.E. (1983) Identification in linear simultaneous equations models with covariance restrictions: An instrumental variables interpretation. Econometrica 51, 15271549.
Hausman, J.A., Newey, W.K., & Taylor, W.E. (1987) Efficient estimation and identification of simultaneous equation models with covariance restrictions. Econometrica 55, 849874.
Hjort, N.L. & Claeskens, G. (2003) Frequentist model average estimators. Journal of the American Statistical Association 98, 879899.
Hsiao, C. (2003) Analysis of Panel Data, 2nd ed.Cambridge University Press.
Liu, T.-C. (1960) Underidentification, structural estimation, and forecasting. Econometrica 28, 855865.
Parzen, E. (1957) On consistent estimates of the spectrum of a stationary time series. Annals of Mathematical Statistics 28, 329348.
Phillips, P.C.B. (1980) The exact distribution of instrumental variable estimators in an equation containing n+1 endogenous variables. Econometrica 48, 861878.
Phillips, P.C.B. (1984) The exact distribution of LIML: I. International Economic Review 25, 249261.
Phillips, P.C.B. (1985) The exact distribution of LIML: II. International Economic Review 26,2136.
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181240.
Phillips, P.C.B. (2006) A remark on bimodality and weak instrumentation in structural equation estimation. Econometric Theory 22, 947960.
Prucha, I.R. (1985) Maximum likelihood and instrumental variable estimation in simultaneous equations systems with error components. International Economic Review 26, 491506.
Sims, C.A. (1980) Macroeconomics and reality. Econometrica 48, 148.
Staiger, D. & Stock, J.H. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557586.
Stock, J.H. & Wright, J.H. (2000) GMM with weak identification. Econometrica 68, 10551096.
Stock, J.H. & Yogo, M. (2005) Asymptotic distributions of instrumental variables statistics and many weak instruments. In Andrews, D.W.K. & Stock, J.H. (eds.), Festschrift in Honor of Thomas Rothenberg. Cambridge University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *