Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T23:14:56.205Z Has data issue: false hasContentIssue false

Some Exact Results for Estimators of the Coefficients on the Exogenous Variables in a Single Equation

Published online by Cambridge University Press:  11 February 2009

Christopher L. Skeels
Affiliation:
Australian National University and University of British Columbia

Abstract

This paper is devoted to a detailed examination of the exact sampling properties of the instrumental variables (IV) estimator of the vector of coefficients on the exogenous variables in a single structural equation. The first two moments of a linear combination of the elements of this estimator and the joint distribution of these elements are considered. Estimable bounds for the first moment that can readily be incorporated into any IV estimation package are provided. The results obtained are in terms of the same special functions as those that characterize other results for this model, allowing a unified treatment of the model.

Type
Articles
Copyright
Copyright © Cambridge University Press 1995

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.)

References

REFERENCES

Chikuse, Y. (1980) Invariant polynomials with matrix arguments and their applications. In Gupta, R.P. (ed.), Multivariate Statistical Analysis, pp. 5368. Amsterdam: North-Holland.Google Scholar
Chikuse, Y. (1982) Some Properties of the Invariant Polynomials with Matrix Arguments and their Application in Econometrics. Mimeo, Kagawa University.Google Scholar
Chikuse, Y. (1987) Methods for constructing top order invariant polynomials. Econometric Theory 3, 195207.CrossRefGoogle Scholar
Chikuse, Y. & Davis, A.W. (1986) A survey on the invariant polynomials with matrix arguments in relation to econometric distribution theory. Econometric Theory 2, 232248.CrossRefGoogle Scholar
Davis, A.W. (1979) Invariant polynomials with two matrix arguments extending the zonal polynomials: Applications to multivariate distribution theory. Annals of the Institute of Statistical Mathematics 31 (part A), 465485.CrossRefGoogle Scholar
Davis, A.W. (1980) Invariant polynomials with two matrix arguments, extending the zonal polynomials. In Krishnaiah, P.R. (ed.), Multivariate Analysis V, pp. 287299. Amsterdam: North-Holland.Google Scholar
Davis, A.W. (1981) On the construction of a class of invariant polynomials in several matrices, extending the zonal polynomials. Annals of the Institute of Statistical Mathematics 33 (part A), 297313.CrossRefGoogle Scholar
Eaton, M.L. (1983) Multivariate Statistics: A Vector Space Approach. New York: John Wiley and Sons.Google Scholar
Herz, C.S. (1955) Bessel functions of matrix argument. Annals of Mathematics 61, 474523.CrossRefGoogle Scholar
Hillier, G.H. (1985a) Marginal Densities of Instrumental Variables Estimators: Further Exact Results. Mimeo, Monash University.Google Scholar
Hillier, G.H. (1985b) On the joint and marginal densities of instrumental variable estimators in a general structural equation. Econometric Theory 1, 5372.CrossRefGoogle Scholar
Hillier, G.H., Kinal, T.W., & Srivastava, V.K. (1984) On the moments of ordinary least squares and instrumental variable estimators in a general structural equation. Econometrica 52, 185202.CrossRefGoogle Scholar
Hillier, G.H. & Skeels, C.L. (1993) Some further exact results for structural equation estimators. In Phillips, P.C.B. (ed.), Models, Methods and Applications of Econometrics: Essays in Honour of A. R. Bergstrom, ch. 9, pp. 117139. Cambridge, Massachusetts: Basil Blackwell.Google Scholar
James, A.T. (1964) Distributions of matrix variates and latent roots derived from normal samples. Annals of Mathematical Statistics 35, 475501.CrossRefGoogle Scholar
Kinal, T.W. (1980) The existence of moments of K-class estimators. Econometrica 48, 241249.CrossRefGoogle Scholar
Kinal, T.W. (1986) The Exact Distribution of OLS and Instrumental Variable Estimates of the Coefficients of the Exogenous Variables in a Structural Equation. Mimeo, State University of New York at Albany.Google Scholar
Mariano, R.S. (1982) Analytical small-sample distribution theory in econometrics: The simultaneous equation case. International Economic Review 23, 503533.CrossRefGoogle Scholar
Phillips, P.C.B. (1980) The exact finite sample density of instrumental variable estimators in an equation with n + 1 endogenous variables. Econometrica 48, 861868.CrossRefGoogle Scholar
Phillips, P.C.B. (1983) Exact small sample theory in the simultaneous equations model. In Griliches, Z. & Intriligator, M.D. (eds.), Handbook of Econometrics, vol. 1, ch. 8, pp. 449516. Amsterdam: North-Holland.CrossRefGoogle Scholar
Phillips, P.C.B. (1984) The exact distribution of exogenous variable coefficient estimators. Journal of Econometrics 26, 387398.CrossRefGoogle Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181240.CrossRefGoogle Scholar
Skeels, C.L. (1995) Instrumental variables estimation in misspecified single equations. Econometric Theory 11, 498529.CrossRefGoogle Scholar
Takeuchi, K. (1970) Exact sampling moments of the ordinary least squares, instrumental variable, and two-stage least squares estimators. International Economic Review 11, 112.CrossRefGoogle Scholar