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On the Joint and Marginal Densities of Instrumental Variable Estimators in a General Structural Equation

Published online by Cambridge University Press:  18 October 2010

Grant H. Hillier
Grant H. Hillier*
Affiliation:
Monash University, Clayton, Australia
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Abstract

Starting from the conditional density of the instrumental variable (IV) estimator given the right-hand-side endogenous variables, we provide an alternative derivation of Phillips' result on the joint density of the IV estimator for the endogenous coefficients, and derive an expression for the marginal density of a linear combination of these coefficients. In addition, we extend Phillips' approximation to the joint density to 0(T−2,) and show how this result can be used to improve the approximation to the marginal density. Explicit formulae are given for the special case of no simultaneity, and the case of an equation with just three endogenous variables. The classical assumptions of independent normal reduced-form errors are employed throughout.

Information

Type
Research Article
Information
Econometric Theory , Volume 1 , Issue 1 , April 1985 , pp. 53 - 72
Copyright
Copyright © Cambridge University Press 1985 

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References

REFERENCES

1. Basmann, R. L. (1974). Exact finite sample distributions for some econometric estimators and test statistics: A survey and appraisal. In Intriligator, M. D. and Kendrick, D. A. (Eds.), Frontiers of Quantitative Economics. (Vol 2). Amsterdam: North-Holland, Chapter 4.Google Scholar
2. Chikuse, Y. (1980). Invariant polynomials with several matrix arguments and their applications. In Gupta, R. P. (Ed.), Multivariate Analysis. Amsterdam: North-Holland, 53–681.Google Scholar
3. Chikuse, Y. (1982). Some properties of the invariant polynomials with matrix arguments and their application in econometrics. Mimeo, Kagawa University, Japan.Google Scholar
4. Constantine, A. G. (1963). Some noncentral distribution problems in multivariate analysis. Annals of Mathematical Statistics, 34, 1270–1285.Google Scholar
5. Davis, A. W. (1979), Invariant polynomials with two matrix arguments extending the zonal polynomials: Applications to multivariate distribution theory. Annals of Institute of Statistical Mathematics, 31, Part A, 465–485.Google Scholar
6. Davis, A. W. (1980). Invariant polynomials with two matrix arguments extending the zonal polynomials. In Krishnaiah, P. R. (Ed.), Multivariate Analysis V 278–299.Google Scholar
7. Davis, A. W. (1981). On the construction of a class of invariant polynomials in several matrices, extending the zonal polynomials. Annals of Institute of Statistical Mathematics, 33, 297–313.Google Scholar
8. Herz, C. S. (1955). Bessel Functions of Matrix Argument. Annals of Mathematics, 61, 474–523.Google Scholar
9. 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. Economet-rica, 52 (1), 185–202.Google Scholar
10. James, A. T. (1954). Normal multivariate analysis and the orthogonal group. Annals of Mathematical Statistics, 25, 40–75.Google Scholar
11. James, A. T. (1960). The distribution of the latent roots of the covariance matrix. Annals of Mathematical Statistics, 31, 151–158.Google Scholar
12. James, A. T. (1961). The distribution of noncentral means with known covariance. Annals of Mathematical Statistics, 32, 874–882.Google Scholar
13. James, A. T. (1961). Zonal polynomials of the real positive definite symmetric matrices. Annals of Mathematics, 34, 456–469.Google Scholar
14. James, A. T. (1964). Distribution of matrix variates and latent roots derived from normal samples. Annals of Mathematical Statistics, 35, 475–501.Google Scholar
15. Kinal, T. W. (1980). The Existence of Moments of k-Class Estimators. Econometrica, 48, 241–249.Google Scholar
16. Mariano, R. S. (1982). Analytical small-sample distribution theory in econometrics: The simultaneous equation case. International Economic Review, 23, 503–533.Google Scholar
17. Muirhead, R. J. (1975). Expressions for some hypergeometric functions of matrix argument with applications. Journal of Multivariate Analysis, 5, 283–293.10.1016/0047-259X(75)90046-9Google Scholar
18. Phillips, P.C.B. (1980). Finite sample theory and the distributions of alternative estimators of the marginal propensity to consume. Review of Economic Studies, 27, 183–224.Google Scholar
19. Phillips, P.C.B. (1980). The exact finite sample density of instrumental variable estimators i n an equation with n + 1 endogenous variables. Econometrica, 48, 861–868.Google Scholar
20. Phillips, P.C.B. (1983). Exact small sample theory in the simultaneous equations model. In Handbook of Econometrics. Amsterdam: North-Holland.Google Scholar
21. Phillips, P.C.B. (1983). Marginal densities of instrumental variable estimators in the general single equation case. Advances in Econometrics (forthcoming).Google Scholar
22. Phillips, P.C.B. (1983). ERA's: A new approach to small sample theory. Econometrica, 51, 1505–1526.Google Scholar
23. Rhodes, G. F. (1981). Exact density functions and approximate critical regions for likelihood ratio identifiability test statistics. Econometrica, 49, 1035–1056.Google Scholar
24. Richardson, D. H. (1968). The exact distribution of a structural coefficient estimator. Journal of the American Statistical Association, 63, 1214–1226.Google Scholar
25. Sargan, J. D. (1976). Econometric estimators and the Edgeworth approximation. Econometrica, 44, 421–428; (1977). Erratum. Econometrica, 45, 272.10.2307/1913972Google Scholar