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Some Exact Distribution Results for the Partially Restricted Reduced form Estimator

Published online by Cambridge University Press:  11 February 2009

Terrence W. Kinal
Affiliation:
State University of New York at Albany
John L. Knight
Affiliation:
University of Western Ontario

Abstract

This paper considers some finite sample properties of the partially restricted reduced form estimators in a general (n + 1) endogenous variable model. In particular, the characteristic functions, density functions, and moments are examined for both the vector of estimators and a linear combination. The approach utilizes both invariant polynomials of matrix argument (see Chikuse and Davis [4] and Davis [6]) and fractional calculus techniques (see Phillips [20,25,26]).

Type
Articles
Copyright
Copyright © Cambridge University Press 1994

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References

1.Amemiya, T.On the use of principal components of independent variables in two-stage least squares estimation. International Economic Review 7 (1966): 283303.CrossRefGoogle Scholar
2.Arvin-Rad, H. Essays in Econometric Theory, unpublished Ph.D. dissertation, University of Pennsylvania, 1990.Google Scholar
3.Chetty, V.K.Bayesian analysis of Haavelmo's models. Econometrica 36 (1968): 582602.CrossRefGoogle Scholar
4.Chikuse, Y. & Davis, A.W.. A survey on the invariant polynomials with matrix arguments in relation to econometric distribution theory. Econometric Theory 2 (1986): 232248.CrossRefGoogle Scholar
5.Court, R.H.Efficient estimation of the reduced form from econometric models. Review of Economic Studies 40 (1973): 411418.CrossRefGoogle Scholar
6.Davis, A.W.Invariant polynomials with two matrix arguments extending the zonal polynomials: Applications to multivariate distribution theory. Annals of the Institute of Statistical Mathematics A31 (1979): 465485.CrossRefGoogle Scholar
7.Goldberger, A.S., Nagar, A.L. & Odeh, H.S.. The covariance matrices of reduced form coefficients and of forecasts for a structural econometric model. Econometrica 29 (1961): 556573.CrossRefGoogle Scholar
8.Hillier, G.H.Exact densities of instrumental variables estimators: An alternative approach. Monash University, mimeo, 1982.Google Scholar
9.Hillier, G.H.On the joint and marginal densities of instrumental variable estimators in a general structural equation. Econometric Theory 1 (1985): 5372.CrossRefGoogle Scholar
10.Hillier, G.H., Kinal, T.W. & Srivastava, V.K.. On the moments of ordinary least squares and instrumental variable estimators in a general structural equation. Econometrica 52 (1984): 185202.CrossRefGoogle Scholar
11.Kakwani, N.C. & Court, R.H.. Reduced form coefficient estimation and forecasting from a simultaneous equation model. Australian Journal of Statistics 14 (1972): 143160.CrossRefGoogle Scholar
12.Knight, J.L.On the existence of moments of the partially restricted reduced-form estimators from a simultaneous-equations model. Journal of Econometrics 5 (1977): 315321.CrossRefGoogle Scholar
13.Knight, J.L.The exact distribution of the PRRF estimator: A fractional calculus approach. University of New South Wales, mimeo, 1985.Google Scholar
14.Maasoumi, E.A modified Stein-like estimator for the reduced form coefficients of simultaneous equations. Econometrica 46 (1978): 695703.CrossRefGoogle Scholar
15.McCarthy, M.D.A note on the forecasting properties of 2SLS restricted reduced forms – the finite sample case. International Economic Review 13 (1972): 756761.CrossRefGoogle Scholar
16.McCarthy, M.D.A note on the moments of partially restricted reduced forms. Journal of Econometrics 17 (1981): 383387.CrossRefGoogle Scholar
17.Nagar, A.L. & Sahay, S.N.. The bias and mean squared error of forecasts from partially restricted reduced form. Journal of Econometrics 7 (1978): 227243.CrossRefGoogle Scholar
18.Phillips, P.C.B.The exact distribution of instrumental variable estimators in an equation containing n + 1 endogenous variables. Econometrica 48 (1980): 861878.CrossRefGoogle Scholar
19.Phillips, P.C.B. Exact small sample theory in the simultaneous equations model. In Griliches, Z. and Intriligator, M.D. (eds.), Handbook of Econometrics, Volume 1, Chapter 8. Amsterdam: North-Holland, 1983.Google Scholar
20.Phillips, P.C.B.The exact distribution of the Stein-rule estimator. Journal of Econometrics 25 (1984): 123131.CrossRefGoogle Scholar
21.Phillips, P.C.B.The exact distribution of the SUR estimator. Econometrica 53 (1985): 745756.CrossRefGoogle Scholar
22.Phillips, P.C.B.The exact distribution of the exogenous variable coefficient estimators. Journal of Econometrics 26 (1985): 385398.Google Scholar
23.Phillips, P.C.B.The exact distribution of the Wald statistic. Econometrica 54 (1986): 881895.CrossRefGoogle Scholar
24.Phillips, P.C.B.The distribution of FIML in the leading case. International Economic Review 27 (1986): 239243.CrossRefGoogle Scholar
25.Phillips, P.C.B. Fractional matrix calculus and the distribution of multivariate tests. In MacNeil, I.B. and Umphrey, G.J. (eds.), Advances in Statistical Sciences: Time Series and Econometric Modelling, Volume 3, pp. 219234. Dordrecht: R. Reidel, 1987.Google Scholar
26.Phillips, P.C.B. Operational algebra and regression t-tests, Cowles Foundation Paper No. 840, Yale University, 1993. Forthcoming in Phillips, P.C.B. (ed.), Models, Methods, and Applications of Econometrics.Google Scholar
27.Swamy, P.A.V.B. & Metha, J.S.. On the existence of moments of the partially restricted reduced form coefficients. Journal of Econometrics 14 (1980): 183194. Also: Comment Journal of Econometrics 17 (1981): 389–392.CrossRefGoogle Scholar
28.Sargan, J.D.Existence of moments of estimated reduced form coefficients. London School of Economics, Discussion Paper A6 (1976).Google Scholar