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CONDITIONAL INFERENCE FOR POSSIBLY UNIDENTIFIED STRUCTURAL EQUATIONS

Published online by Cambridge University Press:  04 August 2003

Giovanni Forchini  and
Grant Hillier
Giovanni Forchini
Affiliation:
University of York
Grant Hillier
Affiliation:
University of Southampton
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Abstract

The possibility that a structural equation may not be identified casts doubt on measures of estimator precision that are usually used. Using the Fieller–Creasy problem for illustration, we argue that an observed identifiability test statistic is directly relevant to the precision with which the structural parameters can be estimated, and hence we argue that inference in such models should be conditioned on the observed value of that statistic (or statistics).

We examine in detail the effects of such conditioning on the properties of the ordinary least squares (OLS) and two-stage least squares (TSLS) estimators for the coefficients of the endogenous variables in a single structural equation. We show that (a) conditioning has very little impact on the properties of the OLS estimator but a substantial impact on those of the TSLS estimator; (b) the conditional variance of the TSLS estimator can be very much larger than its unconditional variance (when the identifiability statistic is small) or very much smaller (when the identifiability statistic is large); and (c) conditional mean-square-error comparisons of the two estimators favor the OLS estimator when the sample evidence only weakly supports the identifiability hypothesis but favor TSLS when that evidence moderately supports identifiability.

Finally, we note that another consequence of our argument is that the statistic upon which Anderson–Rubin confidence sets are based is in fact nonpivotal.We are grateful for the constructive comments offered by Peter Phillips and three anonymous referees that greatly improved the paper. Giovanni Forchini acknowledges support from ESRC grant NR00429424115.

Type
Research Article
Information
Econometric Theory , Volume 19 , Issue 5 , October 2003 , pp. 707 - 743
Copyright
© 2003 Cambridge University Press

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References

REFERENCES

Anderson, T.W., K. Morimune, & T. Sawa (1983) The numerical values of some key parameters in econometric models. Journal of Econometrics 21, 229–243.Google Scholar
Anderson, T.W. & H. Rubin (1949) Estimation of the parameters of a single equation in a complete system of stochastic equations. Annals of Mathematical Statistics 20, 46–63.Google Scholar
Barndorff-Nielsen, O.E. (1980) Conditionality resolutions. Biometrika 67, 293–310.Google Scholar
Basu, D. (1981) On ancillary statistics, pivotal quantities, and confidence statements. In Y.P. Chaubey & T.D. Dwivedi (eds.), Topics in Applied Statistics. Montreal: Concordia University. Reprinted in J.K. Ghosh (ed.), Statistical Information and Likelihood: A Collection of Critical Essays by Dr. D. Basu, Lecture Notes in Statistics (Springer-Verlag), pp. 161–176.
Chao, J.C. & P.C.B. Phillips (1998) Bayesian posterior distributions in limited information analysis of the simultaneous equation model using the Jeffreys' prior. Journal of Econometrics 87, 49–86.Google Scholar
Chikuse, Y. & A.W. Davis (1986) A survey on the invariant polynomials with matrix arguments in relation to econometric distribution theory. Econometric Theory 2, 232–248.Google Scholar
Choi, I. & P.C.B. Phillips (1992) Asymptotic and finite sample distribution theory of IV estimators and tests in partially identified structural equations. Journal of Econometrics 51, 113–150.Google Scholar
Cox, D.R. (1958) Some problems connected with statistical inference. Annals of Mathematical Statistics 29, 357–337.Google Scholar
Cox, D.R. & D.V. Hinkley (1974) Theoretical Statistics. London: Chapman and Hall.
Creasy, M.A. (1954) Limits for the ratio of means. Journal of the Royal Statistical Society, Series B, 16, 186–194.Google 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, 465–485.Google Scholar
Dobrigal, A., D.A.S. Fraser, & R. Gebotys (1987) Linear calibration and conditional inference. Communications in Statistics, Theory and Methods 16, 1037–1048.Google Scholar
Dufour, J.-M. (1997) Some impossibility theorems in econometrics with applications to instrumental variables and dynamic models. Econometrica 65, 1365–1388.Google Scholar
Edwards, A.W.F. (1972) Likelihood. Cambridge: Cambridge University Press.
Efron, B. & D.V. Hinkley (1978) Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information. Biometrika 65, 457–487.Google Scholar
Fieller, E.C. (1954) Some problems in interval estimation. Journal of the Royal Statistical Society, Series B 16, 175–185.Google Scholar
Forchini, G. (1998) Exact Distribution Theory for Some Econometric Problems. Ph.D. Dissertation, University of Southampton.
Ghosh, J.K. (ed.) (1988) Statistical Information and Likelihood: A Collection of Critical Essays by Dr. D. Basu, Lecture Notes in Statistics. New York: Springer-Verlag.
Gleser, L.J. & J.T. Hwang (1987) The nonexistence of 100(1-α) confidence sets of finite expected diameter in errors in variables and related models. The Annals of Statistics 15, 1351–1362.Google Scholar
Goutis, C. & G. Casella (1995) Frequentist post-data inference. International Statistical Review 63, 325–344.Google Scholar
Hall, A.R., G.D. Rudenbusch, & D.W. Wilcox (1996) Judging instrument relevance in instrumental variables estimation. International Economic Review 37, 283–289.Google Scholar
Hillier, G.H. (1985) On the joint and marginal densities of instrumental variable estimators in a general structural equation. Econometric Theory 1, 53–72.Google Scholar
Hillier, G.H. (1990) On the normalization of structural equations: Properties of direction estimators. Econometrica 58, 1181–1194.Google Scholar
Hillier, G.H. & C.L. Skeels (1993) Some further exact results for structural equation estimators. In P.C.B. Phillips (ed.), Models, Methods, and Applications of Econometrics: Essays in Honor of A.R. Bergstrom. New York: Basil Blackwell.
Hoadley, B. (1970) A Bayesian look at inverse linear regression. Journal of the American Statistical Association 65, 356–369.Google Scholar
Hwang, J.T. & L.D. Brown (1991) Estimated confidence under the validity constraint. Annals of Statistics 19, 1964–1977.Google Scholar
Hwang, J.T., G. Casella, C. Robert, M.T. Wells, & R.H. Farrell (1992) Estimation of accuracy in testing. Annals of Statistics 20, 490–509.Google Scholar
James, A.T., G.N. Wilkinson, & W.N. Venables (1974) Interval estimates for a ratio of means. Sankhya 36, Series A, Part 2, 177–183.Google Scholar
Kiefer, J. (1977) Conditional confidence statements and confidence estimators. Journal of the American Statistical Association 72, 789–827.Google Scholar
Kleibergen, F. & H.K. van Dijk (1998) Bayesian simultaneous equations analysis using reduced rank structures. Econometric Theory 14, 701–743.Google Scholar
Koschat, M.A. (1987) A characterization of the Fieller solution. Annals of Statistics 15, 462–468.Google Scholar
Lindsay, B.G. & B. Li (1997) On second-order optimality of the observed Fisher information. Annals of Statistics 25, 2172–2199.Google Scholar
Moreira, M.J. (2002) A Conditional Likelihood Ratio Test for Structural Models. Working paper, Department of Economics, Harvard University.
Muirhead, R.J. (1982) Aspects of Multivariate Statistical Theory. New York: Wiley.
Nelson, C. & R. Startz (1990) Some further results on the exact small sample properties of the instrumental variable estimator. Econometrica 58, 967–976.Google Scholar
Pfanzagl, J. (1998) The nonexistence of confidence sets for discontinuous functionals. Journal of Statistical Planning and Inference 75, 9–20.Google Scholar
Phillips, P.C.B. (1983) Exact small sample theory in the simultaneous equation model. In M.D. Intriligator and Z. Griliches (eds.), Handbook of Econometrics, pp. 449–516. Amsterdam: North Holland.
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181–240.Google Scholar
Sargan, J.D. (1983) Identification and lack of identification. Econometrica 51, 1605–1633.Google Scholar
Scheffé, H. (1970) Multiple testing versus multiple estimation: Improper confidence sets: Estimation of directions and ratios. Annals of Mathematical Statistics 41, 1–29.Google Scholar
Sims, C.A. (1980) Macroeconomics and reality. Econometrica 48, 1–48.Google Scholar
Staiger, D. & J.H. Stock (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557–586.Google Scholar
Stock, J. & M. Yogo (2001) Testing for weak instruments in linear IV regression. In Don Andrews, Jim Powell, Paul Ruud, & Jim Stock (eds.), Identification and Inference for Econometric Models: A Festschrift for Tom Rothenberg, Econometric Society Monograph Series. Cambridge University Press, forthcoming.
Wallace, D.L. (1980) The Behrens-Fisher and the Fieller–Creasy problems. In S.E. Fienberg & D.V. Hinkley (eds.), R.A. Fisher: An Appreciation, Lecture Notes in Statistics I. New York: Springer-Verlag.
Zivot, E., R. Startz, & C. Nelson (1998) Valid confidence intervals and inference in the presence of weak instruments. International Economic Review 39, 1119–1144.Google Scholar