Kociecki, Andrzej
(2010):
*Algebraic theory of identification in parametric models.*

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## Abstract

The article presents the problem of identification in parametric models from an algebraic point of view. We argue that it is not just another perspective but the proper one. That is, using our approach we can see the very nature of the identification problem, which is slightly different than that suggested in the literature. In practice, it means that in many models we can unambiguously estimate parameters that have been thought as unidentifiable. This is illustrated in the case of Simultaneous Equations Model (SEM), where our analysis leads to conclusion that existing identification conditions, although correct, are based on the inappropriate premise: only the structural parameters that are in one–to–one correspondence with the reduced form parameters are identified. We will show that this is not true. In fact, there are other structural parameters, which are identified, but can not be uniquely recovered from the reduced form parameters. Although we apply our theory only to SEM, it can be used in many standard econometric models.

Item Type: | MPRA Paper |
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Original Title: | Algebraic theory of identification in parametric models |

Language: | English |

Keywords: | identification; group theory |

Subjects: | C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics |

Item ID: | 26820 |

Depositing User: | Andrzej Kociecki |

Date Deposited: | 19 Nov 2010 13:14 |

Last Modified: | 08 Oct 2019 12:42 |

References: | Ahn, S.K., and G.C. Reinsel (1988), “Nested Reduced–Rank Autoregressive Models for Multiple Time Series”, Journal of the American Statistical Association, 83, pp. 849–856. Alperin, J.L., and R.B. Bell (1995), Groups and Representations, Springer–Verlag. Aschbacher, M. (1988), Finite Group Theory, Cambridge University Press. Bhattacharjee, M., D. Macpherson, R.G. Möller and P.M. Neumann (1998), Notes on Infinite Permutation Groups, Springer–Verlag. Bourbaki, N. (1968), Elements of Mathematics: Theory of Sets, Addison–Wesley Pub. Co. (Hermann, Paris). Bowden, R. (1973), “The Theory of Parametric Identification”, Econometrica, 41, pp. 1069–1074. Brillinger, D.R. (1963), “Necessary and Sufficient Conditions for a Statistical Problem to be Invariant Under a Lie Group”, The Annals of Mathematical Statistics, 34, pp. 492–500. Dixon, J.D., and B. Mortimer (1996), Permutation Groups, Springer–Verlag. Duhem, P. (1962), The Aim and Structure of Physical Theory, Atheneum. Fraser, D.A.S. (1967), “Statistical Models and Invariance”, The Annals of Mathematical Statistics, 38, pp. 1061–1067. Friedman, M. (1953), “The Methodology of Positive Economics”, in: Essays in Positive Economics, The University of Chicago Press. Haavelmo, T. (1944), “The Probability Approach in Econometrics”, Econometrica, 12, Supplement, pp. 1–115. Hall, M. (1959), The Theory of Groups, The Macmillan Company. Harville, D.A. (1997), Matrix Algebra From a Statistician’s Perspective, Springer–Verlag. Huppert, B. (1967), Endliche Gruppen I, Springer–Verlag. Jacobson, N. (1985), Basic Algebra I, W.H. Freeman & Co. Jöreskog K.G., and A.S. Goldberger (1975), “Estimation of a Model with Multiple Indicators and Multiple Causes of a Single Latent Variable”, Journal of the American Statistical Association, 70, pp. 631–639. Kadane, J.B. (1975), “The Role of Identification in Bayesian Theory”, in: S.E. Fienberg and A. Zellner, eds., Studies in Bayesian Econometrics and Statistics, North–Holland Pub. Co. Koopmans, T.C. (1949), “Identification Problems in Economic Model Construction”, Econometrica, 17, pp. 125–144. Koopmans, T.C. (1953), “Identification Problems in Economic Model Construction”, in: Wm. C. Hood and T.C. Koopmans, eds., Studies in Econometric Method, Cowles Commission Monograph No. 14, John Wiley & Sons. Koopmans, T.C., and Wm.C. Hood (1953), “The Estimation of Simultaneous Linear Economic Relationships”, in: Wm. C. Hood and T.C. Koopmans, eds., Studies in Econometric Method, Cowles Commission Monograph No. 14, John Wiley & Sons. Koopmans, T.C., and O. Reiersøl (1950), “The Identification of Structural Characteristics”, The Annals of Mathematical Statistics, 21, pp. 165–181. Koopmans, T.C., H. Rubin, and R.B. Leipnik (1950), “Measuring the Equation Systems of Dynamic Economics”, in: T.C. Koopmans, ed., Statistical Inference in Dynamic Economic Models, Cowles Commission Monograph No. 10, John Wiley & Sons. Lehmann, E.L. (1986), Testing Statistical Hypotheses, second edition, Springer–Verlag. Mach, E. (1898), Popular Scientific Lectures, The Open Court Pub. Co. MacLane, S, and G. Birkhoff (1993), Algebra, third edition, AMS Chelsea Pub. Mäki, U. (2009), “Realistic Realism about Unrealistic Models”, in: H. Kincaid and D. Ross, eds., Oxford Handbook of the Philosophy of Economics, Oxford University Press. Mäki, U. (2010), “Models and the Locus of Their Truth”, forthcoming, Synthese. Marschak, J. (1953), “Economic Measurements for Policy and Prediction”, in: Wm. C. Hood and T.C. Koopmans, eds., Studies in Econometric Method, Cowles Commission Monograph No. 14, John Wiley & Sons. Prakasa Rao, B.L.S. (1992), Identifiability in Stochastic Models: Characterization of Probability Distributions, Academic Press, Inc. Reinsel, G.C. (1983), “Some Results on Multivariate Autoregressive Index Models”, Biometrika, 70, pp. 145–156. Robinson, D.J.S. (1982), A Course in the Theory of Groups, Springer–Verlag. Rose, J.S. (1978), A Course on Group Theory, Cambridge University Press. Rothenberg, T.J. (1971), “Identification in Parametric Models”, Econometrica, 39, pp. 577–591. Sargent, T.J., and C.A. Sims (1977), “Business Cycle Modeling Without Pretending to Have Too Much a Priori Economic Theory”, in: C.A. Sims, ed., New Methods in Business Cycle Research, Federal Reserve Bank of Minneapolis. Scott, W.R. (1987), Group Theory, Dover Pub., Inc. Sims, C.A. (1981), “An Autoregressive Index Model for the U.S., 1948–1975”, in: J. Kmenta and J.B. Ramsey, eds., Large–Scale Macro–Econometric Models: Theory and Practice, North–Holland Pub. Co. Steinberger, M. (1993), Algebra, Prindle, Weber & Schmidt Pub. Co. Weyl, H. (1952), Symmetry, Princeton University Press. Wielandt, H. (1964), Finite Permutation Groups, Academic Press, Inc. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/26820 |