Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-17T01:36:09.709Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LIMITS OF ECONOMETRICS: NONPARAMETRIC ESTIMATION IN HILBERT SPACES

Published online by Cambridge University Press:  01 August 2009

Graciela Chichilnisky
Graciela Chichilnisky*
Affiliation:
Columbia University
*
*Address correspondence to Graciela Chichilnisky, 335 Riverside Drive, New York, NY, 10025 USA; email: chichilnisky1@gmail.com.
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend Bergstrom's 1985 results on nonparametric (NP) estimation in Hilbert spaces to unbounded sample sets. The motivation is to seek the most general possible framework for econometrics, NP estimation with no a priori assumptions on the functional relations nor on the observed data. In seeking the boundaries of the possible, however, we run against a sharp dividing line, which defines a necessary and sufficient condition for NP estimation. We identify this condition somewhat surprisingly with a classic statistical assumption on the relative likelihood of bounded and unbounded events (DeGroot, 2004). Other equivalent conditions are found in other fields: decision theory and choice under uncertainty (monotone continuity axiom (Arrow, 1970), insensitivity to rare events (Chichilnisky, 2000), and dynamic growth models (dictatorship of the present; Chichilnisky, 1996). When the crucial condition works, NP estimation can be extended to the sample space R+. Otherwise the estimators, which are based on Fourier coefficients, do not converge: the underlying distributions are shown to have “heavy tails” and to contain purely finitely additive measures. Purely finitely additive measures are not constructible, and their existence has been shown to be equivalent to the axiom of choice in mathematics. Statistics and econometrics involving purely finitely additive measures are still open issues, which suggests the current limits of econometrics.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 4: Bergstrom Memorial Dedication Issue , August 2009 , pp. 1070 - 1086
Copyright
Copyright © Cambridge University Press 2009

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

Andrews, D. (1991) Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrica 59, 307–345.CrossRefGoogle Scholar
Arrow, K. (1970) Essays in the Theory of Risk Bearing. North Holland.Google Scholar
Bergstrom, R. (1985) The estimation of nonparametric functions in a Hilbert space. Econometric Theory 1, 7–26.CrossRefGoogle Scholar
Blundell, R., Chen, X., & Kristensen, D.. (2006) Semi-Nonparametric IV Estimation of Shape Invariant Engel Curves. Working paper, Columbia University.CrossRefGoogle Scholar
Chen, X., Hansen, L.P., & Sheinkman, J.. (2000) Principal Components and the Long Run. Working paper, revised November 2005.Google Scholar
Chichilnisky, G. (1976) Manifolds of Preferences and Equilibria. Ph.D. dissertation, UC Berkeley.Google Scholar
Chichilnisky, G. (1977) Non linear functional analysis and optimal economic growth. Journal of Math Analysis and Applications 61, 504–520.CrossRefGoogle Scholar
Chichilnisky, G. (1996) An axiomatic approach to sustainable development. Social Choice and Welfare 13, 231–257.CrossRefGoogle Scholar
Chichilnisky, G. (1997) What is sustainable development? Land Economics 73, 467–491.CrossRefGoogle Scholar
Chichilnisky, G. (1999) An axiomatic approach to the theory of choice under uncertainty with catastrophic risks. In Chichilnisky, G. (ed.), Topology and Markets. American Mathematical Society and The Fields Institute for Mathematical Sciences, Toronto.Google Scholar
Chichilnisky, G. (2002) An axiomatic approach to choice under uncertainty with catastrophic risks. Resource and Energy Economics 22, 221–231.CrossRefGoogle Scholar
Chichilnisky, G. (2002) Catastrophic risks. In El Shaarawi, A.H., & Piergorsch, W. (eds.), Encyclopedia of Environmetrics, pp. 274–279. Wiley.Google Scholar
Chichilnisky, G. (2006a) Non Parametric Estimation in Hilbert Spaces. Invited address to the Conference in Honor of Rex Bergstrom, Colchester, Essex, U.K. May.Google Scholar
Chichilnisky, G. (2006b) Topology of Fear. Working paper, Columbia University. Presented at 2006 NBER GE Conference in Honor of Gerard Debreu, U.C. Berkeley, September.Google Scholar
DeGroot, M. (2004) Optimal Statistical Decisions. Wiley.CrossRefGoogle Scholar
Gallant, A.R. (1981) On the bias of flexible functional forms. Journal of Econometrics 15, 211–246.CrossRefGoogle Scholar
Kolmogorov, A.N. & Fomin, S.V. (1961) Elements of the Theory of Functions and Functional Analysis, Vol. 2. Graylock Press.Google Scholar
Newey, W.K. (1997) Convergence rates and asymptotic normality for series estimators. Journal of Econometrics 79, 147–168.CrossRefGoogle Scholar
Priestley, M.B. & Chao, M.T. (1972) Non parametric function fitting. Journal of Royal Statistical Society B 34, 385–392.Google Scholar
Stinchcombe, M. (2002) Some Genericity Analysis in Nonparametric Statistics. Working paper,December 10.Google Scholar
Yosida, K. (1974) Funcional Analysis, 4th ed.Springer Verlag.CrossRefGoogle Scholar
Yosida, K. & Hewitt, E. (1952) Finitely level independent measures. Transactions of the American Mathematical Society 72, 46–66.CrossRefGoogle Scholar