The Limits of Econometrics: Nonparametric Estimation in Hilbert Spaces
January 1, 2009
Econometric Theory, Vol. 25, 2009
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 constructable, 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.
Number of Pages in PDF File: 17working papers series
Date posted: February 5, 2010
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