The Limits of Econometrics: Nonparametric Estimation in Hilbert Spaces

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 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.