UNIFORM CONVERGENCE OF SERIES ESTIMATORS OVER FUNCTION SPACES

Kyungchul Song

Econometric Theory, 2008, vol. 24, issue 6, 1463-1499

Abstract: This paper considers a series estimator of E[α(Y)|λ(X) = λ̄], (α,λ) ∈ 𝛢 × Λ, indexed by function spaces, and establishes the estimator's uniform convergence rate over λ̄ ∈ R, α ∈ 𝛢, and λ ∈ Λ, when 𝛢 and Λ have a finite integral bracketing entropy. The rate of convergence depends on the bracketing entropies of 𝛢 and Λ in general. In particular, we demonstrate that when each λ ∈ Λ is locally uniformly ℒ2-continuous in a parameter from a space of polynomial discrimination and the basis function vector pK in the series estimator keeps the smallest eigenvalue of E[pK(λ(X))pK(λ(X))‼] above zero uniformly over λ ∈ Λ, we can obtain the same convergence rate as that established by de Jong (2002, Journal of Econometrics 111, 1–9).

Date: 2008
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