Compactness of infinite dimensional parameter spaces Joachim Freyberger Matthew Masten The Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP01/16 Compactness of Infinite Dimensional Parameter Spaces∗ Joachim Freyberger† Matthew A. Masten‡ December 23, 2015 Abstract We provide general compactness results for many commonly used parameter spaces in non- parametric estimation. We consider three kinds of functions: (1) functions with bounded do- mains which satisfy standard norm bounds, (2) functions with bounded domains which do not satisfy standard norm bounds, and (3) functions with unbounded domains. In all three cases we provide two kinds of results, compact embedding and closedness, which together allow one to show that parameter spaces defined by a s-norm bound are compact under a norm c. We apply these results to nonparametric meank ·k regression and nonparametric instrumentalk variables ·k estimation. JEL classification: C14, C26, C51 Keywords: Nonparametric Estimation, Sieve Estimation, Trimming, Nonparametric Instrumental Variables ∗This paper was presented at Duke and the 2015 Triangle Econometrics Conference. We thank audiences at those seminars as well as Bruce Hansen, Jack Porter, Yoshi Rai, and Andres Santos for helpful conversations and comments. †Department of Economics, University of Wisconsin-Madison,
[email protected] ‡Department of Economics, Duke University,
[email protected] 1 1 Introduction Compactness is a widely used assumption in econometrics, for both finite and infinite dimensional parameter spaces. It can ensure the existence of extremum estimators and is an important step in many consistency proofs (e.g. Wald 1949). Even for noncompact parameter spaces, compactness results are still often used en route to proving consistency.