Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Encompassing Tests for Nonparametric Regressions

Elia Lapenta, Pascal Lavergne

TL;DR

The paper develops a formal, nonparametric framework for encompassing nonparametric regressions using the $L^2$ distance and constructs fully nonparametric encompassing tests based on an Integrated Conditional Moment principle. It presents two bias-reduction strategies—bias-corrected kernel estimation via boosting and a locally robust empirical process—to achieve small bias properties and reduce sensitivity to bandwidth, all within a kernel-based estimation setup. A wild bootstrap is proven to provide valid inference, and a data-driven bandwidth (AIC_c) is shown to yield reliable size and power in finite samples; the methods are illustrated with simulations and an empirical example on consumption behavior. Overall, the work delivers a principled approach to nonparametric model comparison and introduces practical tools for inference when nonparametric nuisance components are present.

Abstract

We set up a formal framework to characterize encompassing of nonparametric models through the L2 distance. We contrast it to previous literature on the comparison of nonparametric regression models. We then develop testing procedures for the encompassing hypothesis that are fully nonparametric. Our test statistics depend on kernel regression, raising the issue of bandwidth's choice. We investigate two alternative approaches to obtain a "small bias property" for our test statistics. We show the validity of a wild bootstrap method. We empirically study the use of a data-driven bandwidth and illustrate the attractive features of our tests for small and moderate samples.

Encompassing Tests for Nonparametric Regressions

TL;DR

The paper develops a formal, nonparametric framework for encompassing nonparametric regressions using the distance and constructs fully nonparametric encompassing tests based on an Integrated Conditional Moment principle. It presents two bias-reduction strategies—bias-corrected kernel estimation via boosting and a locally robust empirical process—to achieve small bias properties and reduce sensitivity to bandwidth, all within a kernel-based estimation setup. A wild bootstrap is proven to provide valid inference, and a data-driven bandwidth (AIC_c) is shown to yield reliable size and power in finite samples; the methods are illustrated with simulations and an empirical example on consumption behavior. Overall, the work delivers a principled approach to nonparametric model comparison and introduces practical tools for inference when nonparametric nuisance components are present.

Abstract

We set up a formal framework to characterize encompassing of nonparametric models through the L2 distance. We contrast it to previous literature on the comparison of nonparametric regression models. We then develop testing procedures for the encompassing hypothesis that are fully nonparametric. Our test statistics depend on kernel regression, raising the issue of bandwidth's choice. We investigate two alternative approaches to obtain a "small bias property" for our test statistics. We show the validity of a wild bootstrap method. We empirically study the use of a data-driven bandwidth and illustrate the attractive features of our tests for small and moderate samples.
Paper Structure (20 sections, 8 theorems, 70 equations, 3 figures, 2 tables)

This paper contains 20 sections, 8 theorems, 70 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Encompassing for Nonparametric Regressions
  3. Encompassing and Model Fit
  4. Encompassing and Significance
  5. Example.
  6. Tests Statistics
  7. ICM Statistic
  8. Bias Corrected Estimation
  9. Locally Robust Process
  10. Asymptotic Analysis
  11. Bootstrap Tests
  12. Numerical Results
  13. Small Sample Behavior
  14. Empirical Illustration
  15. Concluding Remarks
  16. ...and 5 more sections

Key Result

Proposition 2.1

If $\mathcal{M}_{W}$ encompasses $\mathcal{M}_{X}$,

Figures (3)

  • Figure 1: Errors in rejection probabilities for $n=400$ using a rule-of-thumb bandwidth $h=C\widehat{\sigma}_W n^{-1/5}$.
  • Figure 2: Errors in rejection probabilities using a data-driven bandwidth: results with trimming on first row, results with no trimming on second row.
  • Figure 3: Power curves for tests at 10% level using a data-driven bandwidth and no trimming.

Theorems & Definitions (9)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 4.1
  • Proposition 4.2
  • Proposition 5.1
  • Lemma 8.1
  • Lemma 8.2
  • Lemma 8.3
  • Lemma 8.4