Empirical Likelihood and Uniform Convergence Rates for Dyadic Kernel Density Estimation

ABSTRACT

This article studies the asymptotic properties of and alternative inference methods for kernel density estimation (KDE) for dyadic data. We first establish uniform convergence rates for dyadic KDE. Second, we propose a modified jackknife empirical likelihood procedure for inference. The proposed test statistic is asymptotically pivotal regardless of presence of dyadic clustering. The results are further extended to cover the practically relevant case of incomplete dyadic data. Simulations show that this modified jackknife empirical likelihood-based inference procedure delivers precise coverage probabilities even with modest sample sizes and with incomplete dyadic data. Finally, we illustrate the method by studying airport congestion in the United States.

KEYWORDS:

• Density estimation
• Dyadic network data
• Empirical likelihood
• Jackknife

Supplementary Materials

The supplementary materials comprise the Supplementary Appendix, application code and data, and simulation code. The Supplementary Appendix includes the proofs of the Theorems in this article and the bandwidth choices for our applications and simulations. The application code includes a Stata do-file for preprocessing the data and a R script that generates Figures 1a, 1b and 1c. The simulation code includes two R scripts, one for the complete case and one for the incomplete case.

Acknowledgments

We thank coeditor Jianqing Fan, the Associate Editor, and three anonymous referees for their constructive comments that have helped us significantly improved the paper. We are indebted to Matias Cattaneo, the anonymous AE, and two anonymous referees for pointing out an error in the earlier version and for suggesting a fix. We benefited from useful comments and discussions with Bruce Hansen, Yuya Sasaki, and the participants at Bristol Econometrics Study Group. We thank Yukitoshi Matsushita for sharing their MATLAB simulation code. All remaining errors are ours.