Fixed-k Inference for Conditional Extremal Quantiles

Abstract

We develop a new extreme value theory for repeated cross-sectional and longitudinal/panel data to construct asymptotically valid confidence intervals (CIs) for conditional extremal quantiles from a fixed number k of nearest-neighbor tail observations. As a by-product, we also construct CIs for extremal quantiles of coefficients in linear random coefficient models. For any fixed k, the CIs are uniformly valid without parametric assumptions over a set of nonparametric data generating processes associated with various tail indices. Simulation studies show that our CIs exhibit superior small-sample coverage and length properties than alternative nonparametric methods based on asymptotic normality. Applying the proposed method to Natality Vital Statistics, we study factors of extremely low birth weights. We find that signs of major effects are the same as those found in preceding studies based on parametric models, but with different magnitudes.

KEYWORDS:

• Conditional extremal quantile
• Confidence interval
• Extreme value theory
• Fixed k
• Random coefficient

Supplementary Materials

The online supplementary appendix presents additional Monte Carlo studies, proofs, details of the computational procedure, discussions about primitive conditions, and additional results about the asymptotic framework under an increasing k.

Acknowledgments

We thank Jianqing Fan (editor), an anonymous associate editor, two anonymous referees, Federico Bugni, Xiaohong Chen, Tim Christensen, Yanqin Fan, Yoonseok Lee, Zhijie Xiao, Yichong Zhang, and participants at the seminar/conference at Boston College, PSU, SMU, EC² Conference 2019, Econometric Society North American Summer Meeting 2019, and Greater New York Metropolitan Area Econometrics Colloquium 2019, for very helpful comments and advice.

Notes

1 See Ding (2016) for the exact expression for the PDF.

Additional information

Funding

Wang gratefully acknowledges the financial support by the Applyby-Mosher Fund.