Computing Kernel-Smoothed Conditional Quantiles from Many Observations

Abstract

Quantiles of a variable Y conditional on another variable X, when plotted against X, can be a useful descriptive tool. These plots give a quick impression of the functional form of the relation between X and the location, spread, and shape of the conditional distribution of Y. If several Y are observed for each X, then sample quantiles could be calculated for each X. The resulting quantile plot may be quite noisy, however, and smoothing across X may be desired. This article presents an algorithm that calculates kernel-smoothed conditional quantiles with a cross-validation choice of bandwidth for X. It is computationally feasible for large data sets when X assumes a small number of values since it requires only one pass through the full data set. The cross-validation does not require a pass through the data because of simplifications arising from the L ₁ loss function being based on absolute values. The technique is illustrated by plotting the conditional quantiles of the net wealth of a sample of Canadian families against the age of the head of the family. The results indicate that the common practice of using age and age squared as regressors for explaining net wealth may be misleading. The inverted-U shape given by the age-and-age-squared regressions is not supported by the quantile plots.

Key Words:

• Cross-validation
• Life cycle models
• Median estimation
• Nonparametric
• Quantile loss function
• Wealth distribution