Fast Inference for Quantile Regression with Tens of Millions of Observations

TL;DR

This paper introduces a fast, online inference method for linear quantile regression on ultra-large datasets, enabling efficient analysis of tens of millions of observations with minimal computational resources.

Contribution

It develops a stochastic subgradient descent-based inference framework for ultra-large datasets, providing a fully online, resampling-free pivotal statistic for quantile regression.

Findings

Handles datasets with up to 10^7 observations and 10^3 regressors.

Outperforms existing methods in computational efficiency.

Applied to U.S. wage data, revealing gender gap trends.

Abstract

Big data analytics has opened new avenues in economic research, but the challenge of analyzing datasets with tens of millions of observations is substantial. Conventional econometric methods based on extreme estimators require large amounts of computing resources and memory, which are often not readily available. In this paper, we focus on linear quantile regression applied to "ultra-large" datasets, such as U.S. decennial censuses. A fast inference framework is presented, utilizing stochastic subgradient descent (S-subGD) updates. The inference procedure handles cross-sectional data sequentially: (i) updating the parameter estimate with each incoming "new observation", (ii) aggregating it as a $\textit{Polyak-Ruppert}$ average, and (iii) computing a pivotal statistic for inference using only a solution path. The methodology draws from time-series regression to create an asymptotically pivotal statistic through random scaling. Our proposed test statistic is calculated in a fully online fashion and critical values are calculated without resampling. We conduct extensive numerical studies to showcase the computational merits of our proposed inference. For inference problems as large as $(n, d) \sim (10^7, 10^3)$, where $n$ is the sample size and $d$ is the number of regressors, our method generates new insights, surpassing current inference methods in computation. Our method specifically reveals trends in the gender gap in the U.S. college wage premium using millions of observations, while controlling over $10^3$ covariates to mitigate confounding effects.