Regression with Partially Observed Ranks on a Covariate: Distribution-Guided Scores for Ranks

TL;DR

This paper introduces a new score function based on moments of order statistics, called D-rank, which optimally correlates with responses in rank-based regression, demonstrated through stock market data analysis.

Contribution

It proposes the D-rank score function for rank regression, showing its asymptotic optimality and providing a method for score diagnosis and selection.

Findings

D-rank maximizes correlation between response and score.

The least-squares estimator with D-rank is consistent and asymptotically normal.

Application to stock data reveals the effect of investor attention on returns.

Abstract

This work is motivated by a hand-collected data set from one of the largest Internet portals in Korea. This data set records the top 30 most frequently discussed stocks on its on-line message board. The frequencies are considered to measure the attention paid by investors to individual stocks. The empirical goal of the data analysis is to investigate the effect of this attention on trading behavior. For this purpose, we regress the (next day) returns and the (partially) observed ranks of frequencies. In the regression, the ranks are transformed into scores, for which purpose the identity or linear scores are commonly used. In this paper, we propose a new class of scores (a score function) that is based on the moments of order statistics of a pre-decided random variable. The new score function, denoted by D-rank, is shown to be asymptotically optimal to maximize the correlation between the response and score, when the pre-decided random variable and true covariate are in the same location-scale family. In addition, the least-squares estimator using the D-rank consistently estimates the true correlation between the response and the covariate, and asymptotically approaches the normal distribution. We additionally propose a procedure for diagnosing a given score function (equivalently, the pre-decided random variable Z) and selecting one that is better suited to the data. We numerically demonstrate the advantage of using a correctly specified score function over that of the identity scores (or other misspecified scores) in estimating the correlation coefficient. Finally, we apply our proposal to test the effects of investors' attention on their returns using the motivating data set.