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

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 thesis, we propose a new class of scores (a score function) that is based on the moments of order statistics of a random variable Z. The new scores are shown to be flexible in modeling the desired features (e.g., monotonicity or convexity) of the scores. In addition, if the true covariate X is drawn from a location-scale family and Z is its standardized distribution, then the least-squares estimator calculated using the proposed scores consistently estimates the true correlation between the response and the covariate and asymptotically approaches the normal distribution. We also propose a procedure for diagnosing a given score function and selecting one that is better suited to the data. We numerically demonstrate the advantage of using a correctly specifed score function over that of the identity scores (or other misspecifed 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.