Causal Interpretation of Regressions With Ranks

TL;DR

This paper clarifies the causal interpretation of regression coefficients when using rank-transformed variables, introducing the Rank Average Treatment Effect (rank-ATE) as a key estimand and proposing methods for more interpretable causal inference.

Contribution

It derives effective causal estimands for regressions with ranks, introduces the rank-ATE, and develops methods for clearer causal interpretation in common econometric designs.

Findings

Rank-ATE serves as the fundamental causal estimand for regressions with ranks.

Direct application of regressions on outcome ranks yields difficult-to-interpret parameters.

Proposed alternative methods improve interpretability of causal estimates.

Abstract

In studies of educational production functions or intergenerational mobility, it is common to transform the key variables into percentile ranks. Yet, it remains unclear what the regression coefficient estimates with ranks of the outcome or the treatment. In this paper, we derive effective causal estimands for a broad class of commonly-used regression methods, including the ordinary least squares (OLS), two-stage least squares (2SLS), difference-in-differences (DiD), and regression discontinuity designs (RDD). Specifically, we introduce a novel primitive causal estimand, the Rank Average Treatment Effect (rank-ATE), and prove that it serves as the building block of the effective estimands of all the aforementioned econometrics methods. For 2SLS, DiD, and RDD, we show that direct applications to outcome ranks identify parameters that are difficult to interpret. To address this issue, we develop alternative methods to identify more interpretable causal parameters.