What Quantile Regression Does and Doesn't Do: A Commentary on Petscher and Logan (2014)

Abstract

Petscher and Logan's (2014) description of quantile regression (QR) might mislead readers to believe it would estimate the relation between an outcome, y, and one or more predictors, x, at different quantiles of the unconditional distribution of y. However, QR models the conditional quantile function of y given x just as linear regression models the conditional mean function. This article's contribution is twofold: First, it discusses potential consequences of methodological misconceptions and formulations of Petscher and Logan's (2014) presentation by contrasting features of QR and linear regression. Second, it reinforces the importance of correct understanding of QR in empirical research by illustrating similarities and differences in various QR estimators and linear regression using simulated data.

Quantile Regression Estimands and Models: Revisiting the Motherhood Wage Penalty Debate

“…Although several studies have emphasized the typical misinterpretation of the CQR model in the last decade (Killewald & Bearak, 2014;Porter, 2015;Wenz, 2018), the crucial distinction between the UQR model and QTE models has received little attention across a range of disciplines. Consequently, there is a mismatch between the quantile regression models used in many studies (the UQR model) and these studies' aim (identify unconditional QTEs).…”

“…Several authors have noted that conditional quantile regression (CQR) is not well-suited to identify unconditional QTEs in the presence of control variables (Firpo, 2007;Killewald & Bearak, 2014;Porter, 2015;Wenz, 2018). There are, however, a few available approaches that identify unconditional QTEs.…”

“…Concerning this point, the seminal paper of Killewald and Bearak (2014), and the later paper of Wenz (2018), was timely and vital in increasing the awareness of how CQR models should be interpreted.…”

“…However, the approach that replaced the CQR model in many research streams is ill-suited to investigate quantile treatment effects. Despite an early proposal for estimating the QTE model by Firpo (2007), the UQR model gained massive popularity across several disciplines because it seemingly tackled the CQR model's problem of localizing outcome differentials across the distribution (Killewald & Bearak, 2014;Wenz, 2018). However, the UQR model was developed to infer how changes in the independent variables influence unconditional quantile values (Firpo et al, 2009).…”

“…The UQR model's coefficients capture these factors' combined effect -called unconditional quantile partial effects (UQPE). Thus, the typical interpretation and use of UQR to estimate QTE within, for example, parts of the sociology literature (Budig & Hodges, 2014;Cooke, 2014;England et al, 2016;Glauber, 2018;Killewald & Bearak, 2014;Lin & Weiss, 2019;Wenz, 2018), the education literature (Porter, 2015), and the econometrics literature (Havnes & Mogstad, 2011;Lindqvist & Vestman, 2011) is insufficient, and in worst case erroneous. Admittedly, our previous work is not exempt from this incorrect interpretation of UQR coefficients as QTEs (Borgen, 2015(Borgen, , 2016.…”