Thread by @sewenz

Very happy that my paper "What Quantile Regression Does and Doesn’t Do: A Commentary on Petscher and Logan (2014)” is out in Child Development: doi.org/10.1111/cdev.1… In this <thread> I summarize its main points, most of which contradict Petscher & Logan (2014). 1/18

The description of Quantile Regression (QR) by Petscher & Logan (2014), henceforth PL, might mislead readers to believe QR would estimate the relation between an outcome, y, and one or more predictors, x, at different quantiles of the unconditional distribution of y, F_Y(y). 2/

However, QR, as introduced by Koenker & Bassett (1978), henceforth KB, models the conditional quantile function of y given x just as linear regression models the conditional mean function. Thus, the quantile regression model (QRM) is a linear conditional quantile model. 3/

Parameters of the linear regression model (LRM) are often estimated using OLS, i.e., by minimizing the sum of squared residuals. To estimate parameters of the QRM, KB suggest to minimize the sum of weighted absolute residuals. 4/

Assumptions of LRM & QRM differ less than it might seem based on PL. Violating the assumption of iid errors, e.g., has similar consequences & remedies in LRM & QRM. PL focus on the linearity assumption that's dropped in the QRM. However, it's also not very important in the LRM 5/

Much more important is the zero conditional quantile assumption, Q_tau(e_tau|x)=0, that has to hold to obtain consistent QRM estimates; it's not mentioned by PL. In the LRM, the zero conditional means assumption, E(e|x)=0, is needed for unbiased & consistent estimates. 6/

The QRM slope coef for x, beta_tau, indicates the amount of change in the conditional quantile tau of y, associated w/ a unit change in x. The change concerns the distribution of y conditional on all right hand side vars, F_Y|X(y,x), not the unconditional distribution F_Y(y). 7/

The simultaneous estimation of QRMs for different quantiles enables researchers to detect and describe the change in shape of F_Y|X(y,x) and, thus, heteroskedasticity but also higher moments such as skewness and kurtosis and how they change with x. 8/

In contrast to what PL suggest, as a linear conditional quantile model, the QRM does not capture nonlinear relations unless explicitly modeled. In fact, the QRM is not any better suited for or in any way superior to the LRM when modeling nonlinear relations. 9/

Furthermore, unless rank-invariance or, at least, rank-similarity is assumed, QRM coefficients tell us nothing about individuals (at particular quantiles) but only about the specified quantile of the conditional distribution F_Y|X(y,x). PL do not mention this assumption. 10/

If quantile treatment effects (QTEs) are defined as the difference between marginal distributions of potential states of the world, the QR estimator (KB) identifies QTEs on/at the specified quantile iff no covariates are included & needed (e.g., using experimental data). 11/

When covariates are included (e.g., for identification purposes or reducing SEs), researchers have to turn to unconditional QR estimators to learn about effects on/at quantiles of the unconditional distribution of y, F_Y(y), or, conditional on causal states D=d, F_Y|D(y,d). 12/

I discuss 2 unconditional QR estimators published before PL: If QTEs are defined as above (eq 6 in my paper), Firpo (2007)'s estimates are unbiased (but restricted to binary D), Firpo et al. (2009)'s are slightly biased, while KB's can be way off when covariates are present. 13/

I suggest avoiding PL as a source of learning about QR. Instead I recommend KB, Koenker (2005, 2017), Koenker et al (2018), and intros by Angrist & Pischke (2009), Cade & Noon (2003), Fitzenberger & Wilke (2015), Hao & Naiman (2007), Koenker & Hallock (2001), Porter (2015). 14/

References: PL: doi.org/10.1111/cdev.1… KB: doi.org/10.2307/1913643 Firpo (2007): doi.org/10.1111/j.1468… Firpo et al (2009): doi.org/10.3982/ECTA68… Koenker (2005): doi.org/10.1017/CBO978… Koenker (2017): doi.org/10.1146/annure… Koenker et al (2018): crcpress.com/Handbook-of-Qu… 15/

References (cont'd): Angrist & Pischke (2009): press.princeton.edu/titles/8769.ht… Cade & Noon (2003): doi.org/10.1890/1540-9… Fitzenberger & Wilke (2015): doi.org/10.1002/978111… Hao & Naiman (2007): doi.org/10.4135/978141… 16/

References (cont'd): Koenker & Hallock (2001): jstor.org/stable/2696522 Porter (2015): doi.org/10.1007/978-3-… 17/

Last, not least: If you wish to replicate the findings I present in Wenz (2018), please see the associated OSF project (incl a preprint) at osf.io/79gpf/ or go directly to the replication package featuring Stata syntax at osf.io/wn68j/ 18/18 </thread>

I just discovered @threadreaderapp that does unroll threads, so they are easier to read. Nice.