# Quantile Treatment Effects

The $42,000 question regarding any set of regression estimates is whether they have a causal interpretation. This is no less true for quantile regression than Ordinary Least Squares. Suppose we are interested in estimating the effect of a training program on earnings. OLS regression estimates measure the impact of the program on average earnings while quantile regression estimates can be used to measure the impact of the program on median earnings. In both cases, we must worry about whether the estimated program effects are contaminated by omitted variables bias.

Here too, omitted variables problems can be solved using instrumental variables, though IV methods for quantile models are a relatively new development, and are not yet as flexible as conventional 2SLS. We discuss an approach that captures the causal effect of a binary variable on quantiles (i. e., a treatment effect) using a binary instrument. The Quantile Treatment Effects (QTE) estimator, introduced in Abadie, Angrist, and Imbens (2002), relies on essentially the same assumptions as the LATE framework for average causal effects. The result is an Abadie-type weighting estimator of the causal effect of treatment on quantiles for compliers.[106]

Our discussion of the QTE estimator is based on an additive model for conditional quantiles, so that a single treatment effect is estimated. The resulting estimator simplifies to Koenker and Bassett (1978) linear quantile regression when there is no instrumenting. The relationship between QTE and quantile regression is therefore analogous to that between conventional 2SLS and OLS when the regressor of interest is a dummy.

The parameters of interest are defined as follows. For т 2 (0, 1), we assume there exist aT 2 R and PT 2 Rr such that

Qr (Yi|Xi; Di; D1i>Doi) — ^r Di + XiPT; (7.2.1)

where QT(Yi|Xi, Di, Dii>Doi) denotes the т-quantile of Yi given Xi and Di for compliers. Thus, aT and PT

are quantile regression coefficients for compliers.

Recall that Di is independent of potential outcomes conditional on Xi and D1i>Doi, as we discussed in

(4.5.2) . The parameter aT in this model therefore gives the difference in the conditional-on-Xi quantiles of Yii and Yoi for compliers. In other words,

Qt (y1i|Xi; D1i >D0i) Qt (y0i|Xi; D1i>D0i) — ^t (7.2.2)

This tells us, for example, whether a training program changed the conditional median or lower decile of earnings for compliers. Note that the parameter aT does not tell us whether treatment changed the quantiles of the unconditional distributions of Y1i and Yoi. For that, we have to integrate families of quantile regression results using procedures like the one described in Section 7.1.3.

It also bears emphasizing that aT is not the conditional quantile of the individual treatment effects, (y 1i — Yoi). You might want to know, for example, whether the median treatment effect is positive. Unfortunately, questions like this are very hard to answer without making the assumptions usually invoked for causal inference.[107] Even a randomized trial with perfect compliance fails to reveal the distribution of

(Yij—Yoj). This does not matter for average treatment effects since the mean of a difference is the difference in means. But all other features of the distribution of Y1i—Yoi are hidden because we never get to see both Yii and Yoi for any one person. The good news for applied econometricians is that the difference in marginal distributions, (7.2.2), is usually more important than the distribution of treatment effects because comparisons of aggregate economic welfare typically require only the marginal distributions of Y1i and Yoi and not the distribution of their difference (see, e. g., Atkinson (1970), for the traditional view). This point can be made by example without reference to quantiles. When evaluating an employment program, we are inclined to view the program favorably if it increases overall employment rates. In other words, we are happy if the average Y1i is higher than the average Yoi. The number of individuals who gain jobs (Y1i—Yoi = 1) or lose jobs (Yii—Yoi = 0) seems like it should be of secondary interest since a good program will necessarily have more gainers than losers.

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