Abstract
An empirical approach for the analysis of treatment effect at various quantiles in the case of multiple treatment conditions is here proposed. Outcome changes under multiple treatment conditions are computed using (a) inverse propensity score weights and (b) unconditional outcome distribution within each group. Through (a) and (b), the standard double robust estimator is extended to evaluate treatment effect not only on average but also in the tails (quantiles). A Monte Carlo study designed to examine and assess the performance of the proposed approach and two empirical applications conclude the analysis.
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Notes
In the OLS case, the conditional mean, E[Y|X], averages up to the unconditional mean, E[Y]. As a result, the OLS linear model for conditional means E[Y|X] = Xα implies that E[Y] = E[X]α. When looking at a given quantile Q(), Q(Y) differs from Q(X)α, and the fitted values yielded by Xα are no longer appropriate to compute the quantile of the outcome. The unconditional distribution of Y is needed.
The Machado and Mata approach is computationally intensive and may be troublesome in very large datasets (Fortin et al. 2010). Albaek and Thomsen (2014) suggest replacing the bootstrap with a non-replacement subsampling scheme. Alternatively, Fortin et al. (2010) simplify the Machado and Mata routine by computing many quantile regressions instead of bootstrapping them.
All the empirical analyses were performed using Stata software (Version SE15, http://www.stata.com). The same random numbers generator has been considered in each experiment. DR, EIF and IPW/IPWQ estimators were implemented using the user-written Stata module “poparms” by Cattaneo et al. (2013), while the Machado and Mata (2005) decomposition approach was implemented using the “mmsel” Stata program by Frolich and Melly (2010).
References
Abrevaya J (2001) The effects of demographics and maternal behavior on the distribution of birth outcomes. Empir Econ 26:247–257
Albaek K, Thomsen L (2014) Decomposing wage distribution on a large data set—a quantile regression analysis of the gender wage gap, Discussion Paper, The Danish National Centre for Social Research
Caracciolo F, Furno M (2017) Quantile treatment effect and double robust estimators: an appraisal on the Italian labor market. J Econ Stud 44(4):585–604
Cattaneo M (2010) Efficient semiparametric estimation of multi-valued treatment effects under ignorability. J Econom 155(2):138–154
Cattaneo M, Drukker D, Holland A (2013) Estimation of multivalued treatment effects under conditional independence. Stata J 13(3):407–450
Chernozhukov V, Fernandez-Val I (2011) Inference for extremal conditional quantile models, with an application to market and birthweight risks. Rev Econ Stud 78:559–589
Chernozhukov V, Fernandez-Val I, Melly B (2013) Inference on counterfactual distributions. Econometrica 81:2205–2268
Firpo S (2007) Efficient semi-parametric estimation of quantile treatment effect. Econometrica 75:259–276
Fortin N, Lemieux T, Firpo S (2010) Decomposition methods in econometrics. NBER w.p. 16045
Frolich M, Melly B (2010) Estimation of quantile treatment effects with Stata. Stata J 10:423–457
Guo S, Fraser MW (2010) Propensity score analysis: statistical methods and applications. Sage, Thousand Oaks
Heckman J, Smith J, Clements N (1997) Making the most out of programme evaluations and social experiments: accounting for heterogeneity in programme impacts. Rev Econ Stud 64:487–535
Huber M, Lechner M, Wunsch C (2013) The performance of estimators based on the propensity score. J Econom 175:1–21
Imai K, van Dyk D (2004) Causal inference with general treatment regimes: generalizing the propensity score. J Am Stat Assoc 99:854–866
Imbens G (2000) The role of propensity score in estimating dose-response functions. Biometrika 87(3):706–710
Koenker R (2005) Quantile regression. Cambridge University Press, Cambridge
Koenker R, Hallock K (2001) Quantile regression: an introduction. J Econ Perspect 15:143–156
Linden A, Uysal D, Rayan A, Adams J (2016) Estimating casual effects for multivalued treatments: a comparison of approaches. Stat Med 35:534–552
Lunceford J, Davidian M (2004) Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. Stat Med 23:2937–2960
Machado J, Mata J (2005) Counterfactual decomposition of changes in wage distributions using quantile regression. J Appl Econom 20:445–465
Melly B (2006) Estimation of counterfactual distributions using quantile regressions. University of St. Gallen w.p
Nichols A (2008) Erratum and discussion of the propensity-score reweighting. Stata J 8:532–539
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Furno, M., Caracciolo, F. Multi-valued Double Robust quantile treatment effect. Empir Econ 58, 2545–2571 (2020). https://doi.org/10.1007/s00181-018-1584-7
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DOI: https://doi.org/10.1007/s00181-018-1584-7