Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T08:12:15.618Z Has data issue: false hasContentIssue false
  • English
  • Français

TESTING FOR UNOBSERVED HETEROGENEOUS TREATMENT EFFECTS WITH OBSERVATIONAL DATA

Published online by Cambridge University Press:  28 April 2022

Yu-Chin Hsu ,
Ta-Cheng Huang  and
Haiqing Xu
Yu-Chin Hsu
Affiliation:
Academia Sinica, National Central University, National Chengchi University, and National Taiwan University
Ta-Cheng Huang*
Affiliation:
National University of Singapore
Haiqing Xu
Affiliation:
University of Texas at Austin
*
Address correspondence to Ta-Cheng Huang, Global Asia Institute, National University of Singapore, Singapore; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Unobserved heterogeneous treatment effects have been emphasized in the recent policy evaluation literature (see, e.g., Heckman and Vytlacil (2005, Econometrica 73, 669–738)). This paper proposes a nonparametric test for unobserved heterogeneous treatment effects in a treatment effect model with a binary treatment assignment, allowing for individuals’ self-selection to the treatment. Under the standard local average treatment effects assumptions, i.e., the no defiers condition, we derive testable model restrictions for the hypothesis of unobserved heterogeneous treatment effects. Furthermore, we show that if the treatment outcomes satisfy a monotonicity assumption, these model restrictions are also sufficient. Then, we propose a modified Kolmogorov–Smirnov-type test which is consistent and simple to implement. Monte Carlo simulations show that our test performs well in finite samples. For illustration, we apply our test to study heterogeneous treatment effects of the Job Training Partnership Act on earnings and the impacts of fertility on family income, where the null hypothesis of homogeneous treatment effects gets rejected in the second case but fails to be rejected in the first application.

Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 3 , June 2023 , pp. 582 - 622
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

The authors are grateful to the Editor (Peter C.B. Phillips), the Co-Editor (Yoon-Jae Whang), three anonymous referees, Jason Abrevaya, Qi Li, Xiaojun Song, and Quang Vuong for valuable comments and suggestions, which have considerably improved the presentation of the paper. Yu-Chin Hsu gratefully acknowledges the research support from the Ministry of Science and Technology of Taiwan (MOST2628-H-001-007, MOST110-2634-F-002-045), Academia Sinica Investigator Award of Academia Sinica (AS-IA-110-H01), and Center for Research in Econometric Theory and Applications (110L9002) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education of Taiwan. Ta-Cheng Huang is indebted to Qi Li for his continued inspiration, guidance, and support. Haiqing Xu would like to dedicate this paper to the memory of Professor Halbert White.

References

REFERENCES

Abadie, A. (2002) Bootstrap tests for distributional treatment effects in instrumental variable models. Journal of the American Statistical Association 97(457), 284–292.CrossRefGoogle Scholar
Abadie, A. (2003) Semiparametric instrumental variable estimation of treatment response models. Journal of Econometrics 113(2), 231–263.CrossRefGoogle Scholar
Abadie, A., Angrist, J., & Imbens, G. (2002) Instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings. Econometrica 70(1), 91–117.CrossRefGoogle Scholar
Abrevaya, J., Hsu, Y.-C., & Lieli, R.P. (2015) Estimating conditional average treatment effects. Journal of Business & Economic Statistics 33(4), 485–505.CrossRefGoogle Scholar
Andrews, D.W. (1994) Empirical process methods in econometrics. In F. Engle, & D.L. McFadden (eds.), Handbook of Econometrics , vol. 4, pp. 2247–2294. Elsevier.Google Scholar
Andrews, D.W. (1997) A conditional Kolmogorov test. Econometrica 65(5), 1097–1128.CrossRefGoogle Scholar
Andrews, D.W. & Shi, X. (2013) Inference based on conditional moment inequalities. Econometrica 81(2), 609–666.Google Scholar
Angrist, J.D. & Evans, W.N. (1998) Children and their parents’ labor supply: Evidence from exogenous variation in family size. American Economic Review 88(3), 450–477.Google Scholar
Angrist, J.D. & Krueger, A.B. (1991) Does compulsory school attendance affect schooling and earnings? The Quarterly Journal of Economics 106(4), 979–1014.CrossRefGoogle Scholar
Athey, S. & Imbens, G. (2016) Recursive partitioning for heterogeneous causal effects. Proceedings of the National Academy of Sciences of the United States of America 113(27), 7353–7360.CrossRefGoogle ScholarPubMed
Barrett, G.F. & Donald, S.G. (2003) Consistent tests for stochastic dominance. Econometrica 71(1), 71–104.CrossRefGoogle Scholar
Bloom, H.S., Orr, L.L., Bell, S.H., Cave, G., Doolittle, F., Lin, W., & Bos, J.M. (1997) The benefits and costs of JTPA Title II-A programs: Key findings from the National Job Training Partnership Act study. The Journal of Human Resources 32(3), 549–576.CrossRefGoogle Scholar
Blundell, R. & Powell, J.L. (2003) Endogeneity in nonparametric and semiparametric regression models. In Dewatripont, M., Hansen, L., & Turnovsky, S. (eds.), Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress . Econometric Society Monographs, vol. 36, pp. 312–357. Cambridge University Press.CrossRefGoogle Scholar
Bouezmarni, T. & Taamouti, A. (2014) Nonparametric tests for conditional independence using conditional distributions. Journal of Nonparametric Statistics 26(4), 697–719.CrossRefGoogle Scholar
Bronars, S.G. & Grogger, J. (1994) The economic consequences of unwed motherhood: Using twin births as a natural experiment. American Economic Review 32(5), 1141–1156.Google Scholar
Chang, M., Lee, S., & Whang, Y.-J. (2015) Nonparametric tests of conditional treatment effects with an application to single-sex schooling on academic achievements. The Econometrics Journal 18(3), 307–346.CrossRefGoogle Scholar
Chernozhukov, V. & Hansen, C. (2005) An IV model of quantile treatment effects. Econometrica 73(1), 245–261.CrossRefGoogle Scholar
Chesher, A. (2003) Identification in nonseparable models. Econometrica 71(5), 1405–1441.CrossRefGoogle Scholar
Chesher, A. (2005) Nonparametric identification under discrete variation. Econometrica 73(5), 1525–1550.CrossRefGoogle Scholar
Crump, R.K., Hotz, V.J., Imbens, G.W., & Mitnik, O.A. (2008) Nonparametric tests for treatment effect heterogeneity. The Review of Economics and Statistics 90(3), 389–405.CrossRefGoogle Scholar
D’Haultfœuille, X. & Février, P. (2015) Identification of nonseparable triangular models with discrete instruments. Econometrica 83(3), 1199–1210.CrossRefGoogle Scholar
Dauxois, J. & Nkiet, G.M. (1998) Nonlinear canonical analysis and independence tests. Annals of Statistics 26(4), 1254–1278.CrossRefGoogle Scholar
Delgado, M.A. & Manteiga, W.G. (2001) Significance testing in nonparametric regression based on the bootstrap. Annals of Statistics 29(5), 1469–1507.CrossRefGoogle Scholar
Donald, S.G. & Hsu, Y.-C. (2014) Estimation and inference for distribution functions and quantile functions in treatment effect models. Journal of Econometrics 178(3), 383–397.CrossRefGoogle Scholar
Donald, S.G., Hsu, Y.-C., & Barrett, G.F. (2012) Incorporating covariates in the measurement of welfare and inequality: Methods and applications. The Econometrics Journal 15(1), C1–C30.CrossRefGoogle Scholar
Florens, J.-P., Heckman, J.J., Meghir, C., & Vytlacil, E. (2008) Identification of treatment effects using control functions in models with continuous, endogenous treatment and heterogeneous effects. Econometrica 76(5), 1191–1206.Google Scholar
Frölich, M. & Melly, B. (2013) Unconditional quantile treatment effects under endogeneity. Journal of Business & Economic Statistics 31(3), 346–357.CrossRefGoogle Scholar
Heckman, J.J., Schmierer, D., & Urzua, S. (2010) Testing the correlated random coefficient model. Journal of Econometrics 158(2), 177–203.CrossRefGoogle ScholarPubMed
Heckman, J.J., Smith, J., & Clements, N. (1997) Making the most out of programme evaluations and social experiments: Accounting for heterogeneity in programme impacts. The Review of Economic Studies 64(4), 487–535.CrossRefGoogle Scholar
Heckman, J.J. & Vytlacil, E. (2001) Policy-relevant treatment effects. American Economic Review 91(2), 107–111.CrossRefGoogle Scholar
Heckman, J.J. & Vytlacil, E. (2005) Structural equations, treatment effects, and econometric policy evaluation. Econometrica 73(3), 669–738.CrossRefGoogle Scholar
Hoderlein, S. & Mammen, E. (2009) Identification and estimation of marginal effects in nonseparable, nonmonotonic models. The Econometrics Journal 12(1), 1–25.CrossRefGoogle Scholar
Hoderlein, S. & White, H. (2012) Nonparametric identification in nonseparable panel data models with generalized fixed effects. Journal of Econometrics 168(2), 300–314.CrossRefGoogle Scholar
Hsu, Y.-C. (2017) Consistent tests for conditional treatment effects. The Econometrics Journal 20(1), 1–22.CrossRefGoogle Scholar
Hsu, Y.-C., Liu, C.-A., & Shi, X. (2019) Testing generalized regression monotonocity. Econometric Theory , 1146–1200.CrossRefGoogle Scholar
Huang, M., Sun, Y., & White, H. (2016) A flexible nonparametric test for conditional independence. Econometric Theory 32(6), 1434–1482.CrossRefGoogle Scholar
Huang, T.-M. (2010) Testing conditional independence using maximal nonlinear conditional correlation. Annals of Statistics 38(4), 2047–2091.CrossRefGoogle Scholar
Imbens, G.W. (2010) Better LATE than nothing: Some comments on Deaton (2009) and Heckman and Urzua (2009). Journal of Economic Literature 48(2), 399–423.CrossRefGoogle Scholar
Imbens, G.W. & Angrist, J.D. (1994) Identification and estimation of local average treatment effects. Econometrica 62(2), 467–475.CrossRefGoogle Scholar
Imbens, G.W. & Newey, W.K. (2009) Identification and estimation of triangular simultaneous equations models without additivity. Econometrica 77(5), 1481–1512.Google Scholar
Imbens, G.W. & Rubin, D.B. (1997) Estimating distributions for outcome compliers models in instrumental variables. The Review of Economic Studies 64(4), 555–574.CrossRefGoogle Scholar
Jacobsen, J.P., Pearce, J.W., & Rosenbloom, J.L. (1999) The effects of childbearing on married women’s labor supply and earnings: Using twin births as a natural experiment. The Journal of Human Resources 34(3), 449–474.CrossRefGoogle Scholar
Jun, S.J., Lee, Y., & Shin, Y. (2016) Treatment effects with unobserved heterogeneity: A set identification approach. Journal of Business & Economic Statistics 34(2), 302–311.CrossRefGoogle Scholar
Kosorok, M. (2008) Introduction to Empirical Processes and Semiparametric Inference . Springer.CrossRefGoogle Scholar
Lee, S., Okui, R., & Whang, Y.-J. (2017) Doubly robust uniform confidence band for the conditional average treatment effect function. Journal of Applied Econometrics 32(7), 1207–1225.CrossRefGoogle Scholar
Lee, S. & Whang, Y.-J. (2009) Nonparametric Tests of Conditional Treatment Effects. Manuscript.CrossRefGoogle Scholar
Linton, O. & Gozalo, P. (2014) Testing conditional independence restrictions. Econometric Reviews 33(5–6), 523–552.CrossRefGoogle Scholar
Lu, X. & White, H. (2014) Testing for separability in structural equations. Journal of Econometrics 182(1), 14–26.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17(6), 571–599.CrossRefGoogle Scholar
Matzkin, R.L. (2003) Nonparametric estimation of nonadditive random functions. Econometrica 71(5), 1339–1375.CrossRefGoogle Scholar
Nolan, D. & Pollard, D. (1988) Functional limit theorems for U-processes. Annals of Probability 16(3), 1291–1298.CrossRefGoogle Scholar
Pagan, A. & Ullah, A. (1999) Nonparametric Econometrics . Cambridge University Press.CrossRefGoogle Scholar
Powell, J.L., Stock, J.H., & Stoker, T.M. (1989) Semiparametric estimation of index coefficients. Econometrica 57(6), 1403–1430.CrossRefGoogle Scholar
Rosenzweig, M.R. & Wolpin, K.I. (1980) Testing the quantity-quality fertility model: The use of twins as a natural experiment. Econometrica 48(1), 227–240.CrossRefGoogle ScholarPubMed
Rudin, W. (1962) Fourier Analysis on Groups . Interscience.Google Scholar
Stinchcombe, M.B. & White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14(3), 295–325.CrossRefGoogle Scholar
Su, L., Tu, Y., & Ullah, A. (2015) Testing additive separability of error term in nonparametric structural models. Econometric Reviews 34(6–10), 1057–1088.CrossRefGoogle Scholar
Su, L. & White, H. (2007) A consistent characteristic function-based test for conditional independence. Journal of Econometrics 141(2), 807–834.CrossRefGoogle Scholar
Su, L. & White, H. (2008) A nonparametric Hellinger metric test for conditional independence. Econometric Theory 24(4), 829–864.CrossRefGoogle Scholar
Su, L. & White, H. (2014) Testing conditional independence via empirical likelihood. Journal of Econometrics 182(1), 27–44.CrossRefGoogle Scholar
Torgovitsky, A. (2015) Identification of nonseparable models using instruments with small support. Econometrica 83(3), 1185–1197.CrossRefGoogle Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes: With Applications to Statistics , pp. 16–28. Springer.CrossRefGoogle Scholar
van der Vaart, A.W. & Wellner, J.A. (2007) Empirical processes indexed by estimated functions. In Cator, E.A., Jongbloed, G., Kraaikamp, C., Lopuhaä, H.P., & Wellner, J.A. (eds.), Asymptotics: Particles, Processes and Inverse Problems . Lecture Notes–Monograph Series, vol. 55, pp. 234–252. Institute of Mathematical Statistics.CrossRefGoogle Scholar
Vuong, Q. & Xu, H. (2017) Counterfactual mapping and individual treatment effects in nonseparable models with binary endogeneity. Quantitative Economics 8(2), 589–610.CrossRefGoogle Scholar
Vytlacil, E. (2002) Independence, monotonicity, and latent index models: An equivalence result. Econometrica 70(1), 331–341.CrossRefGoogle Scholar