Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T19:33:56.984Z Has data issue: false hasContentIssue false
  • English
  • Français

ARE UNOBSERVABLES SEPARABLE?

Published online by Cambridge University Press:  26 January 2024

Andrii Babii Open the ORCID record for Andrii Babii [Opens in a new window]  and
Jean-Pierre Florens Open the ORCID record for Jean-Pierre Florens [Opens in a new window]
Andrii Babii*
Affiliation:
University of North Carolina at Chapel Hill
Jean-Pierre Florens
Affiliation:
Toulouse School of Economics
*
Address correspondence to Andrii Babii, Department of Economics, University of North Carolina at Chapel Hill—Gardner Hall, CB 3305, Chapel Hill, NC 27599-3305, USA; e-mail: babii.andrii@gmail.com.
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.

It is common to assume in empirical research that observables and unobservables are additively separable, especially when the former are endogenous. This is because it is widely recognized that identification and estimation challenges arise when interactions between the two are allowed for. Starting from a nonseparable IV model, where the instrumental variable is independent of unobservables, we develop a novel nonparametric test of separability of unobservables. The large-sample distribution of the test statistics is nonstandard and relies on a Donsker-type central limit theorem for the empirical distribution of nonparametric IV residuals, which may be of independent interest. Using a dataset drawn from the 2015 U.S. Consumer Expenditure Survey, we find that the test rejects the separability in Engel curves for some commodities.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 33
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

This work was supported by the French National Research Agency under Grant ANR-19-CE40-0013-01/ExtremReg project. We thank the seminar participants at Yale University, and the editorial team for helpful comments. We also thank Ivan Canay, Xiaohong Chen, Tim Christensen, Elia Lapenta, Pascal Lavergne, Thierry Magnac, Nour Meddahi, Ingrid Van Keilegom, and Edward Vytlacil for insightful discussions. All remaining errors are ours.

References

REFERENCES

Akritas, M., & Van Keilegom, I. (2001). Non-parametric estimation of the residual distribution. Scandinavian Journal of Statistics , 28(3), 549–567.CrossRefGoogle Scholar
Almås, I. (2012). International income inequality: Measuring PPP bias by estimating Engel curves for food. American Economic Review , 102(2), 1093–1117.Google Scholar
Andrews, D. (1994). Chapter 37: Empirical process methods in econometrics. In Handbook of econometrics (vol. 4, pp. 2247–2294). Elsevier.Google Scholar
Babii, A. (2020). Honest confidence sets in nonparametric IV regression and other ill-posed models. Econometric Theory , 36(4), 658–706.CrossRefGoogle Scholar
Babii, A. (2022). High-dimensional mixed-frequency IV regression. Journal of Business and Economic Statistics , 40(3), 1470–1483.CrossRefGoogle Scholar
Babii, A., & Florens, J.-P. (2020). Is completeness necessary? Estimation in nonidentified linear models. Preprint. arXiv:1709.03473.Google Scholar
Banks, J., Blundell, R., & Lewbel, A. (1997). Quadratic Engel curves and consumer demand. Review of Economics and Statistics , 79(4), 527–538.Google Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007). Semi-nonparametric IV estimation of shape-invariant Engel curves. Econometrica , 75(6), 1613–1669.CrossRefGoogle Scholar
Blundell, R., Horowitz, J., & Parey, M. (2017). Nonparametric estimation of a nonseparable demand function under the Slutsky inequality restriction. Review of Economics and Statistics , 99(2), 291–304.Google Scholar
Breunig, C. (2020). Specification testing in nonparametric instrumental quantile regression. Econometric Theory , 36(4), 583–625.CrossRefGoogle Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007). Chapter 77: Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. In Handbook of econometrics (vol. 6B, pp. 5633–5751). Elsevier.Google Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2014). Asymptotic normal inference in linear inverse problems. In Racine, J. S., Su, L., & Ullah, A. (Eds.), The Oxford handbook of applied nonparametric and semiparametric econometrics and statistics (pp. 64–96). Oxford Academic.Google Scholar
Cavalier, L. (2011). Inverse problems in statistics. In Alquier, P., Gautier, E., & Stoltz, G. (Eds.), Inverse problems and high-dimensional estimation (pp. 3–96). Springer.Google Scholar
Chen, X. (2007). Chapter 76: Large sample sieve estimation of semi-nonparametric models. In Handbook of econometrics (vol. 6, pp. 5549–5632). Elsevier.Google Scholar
Chen, X., Chernozhukov, V., Lee, S., & Newey, W. (2014). Local identification of nonparametric and semiparametric models. Econometrica , 82(2), 785–809.Google Scholar
Chen, X., & Christensen, T. (2018). Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression. Quantitative Economics , 9(1), 39–84.CrossRefGoogle Scholar
Chen, X., Linton, O., & Van Keilegom, I. (2003). Estimation of semiparametric models when the criterion function is not smooth. Econometrica , 71(5), 1591–1608.Google Scholar
Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., & Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters. Econometrics Journal , 21(1), C1–C68.Google Scholar
Chernozhukov, V., Escanciano, J. C., Ichimura, H., Newey, W. K., & Robins, J. M. (2022). Locally robust semiparametric estimation. Econometrica , 90(4), 1501–1535.Google Scholar
Chernozhukov, V., Fernández-Val, I., Newey, W., Stouli, S., & Vella, F. (2020). Semiparametric estimation of structural functions in nonseparable triangular models. Quantitative Economics , 11(2), 503–533.Google Scholar
Chernozhukov, V., & Hansen, C. (2005). An IV model of quantile treatment effects. Econometrica , 73(1), 245–261.Google Scholar
Chernozhukov, V., & Hansen, C. (2013). Quantile models with endogeneity. Annual Review of Economics , 5, 57–81.Google Scholar
Chernozhukov, V., Imbens, G., & Newey, W. (2007). Instrumental variable estimation of nonseparable models. Journal of Econometrics , 139(1), 4–14.CrossRefGoogle Scholar
D’HaultfŒuille, X., & Février, P. (2015). Identification of nonseparable triangular models with discrete instruments. Econometrica , 83(3), 1199–1210.CrossRefGoogle Scholar
Darolles, S., Fan, Y., Florens, J.-P., & Renault, E. (2011). Nonparametric instrumental regression. Econometrica , 79(5), 1541–1565.Google Scholar
Das, M. (2005). Instrumental variables estimators of nonparametric models with discrete endogenous regressors. Journal of Econometrics , 124(2), 335–361.CrossRefGoogle Scholar
Devroye, L. (1986). Non-uniform random variate generation . Springer.Google Scholar
Dunker, F., Florens, J.-P., Hohage, T., Johannes, J., & Mammen, E. (2014). Iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression. Journal of Econometrics , 178(3), 444–455.CrossRefGoogle Scholar
Durbin, J. (1973). Weak convergence of the sample distribution function when parameters are estimated. Annals of Statistics 1(2), 279–290.CrossRefGoogle Scholar
Einmahl, J., & Van Keilegom, I. (2008). Specification tests in nonparametric regression. Journal of Econometrics , 143(1), 88–102.Google Scholar
Engl, H. W., Hanke, M., & Neubauer, A. (2000). Regularization of inverse problems . Springer.Google Scholar
Escanciano, J., Pardo-Fernández, J., & Van Keilegom, I. (2018). Asymptotic distribution-free tests for semiparametric regressions with dependent data. Annals of Statistics , 46(3), 1167–1196.Google Scholar
Evans, L. C. (2010). Partial differential equations . Graduate Studies in Mathematics (vol. 19). American Mathematical Society.Google Scholar
Fève, F., Florens, J.-P., & Van Keilegom, I. (2018). Estimation of conditional ranks and tests of exogeneity in nonparametric nonseparable models. Journal of Business and Economic Statistics , 36(2), 334–345.CrossRefGoogle Scholar
Florens, J.-P. (2003). Inverse problems and structural econometrics: The example of instrumental variables. In Dewatripont, M., Hansen, L. P., & Turnovsky, S. J. (Eds.), Advances in economics and econometrics: Theory and applications, eighth World Congress (vol. II, pp. 284–311). Cambridge University Press.Google Scholar
Florens, J.-P., Heckman, 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
Florens, J.-P., Johannes, J., & Van Bellegem, S. (2011). Identification and estimation by penalization in nonparametric instrumental regression. Econometric Theory , 27(3), 472–496.Google Scholar
Gagliardini, P., & Scaillet, O. (2012). Tikhonov regularization for nonparametric instrumental variable estimators. Journal of Econometrics , 167(1), 61–75.Google Scholar
Gagliardini, P., & Scaillet, O. (2017). A specification test for nonparametric instrumental variable regression. Annals of Economics and Statistics/Annales d’Economie et de Statistique 128, 151–202.Google Scholar
Giné, E., & Nickl, R. (2015). Mathematical foundations of infinite-dimensional statistical models . Cambridge University Press.CrossRefGoogle Scholar
Hall, P., & Horowitz, J. (2005). Nonparametric methods for inference in the presence of instrumental variables. Annals of Statistics , 33(6), 2904–2929.Google Scholar
Heckman, J. J., & Vytlacil, E. (2001). Policy-relevant treatment effects. American Economic Review , 91(2), 107–111.Google Scholar
Hoderlein, S., Su, L., White, H., & Yang, T. (2016). Testing for monotonicity in unobservables under unconfoundedness. Journal of Econometrics , 193(1), 183–202.Google Scholar
Horowitz, J., & Lee, S. (2007). Nonparametric instrumental variables estimation of a quantile regression model. Econometrica , 75(4), 1191–1208.Google Scholar
Imbens, G. (2010). Nonadditive models with endogenous regressors. In Advances in economics and econometrics: Theory and applications, ninth World Congress (vol. III, pp. 17–46). Cambridge University Press.Google Scholar
Imbens, G., & Newey, W. (2009). Identification and estimation of triangular simultaneous equations models without additivity. Econometrica , 77(5), 1481–1512.Google Scholar
Imbens, G. W., & Angrist, J. D. (1994). Identification and estimation of local average treatment effects. Econometrica , 62, 467–475.Google Scholar
Krein, S. G., & Petunin, Y. I. (1966). Scales of Banach spaces. Russian Mathematical Surveys , 21(2), 85–159.Google Scholar
Loynes, R. (1980). The empirical distribution function of residuals from generalised regression. Annals of Statistics , 8(2), 285–298.Google Scholar
Lu, X., & White, H. (2014). Testing for separability in structural equations. Journal of Econometrics , 182(1), 14–26.Google Scholar
Mammen, E. (1996). Empirical process of residuals for high-dimensional linear models. Annals of Statistics , 24(1), 307–335.Google Scholar
Matzkin, R. (2013). Nonparametric identification in structural economic models. Annual Review of Economics , 5, 457–486.Google Scholar
Nair, M. (2015). Role of Hilbert scales in regularization theory. In Romeo, P. G., Meakin, J. C., & Rajan, A. R. (Eds.), Semigroups, algebras and operator theory: Kochi, India, February 2014 (pp. 159–176). Springer.Google Scholar
Nakamura, E., Steinsson, J., & Liu, M. (2016). Are Chinese growth and inflation too smooth? Evidence from Engel curves. American Economic Journal: Macroeconomics , 8(3), 113–144.Google Scholar
Neumeyer, N. (2009). Smooth residual bootstrap for empirical processes of non-parametric regression residuals. Scandinavian Journal of Statistics , 36(2), 204–228.Google Scholar
Neumeyer, N., & Van Keilegom, I. (2019). Bootstrap of residual processes in regression: To smooth or not to smooth? Biometrika , 106(2), 385–400.CrossRefGoogle Scholar
Newey, W., & Powell, J. (2003). Instrumental variable estimation of nonparametric models. Econometrica , 71(5), 1565–1578.Google Scholar
Nickl, R., & Pötscher, B. (2007). Bracketing metric entropy rates and empirical central limit theorems for function classes of Besov- and Sobolev-type. Journal of Theoretical Probability , 20(2), 177–199.Google Scholar
Pardo-Fernández, J., Van Keilegom, I., & González-Manteiga, W. (2007). Testing for the equality of $k$ regression curves. Statistica Sinica , 17(3), 1115–1137.Google 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.Google Scholar
Torgovitsky, A. (2015). Identification of nonseparable models using instruments with small support. Econometrica , 83(3), 1185–1197.Google Scholar
Torgovitsky, A. (2017). Minimum distance from independence estimation of nonseparable instrumental variables models. Journal of Econometrics , 199(1), 35–48.Google Scholar
van der Vaart, A., & Wellner, J. (1996). Weak convergence and empirical processes: With applications to statistics . Springer.Google Scholar