Minimum distance from independence estimation of nonseparable instrumental variables models. (English) Zbl 1452.62255
Summary: I develop a semiparametric minimum distance from independence estimator for a nonseparable instrumental variables model. An independence condition identifies the model for many types of discrete and continuous instruments. The estimator is taken as the parameter value that most closely satisfies this independence condition. Implementing the estimator requires a quantile regression of the endogenous variables on the instrument, so the procedure is two-step, with a finite or infinite-dimensional nuisance parameter in the first step. I prove consistency and establish asymptotic normality for a parametric, but flexibly nonlinear outcome equation. The consistency of the nonparametric bootstrap is also shown. I illustrate the use of the estimator by estimating the returns to schooling using data from the 1979 National Longitudinal Survey.
MSC:
| 62G05 | Nonparametric estimation |
| 62F12 | Asymptotic properties of parametric estimators |
| 62E20 | Asymptotic distribution theory in statistics |
| 62G09 | Nonparametric statistical resampling methods |
| 62P20 | Applications of statistics to economics |
Keywords:
nonseparable models; instrumental variables; quantile regression; two-step estimation; minimum distance from independence; unobserved heterogeneitySoftware:
TOMLABReferences:
| [1] | Abrevaya, J., Computation of the maximum rank correlation estimator, Econom. Lett., 62, 279-285 (1999) · Zbl 0917.90057 |
| [2] | Andrews, D. W.K., Asymptotics for semiparametric econometric models via stochastic equicontinuity, Econometrica, 62, 43-72 (1994) · Zbl 0798.62104 |
| [3] | Bickel, P. J.; Freedman, D. A., Some asymptotic theory for the bootstrap, Ann. Statist., 9, 1196-1217 (1981) · Zbl 0449.62034 |
| [4] | Billingsley, P., Probability and Measure (1995), Wiley: Wiley New York [u.a.] · Zbl 0822.60002 |
| [6] | Brown, D. J.; Wegkamp, M. H., Weighted minimum mean-square distance from independence estimation, Econometrica, 70, 2035-2051 (2002) · Zbl 1101.62013 |
| [7] | Card, D., Using geographic variation in college proximity to estimate the return to schooling, (Christofides, L. N.; Grant, K. E.; Swidinsky, R., Aspects of Labour Market Behaviour: Essays in Honour of John Vanderkamp (1995), University of Toronto Press: University of Toronto Press Toronto), 201-222 |
| [8] | Card, D., Estimating the return to schooling: Progress on some persistent econometric problems, Econometrica, 69, 1127-1160 (2001) |
| [9] | Carrasco, M.; Florens, J.-P., Generalization of GMM to a continuum of moment conditions, Econometric Theory, 16, 797-834 (2000) · Zbl 0968.62028 |
| [10] | Chaudhuri, P.; Doksum, K.; Samarov, A., On average derivative quantile regression, Ann. Statist., 25, 715-744 (1997) · Zbl 0898.62082 |
| [11] | Chen, X.; Linton, O.; van Keilegom, I., Estimation of semiparametric models when the criterion function is not smooth, Econometrica, 71, 1591-1608 (2003) · Zbl 1154.62325 |
| [12] | Chen, X.; Pouzo, D., Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals, Econometrica, 80, 277-321 (2012) · Zbl 1274.62232 |
| [14] | Chesher, A., Identification in nonseparable models, Econometrica, 71, 1405-1441 (2003) · Zbl 1154.62413 |
| [15] | D’Haultfœuille, X.; Février, P., Identification of nonseparable triangular models with discrete instruments, Econometrica, 3, 1199-1210 (2015) · Zbl 1410.62060 |
| [16] | Dominguez, M. A.; Lobato, I. N., Consistent estimation of models defined by conditional moment restrictions, Econometrica, 72, 1601-1615 (2004) · Zbl 1141.91635 |
| [17] | Florens, J. P.; Heckman, J. J.; Meghir, C.; Vytlacil, E., Identification of treatment effects using control functions in models with continuous, endogenous treatment and heterogeneous effects, Econometrica, 76, 1191-1206 (2008) · Zbl 1152.91746 |
| [18] | Giné, E.; Zinn, J., Bootstrapping general empirical measures, Ann. Probab., 18, 851-869 (1990) · Zbl 0706.62017 |
| [19] | Hahn, J., Three Essays in Econometrics (Ph.D. thesis) (1993), Harvard University |
| [20] | Hahn, J., A note on bootstrapping generalized method of moments estimators, Econometric Theory, 12, 187-197 (1996) |
| [21] | Heckman, J. J., Micro data, heterogeneity, and the evaluation of public policy: Nobel lecture, J. Polit. Econ., 109, 673-748 (2001) |
| [23] | Imbens, G. W., Nonparametric estimation of average treatment effects under exogeneity: A review, Rev. Econ. Stat., 86, 4-29 (2004) |
| [24] | Imbens, G. W.; Newey, W. K., Identification and estimation of triangular simultaneous equations models without additivity, Econometrica, 77, 1481-1512 (2009) · Zbl 1182.62215 |
| [25] | Jones, D. R.; Perttunen, C. D.; Stuckman, B. E., Lipschitzian optimization without the Lipschitz constant, J. Optim. Theory Appl., 79, 157-181 (1993) · Zbl 0796.49032 |
| [26] | Koenker, R., Quantile Regression (2005), Cambridge University Press · Zbl 1111.62037 |
| [27] | Koenker, R.; Bassett, G., Regression quantiles, Econometrica, 46, 33-50 (1978) · Zbl 0373.62038 |
| [28] | Komunjer, I.; Santos, A., Semi-parametric estimation of non-separable models: a minimum distance from independence approach, Econom. J., 13, S28-S55 (2010) · Zbl 07708397 |
| [29] | Kosorok, M. R., Introduction to Empirical Processes and Semiparametric Inference (2008), Springer: Springer New York · Zbl 1180.62137 |
| [30] | Linton, O.; Sperlich, S.; van Keilegom, I., Estimation of a semiparametric transformation model, Ann. Statist., 36, 686-718 (2008) · Zbl 1133.62029 |
| [31] | Luenberger, D. G., Optimization by Vector Space Methods (1968), Wiley: Wiley New York · Zbl 0184.44502 |
| [32] | Manski, C. F., Closest empirical distribution estimation, Econometrica, 51, 305-319 (1983) · Zbl 0559.62024 |
| [33] | Matzkin, R. L., Nonparametric estimation of nonadditive random functions, Econometrica, 71, 1339-1375 (2003) · Zbl 1154.62327 |
| [34] | Nelsen, R., An Introduction to Copulas (2006), Springer: Springer New York · Zbl 1152.62030 |
| [35] | Newey, W. K., The asymptotic variance of semiparametric estimators, Econometrica, 62, 1349-1382 (1994) · Zbl 0816.62034 |
| [36] | Newey, W. K.; McFadden, D., Large sample estimation and hypothesis testing, (Engle, R. F.; McFadden, D. L., Handbook of Econometrics, vol. 4 (1994), Elsevier), 2111-2245, (Chapter 36) |
| [37] | Pakes, A.; Pollard, D., Simulation and the asymptotics of optimization estimators, Econometrica, 57, 1027-1057 (1989) · Zbl 0698.62031 |
| [38] | Pollard, D., Convergence of Stochastic Processes (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0544.60045 |
| [39] | Resnick, S. I., A Probability Path (1999), Birkhäuser: Birkhäuser Boston · Zbl 0944.60002 |
| [42] | Torgovitsky, A., Identification of nonseparable models using instruments with small support, Econometrica, 83, 1185-1197 (2015) · Zbl 1410.62210 |
| [43] | van der Vaart, A. W., Asymptotic Statistics (1998), Cambridge University Press: Cambridge University Press New York · Zbl 0910.62001 |
| [44] | van der Vaart, A. W.; Wellner, J. A., Weak Convergence and Empirical Processes: With Applications to Statistics (1996), Springer: Springer New York · Zbl 0862.60002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
