zbMATH — the first resource for mathematics

Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing. (English) Zbl 1293.62106
Summary: A new uniform expansion is introduced for sums of weighted kernel-based regression residuals from nonparametric or semiparametric models. This expansion is useful for deriving asymptotic properties of semiparametric estimators and test statistics with data-dependent bandwidths, random trimming, and estimated efficiency weights. Provided examples include a new estimator for a binary choice model with selection and an associated directional test for specification of this model’s average structural function. An appendix contains new results on uniform rates for kernel estimators and primitive sufficient conditions for high level assumptions commonly used in semiparametric estimation.

MSC:
 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation 62G08 Nonparametric regression and quantile regression
Full Text:
References:
 [1] Ahn, H., Semiparametric estimation of a single-index model with nonparametrically generated regressors, Econometric Theory, 13, 1, 3-31, (1997) [2] Ahn, H.; Manski, C. F., Distribution theory for the analysis of binary choice under uncertainty with nonparametric estimation of expectations, Journal of Econometrics, 56, 3, 291-321, (1993) · Zbl 0792.62100 [3] Ahn, H.; Powell, J. L., Semiparametric estimation of censored selection models with a nonparametric selection problem, Journal of Econometrics, 58, 1-2, 3-29, (1993) · Zbl 0772.62063 [4] Ai, C.; Chen, X., Efficient estimation of models with conditional moment restrictions containing unknown functions, Econometrica, 71, 6, 1795-1843, (2003) · Zbl 1154.62323 [5] Akritas, M. G.; van Keilegom, I., Non-parametric estimation of the residual distribution, Scandinavian Journal of Statistics, 28, 3, 549-567, (2001) · Zbl 0980.62027 [6] Andrews, D. W.K., Asymptotics for semiparametric models via stochastic equicontinuity, Econometrica, 62, 1, 43-72, (1994) · Zbl 0798.62104 [7] Andrews, D. W.K., Nonparametric kernel estimation for semiparametric models, Econometric Theory, 11, 3, 560-596, (1995) [8] Bickel, P. J.; Klaassen, C. A.J.; Ritov, Y.; Wellner, J. A., Efficient and adaptive estimation for semiparametric models, (1993), Springer-Verlag New York · Zbl 0786.62001 [9] Blundell, R. W.; Powell, J. L., Endogeneity in semiparametric binary response models, Review of Economic Studies, 71, 7, 655-679, (2004) · Zbl 1103.91400 [10] Chen, X., Large sample sieve estimation of semi-nonparametric models, (Heckman, J. J.; Leamer, E. E., Handbook of Econometrics, Volume 6, (2007), Elsevier Amsterdam), 5549-5632 [11] Chen, X.; Linton, O. B.; van Keilegom, I., Estimation of semiparametric models when the criterion function is not smooth, Econometrica, 71, 5, 1591-1608, (2003) · Zbl 1154.62325 [12] Cragg, J. G., Some statistical models for limited dependent variables with application to the demand for durable goods, Econometrica, 39, 5, 829-844, (1971) · Zbl 0231.62040 [13] Das, M.; Newey, W. K.; Vella, F., Nonparametric estimation of sample selection models, The Review of Economic Studies, 70, 1, 33-58, (2003) · Zbl 1060.62132 [14] Delgado, M. A.; González Manteiga, W., Significance testing in nonparametric regression based on the bootstrap, Annals of Statistics, 29, 5, 1469-1507, (2001) · Zbl 1043.62032 [15] Einmahl, J. H.J.; Mason, D. M., Uniform in bandwidth consistency of kernel-type function estimators, Annals of Statistics, 33, 3, 1380-1403, (2005) · Zbl 1079.62040 [16] Escanciano, J.C., Jacho-Chávez, D.T., Lewbel, A., 2012. Identification and estimation of semiparametric two step models. Unpublished Manuscript. · Zbl 1398.62090 [17] Escanciano, J. C.; Song, K., Testing single-index restrictions with a focus on average derivatives, Journal of Econometrics, 156, 2, 377-391, (2010) · Zbl 1431.62610 [18] Hahn, J.; Ridder, G., The asymptotic variance of semiparametric estimators with generated regressors, Econometrica, 81, 1, 315-340, (2013) · Zbl 1274.62338 [19] Hansen, B., Uniform convergence rates for kernel estimation with dependent data, Econometric Theory, 24, 3, 726-748, (2008) · Zbl 1284.62252 [20] Heckman, J. J., Sample selection bias as a specification error, Econometrica, 47, 1, 153-161, (1979) · Zbl 0392.62093 [21] Heckman, J. J.; Ichimura, H.; Todd, P., Matching as an econometric evaluation estimator, Review of Economic Studies, 65, 2, 261-294, (1998) · Zbl 0908.90059 [22] Heckman, J. J.; Vytlacil, E., Structural equations, treatment effects, and econometric policy evaluation, Econometrica, 73, 3, 669-738, (2005) · Zbl 1152.62406 [23] Horowitz, J. L.; Spokoiny, V. G., An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative, Econometrica, 69, 3, 599-631, (2001) · Zbl 1017.62012 [24] Ichimura, H., Semiparametric least squares (SLS) and weighted SLS estimation of single index models, Journal of Econometrics, 58, 1-2, 71-120, (1993) · Zbl 0816.62079 [25] Ichimura, H.; Lee, L., Semiparametric least squares estimation of multiple index models: single equation estimation, (Barnett, W. A.; Powell, J.; Tauchen, G., Nonparametric and Semiparametric Methods in Econometrics and Statistics, (1991), Cambridge University Press), 3-49 · Zbl 0766.62065 [26] Ichimura, H.; Lee, S., Characterization of the asymptotic distribution of semiparametric $$M$$-estimators, Journal of Econometrics, 159, 2, 252-266, (2010) · Zbl 1395.62032 [27] Imbens, G.; Newey, W., Identification and estimation of triangular simultaneous equations models without additivity, Econometrica, 77, 5, 1481-1512, (2009) · Zbl 1182.62215 [28] Klein, R., Shen, C., Vella, F., 2012. Semiparametric selection models with binary outcomes. Unpublished Manuscript. · Zbl 1331.62476 [29] Klein, R.; Spady, R., An efficient semiparametric estimator for discrete choice models, Econometrica, 61, 2, 387-421, (1993) · Zbl 0783.62100 [30] Lewbel, A., Endogenous selection or treatment model estimation, Journal of Econometrics, 141, 777-8067, (2007) · Zbl 1418.62507 [31] Lewbel, A.; Linton, O. B., Nonparametric matching and efficient estimators of homothetically separable functions, Econometrica, 75, 4, 1209-1227, (2007) · Zbl 1134.91548 [32] Li, D.; Li, Q., Nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters, Journal of Econometrics, 157, 1, 179-190, (2010) · Zbl 1431.62646 [33] Li, Q.; Wooldridge, J. M., Semiparametric estimation of partially linear models for dependent data with generated regressors, Econometric Theory, 18, 3, 625-645, (2002) · Zbl 1109.62314 [34] Mammen, E.; Rothe, C.; Schienle, M., Nonparametric regression with nonparametrically generated covariates, Annals of Statistics, 40, 2, 1132-1170, (2012) · Zbl 1274.62294 [35] Mammen, E., Rothe, C., Schienle, M., 2013. Semiparametric estimation with generated covariates. Unpublished Manuscript. · Zbl 1441.62808 [36] Matzkin, R. L., Nonparametric and distribution-free estimation of the binary threshold crossing and the binary choice models, Econometrica, 60, 2, 239-270, (1992) · Zbl 0747.62034 [37] Meng, C.-L.; Schmidt, P., On the cost of partial observability in the bivariate probit model, International Economic Review, 26, 1, 71-85, (1985) · Zbl 0557.62096 [38] Neumeyer, N., A central limit theorem for two-sample $$U$$-processes, Statistics & Probability Letters, 67, 73-85, (2004) · Zbl 1079.60026 [39] Neumeyer, N.; Van Keilegom, I., Estimating the error distribution in nonparametric multiple regression with applications to model testing, Journal of Multivariate Analysis, 101, 5, 1067-1078, (2010) · Zbl 1185.62078 [40] Newey, W. K., The asymptotic variance of semiparametric estimators, Econometrica, 62, 6, 1349-1382, (1994) · Zbl 0816.62034 [41] Newey, W. K., Nonparametric continuous/discrete choice models, International Economic Review, 48, 4, 1429-1439, (2007) [42] Newey, W. K.; McFadden, D., Large sample estimation and hypothesis testing, (McFadden, D.; Engle, R. F., Handbook of Econometrics, Vol. IV, (1994), Elsevier, North-Holland Amsterdam), 2111-2245 [43] Newey, W.; Powell, J.; Vella, F., Nonparametric estimation of triangular simultaneous equations models, Econometrica, 67, 3, 565-603, (1999) · Zbl 1035.62031 [44] Nickl, R.; Pötscher, B. M., 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, (2007) · Zbl 1130.46020 [45] Olley, S.; Pakes, A., The dynamics of productivity in the telecommunications equipment industry, Econometrica, 64, 6, 1263-1297, (1996) · Zbl 0862.90024 [46] Pagan, A., Econometric issues in the analysis of regressions with generated regressors, International Economic Review, 25, 1, 221-247, (1984) · Zbl 0547.62078 [47] Pinkse, J., 2001. Nonparametric regression estimation using weak separability. Unpublished Manuscript. [48] Rothe, C., Semiparametric estimation of binary response models with endogenous regressors, Journal of Econometrics, 153, 1, 51-64, (2009) · Zbl 1431.62663 [49] Song, K., Uniform convergence of series estimators over function spaces, Econometric Theory, 24, 6, 1463-1499, (2008) · Zbl 1277.62130 [50] Sperlich, S., A note on non-parametric estimation with predicted variables, The Econometrics Journal, 12, 2, 382-395, (2009) · Zbl 1206.62064 [51] Stock, J. H., Nonparametric policy analysis, Journal of the American Statistical Association, 84, 406, 567-575, (1989) [52] Talagrand, M., Sharper bounds for Gaussian and empirical processes, Annals of Probability, 22, 1, 28-76, (1994) · Zbl 0798.60051 [53] van de Ven, W.; van Praag, B., The demand for deductibles in private health insurance: a probit model with sample selection, Journal of Econometrics, 17, 2, 229-252, (1981) [54] van der Vaart, A. W.; Wellner, J. A., (Weak Convergence and Empirical Processes with Applications to Statistics, Springer Series in Statistics, (1996), Springer-Verlag 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.