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Consistent model specification tests based on \(k\)-nearest-neighbor estimation method. (English) Zbl 1431.62189
Summary: We propose a simple consistent test for a parametric regression functional form based on \(k\)-nearest-neighbor (\(k\)-nn) method. We derive the null distribution of the test statistic and show that the test achieves the minimax rate optimality against smooth alternatives. A wild bootstrap method is used to better approximate the null distribution of the test statistic. We also propose a \(k\)-nn statistic which tests for omitted variables nonparametrically. Simulations and an empirical application using US economics new Ph.D. job market matching data show that the \(k\)-nn method is more appropriate than the kernel method to analyze unevenly distributed data.
Reviewer: Reviewer (Berlin)

MSC:
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62G09 Nonparametric statistical resampling methods
62P20 Applications of statistics to economics
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[1] Abadie, A.; Imbens, G., Large sample properties of matching estimators for average treatment effects, Econometrica, 74, 235-267, (2006) · Zbl 1112.62042
[2] Abadie, A.; Imbens, G., Bias corrected matching estimators for average treatment effects, J. Bus. Econom. Statist., 29, 1-11, (2011) · Zbl 1214.62031
[3] Andrews, D. W.K., A conditional Kolmogorov test, Econometrica, 65, 1097-1128, (1997) · Zbl 0928.62019
[4] Azzalini, A.; Bowman, A., On the use of nonparametric regression for checking linear relationships, J. R. Stat. Soc. Ser. B Stat. Methodol., 55, 549-557, (1993) · Zbl 0800.62222
[5] Bierens, H., Consistent model specification tests, J. Econometrics, 20, 105-134, (1982) · Zbl 0549.62076
[6] Bierens, H., A consistent conditional moment test of functional form, Econometrica, 58, 1443-1458, (1990) · Zbl 0737.62058
[7] Bierens, H. J.; Ploberger, W., Asymptotic theory of integrated conditional moment tests, Econometrica, 65, 1129-1151, (1997) · Zbl 0927.62085
[8] Bravo, F.; Huynh, K. P.; Jacho-Chávez, D. T., Average derivative estimation with missing responses, (Drukker, D. M., Missing Data Methods: Cross-sectional Methods and Applications, Advances in Econometrics, vol. 27A, (2011), Emerald Group Publishing Limited), 129-154, (Chapter 5)
[9] Breunig, C., Goodness-of-fit tests based on series estimators in nonparametric instrumental regressions, J. Econometrics, 184, 328-346, (2015) · Zbl 1331.62226
[10] Casas, I.; Gao, J., Econometric estimation in long-range dependent volatility models: theory and practice, J. Econometrics, 147, 72-83, (2008) · Zbl 1429.62463
[11] Cheng, G.; Huang, J. Z., Bootstrap consistency for general semiparametric M-estimation, Ann. Statist., 38, 2884-2915, (2010) · Zbl 1200.62042
[12] Chu, B. M.; Huynh, K. P.; Jacho-Chávez, D. T., Functionals of order statistics and their multivariate concomitants with application to semiparametric estimation by nearest neighbors, Sankhya B, 75, 238-292, (2013) · Zbl 1282.62128
[13] Chu, B. M.; Jacho-Chávez, D. T., K-nearest neighbour estimation of inverse-density-weighted expectations with dependent data, Econometric Theory, 28, 769-803, (2012) · Zbl 1419.62072
[14] Delgado, M. A., Semiparametric generalized least squares in the multivariate nonlinear regression model, Econometric Theory, 8, 203-222, (1992)
[15] Delgado, M. A.; Manteiga, W. G., Significance testing in nonparametric regression based on the bootstrap, Ann. Statist., 29, 1469-1507, (2001) · Zbl 1043.62032
[16] Delgado, M. A.; Stengos, T., Semiparametric testing of non-nested econometric models, Rev. Econom. Stud., 75, 345-367, (1994)
[17] Dette, H., A consistent test for the functional form of a regression based on a difference of variance estimators, Ann. Statist., 27, 1012-1050, (1999) · Zbl 0957.62036
[18] Escanciano, J. C.; Jacho-Chávez, D. T.; Lewbel, A., Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing, J. Econometrics, 178, 426-443, (2014) · Zbl 1293.62106
[19] Eubank, R. L.; Hart, J. D., Testing goodness-of-fit in regression via order selection criteria, Ann. Statist., 20, 1412-1425, (1992) · Zbl 0776.62045
[20] Eubank, R. L.; Spiegelman, C. H., Testing the goodness of fit of a linear model via non-parametric regression, J. Amer. Statist. Assoc., 85, 387-392, (1990) · Zbl 0702.62037
[21] Fan, Y.; Li, Q., Consistent model specification tests: omitted variables and semiparametric functional forms, Econometrica, 64, 865-890, (1996) · Zbl 0854.62038
[22] Fan, Y.; Li, Q., Central limit theorem for degenerate \(U\)-statistics of absolutely regular processes with applications to model specification testing, J. Nonparametr. Stat., 10, 245-271, (1999) · Zbl 0974.62044
[23] Fan, Y.; Li, Q., Consistent model specification tests: kernel-based tests versus bierens’ ICM tests, Econometric Theory, 16, 1016-1041, (2000) · Zbl 1180.62071
[24] Fan, Y.; Liu, R., Symmetrized multivariate \(k\)-nn estimators, Econometric Rev., 34, 828-848, (2015)
[25] Fix, E.; Hodges, J. L., Discriminatory analysis, nonparametric discrimination consistency properties. technical report 4, (1951), USAF School of Aviation Medicine Randolph Field, Texas
[26] Gan, L.; Li, Q., Efficiency of thin and thick market, J. Econometrics, 192, 40-54, (2016) · Zbl 1419.62506
[27] Gao, J.; Gijbels, I., Bandwidth selection in nonparametric kernel testing, J. Amer. Statist. Assoc., 103, 1584-1594, (2008) · Zbl 1286.62043
[28] Gao, J.; King, M.; Lu, Z.; Tjøstheim, D., Nonparametric specification testing for nonlinear time series with nonstationarity, Econometric Theory, 25, 1869-1892, (2009) · Zbl 1179.62055
[29] González-Manteiga, W.; Crujeiras, R. M., An updated review of goodness-of-fit tests for regression models, Test, 22, 361-411, (2013) · Zbl 1273.62086
[30] Guerre, E.; Lavergne, P., Optimal minimax rates for nonparametric specification testing in regression models, Econometric Theory, 18, 1139-1171, (2002) · Zbl 1033.62042
[31] Guerre, E.; Lavergne, P., Rate-optimal data-driven specification testing for regression models, Ann. Statist., 33, 840-870, (2005) · Zbl 1068.62055
[32] Härdle, W.; Hall, P.; Marron, J. S., How far are automatically chosen regression smoothing parameters from their optimum?, J. Amer. Statist. Assoc., 83, 86-99, (1988) · Zbl 0644.62048
[33] Härdle, W.; Hall, P.; Marron, J. S., Regression smoothing parameters that are not far from their optimum, J. Amer. Statist. Assoc., 87, 227-233, (1992) · Zbl 0850.62352
[34] Härdle, W.; Mammen, E., Comparing non-parametric versus parametric regression fits, Ann. Statist., 21, 1926-1947, (1993) · Zbl 0795.62036
[35] Hart, J., Nonparametric smoothing and lack-of-fit tests, (1997), Springer Berlin · Zbl 0886.62043
[36] Heckman, J. J.; Ichimura, H.; Todd, P., Matching as an econometric evaluation estimator, Rev. Econom. Stud., 65, 261-294, (1998) · Zbl 0908.90059
[37] Hong, Y.; White, H., Consistent specification testing via nonparametric series regression, Econometrica, 63, 1133-1159, (1995) · Zbl 0941.62125
[38] Horowitz, J. L., Specification testing in nonparametric instrumental variable estimation, J. Econometrics, 167, 383-396, (2012) · Zbl 1441.62740
[39] Horowitz, J. L.; Spokoiny, V., An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative, Econometrica, 69, 599-631, (2001) · Zbl 1017.62012
[40] Hsiao, C.; Li, Q.; Racine, J., Consistent model specification tests with mixed discrete and continuous variables, J. Econometrics, 140, 802-826, (2007) · Zbl 1247.62126
[41] Jacho-Chávez, D. T., K nearest-neighbor estimation of inverse density weighted expectations, Econ. Bull., 3, 1-6, (2008)
[42] Jun, S. J.; Pinkse, J., Efficient semiparametric seemingly unrelated quantile regression estimation, Econometric Theory, 25, 1392-1414, (2009) · Zbl 1284.62419
[43] Jun, S. J.; Pinkse, J., Semiparametric tests of conditional moment restrictions under weak or partial identification, J. Econometrics, 152, 3-18, (2009) · Zbl 1431.62188
[44] Jun, S. J.; Pinkse, J., Testing under weak identification with conditional moment restrictions, Econometric Theory, 28, 1229-1282, (2012) · Zbl 1281.62114
[45] Lavergne, P.; Maistre, S.; Patilea, V., A significance test for covariates in nonparametric regression, Electron. J. Stat., 9, 643-678, (2015) · Zbl 1309.62076
[46] Lavergne, P.; Patilea, V., Breaking the curse of dimensionality in nonparametric testing, J. Econometrics, 143, 103-122, (2008) · Zbl 1418.62199
[47] Lavergne, P.; Vuong, Q., Nonparametric selection of regressors: the nonnested case, Econoemtrica, 64, 207-219, (1996) · Zbl 0860.62039
[48] Lavergne, P.; Vuong, Q., Nonparametric significance testing, Econometric Theory, 16, 576-601, (2000) · Zbl 0968.62047
[49] Lee, J., \(U\)-statistics: theory and practice, (1990), Marcel Dekker New York · Zbl 0771.62001
[50] Lewbel, A., Consistent nonparametric hypothesis tests with an application to slutsky symmetry, J. Econometrics, 67, 379-401, (1995) · Zbl 0820.62042
[51] Lewbel, A.; Lu, X.; Su, L., Specification testing for transformation models with an application to generalized accelerated failure-time models, J. Econometrics, 184, 81-96, (2015) · Zbl 1331.62247
[52] Li, K. C., Asymptotic optimality for \(C_p\), \(C_L\), cross-validation and generalized cross-validation: discrete index set, Ann. Statist., 15, 958-975, (1987) · Zbl 0653.62037
[53] Li, Q.; Hsiao, C.; Zinn, J., Consistent specification tests for semiparametric/nonparametric models based on series estimation methods, J. Econometrics, 112, 295-325, (2003) · Zbl 1027.62027
[54] Li, D.; Li, Q., Nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters, J. Econometrics, 157, 179-190, (2010) · Zbl 1431.62646
[55] Li, Q.; Wang, S., A simple consistent bootstrap test for a parametric regression function, J. Econometrics, 87, 145-165, (1998) · Zbl 0943.62031
[56] Lin, Z.; Li, Q.; Sun, Y., A consistent nonparametric test of parametric regression functional form in fixed effects panel data models, J. Econometrics, 178, 167-179, (2014) · Zbl 1293.62196
[57] Liu, Z.; Lu, X., Root-N-consistent estimation of partially linear model by \(K\)-nn method, Econometric Rev., 16, 411-420, (1997) · Zbl 0914.62028
[58] Lu, X.; White, H., Testing for separability in structural equations, J. Econometrics, 182, 14-26, (2014) · Zbl 1311.62068
[59] Mallows, C. L., Some comments on \(c_p\), Technometrics, 15, 661-675, (1973) · Zbl 0269.62061
[60] Neumeyer, N.; Dette, H., Nonparametric comparison of regression curves: an empirical process approach, Ann. Statist., 31, 880-920, (2003) · Zbl 1032.62037
[61] Neumeyer, N.; Van Keilegom, I., Estimating the error distribution in nonparametric multiple regression with applications to model testing, J. Multivariate Anal., 101, 1067-1078, (2010) · Zbl 1185.62078
[62] Newey, W. K., Efficient instrumental variables estimation of nonlinear models, Econometrica, 58, 809-837, (1990) · Zbl 0728.62107
[63] Ouyang, D.; Li, D.; Li, Q., Cross-validation and nonparametric K nearest neighbor estimation, Econom. J., 9, 448-471, (2006) · Zbl 1106.62043
[64] Robinson, P., Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form, Econometrica, 875-891, (1987) · Zbl 0651.62107
[65] Robinson, P., Root-N consistent semi-parametric regression, Econometrica, 156, 931-954, (1988) · Zbl 0647.62100
[66] Robinson, P., Hypothesis testing in semiparametric and nonparametric models for econometric time series, Rev. Econom. Stud., 56, 511-534, (1989) · Zbl 0681.62101
[67] Robinson, P., Nearest-neighbour estimation of semiparametric regression models, J. Nonparametr. Stat., 5, 33-41, (1995) · Zbl 0873.62043
[68] Robinson, P., Consistent nonparametric entropy-based testing, Rev. Econom. Stud., 58, 437-453, (1991) · Zbl 0719.62055
[69] Stengos, T.; Sun, Y., A consistent model specification test for a regression function based on nonparametric wavelet estimation, Econometric Rev., (2000)
[70] Stone, C. J., Consistent nonparametric regression, Ann. Statist., 595-620, (1977) · Zbl 0366.62051
[71] Stute, W., Nonparametric model check for regression, Ann. Statist., 22, 1346-1370, (1997)
[72] Stute, W.; González-Manteiga, W. G., NN goodness-of-fit tests for linear models, J. Statist. Plann. Inference, 53, 75-92, (1996) · Zbl 0847.62036
[73] Stute, W.; Zhu, L. X., Nonparametric checks for single-index models, Ann. Statist., 33, 1048-1083, (2005) · Zbl 1080.62023
[74] Su, L.; Jin, S.; Zhang, Y., Specification test for panel data models with interactive fixed effects, J. Econometrics, 186, 222-244, (2015) · Zbl 1331.62485
[75] Su, L.; Lu, X., Nonparametric dynamic panel data models: kernel estimation and specification testing, J. Econometrics, 176, 112-133, (2013) · Zbl 1284.62268
[76] Su, L.; Tu, Y.; Ullah, A., Testing additive separability of error term in nonparametric structural models, Econometric Rev., 34, 1088-2015, (2015)
[77] Su, L.; Ullah, A., A nonparametric goodness-of-fit-based test for conditional heteroskedasticity, Econometric Theory, 29, 187-212, (2013) · Zbl 1316.62058
[78] Sun, Y.; Cai, Z.; Li, Q., A consistent nonparametric test on semiparametric smooth coefficient models with integrated time, Econometric Theory, (2015), (forthcoming)
[79] Sun, Y.; Li, Q., An alternative series based consistent model specification test, Econom. Lett., 93, 37-44, (2006) · Zbl 1255.62119
[80] Ullah, A., Specification analysis of econometric models, J. Quant. Econ., 2, 187-209, (1985)
[81] van der Vaart, A., Asymptotic statistics, (1998), Cambridge University Press · Zbl 0910.62001
[82] Wang, Q.; Phillips, P. C., A specification test for nonlinear nonstationary models, Ann. Statist., 40, 727-758, (2012) · Zbl 1273.62228
[83] Wooldridge, J. M., A test for functional form against non-parametric alternatives, Econometric Theory, 8, 452-475, (1992)
[84] Yatchew, A. J., Nonparametric regression tests based on least squares, Econometric Theory, 8, 435-451, (1992)
[85] Zheng, X., A consistent test of functional form via nonparametric estimation technique, J. Econometrics, 75, 263-289, (1996) · Zbl 0865.62030
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