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A NONPARAMETRIC TEST OF SIGNIFICANT VARIABLES IN GRADIENTS

Published online by Cambridge University Press:  20 November 2020

Feng Yao  and
Taining Wang
Feng Yao
Affiliation:
Shenzhen University West Virginia University Guangdong University of Foreign Studies
Taining Wang*
Affiliation:
Capital University of Economics and Business
*
Address correspondence to Taining Wang, International School of Economics and Management, Capital University of Economics and Business, Beijing, China; e-mail: taining.wang@cueb.edu.cn.
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Abstract

We propose a nonparametric test of significant variables in the partial derivative of a regression mean function. The derivative is estimated by local polynomial estimation and the test statistic is constructed through a variation-based measure of the derivative in the direction of variables of interest. We establish the asymptotic null distribution of the test statistic and demonstrate that it is consistent. Motivated by the null distribution, we propose a wild bootstrap test, and show that it exhibits the same null distribution, whether the null is valid or not. We perform a Monte Carlo study to demonstrate its encouraging finite sample performance. An empirical application is conducted showing how the test can be applied to infer certain aspects of regression structures in a hedonic price model.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 5 , October 2021 , pp. 959 - 1003
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

We would like to express our sincere thanks to the Editor, Peter C. B. Phillips, the Co-Editor, Liangjun Su, and three anonymous referees for their constructive suggestions and comments that improved the paper substantially. We also thank Carlos Martins-Filho, the participants in the 2017 Asian Meeting of the Econometric Society and the 4th Econometric Workshop in Dongbei University of Finance and Economics (DUFE) for valuable comments. Any remaining errors are the authors’ responsibility.

References

REFERENCES

Aït-Sahalia, Y., Bickel, P. J., & Stoker, T. M. (2001) Goodness-of-fit tests for kernel regression with an application to option implied volatilities. Journal of Econometrics 105, 363–412.CrossRefGoogle Scholar
Altonji, J. G., Ichimura, H., & Otsu, T. (2012) Estimating derivatives in nonseparable models with limited dependent variables. Econometrica 80(4), 1701–1719.Google Scholar
Azzalini, A. & Bowman, A. W. (1993) On the use of nonparametric regression for checking linear relationships . Journal of the Royal Statistical Society Series B (Methodological) 55, 549–557.CrossRefGoogle Scholar
Banerjee, A. N. (2007) A method of estimating the average derivative. Journal of Econometrics 136(1), 65–88.CrossRefGoogle Scholar
Bierens, H. & Ploberger, W. (1997) Asymptotic theory of integrated conditional moment tests. Econometric Theory 65, 1129–1151.CrossRefGoogle Scholar
Bontemps, C., Simioni, M., & Surry, Y. (2008) Semiparametric hedonic price models: Assessing the effects of agricultural nonpoint source pollution. Journal of Applied Econometrics 23(6), 825–842.CrossRefGoogle Scholar
Calonico, S., Cattaneo, M. D., & Farrell, M. H. (2018) On the effect of bias estimation on coverage accuracy in nonparametric inference. Journal of the American Statistical Association 113(522), 767–779.CrossRefGoogle Scholar
Calonico, S., Cattaneo, M. D., & Farrell, M. H. (2019) Coverage Error Optimal Confidence Intervals for Local Polynomial Regression. Working Paper. https://arxiv.org/abs/1808.01398.Google Scholar
Calonico, S., Cattaneo, M. D., & Titiunik, R. (2014) Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica 82(6), 2295–2326.CrossRefGoogle Scholar
Cattaneo, M. D., Crump, R. K., & Jansson, M. (2014) Small bandwidth asymptotics for density-weighted average derivatives. Econometric Theory 30(1), 176–200.CrossRefGoogle Scholar
Cattaneo, M. D., Farrell, M. H., & Feng, Y. (2019a) Large sample properties of partitioning-based series estimators. Annals of Statistics 48, 1718.Google Scholar
Cattaneo, M. D. & Jansson, M. (2018) Kernel-based semiparametric estimators: Small bandwidth asymptotics and bootstrap consistency. Econometrica 86(3), 955–995.CrossRefGoogle Scholar
Cattaneo, M. D., Jansson, M., & Ma, X. (2019b) Two-step estimation and inference with possibly many included covariates. Review of Economic Studies 86(3), 1095–1122.CrossRefGoogle Scholar
Chaudhuri, P., Doksum, K., & Samarov, A. (1997) On average derivative quantile regression. Annals of Statistics 25, 715–744.CrossRefGoogle Scholar
Chen, R., Liang, H., & Wang, J. (2011) Determination of linear components in additive models. Journal of Nonparametric Statistics 23(2), 367–383.CrossRefGoogle Scholar
Chen, X. & Fan, Y. (1999) Consistent hypothesis testing in semiparametric and nonparametric models for econometric time series. Journal of Econometrics 91, 373–401.CrossRefGoogle Scholar
Coppejans, M. & Sieg, H. (2005) Kernel estimation of average derivatives and differences. Journal of Business and Economic Statistics 23(2), 211–225.CrossRefGoogle Scholar
Delgado, M. & Manteiga, W. G. (2001) Significance testing in nonparametric regression based on the bootstrap. Annals of Statistics 29, 1469–1507.CrossRefGoogle Scholar
Dette, H. (1999) A consistent test for the functional form of a regression based on a difference of variance estimators. Annals of Statistics 27, 1012–1040.CrossRefGoogle Scholar
Ekeland, I., Heckman, J. J., & Nesheim, L. (2004) Identification and estimation of hedonic models. Journal of Political Economy, 112(S1), S60–S109.CrossRefGoogle Scholar
Eubank, R. L. & Spiegelman, C. H. (1990) Testing the goodness of fit of a linear model via nonparametric regression techniques. Journal of the American Statistical Association 85, 387–392.CrossRefGoogle Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modeling and its Applications. Chapman & Hall.Google Scholar
Fan, J. & Huang, L. (2001) Goodness-of-Fit tests for parametric regression models. Journal of the American Satistical Association 96, 640–652.CrossRefGoogle Scholar
Fan, Y. & Guerre, E. (2016) Multivariate local polynomial estimators: Uniform boundary properties and asymptotic linear representation. In Essays in Honor of Aman Ullah, Emerald Group Publishing Limited 36, 489–537.Google Scholar
Fan, Y. & Li, Q. (1999) Central limit theorem for degenerate U-statistic of absolutely regular processes with applications to model specification tests. Journal of Nonparametric Statistics 10, 245–271.CrossRefGoogle Scholar
Fan, Y. & Li, Q. (2002) A consistent model specification test based on the kernel sum of squares of residuals. Econometric Reviews 21, 337–352.CrossRefGoogle Scholar
Fan, Y., Li, Q., & Min, I. (2006) A nonparametric bootstrap test of conditional distributions. Econometric Theory 22, 587–613.CrossRefGoogle Scholar
Fan, Y., Li, Q., & Weersink, A. (1996) Semiparametric estimation of stochastic production frontier models. Journal of Business and Economic Statistics 14, 460–468.Google Scholar
Gao, J., King, M., Lu, Z., & Tjøstheim, D. (2009) Nonparametric specification testing for nonlinear time series with nonstationarity. Econometric Theory 25(6), 1869–1892.CrossRefGoogle Scholar
Gozalo, P. L. (1993) A consistent model specification test for nonparametric estimation of regression function models. Econometric Theory 9, 451–477.CrossRefGoogle Scholar
Gozalo, P. L. & Linton, O. B. (2001) Testing additivity in generalized nonparametric regression models with estimated parameters. Journal of Econometrics 104(1), 1–48.CrossRefGoogle Scholar
Gu, J., Li, D., & Liu, D. (2007) Bootstrap non-parametric significance test. Journal of Nonparametric Statistics 19, 215–230.CrossRefGoogle Scholar
Hall, P. (1984) Central limit theorem for integrated square error of multivariate nonparametric density estimators. Journal of Multivariate Analysis 14, 1–16.CrossRefGoogle Scholar
Hall, P. & Wilson, S. R. (1991) Two guidelines for bootstrap hypothesis testing. Biometrics 47(2), 757–762.CrossRefGoogle Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Härdle, W. & Stoker, T. (1989) Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association 84(408), 986–995.Google Scholar
Hart, J. D. (1997) Nonparametric Smoothing and Lack-of-Fit Test. Springer.CrossRefGoogle Scholar
Hjellvik, V. & Tjøstheim, D. (1995) Nonparametric tests of linearity for time series. Biometrika 82, 351–368.CrossRefGoogle Scholar
Hjellvik, V., Yao, Q., & Tjøstheim, D. (1998) Linearity testing using local polynomial approximation. Journal of Statistical Planning and Inference 68, 295–321.CrossRefGoogle Scholar
Hong, Y. & Lee, Y. J. (2013) A loss function approach to model specification testing and its relative efficiency. Annals of Statistics 41, 1166–1203.CrossRefGoogle Scholar
Hong, Y. M. & White, H. L. (1995) Consistent specification testing via nonparametric series regression. Econometrica 63, 1133–1159.CrossRefGoogle Scholar
Horowitz, J. (1997) Bootstrap methods in econometrics: Theory and numerical performance. In Kreps, D. and Wallis, K. W. (eds.), Advances in Economics and Econometrics, 7th World Congress, vol. 3, pp. 188–222. Cambridge University Press.Google Scholar
Horowitz, J. L. & Spokoiny, V. G. (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69, 599–631.CrossRefGoogle Scholar
Hsiao, C., Li, Q., & Racine, J. (2007) A consistent model specification test with mixed discrete and continuous data. Journal of Econometrics 140, 802–826.CrossRefGoogle Scholar
Kaido, H. (2017) Asymptotically efficient estimation of weighted average derivatives with an interval censored variable. Econometric Theory 33 (5), 1218–1241.CrossRefGoogle Scholar
Korolev, I. (2018) Consistent LM Type Specification Tests for Semiparametric Models. Working Paper.CrossRefGoogle Scholar
Kress, J. P., Neumann, M. H., & Yao, Q. (2008) Bootstrap tests for simple structures in nonparametric time series regression. Statistics and Its Interface 1(2), 367–380.CrossRefGoogle Scholar
Lavergne, P. & Vuong, Q. (2000) Nonparametric significance testing. Econometric Theory 16(4), 576–601.CrossRefGoogle Scholar
Lewbel, A. (1995) Consistent nonparametric hypothesis tests with an application to Slutsky symmetry. Journal of Econometrics 67(2), 379–401.CrossRefGoogle Scholar
Li, H., Li, Q., & Liu, R. (2016) Consistent model specification tests based on k-nearest-neighbor estimation method. Journal of Econometrics 194(1), 187–202.CrossRefGoogle Scholar
Li, Q. (1999) Consistent model specification tests for time series econometric models. Journal of Econometrics 92, 101–147.CrossRefGoogle Scholar
Li, Q. (2000) Efficient estimation of additive partially linear models. International Economic Review, 41(4), 1073–1092.CrossRefGoogle Scholar
Li, Q., Hsiao, C., & Zinn, J. (2003a). Consistent specification tests for semiparametric/nonparametric models based on series estimation methods. Journal of Econometrics 112, 295–325.CrossRefGoogle Scholar
Li, Q., Lu, X., & Ullah, A. (2003b) Multivariate local polynomial regression for estimating average derivatives. Journal of Nonparametric Statistics 15(4–5), 607–624.CrossRefGoogle Scholar
Li, Q. & Racine, J. (2007) Nonparametric Econometrics: Theory and Practice. Princeton University Press.Google Scholar
Li, Q. & Wang, S. (1998) A simple consistent bootstrap test for a parametric regression function. Journal of Econometrics, 145–165.CrossRefGoogle Scholar
Lin, Z., Li, Q., & Sun, Y. (2014) A consistent nonparametric test of parametric regression functional form in fixed effects panel data models. Journal of Econometrics 178, 167–179.CrossRefGoogle Scholar
Linton, O. B. & Nielsen, J. P. (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82, 93–100.CrossRefGoogle Scholar
Lu, Z. Q. (1996) Multivariate locally weighted polynomial fitting and partial derivative estimation. Journal of Multivariate Analysis 59 (2), 187–205.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17, 571–599.CrossRefGoogle Scholar
Powell, J. L., Stock, J. H., Stoker, T. M. (1989) Semiparametric estimation of index coefficients. Econometrica 57, 1403–1430.CrossRefGoogle Scholar
Racine, J. (1997) Consistent significance testing for nonparametric regression. Journal of Business and Economic Statistics 15, 369–378.Google Scholar
Robinson, P. M. (1988) Root-N-consistent semiparametric regression. Econometrica 56, 931–954.CrossRefGoogle Scholar
Rosen, S. (1974) Hedonic prices and implicit markets: Product differentiation in pure competition. Journal of Political Economy 82(1), 34–55.CrossRefGoogle Scholar
Ruppert, D. & Wand, M. P. (1994) Multivariate locally weighted least squares regression. Annals of Statistics 22, 1346–1370.CrossRefGoogle Scholar
Sperlich, S., Tjøstheim, D., & Yang, L. (2002) Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18, 197–251.CrossRefGoogle Scholar
Staniswalis, J. G., & Severini, T. A. (1991) Diagnostics for assessing regression models. Journal of the American Satistical Association 86, 684–692.CrossRefGoogle Scholar
Stengos, T. & Sun, Y. (2001) A consistent model specification test for a regression function based on nonparametric wavelet estimation. Econometric Reviews 20(1), 41–60.CrossRefGoogle Scholar
Su, L., & Lu, X. (2013) Nonparametric dynamic panel data models: Kernel estimation and specification testing. Journal of Econometrics 176(2), 112–133.CrossRefGoogle Scholar
Su, L. & Ullah, A. (2008) Local polynomial estimation of nonparametric simultaneous equations models. Journal of Econometrics 144, 193–218.CrossRefGoogle Scholar
Su, L. & Ullah, A. (2013) A nonparametric goodness-of-fit-based test for conditional heteroskedasticity. Econometric Theory 29(1), 187–212.CrossRefGoogle Scholar
Ullah, A. (1985) Specification analysis of econometric models. Journal of Quantitative Economics 1, 187–209.Google Scholar
Wang, J. & Yang, L. (2009) Efficient and fast spline-backfitted kernel smoothing of additive models. Annals of the Institute of Statistical Mathematics 61(3), 663–690.CrossRefGoogle Scholar
Wang, X. & Carriere, K. C. (2011) Assessing additivity in nonparametric models – a kernel-based method. Canadian Journal of Statistics 39(4), 632–655.CrossRefGoogle Scholar
Whang, Y. J. (2000) Consistent bootstrap tests of parametric regression functions. Journal of Econometrics 98(1), 27–46.CrossRefGoogle Scholar
Wooldridge, J. M. (1992) A test for functional form against nonparametric alternatives. Econometric Theory 8, 452–475.CrossRefGoogle Scholar
Yao, F. & Martins-Filho, C. (2015) An asymptotic characterization of finite degree U-statistics with sample size-dependent kernels: Applications to nonparametric estimators and test statistics. Communications in Statistics–Theory and Methods 44(15), 3251–3265.CrossRefGoogle Scholar
Yao, F. & Ullah, A. (2013) A nonparametric R-square test for the presence of relevant variables. Journal of Statistical Planning and Inference 143(9), 1527–1547.CrossRefGoogle Scholar
Yao, F., Zhang, F., & Kumbhakar, S. C. (2019) Semiparametric smooth coefficient stochastic frontier model with panel data. Journal of Business and Economic Statistics 37(3), 556–572.CrossRefGoogle Scholar
Yatchew, A. (1992) Nonparametric regression tests based on least squares. Econometric Theory 8(4), 435–451.CrossRefGoogle Scholar
Zheng, J. X. (1996) A consistent test of functional form via nonparametric estimation techniques. Journal of Econometrics 75, 263–289.CrossRefGoogle Scholar
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