One for all and all for one: regression checks with many regressors

Lavergne, Pascal and Patilea, Valentin (2011): One for all and all for one: regression checks with many regressors.

Abstract

We develop a novel approach to build checks of parametric regression models when many regressors are present, based on a class of sufficiently rich semiparametric alternatives, namely single-index models. We propose an omnibus test based on the kernel method that performs against a sequence of directional nonparametric alternatives as if there was one regressor only, whatever the number of regressors. This test can be viewed as a smooth version of the integrated conditional moment (ICM) test of Bierens. Qualitative information can be easily incorporated into the procedure to enhance power. In an extensive comparative simulation study, we find that our test is little sensitive to the smoothing parameter and performs well in multidimensional settings. We then apply it to a cross-country growth regression model.

• Item Type: MPRA Paper
• Language: English
• Keywords: Dimensionality, Hypothesis testing, Nonparametric methods
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General

References

Aerts, M., G. Claeskens and J.D. Hart (1999). Testing the fit of a parametric function. J. Amer. Statist. Assoc. 94 (447), 869-879.

Aerts, M., G. Claeskens and J.D. Hart (2000). Testing lack of fit in multiple regression. Biometrika 87 (2), 405-4242.

Azzalini, A., A.W. Bowman and W. H�ardle (1989). On the use of nonparametric regression for model checking. Biometrika 76 (1), 1-11.

Baraud, Y., S. Huet and B. Laurent (2003). Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31 (1), 225-251.

Barro, R.J. (1991). Economic growth in a cross-section of countries. Quart. J. Econ. 106 (2), 407-443.

Bierens, H.J. (1982). Consistent model specification tests. J. Econometrics 20, 105-134.

Bierens, H.J. (1990). A consistent conditional moment test of functional form. Econometrica 58 (6), 1443-1458.

Bierens, H.J., and W. Ploberger (1997). Asymptotic theory of integrated conditional moment tests. Econometrica 65 (5), 1129-1151.

Cook, R.D. (1993). Exploring partial residual plots. Technometrics 35 (4), 351-362.

Cox, D., E. Koh, G. Wahba and B.S. Yandell (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Statist. 16 (1), 113-119.

Dette, H. (1999). A consistent test for the functional form of a regression based on a difference of variance estimators. Ann. Statist. 27 (3), 1012-1040.

Delgado, M.A., M.A. Dominguez, and P. Lavergne (2006). Consistent tests of conditional moment restrictions. Ann. Econom. Statist. 81 (1), 33-67.

Dominguez, M.A. (2004). On the power of boootstrapped specification tests. Econometric Rev. 23 (3), 215-228.

Donald, S.G., G.W. Imbens, and W.K. Newey (2003). Empirical likelihood estimation and consistent tests with conditional moment restrictions. J. Econometrics 117 (1), 55-93.

Eubank, R.L. and C.H. Spiegelman (1990). Testing the goodness of fit of a linear model via nonparametric regression techniques. J. Amer. Statist. Assoc. 85 (410), 387-392.

Eubank, R.L. and J.D. Hart (1993). Commonality of cusum, von Neumann and smoothing- based goodness-of-t tests. Biometrika 80 (1), 89-98.

Escanciano, J.C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory 22 (6) 1030-1051.

Escanciano, J.C. (2006). Goodness-of-fit tests for linear and nonlinear rime series models. J. Amer. Statist. Assoc. 101 (474) 531-541. 23

Fan, J. and L.S. Huang (2001). Goodness-of-fit tests for parametric regression m odels. J. Amer. Statist. Assoc. 96 (454), 640-652.

Fan, J., C. Zhang and J. Zhang (2001). Generalized lihelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 (1) ,153-193.

Godfrey, L.G., and C.D. Orme (2004). Controlling the finite sample significance levels of heteroskedasticity-robust tests of several linear restrictions on regression coefficients. Economics Letters 82 (2), 282-287.

Gozalo, P.L. (1997). Nonparametric bootstrap analysis with applications to demographic effects in demand functions. J. Econometrics 81 (2), 357-393.

Guerre, E., and P. Lavergne (2002). Optimal minimax rates for nonparametric specification testing in regression models. Econometric Theory 18 (5), 1139-1171.

Guerre, E., and P. Lavergne (2005). Data-driven rate-optimal specification testing in regression models. Ann. Statist. 33 (2), 840-870.

Hardle, W., and E. Mammen (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 (4), 1296-1947.

Hart, J.D., and T.E. Wehrly (1992). Kernel regression when the boundary region is large, with an application to testing the adequacy of polynomial models. J. Amer. Statist. Assoc. 87 (420), 1018-1024.

Hart, J.D. (1997). Nonparametric smoothing and lack-of-fit tests. Springer-Verlag, New- York.

Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (301), 13-30.

King, R.G., and R. Levine (1993). Finance and growth: Schumpeter might be right. Quart. J. Econ. 108 (3), 717-737.

Lavergne, Q., and V. Patilea (2008). Breaking the curse of dimensionality in nonparametric testing. J. Econometrics 143 (1), 103-122.

Lewbel, A. (1995). Consistent nonparametric hypothesis tests with an application to Slutsky symmetry. J. Econometrics 67 (1), 379-401.

Li, Q., and S. Wang (1998). A simple consistent bootstrap test for a parametric regression function. J. Econometrics 87 (1), 145-165. 24

Liu, Z., and T. Stengos (1999). Non-linearities in cross-country growth regressions: a semi- parametric approach. J. Appl. Econometrics 14 (5), 527-538.

Mankiw, N.G., D. Romer, and D.N. Weil (1992). A contribution to the empirics of economic growth. Quart. J. Econ. 107 (2), 407-437.

Miles, D., and Mora, J. (2003). On the performance of nonparametric specification tests in regression models. Comput. Statist. Data Anal. 42 (3), 477-490.

Ramsey, J.B. (1969). Tests for specication errors in classical linear least-squares regression analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 31 (2), 511-534.

Rudin, W. (1987). Real and complex analysis. McGraw-Hill.

Spokoiny, V. (2001). Data-driven testing the fit of linear models. Math. Methods Statist. 10 (4), 465-497.

Stone, C.J. (1980). Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 (6), 1348-1360.

Stute, W., W. Gonzalez Manteiga and M. Presedo Quindimil (1998). Bootstrap approximations in model checks for regression. J. Amer. Statist. Assoc. 93 (441), 141-149.

Wu, C.F.J. (1986). Jacknife, bootstrap and other resampling methods in regression analysis (with discussion). Ann. Statist. 14 (4), 1261-1350.

Zheng, J.X. (1996). A consistent test of functional form via nonparametric estimation tech- niques. J. Econometrics 75 (2), 263-289.

Zhu, L.X., and R. Li (1998). Dimension-reduction type test for linearity of a stochastic model. Acta Math. Appli. Sinica 14 (2), 165-175.

Zhu, L.X. (2003). Model checking of dimension-reduction type for regression. Statist. Sinica 13 (2), 283-296.