Tests for validity of the semiparametric heteroskedastic transformation model

Abstract There exist a number of tests for assessing the nonparametric heteroskedastic location-scale assumption. The goodness-of-fit tests considered are for the more general hypothesis of the validity of this model under a parametric functional transformation on the response variable, specifically testing for independence between the regressors and the errors in a model where the transformed response is just a location/scale shift of the error is considered. The proposed criteria use the familiar factorization property of the joint characteristic function under independence. The difficulty is that the errors are unobserved and hence one needs to employ properly estimated residuals in their place. The limit distribution of the test statistics is studied under the null hypothesis as well as under alternatives, and also a resampling procedure is suggested in order to approximate the critical values of the tests. This resampling is subsequently employed in a series of Monte Carlo experiments that illustrate the finite-sample properties of the new test. The performance of related test statistics for normality and symmetry of errors is also investigated, and application of our methods on real data sets is provided.

[1]  Ingrid Van Keilegom,et al.  Bootstrap of residual processes in regression: to smooth or not to smooth? , 2017, Biometrika.

[2]  I. Keilegom,et al.  Specification tests in nonparametric regression , 2008 .

[3]  Shakeeb Khan,et al.  Nonparametric Identification and Estimation of a Censored Location-Scale Regression Model , 2005 .

[4]  Maria L. Rizzo,et al.  Measuring and testing dependence by correlation of distances , 2007, 0803.4101.

[5]  Marie Husková,et al.  Tests for Symmetric Error Distribution in Linear and Nonparametric Regression Models , 2012, Commun. Stat. Simul. Comput..

[6]  Ingrid Van Keilegom,et al.  Goodness-of-fit tests in semiparametric transformation models using the integrated regression function , 2017, J. Multivar. Anal..

[7]  Wenceslao González Manteiga,et al.  Significance testing in nonparametric regression based on the bootstrap , 2001 .

[8]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[9]  Testing the adequacy of semiparametric transformation models , 2018 .

[10]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[11]  Jeffrey S. Racine,et al.  A Smooth Nonparametric, Multivariate, Mixed-Data Location-Scale Test , 2017, Journal of Business & Economic Statistics.

[12]  On a data based power transformation for reducing skewness , 1993 .

[13]  Wenceslao González-Manteiga,et al.  An updated review of Goodness-of-Fit tests for regression models , 2013, TEST.

[14]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[15]  Lixing Zhu,et al.  Nonparametric Monte Carlo tests for multivariate distributions , 2000 .

[16]  Feifei Chen,et al.  On some characterizations and multidimensional criteria for testing homogeneity, symmetry and independence , 2019, J. Multivar. Anal..

[17]  Raffaella Giacomini,et al.  A WARP-SPEED METHOD FOR CONDUCTING MONTE CARLO EXPERIMENTS INVOLVING BOOTSTRAP ESTIMATORS , 2013, Econometric Theory.

[18]  A. Quiroz,et al.  Estimation of a multivariate Box-Cox transformation to elliptical symmetry via the empirical characteristic function , 1996 .

[19]  R. Sakia.  The Box-Cox transformation technique: a review , 1992 .

[20]  W. Stute,et al.  Nonparametric checks for single-index models , 2005, math/0507416.

[21]  Richard A. Johnson,et al.  A new family of power transformations to improve normality or symmetry , 2000 .

[22]  John P. Nolan,et al.  Multivariate elliptically contoured stable distributions: theory and estimation , 2013, Computational Statistics.

[23]  Marie Hušková,et al.  Tests for the error distribution in nonparametric possibly heteroscedastic regression models , 2010 .

[24]  Michael A. Stephens,et al.  Box‐Cox transformations in linear models: Large sample theory and tests of normality , 2002 .

[25]  J. Friedman,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation. , 1985 .

[26]  A. Azzalini.  The Skew‐normal Distribution and Related Multivariate Families * , 2005 .

[27]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[28]  Jeffrey S. Racine,et al.  Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach , 2017 .

[29]  Heteroscedastic semiparametric transformation models: estimation and testing for validity , 2014, 1411.0393.

[30]  Jeffrey S. Racine,et al.  Nonparametric Econometrics: The np Package , 2008 .

[31]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[32]  J. Horowitz.  Semiparametric and Nonparametric Methods in Econometrics , 2007 .

[33]  Simos G. Meintanis,et al.  Invariant tests for symmetry about an unspecified point based on the empirical characteristic function , 2003 .

[34]  Simos G. Meintanis,et al.  Transformations to symmetry based on the probability weighted characteristic function , 2015, Kybernetika.

[35]  Natalie Neumeyer,et al.  Smooth Residual Bootstrap for Empirical Processes of Non‐parametric Regression Residuals , 2009 .

[36]  I. Keilegom,et al.  Estimation of a semiparametric transformation model , 2008, 0804.0719.

[37]  I. Keilegom,et al.  Goodness-of-fit tests in semiparametric transformation models , 2016 .

[38]  M. C. Jones,et al.  A reliable data-based bandwidth selection method for kernel density estimation , 1991 .

[39]  Richard A. Johnson,et al.  An Empirical Characteristic Function Approach to Selecting a Transformation to Normality , 2014 .

[40]  M. Wand.  Functions for Kernel Smoothing Supporting Wand & Jones (1995) , 2015 .

[41]  Ingrid Keilegom,et al.  Goodness-of-fit tests in parametric regression based on the estimation of the error distribution , 2008 .

[42]  M. Wand,et al.  An Effective Bandwidth Selector for Local Least Squares Regression , 1995 .

[43]  Krzysztof Podgórski,et al.  Multivariate generalized Laplace distribution and related random fields , 2013, J. Multivar. Anal..

[44]  Lawrence D. Brown,et al.  Variance estimation in nonparametric regression via the difference sequence method , 2007, 0712.0898.

[45]  Konstantinos Fokianos,et al.  An Updated Literature Review of Distance Correlation and Its Applications to Time Series , 2017, International Statistical Review.

[46]  Simos G. Meintanis,et al.  A Class of Omnibus Tests for the Laplace Distribution based on the Empirical Characteristic Function , 2005 .

[47]  I. Keilegom,et al.  Tests for Independence in Nonparametric Regression , 2006 .

[48]  Xuming He,et al.  Power Transformation Toward a Linear Regression Quantile , 2007 .

[49]  I. Van Keilegom,et al.  Estimation of the error density in a semiparametric transformation model , 2011, 1110.1846.