A Nonparametric Goodness-of-Fit-Based Test for Conditional Heteroskedasticity

Abstract

In this paper we propose a nonparametric test for conditional heteroskedasticity based on a new measure of nonparametric goodness-of-fit (R 2 ). In analogy with the ANOVA tools for classical linear regression models, the nonparametric R 2 is obtained for the local polynomial regression of the residuals from a parametric regression on some covariates. It is close to 0 under the null hypothesis of conditional homoskedasticity and stays away from 0 otherwise. Unlike most popular parametric tests in the literature, the new test does not require the correct specification of parametric conditional heteroskedasticity form and thus is able to detect all kinds of conditional heteroskedasticity of unknown form. We show that after being appropriately centered and standardized, the nonparametric R 2 is asymptotically normally distributed under the null hypothesis of conditional homoskedasticity and a sequence of Pitman local alternatives.We also prove the consistency of the test, propose a bootstrap method to obtain the critical values or bootstrap p-values, and justify the validity of the bootstrap method. We conduct a small set of Monte Carlo simulations and compare our test with some popular parametric and nonparametric tests in the literature. Applications to the U.S. real GDP growth rate data indicate that our nonparametric test can reveal certain conditional heteroskedasticity which the parametric tests fail to detect.

Local Linear GMM Estimation of Functional Coefficient IV Models With an Application to Estimating the Rate of Return to Schooling

“…So we suggest using a bootstrap method to obtain the bootstrap approximation to the finite-sample distribution of our test statistic under the null. We find that it is easy to adopt the fixed-design wild bootstrap method in the spirit of Hansen (2000) in our framework; see also Su and White (2010) and Su and Ullah (2012). The great advantage of this method lies in the fact that we do not need to mimic some important features (such as dependence or endogeneity structure) in the data generating process and can still justify its asymptotic validity.…”

“…To show that the bootstrap statistic J * n can be used to approximate the asymptotic null distribution of J n , we follow Li, Hsiao and Zinn (2003) and Su and Ullah (2012) and rely on the notion of convergence in distribution in probability, which generalizes the usual convergence in distribution to allow for conditional (random) distribution functions. The following theorem establishes the asymptotic validity of the above bootstrap procedure.…”

Specification test for panel data models with interactive fixed effects

“…As a result, the projection of the parametric residual to the regressor space is expected to be zero under the null and nonzero under the alternative. This motivates our residual-based test, like many other residual-based tests in the literature (e.g., Fan and Li (1996), Zheng (1996), Hsiao et al (2007), and Su and Ullah (2013)). We show that after being appropriately centered and standardized, our test statistic is asymptotically normally distributed under the null hypothesis and a sequence of Pitman local alternatives.…”

“…The RESET test of Ramsey (1969) is the common used specification test for the linear regression model but it is not consistent. Since Hausman (1978) a large literature on testing for the correct specification of functional forms has developed; see Bierens (1982Bierens ( , 1990, Wooldridge (1992), Yatchew (1992), Härdle and Mammen (1993), Hong and White (1995), Fan and Li (1996), Zheng (1996), Li and Wang (1998), Stinchcombe and White (1998), Chen and Gao (2007), Hsiao et al (2007), and Su and Ullah (2013), to name just a few. In addition, Hjellvik and Tjøstheim (1995) and Hjellvik et al (1998) derive tests for linearity specification in nonparametric regressions and Hansen (1999) reviews the problem of testing for linearity in the context of self-exciting threshold autoregressive (SETAR) models.…”

Parametric and Nonparametric Frequentist Model Selection and Model Averaging

“…R 2 2 is introduced in [89,90]. For the variables selection it may be more appropriate to consider an adjusted R 2 1 as…”