Bootstrap Tests for the Error Distribution in Linear and Nonparametric Regression Models

Abstract

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and nonparametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution-free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness-of-fit testing of assumptions regarding the error distribution in linear and nonparametric regression models.

Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density

“…However, the estimation of error density is important to understand the residual behavior and to assess the adequacy of error distribution assumption (see for example, Akritas and Van Keilegom, 2001;Cheng and Sun, 2008); the estimation of error density is also useful to test the symmetry of the residual distribution (see for example, Ahmad and Li, 1997;Dette et al, 2002;Neumeyer and Dette, 2007); the estimation of error density is important to statistical inference, prediction and model validation (see for example, Efromovich, 2005;Muhsal and Neumeyer, 2010); and the estimation of error density is also useful for the estimation of the density of the response variable (see for example, Escanciano and Jacho-Chávez, 2012). In the realm of financial asset return, an important use of the estimated error density is to estimate value-at-risk for holding an asset.…”

Comments on: An updated review of Goodness-of-Fit tests for regression models

“…For instance, Fan and Gencay (1995) and Dette et al (2002) use kernel density estimation to construct their test statistic. In the same spirit, classical Kolmogorov-Smirnov and Cramér-von Mises statistics for the null hypothesis H 0S have been proposed by Neumeyer et al (2005) and Neumeyer and Dette (2007). On the other hand Hušková and Meintanis (2012) use the equivalent formulation of (1.2) as H 0S : (ϕ ε ) ≡ 0, where (·) denotes the imaginary part in complex arithmetic and ϕ · is a generic symbol for the CF of an arbitrary distribution.…”

Nonparametric tests for conditional symmetry in dynamic models

“…The wild bootstrap (WB) introduced in Wu (1986) and Liu (1988) appears to be relevant for respecting the underlaying relation between innovations and explanatory variables. We adapt the WB approach to test for conditional symmetry with time series data, extending the method proposed by Neumeyer and Dette (2003) for the iid linear regression case. Other proposal, only valid for linear processes, is that of Psaradakis (2003) who considered a sieve bootstrap procedure for testing unconditional symmetry based on residuals resampled from an autoregressive approximation of the given process.…”

“…Smirnov (1947) first proposed an omnibus test for the simple hypothesis of symmetry around a known value based on the standard empirical process. See also the related works by Butler (1969), Nadaraya (1975), Rao (1977, 1981), Aki (1981), Antille et al (1982), Bhattacharya et al (1982) or more recently Neumeyer and Dette (2003).…”