Model checking in errors-in-variables regression

Abstract

This paper discusses a class of minimum distance tests for fitting a parametric regression model to a class of regression functions in the errors-in-variables model. These tests are based on certain minimized distances between a nonparametric regression function estimator and a deconvolution kernel estimator of the conditional expectation of the parametric model being fitted. The paper establishes the asymptotic normality of the proposed test statistics under the null hypothesis and that of the corresponding minimum distance estimators. We also prove the consistency of the proposed tests against a fixed alternative and obtain the asymptotic distributions for general local alternatives. Simulation studies show that the testing procedures are quite satisfactory in the preservation of the finite sample level and in terms of a power comparison.

Specification Testing for Errors-in-Variables Models

“…However, as is the case without measurement error, these tests are generally inconsistent for some fixed alternatives. Song (2008) proposed a consistent specification test for linear errors-in-variables regression models by comparing nonparametric and model-based estimators on the conditional mean function of the dependent variable Y given the mismeasured observable covariates W , that is E[Y |W ]. As we clarify at the end of the next section, this approach may not have sensible local power for the original hypothesis on E[Y |X], where X is a vector of error-free unobservable covariates.…”

“…However, the test does not allow the number of basis functions to increase with the sample size and so is not strictly a nonparametric test. They also discuss that as a result of the way the test is constructed, it has low power against high-frequency alternatives -similar to the tests of Song (2008) and Hall and Ma (2007).…”

An Adaptive-to-Model Test for Parametric Single-Index Errors-in-Variables Models

“…Zhu & Cui (2005) proposed a test for a general linear model β T 0 h(x), where h is a vector of known functions. Song (2008) proposed a test for β T 0 h(x) based on a deconvolution kernel density estimator. Koul & Song (2009) developed an analog of the minimum distance tests of Koul & Ni (2004) to fit a parametric form to the regression function for the Berkson measurement error models.…”

“…Koul & Song (2010) developed tests for fitting a parametric function to the nonparametric part in a partial linear regression Berkson measurement error model. All of these authors, except Hall & Ma (2007) and Song (2008), employ the calibration methodology and test for fitting the parameter form of the regression function E[Y |W ] implied by H 0 .…”

Consistent test of error-in-variables partially linear model with auxiliary variables

“…We can refer such methods to Fan and Li [6], Song [15], Gao and Gijbels [7], , and so on. Comparatively, the proposed empiricalprocess-based method has the following merits: (i) The proposed test is consistent; (ii) The effect of the proposed test depends slightly on the choice of the smoothing parameters; (iii) It can detect the local alternative models close to the null hypothetical model at the rate n −1/2 .…”