Specification Testing for Errors-in-Variables Models

Abstract

This paper considers specification testing for regression models with errors-in-variables and proposes a test statistic comparing the distance between the parametric and nonparametric fits based on deconvolution techniques. In contrast to the methods proposed by Hall and Ma (2007, Annals of Statistics, 35, 2620–2638) and Song (2008, Journal of Multivariate Analysis, 99, 2406–2443), our test allows general nonlinear regression models and possesses complementary local power properties. We establish the asymptotic properties of our test statistic for the ordinary and supersmooth measurement error densities. Simulation results endorse our theoretical findings: our test has advantages in detecting high-frequency alternatives and dominates the existing tests under certain specifications.

On the Uniform Convergence of Deconvolution Estimators From Repeated Measurements

“…In this case, our faster convergence rates are useful to allow a larger size of the subsample, which yields better power properties. (ii) Otsu and Taylor (2019) proposed a specification test for errors-in-variables models, where the measurement errors are required to be symmetrically distributed. To extend their approach to allow possibly asymmetric distributions on the measurement errors, one may plug in the LV-type estimators for the characteristic functions of the measurement errors to their test statistic.…”

“…To extend their approach to allow possibly asymmetric distributions on the measurement errors, one may plug in the LV-type estimators for the characteristic functions of the measurement errors to their test statistic. In this case, if we wish to guarantee that the estimation errors for the plugin LV-type estimators are dominated by the main term considered in Otsu and Taylor (2019), our faster convergence rates are useful to establish such asymptotic negligibility under weaker conditions. Even if the convergence rates of Otsu and Taylor's (2019) statistic are not sufficiently fast, we can adapt a subsample-based modification as in Adusumilli et al (2020) to Otsu and Taylor's (2019) statistic so that our main theorems can be applied in an analogous way.…”

“…The semiparametric estimators by Li (2002) and Li and Hsiao (2004) are constructed as functionals of the LV estimator. Also, other nonparametric measurement error problems often call for estimation of the characteristic function of the measurement error, such as Adusumilli and Otsu (2018) for nonparametric instrumental regression with errors in variables, Otsu and Taylor (2019) for specification testing on errors-invariables regressions, and Adusumilli et al (2020) for inference on distribution functions under measurement errors. For those purposes, the LV-type estimators play the same roles as primitive nonparametric estimators for semiparametric problems.…”

Measurement systems

“…It should be noted that it is of course possible for a dataset to exhibit both measurement error and endogeneity at the same time. In general, in a nonlinear model, a single instrument is unfortunately not sufficient to handle both problems simultaneously and dedicated methods that address each problem individually have to be used (Otsu and Taylor (2016), Song, Schennach, and White (2015), Schennach, White, and Chalak (2012)).…”

Average Derivative Estimation Under Measurement Error

“…Extending these results to supersmooth measurement error is not a trivial extension, and it is not clear a priori whether a √ n-rate can be achieved in this case. Indeed, in many estimation and testing problems, convergence rates and asymptotic distributions are fundamentally different between ordinary smooth and supersmooth error densities (see, for example, Fan, 1991, van Es and Uh, 2005, Dong and Otsu, 2018, and Otsu and Taylor, 2020. Furthermore, no result has been provided regarding the asymptotic properties of average derivative estimators in the more realistic situation where the measurement error density is unknown.…”