Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T18:14:26.970Z Has data issue: false hasContentIssue false
  • English
  • Français

SPECIFICATION TESTING FOR ERRORS-IN-VARIABLES MODELS

Published online by Cambridge University Press:  19 June 2020

Taisuke Otsu  and
Luke Taylor
Taisuke Otsu*
Affiliation:
London School of Economics
Luke Taylor
Affiliation:
Aarhus University
*
Address correspondence to Taisuke Otsu, Department of Economics, London School of Economics, Houghton Street, London WC2A 2AE, UK; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

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.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 4 , August 2021 , pp. 747 - 768
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The authors would like to thank anonymous referees and a co-editor for helpful comments, and acknowledge financial support from the ERC Consolidator Grant (SNP 615882) (T.O.) and the ESRC (L.T.).

References

REFERENCES

Bickel, P.J. & Ritov, A.J.C. (1987) Efficient estimation in the errors-in-variables model. Annals of Statistics 15, 513–540.CrossRefGoogle Scholar
Butucea, C. (2007) Goodness-of-fit testing an quadratic functional estimation from indirect observations. Annals of Statistics 35, 1907–1930.CrossRefGoogle Scholar
Butucea, C. & Taupin, M.-L. (2008) New M-estimators in semi-parametric regression with errors in variables. Annales de l’Institut Henri Poincarrobabilitt Statistiques 44, 393–421.Google Scholar
Carrasco, M. & Florens, J.-P. (2011) A spectral method for deconvolving a density. Econometric Theory 27, 546–581.CrossRefGoogle Scholar
Cheng, C.-L. & Kukush, A.G. (2004) A goodness-of-fit test for a polynomial errors-in-variables model. Ukrainian Mathematical Journal 56(4), 641–661.CrossRefGoogle Scholar
Cheng, C.-L. & Schneeweiss, H. (1998) Polynomial regression with errors in the variables. Journal of the Royal Statistical Society B 60, 189–199.CrossRefGoogle Scholar
Delaigle, A. & Gijbels, I. (2004) Practical bandwidth selection in deconvolution kernel density estimation. Computational Statistics & Data Analysis 45, 249–267.CrossRefGoogle Scholar
Delaigle, A., Hall, P., & Jamshidi, F. (2015) Confidence bands in non-parametric errors-in-variables regression. Journal of the Royal Statistical Society B 77, 149–169.CrossRefGoogle Scholar
Delaigle, A., Hall, P., & Meister, A. (2008) On deconvolution with repeated measurements. Annals of Statistics 36, 665–685.CrossRefGoogle Scholar
Fan, J. (1992) Deconvolution with supersmooth distributions. Canadian Journal of Statistics 20, 155–169.CrossRefGoogle Scholar
Fan, J. & Truong, Y.K. (1993) Nonparametric regression with errors in variables. Annals of Statistics 4, 1900–1925.Google Scholar
Fan, Y. & Li, Q. (2000) Consistent model specification tests: Kernel-based versus Bierens’ ICM tests. Econometric Theory 16, 1016–1041.CrossRefGoogle Scholar
Gleser, L.J. (1981) Estimation in a multivariate “errors in variables” regression model: large sample results. Annals of Statistics 9, 24–44.CrossRefGoogle Scholar
Hall, P. (1984) Central limit theorem for integrated square error of multivariate nonparametric density estimators. Journal of Multivariate Analysis 14, 1–16.CrossRefGoogle Scholar
Hall, P. & Ma, Y. (2007) Testing the suitability of polynomial models in error-in-variables problems. Annals of Statistics 35, 2620–2638.CrossRefGoogle Scholar
Hall, P. & Meister, A. (2007) A ridge-parameter approach to deconvolution. Annals of Statistics 35, 1535–1558.CrossRefGoogle Scholar
Hardle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Holzmann, H., Bissantz, N., & Munk, A. (2007) Density testing in a contaminated sample. Journal of Multivariate Analysis 98, 57–75.CrossRefGoogle Scholar
Holzmann, H. & Boysen, L. (2006) Integrated square error asymptotics for supersmooth deconvolution. Scandinavian Journal of Statistics 33, 849–860.CrossRefGoogle Scholar
Koul, H.L. & Song, W.-X. (2009) Minimum distance regression model checking with Berkson measurement errors. Annals of Statistics 37, 132–156.CrossRefGoogle Scholar
Koul, H.L. & Song, W.-X. (2010) Model checking in partial linear regression models with Berkson measurement errors. Statistica Sinica 20, 1551–1579.Google Scholar
Li, T. & Vuong, Q. (1998) Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis 65, 139–165.CrossRefGoogle Scholar
Ma, Y., Hart, J.D., Janicki, R., & Carroll, R.J. (2011) Local and omnibus goodness-of-fit tests in classical measurement error models. Journal of the Royal Statistical Society B 73, 81–98.CrossRefGoogle ScholarPubMed
Masry, E. (1993) Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stochastic Processes and their Applications 47(1), 53–74.CrossRefGoogle Scholar
Meister, A. (2009) Deconvolution Problems in Nonparametric Statistics. Springer.CrossRefGoogle Scholar
Rosenblatt, M. (1975) A quadratic measure of deviation of two-dimensional density estimates and a test of independence . Annals of Statistics 3, 1–14.CrossRefGoogle Scholar
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Song, W.-X. (2008) Model checking in errors-in-variables regression. Journal of Multivariate Analysis 99, 2406–2443.CrossRefGoogle Scholar
Song, W.-X. (2009) Lack-of-fit testing in errors-in-variables regression model with validation data. Statistics & Probability Letters 79, 765–773.CrossRefGoogle Scholar
Taupin, M.-L. (2001) Semi-parametric estimation in the nonlinear structural errors-in-variables model. Annals of Statistics 29, 66–93.CrossRefGoogle Scholar
van der Vaart, A.W. (1988) Estimating a real parameter in a class of semiparametric models. Annals of Statistics 16, 1450–1474.CrossRefGoogle Scholar
van Es, A. J. & Uh, H.-W. (2005) Asymptotic normality for kernel type deconvolution estimators. Scandinavian Journal of Statistics 32, 467–483.Google Scholar
Xu, W.-L. & Zhu, L.-X. (2015) Nonparametric check for partial linear errors-in-covariables models with validation data. Annals of Institute of Statistical Mathematics 67, 793–815.CrossRefGoogle Scholar
Zhu, L.-X. & Cui, H.-J. (2005) Testing the adequacy for a general linear errors-in-variables model. Statistica Sinica 15, 1049–1068.Google Scholar
Zhu, L.-X., Song, W.-X., & Cui, H.-J. (2003) Testing lack-of-fit for a polynomial errors-in-variables model. Acta Mathematicae Applicatae Sinica 19, 353–362.CrossRefGoogle Scholar