Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T18:06:02.408Z Has data issue: false hasContentIssue false
  • English
  • Français

MEASUREMENT ERROR AND DECONVOLUTION IN SPACES OF GENERALIZED FUNCTIONS

Published online by Cambridge University Press:  16 April 2014

Victoria Zinde-Walsh
Victoria Zinde-Walsh*
Affiliation:
McGill University and CIREQ
*
*Address correspondence to Victoria Zinde-Walsh, Department of Economics, McGill University, 855 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2T7; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers convolution equations that arise from problems such as measurement error and nonparametric regression with errors in variables with independence conditions. The equations are examined in spaces of generalized functions to account for possible singularities; this makes it possible to consider densities for arbitrary and not only absolutely continuous distributions, and to operate with Fourier transforms for polynomially growing regression functions. Results are derived for identification and well-posedness in the topology of generalized functions for the deconvolution problem and for some regression models. Conditions for consistency of plug-in estimation for these models are provided.

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 6 , December 2014 , pp. 1207 - 1246
Copyright
Copyright © Cambridge University Press 2014 

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.)

References

REFERENCES

An, Y. & Hu, Y. (2012) Well-posedness of measurement error models for self-reported data. Journal of Econometrics 168, 259–269.Google Scholar
Antosik, P., Mikusinski, J., & Sikorski, R. (1973) Theory of Distributions. The Sequential Approach. Elsevier/PWN.Google Scholar
Carrasco, M. & Florens, J.-P. (2010) A spectral method for deconvolving a density. Econometric Theory 27, 546–581.CrossRefGoogle Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007) Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. In Heckman, J. & Leamer, E. (eds.), Handbook of Econometrics, vol. 6B, pp. 5633–5751. North-Holland.CrossRefGoogle Scholar
Carroll, R.J., Ruppert, D., Stefanski, L.A., & Crainiceanu, C.M. (2006) Measurement Error in Nonlinear Models: A Modern Perspective. Chapman & Hall.CrossRefGoogle Scholar
Chen, X., Hong, H., & Nekipelov, D. (2011) Nonlinear models of measurement errors. Journal of Economic Literature 49, 901–937.Google Scholar
Cunha, F., Heckman, J.J., & Schennach, S.M. (2010) Estimating the technology of cognitive and noncognitive skill formation. Econometrica 78, 883–933.Google Scholar
Devroye, L. (1978) The uniform convergence of the Nadaraya-Watson regression function estimate. The Canadian Journal of Statistics 6, 179–191.CrossRefGoogle Scholar
Devroye, L. & Győrfi, L. (1985) Nonparametric Density Estimation: The L1 View. Wiley.Google Scholar
Fan, J.Q. (1991) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19, 1257–1272.Google Scholar
Gelfand, I.M. & Shilov, G.E. (1964a) Generalized Functions, vol. 1, Properties and Operations. Academic Press.Google Scholar
Gelfand, I.M. & Shilov, G.E. (1964b) Generalized Functions, vol. 2, Spaces of Test Functions and Generalized Functions. Academic Press.Google Scholar
Gelfand, I.M. & Ya Vilenkin, N. (1964c) Generalized Functions, vol. 4, Applications of Harmonic Analysis. Academic Press.Google Scholar
Hadamard, J. (1923) Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press.Google Scholar
Hausman, J.A., Newey, W.K., Ichimura, H., & Powell, J. (1991) Identification and estimation of polynomial errors-in-variables models. Journal of Econometrics 50(3), 273–295.Google Scholar
Hirata, Y. & Ogata, H. (1958) On the exchange formula for distributions. Journal of Science of the Hiroshima University, Series A 22, 147–152.Google Scholar
Hu, Y. (2008) Identification and estimation of nonlinear models with misclassification error using instrumental variables: A general solution. Journal of Econometrics 144, 27–61.CrossRefGoogle Scholar
Kirillov, A.A. & Gvishiani, A.D. (1982) Theorems and Problems in Functional Analysis. Springer-Verlag.CrossRefGoogle Scholar
Klann, E., Kuhn, M., Lorenz, D.A., Maass, P., & Thiele, H. (2007) Shrinkage versus deconvolution. Inverse Problems 23, 2231–2248.CrossRefGoogle Scholar
Koralov, L. & Sinai, Ya. (2007) Theory of Probability and Random Processes. Springer.Google Scholar
Kotlyarski, I. (1967) On characterizing the gamma and normal distribution. Pacific Journal of Mathematics 20, 69–76.Google Scholar
Meister, A. (2009) Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics. Springer-Verlag.Google Scholar
Newey, W. (2001) Flexible simulated moment estimation of nonlinear errors-in-variables models. Review of Economics and Statistics 83, 616–627.CrossRefGoogle Scholar
Schennach, S. (2007) Instrumental variable estimation in nonlinear errors-in-variables models. Econometrica 75, 201–239.Google Scholar
Schwartz, L. (1966) Théorie des distributions. Hermann.Google Scholar
Sobolev, S.L. (1992) Cubature Formulas and Modern Analysis. Gordon and Breach Science Publishers.Google Scholar
Wang, L. & Hsiao, Ch. (2011) Method of moments estimation and identifiability of semiparametric nonlinear errors-in-variables models. Journal of Econometrics, 30–44.Google Scholar
Zinde-Walsh, V. (2008) Kernel estimation when density may not exist. Econometric Theory 24, 696–725.Google Scholar
Zinde-Walsh, V. (2009) Errors-In-Variables Models: A Generalized Functions Approach. McGill University Working paper, arXiv:0909.5390v1 [stat.ME].Google Scholar
Zinde-Walsh, V. (2011) Presidential address: Mathematics in economics and econometrics. Canadian Journal of Economics 44, 1052–1068.Google Scholar
Zinde-Walsh, (2014) Identification and well-posedness in nonparametric models with independence conditions. In Racine, J., Su, L., & Ullah, A. (eds.), Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics. Oxford University Press, 97–128.Google Scholar