×

Estimating nonlinear structural relationships. (English) Zbl 1422.62090

Summary: We consider the estimation of a general nonlinear structural relationship when the model is identifiable. In contrast to traditional approaches, the estimation method proposed does not require knowledge of the error variances provided that the model is identifiable. The model is effectively parametric in nature, but that the practitioner is not required to come up with a complete distributional formulation of the underlying latent variable. This is made possible using the general approximation to probability distributions given in our work [Comput. Stat. 28, No. 5, 2211–2230 (2013; Zbl 1306.65094)]. Since the proposed approach is likelihood based, the resulting estimators are approximately consistent and efficient. As a by-product, the entire distribution of the latent variable can be obtained and that a simple test for normality can also be easily derived.

MSC:

62F10 Point estimation
62E17 Approximations to statistical distributions (nonasymptotic)

Citations:

Zbl 1306.65094
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Y. Amemiya, Instrumental variable estimator for the nonlinear errors-in-variables model, J. Econometrics 28 (1985), 273-289. · Zbl 0581.62096
[2] V. Barnett, Simultaneous pairwise linear structural relationships, Biometrics 25 (1970), 129-142.
[3] S. Bonhomme and J. M. Robin, Generalized non-parametric deconvolution with an application to earnings dynamics, Rev. Econom. Stud. 77 (2010), 491-533. · Zbl 1187.62071
[4] J. Buzas, Instrumental variable estimation in nonlinear measurement error models, Comm. Statist. Theory Methods 26 (1997), 2861-2877. · Zbl 0954.62552
[5] L. K. Chan and T. K. Mak, Maximum likelihood estimation of a linear structural relationship with replication, J. R. Statist. Soc. B 41 (1979), 263-268. · Zbl 0408.62026
[6] L. K. Chan and T. K. Mak, Two adaptive procedures for the estimation of a linear structural relationship, Scand. J. Statist. 9 (1982), 223-228. · Zbl 0503.62029
[7] W. A. Fuller, Measurement Error Models, Wiley, New York, 1987. · Zbl 0800.62413
[8] P. Hall and Y. Ma, Semiparametric estimators of functional measurement error models with unknown error, J. R. Statist. Soc. B 69 (2004), 429-446. · Zbl 07555360
[9] H. Hong and E. Tamer, A simple estimator for nonlinear error in variable models, J. Econometrics 117 (2003), 1-19. · Zbl 1022.62047
[10] C. Hsiao, Consistent estimation for some nonlinear errors-in-variables models, J. Econometrics 41 (1989), 159-185. · Zbl 0705.62105
[11] Y. Hu and G. Ridder, On deconvolution as a first stage nonparametric estimator, Econometric Rev. 29 (2010), 365-396. · Zbl 1192.62117
[12] L. F. Lee and J. H. Sepanski, Estimation of linear and nonlinear errors-invariables models using validation data, J. Amer. Statist. Assoc. 90 (1995), 130-140. · Zbl 0818.62059
[13] T. K. Mak and F. Nebebe, On a general class of probability distributions and its applications, Comput. Statist. 28 (2013), 2211-2230. · Zbl 1306.65094
[14] P. A. P. Moran, Estimating structural and functional relationships, J. Multivariate Anal. 1 (1971), 232-255. · Zbl 0219.62011
[15] S. M. Schennach, Measurement error in nonlinear models - a review, Cemmap working paper, Centre for Microdata Methods and Practice, No. CWP41/12, 2012. http://dx.doi.org/10.1920/wp.cem.2012.4112.
[16] J. H. Sepanski and R. J. Carroll, Semiparametric quasilikelihood and variance function estimation in measurement error models, J. Econometrics 58 (1993), 223-256. · Zbl 0780.62038
[17] J. Wu and W. Song, On Hong-Tamer’s estimator in nonlinear errors-in-variable regression models, Statist. Probab. Lett. 97 (2015), 165-175. · Zbl 1312.62071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.