Instrumental weighted variables under heteroscedasticity. Part II – Numerical study

Víšek, Jan Ámos

Kybernetika, Tome 53 (2017) no. 1, pp. 26-58

Abstract (VO)

Results of a numerical study of the behavior of the instrumental weighted variables estimator – in a competition with two other estimators – are presented. The study was performed under various frameworks (homoscedsticity/heteroscedasticity, several level and types of contamination of data, fulfilled/broken orthogonality condition). At the beginning the optimal values of eligible parameters of estimatros in question were empirically established. It was done under the various sizes of data sets and various levels of the contamination of data. These values were then utilized in the numerical study. Its results indicate that instrumental weighted variables are as good as $S$- and $W$-estimators and under heteroscedasticity even better. The weight function of Tukey's type was used.

Détail

DOI : 10.14736/kyb-2017-1-0026

Classification : 62F35, 62J02
Keywords: heteroscedasticity of disturbances; numerical study of instrumental weighted variables.

Bibliographie

[1] Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H., Tukey, J. W.: Robust Estimates of Location: Survey and Advances. Princeton University Press, Princeton 1972. | MR | Zbl

[2] Antoch, J., Víšek, J. Á.: Robust estimation in linear models and its computational aspects. In: Contributions to Statistics: Computational Aspects of Model Choice (J. Antoch, ed.), Springer Verlag 1992, pp. 39-104. | DOI | MR

[3] Atkinson, A. C., Riani, M.: Robust Diagnostic Regresion Analysis. | MR

[4] Barndorff-Nielsen, O., Cox, D. R.: Edgeworth and saddle-point approximations with statistical applications. J. Royal Statist. Soc. B 41 (1979), 3, 279-312. | MR | Zbl

[5] Benáček, V., Víšek, J. Á.: Determining factors of trade specialization and growth of a small economy in transition. Impact of the EU opening-up on Czech exports and imports. IIASA, IR series, no. IR-03-001, 2002, pp. 1-41.

[6] Bickel, P. J.: One-step Huber estimates in the linear model. JASA 70 (1975), 428-433. | DOI | MR | Zbl

[7] Campbell, N. A., Lopuhaa, H. P., Rousseeuw, P. J.: On calculation of a robust $S$-estimator if a covariance matrix. Statist. Medcine 17 (1998), 2685-2695. | DOI

[8] Chatterjee, S., Hadi, A. S.: Sensitivity Analysis in Linear Regression. J. Wiley and Sons, New York 1988. | DOI | MR | Zbl

[9] Čížek, P.: Generalized Method of Trimmed Moments. Center Discussion Paper 2009-25, Tilburg University 2009. | DOI | MR | Zbl

[10] Cohen-Freue, G. V., Ortiz-Molina, H., Zamar, R. H.: Natural robustification of the ordinary instrumental variables estimator. Biometrics 69 (2013), 641-650. | DOI | MR

[11] Desborges, R., Verardi, V.: A robust instrumental-variable estimator. The Stata J. 12 (2012), 169-181.

[12] Edgeworth, F. Y.: On observations relating to several quantities. Phil. Mag. (5th Series) 24 (1887), 222.

[13] Field, C. A., Ronchetti, E. M.: Small Sample Asymptotics. Institute of Mathematical Statistics Monograph Series, Hayward 1990. | MR | Zbl

[14] J., D. Gervini V., Yohai: A class of robust and fully efficient regression estimators. Annal Statist. 30 (2002), 583-616. | DOI | MR | Zbl

[15] Halmos, P. R.: Applied mathematics is a bad mathematics. In: Mathematics Tomorrow (L. Steen, ed.), Springer Verlag, New York 1981, pp. 9-20. | DOI | MR

[16] Jurečková, J.: Robust estimation of location and regression parameters and their second order asymptotic relations. In: 9th Prague Conf. on Inform. Theory, Reidel 1983, pp. 19-32. | DOI | MR

[17] Jurečková, J., Sen, P. K.: Uniform second order asymptotic linearity of $M$-statistics in linear models. Statistics and Decisions 7 (1989), 263-276. | DOI | MR | Zbl

[18] Jurečková, J., Malý, M.: The asymptotics for studentized $k$-step $M$-estimators of location. Sequential Analysis 14 (1995), 229-245. | DOI | MR | Zbl

[19] Jurečková, J., Welsh, A. H.: Asymptotic relations between $L$- and $M$-estimators in the linear model. Ann. Inst. Statist. Math. 42 (1990), 671-698. | DOI | MR | Zbl

[20] Koenker, R., Bassett, G.: Regression quantiles. Econometrica 46 (1978), 33-50. | DOI | MR | Zbl

[21] Maronna, R. A., Yohai, V. J.: Asymptotic behaviour of general $M$-estimates for regression and scale with random carriers. Z. Wahrscheinlichkeitstheorie verw. Gebiete 58 (1981), 7-20. | DOI | MR

[22] Mašíček, L.: Optimality of the least weighted squares estimator. Kybernetika 40 (2004), 715-734. | MR | Zbl

[23] Robinson, J.: Saddle point approximations for permutation tests and confidence intervals. J. Roy. Statist. Soc. B 44 (1982), 1, 91-101. | MR

[24] Rousseeuw, P. J.: Multivariate estimation with high breakdown point. In: Proc. Fourth Pannonian Symposium on Mathematical Statistics, Bad Tatzmannsdorf 1983, pp. 283-297. | DOI | MR | Zbl

[25] Rousseeuw, P. J.: Least median of square regression. J. Amer. Statist. Association 79 (1984), 871-880. | DOI | MR

[26] Rousseeuw, P. J., Leroy, A. M.: Robust Regression and Outlier Detection. J. Wiley and Sons, New York 1987. | DOI | MR | Zbl

[27] Rousseeuw, P. J., Yohai, V.: Robust regressiom by means of $S$-estimators. In: Robust and Nonlinear Time Series Analysis (J. Franke, W. Härdle, and R. D. Martin, eds.), Lecture Notes in Statistics No. 26, Springer Verlag, New York 1984, pp. 256-272. | DOI | MR

[28] Rubio, A. M., Víšek, J. Á.: A note on asymptotic linearity of $M$-statistics in non-linear models. Kybernetika 32 (1996), 353-374 | MR

[29] Siegel, A. F.: Robust regression using repeated medians. Biometrica 69 (1982), 242-244. | DOI | Zbl

[30] Verardi, V., McCathie, A.: The $S$-estimator of multivariate location and scatter in Stata. The Stata J. 12 (2012), 299-307.

[31] Welsh, A. H.: Bahadur representation for robust scale estimators based on regression residuals. Ann. Statist. 14 (1986), 1246-1251. | DOI | MR

[32] Víšek, J. Á.: On second order efficiency of a robust test and approximations of its error probabilities. Kybernetika 8 (1983), 3, 387-407. | MR | Zbl

[33] Víšek, J. Á.: Estimating contamination level. In: Proc. Fifth Pannonian Symposium on Mathematical Statistics, Visegrád 1985, pp. 401-414. | MR | Zbl

[34] Víšek, J. Á.: Robust instruments. In: Robust'98 (J. Antoch and G. Dohnal, eds.), Union of Czechoslovak Mathematicians and Physicists 1998, pp. 195-224.

[35] Víšek, J. Á.: The robust regression and the experiences from its application on estimation of parameters in a dual economy. In: Proc. Conference Macromodels'99, Wroclaw University 1999, pp. 424-445.

[36] Víšek, J. Á.: Regression with high breakdown point. In: ROBUST 2000 (J. Antoch and G. Dohnal, eds.), The Union of the Czech Mathematicians and Physicists and the Czech Statistical Society 2001, pp. 324-356.

[37] Víšek, J. Á.: Character of the Czech economy in transition. In: Proc. The Czech society on the break of the third millennium, Karolinum, Publishing House of the Charles University 2000, pp. 181-205.

[38] Víšek, J. Á.: The least trimmed squares. Part I - Consistency. Kybernetika 42 (2006), 1-36. | MR

[39] Víšek, J. Á.: Robust error-term-scale estimate. IMS Coll., Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: Festschrift for Jana Jurečková 7 (2010), 254-267. | MR

[40] Víšek, J. Á.: Least weighted squares with constraints and under heteroscedasticity. Bull. Czech Econometr. Soc. 20 (2012), 31, 21-54.

[41] Wooldridge, J. M.: Introductory Econometrics. A Modern Approach. MIT Press, Cambridge 2006. Second edition 2009.