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Consistency of the instrumental weighted variables. (English) Zbl 1332.62246

Summary: A robust version of method of Instrumental Variables accommodating the idea of an implicit weighting the residuals is proposed and its properties studied. Firstly, it is shown that all solutions of the corresponding normal equations are bounded in probability. Then the weak consistency of them is proved. The algorithm, evaluating the estimate, is described and results of small MC study discussed.

MSC:

62J05 Linear regression; mixed models
62F35 Robustness and adaptive procedures (parametric inference)

Software:

car; alr3
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[1] Arellano M., Bond S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies 58:277–297 · Zbl 0719.62116
[2] Arellano M., Bover O. (1995). Another look at the instrumental variables estimation of error components models. Journal of Econometrics 68(1):29–52 · Zbl 0831.62099
[3] Bickel P.J. (1975). One-step Huber estimates in the linear model. Journal of American Statistical Association 70:428–433 · Zbl 0322.62038
[4] Boček P., Lachout P. (1995). Linear programming approach to LMS-estimation. Memorial Volume of Computational Statistics and Data Analysis 19:129–134 · Zbl 0875.62292
[5] Bowden R.J., Turkington D.A. (1984). Instrumental Variables. Cambridge, Cambridge University Press · Zbl 0606.62130
[6] Breiman L. (1968). Probability. London, Addison-Wesley · Zbl 0174.48801
[7] Chatterjee S., Hadi A.S. (1988). Sensitivity Analysis in Linear Regression. New York, Wiley · Zbl 0648.62066
[8] Čížek, P. (2002). Robust estimation with discrete explanatory variables. COMPSTAT 2002, Berlin, pp. 509–514.
[9] Der G., Everitt B.S. (2002). A handbook of statistical analyses using SAS. Boca Raton, Chapman and Hall/CRC Press · Zbl 1038.62001
[10] Erickson T. (2001). Constructing instruments for regression with measurement error when no additional data are available. Econometrica, Notes and Comments, 69:221–222
[11] Fisher R.A. (1920). A mathematical examination of the methods of determining the accuracy of an observation by the mean error and by the mean squares error. Monthly Notes of Royal Astrophysical Society. 80:758–770
[12] Fisher R.A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of Royal Society London Series 222:309–368 · JFM 48.1280.02
[13] Fox J. (2002). An R and S-PLUS companion to applied regression. Thousand Oaks, SAGE Publications
[14] Glivenko V.I. (1933). Sulla determinazione empirica delle leggi di probabilita. Giornale Istituto Italiano Attuari 4:92 · Zbl 0006.17403
[15] Hahn J., Hausman J. (2002). A new specification test for the validity of instrumental variables. Econometrica 70:163–189 · Zbl 1104.62332
[16] Hájek J., Šidák Z. (1967). Theory of rank test. New York, Academic Press
[17] Hampel F.R., Ronchetti E.M., Rousseeuw P.J., Stahel W.A. (1986). Robust statistics–The approach based on influence functions. New York, Wiley · Zbl 0593.62027
[18] Heckman J.J. (1996). Randomization as instrumental variables. The Review of Economics and Statistics 78:336–341
[19] Hettmansperger T.P., Sheather S.J. (1992). A cautionary note on the method of least median squares. The American Statistician 46:79–83 · Zbl 04510711
[20] Judge G.G., Griffiths W.E., Hill R.C., Lutkepohl H., Lee T.C. (1985). The theory and practice of econometrics (2nd edn). New York, Wiley
[21] Jurečková, J., Sen, P. K. (1993). Regression rank scores scale statistics and studentization in linear models. In Proceedings of the fifth Prague symposium on asymptotic statistics (pp. 111–121). Heidelberg: Physica-Verlag/Springer.
[22] Kalina, J. (2004). Durbin–Watson test for least weighted squares. In Proceedings of COMPSTAT 2004 (pp. 1287–1294). Heidelberg: Physica-Verlag/Springer.
[23] Manski F.C., Pepper J.V. (2000). Monotone instrumental variables: With application to the return to scholling. Econometrica 68:997–1010 · Zbl 1016.62133
[24] Mašíček, L. (2003). Diagnostika a sensitivita robustních odhadu.(Diagnostics and sensitivity of robust estimators, in Czech.) Disertační práce (PhD-disertation).
[25] Mašíček, L. (2004a). Consistency of the least weighted squares estimator. In Statistics for industry and technology (pp. 183–194). Basel: Birkhaser Verlag. · Zbl 1088.62037
[26] Mašíček L. (2004b). Optimality of the least weighted squares estimator. Kybernetika 40:715–734 · Zbl 1245.62013
[27] Plát, P. (2004a). Modifikace Whiteova testu pro nejmenší vážené čtverce. (Modification of White’s test for the least weighted squares, in Czech.) In J. Antoch, G. Dohnal (Eds.), ROBUST 2004 (pp. 291–298).
[28] Plát, P. (2004b). The least weighted squares estimator. Proceedings of COMPSTAT 2004 (pp. 1653–1660). Heidelberg: Physica-Verlag/Springer.
[29] Rousseeuw P.J. (1984). Least median of square regression. Journal of American Statistical Association, 79:871–880 · Zbl 0547.62046
[30] Rousseeuw P.J., Leroy A.M. (1987). Robust regression and outlier detection. New York, Wiley · Zbl 0711.62030
[31] Sargan J.D. (1988). Testing for misspecification after estimating using instrumental variables. In: Massouumi E. (ed). Contribution to econometrics: John Denis Sargan (Vol. 1). Cambridge, Cambridge University Press
[32] Stock J.H., Trebbi F. (2003). Who invented instrumental variable regression? Journal of Economic Perspectives 17:177–194
[33] Van Huffel S. (2004). Total least squares and error-in-variables modelling: Bridging the gap between statistics, computational mathematics and enginnering. In: Antoch J. (ed). Proceedings in Computational Statistics, COMPSTAT 2004. Heidelberg, Physica-Verlag/Springer, pp. 539–555 · Zbl 1153.62335
[34] Víšek J.Á. (1992). Stability of regression model estimates with respect to subsamples. Computational Statistics 7:183–203 · Zbl 0775.62182
[35] Víšek, J. Á. (1994). A cautionary note on the method of the Least Median of Squares reconsidered. In Transactions of the twelth prague conference on information theory, statistical decision functions and random processes (pp. 254–259).
[36] Víšek J.Á. (1996a). Sensitivity analysis of M-estimates. Annals of the Institute of Statistical Mathematics 48:469–495 · Zbl 0925.62131
[37] Víšek J.Á. (1996b). On high breakdown point estimation. Computational Statistics 11:137–146 · Zbl 0933.62015
[38] Víšek J.Á. (1998a). Robust instruments. In: Antoch J., Dohnal G., (eds). (published by Union of Czech Mathematicians and Physicists), Robust’98. Prague, MatFyz Press, pp. 195–224
[39] Víšek, J. Á. (1998b). Robust specification test. In M. Hušková, P. Lachout, J. Á. Víšek, Union of Czech Mathematicians and Physicists (Eds.), Proceedings of Prague Stochastics’98 (pp. 581–586). Prague: MatFyz Press.
[40] Víšek J.Á. (2000a). On the diversity of estimates. Computational Statistics and Data analysis 34:67–89 · Zbl 1052.62509
[41] Víšek, J. Á. (2000b). A new paradigm of point estimation. Data Analysis 2000/II, Modern statistical methods–modelling, regression, classification and data mining (pp. 195–230). Pardubice: Trilobyte.
[42] Víšek J.Á. (2000c): Regression with high breakdown point. In: Antoch J., Dohnal G. (eds). Union of Czech Mathematicians and Physicists, Robust 2000. Prague, MatFyz Press, pp. 324–356
[43] Víšek J.Á. (2000d). Over- and underfitting the M-estimates. Bulletin of the Czech Econometric Society 7:53–83
[44] Víšek, J. Á. (2000e). Character of the Czech economy in transition. In Proceedings of the conference ”The Czech society on the break of the third millennium” (pp. 181–205). Karolinum: Publishing House of the Charles University. ISBN 80-7184-825-5.
[45] Víšek J.Á. (2002a). The least weighted squares I. The asymptotic linearity of normal equations. Bulletin of the Czech Econometric Society 9:31–58
[46] Víšek J.Á. (2002b). The least weighted squares II. Consistency and asymptotic normality. Bulletin of the Czech Econometric Society 9:1–28
[47] Víšek J.Á. (2002c). Sensitivity analysis of M-estimates of nonlinear regression model: Influence of data subsets. Annals of the Institute of Statistical Mathematics 54:261–290 · Zbl 1013.62072
[48] Víšek J.Á. (2002d). White test for the least weigthed squares. In: Klinke S., Ahrend P., Richter L. (eds). COMPSTAT 2002, Proceedings of the conference CompStat 2002–Short communications and poster (CD). Berlin, Springer
[49] Víšek J.Á. (2003a). Durbin-Watson statistic in robust regression. Probability and Mathematical Statistics 23:435–483 · Zbl 1046.62075
[50] Víšek, J. Á. (2003b). Development of the Czech export in nineties. In Konsolidace vládnutí a podnikání v České republice a v Evropské unii I. Umění vládnout, ekonomika, politika, 2003 (pp. 193–220). Prague: MatFyz Press. ISBN 80-86732-00-2.
[51] Víšek J.Á. (2006a). Kolmogorov–Smirnov statistics in multiple regression. In: Antoch J., Dohnal G. (eds). Proceedings of the ROBUST 2006, JČMF and KPMS MFF UK. Prague, MatFyz Press, pp. 367–374
[52] Víšek, J. Á. (2006b). The least trimmed squares. Part I–Consistency. Part II– \(\sqrt{n}\) -consistency. Part III–Asymptotic normality and Bahadur representation. Kybernetika, 42, 1–36, 181–202, 203–224.
[53] Víšek J.Á. (2006c). Instrumental weighted variables–algorithm. In: Rizzi A., Vichi M. (eds). Proceedings of the COMPSTAT 2006. Heidelberg, Physica-Verlag/Springer, pp. 777–786
[54] Víšek J.Á. (2006d). The least trimmed squares. Sensitivity study. In: Hušková M., Janžura M. (eds). Proceedings of the Prague stochastics 2006. Prague, MatFyz Press, pp. 728–738
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