Durbin-Watson statistic in robust regression

• Autorzy: Višek J. Á.
• Języki publikacji: EN

Abstrakty

EN

It is shown that the lower and upper critical values of the Durbin-Watson (D-W) statistic are asymptotically the same for the analysis based on M-estimators as for the classical least squares analysis. Moreover, the paper offers a possibility to make an idea when the asymptotics may start to work. Considering the B-robust optimal ψ-function, we demonstrate that the differences between the precise critical values of Durbin-Watson statistics evaluated for residuals corresponding to the M-estimate and critical values which were found by Durbin and Watson for the least squares analysis are rather small even for moderate sample size.

Słowa kluczowe

EN

regression; diagnostics; M-estimators; robustified D-W statistics; critical values

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2003
• Tom: Vol. 23, Fasc. 2
• Strony: 435--483
• Opis fizyczny: Bibliogr. 92 poz., tab.

Twórcy

autor

Višek J. Á.

• Department of Macroeconomics and Econometrics, Institute of Economic Studies, Faculty of Social Sciences, Charles University
• Department of Stochastic Informatics, Institute of Information Theory and Automation, Academy of Sciences of Czech Republic, Opletalova ulice 26, CZ - 11000 Prague 1, Czech Republic

Bibliografia

• [1] T. W. Anderson, On the theory of testing serial correlation, Skandinavisk Aktuarietidskrift 31 (1948), pp. 88-116.
• [2] I. Antoch and J. Á. Víšek, Robust estimation in linear models and its computational aspects, in: Contributions to Statistics: Computational Aspects of Model Choice, J. Antoch (Ed.), Springer, 1991, pp. 39-104.
• [3] J. Antoch and D. Vorlíčková, Vyrané metody statistické analýzy dat (Selected Methods of Statistical Analysis - in Czech) Academia, Praha 1992.
• [4] O. Arslan, A simple test to identify good solution of redescending M-estimating equations for regression, in: Developments in Robust Statistics, R. Dutter, P. Filzmoser, U. Gather and P. J. Rousseeuw (Eds.), Physica-Verlag, Springer-Verlag Company 2003, pp. 50-61.
• [5] A. C. Atkinson, Plots, Transformations and Regression: An introduction to Graphical Methods of Diagnostic Regression Analysis, Claredon Press, Oxford -1985.
• [6] D. A. Belsley, E. Kuh and R. E. Welsch, Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, Wiley, New York 1980.
• [7] R. Beran, An efficient and robust adaptive estimator of location, Ann. Statist. 6 (1978), pp. 292-313.
• [8] P. J. Bickel, One-step Huber estimates in the linear model, J. Amer. Statist. Assoc. 70 (1975), pp. 428-433.
• [9] G. Boente and R. Fraiman, A functional approach to robust nonparametric regression, in: Directions in Robust Statistics and Diagnostics, Part I, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 35-46.
• [10] L. Breiman, Probability, Addison-Wesley Publishing Company, London 1968.
• [11] S. Chatterjee and A. S. Hadi, Sensitivity Analysis in Linear Regression, Wiley, New York 1988.
• [12] R. D. Cook and S. Weisberg, Diagnostics of heteroscedasticity in regression, Biometrika 70 (1983), pp. 1-10.
• [13] C. Croux and G. Haesbroeck, Influence function and effciency of the minimum covariance determinant scatter matrix estimation, J. Multivariate Anal. 71 (1999), pp. 161-190.
• [14] M. Csörgő and P. Révész, Strong Approximation in Probability and Statistics, Akadémiai Kiadó, Budapest 1981.
• [15] M. B. Dollinger and R. G. Staudte, Efficiency of reweighted least squares iterates, in: Directions in Robust Statistics and Diagnostics, Part I, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 61-66.
• [16] N. R. Draper and H. Smith, Applied Regression Analysis, 3rd edition, Wiley, New York 1998.
• [17] J. Durbin and G. S. Watson, Testing for serial correlation in least squares regression. I, Biometrika 37 (1952), pp. 409-428.
• [18] S. Geisser, Diagnostics, divergence and perturbation analysis, in: Directions in Robust Statistics and Diagnostics, Part I, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 89-100.
• [19] W. H. Greene, Econometric Analysis, Macmillan Press, New York 1993.
• [20] A. S. Hadi, Identifying multiple outliers in multivariate data, J. Roy. Statist. Soc., Ser. B, 54 (1992), pp. 761-777.
• [21] A. S. Hadi, A modification of a method for the detection of outliers in multivariate samples, J. Roy. Statist. Soc., Ser. B, 56 (1994), pp. 393-396.
• [22] F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw and W. A. Stahel, Robust Statistics - The Approach Based on Influence Functions, Wiley, New York 1986.
• [23] M. A. Hauser, Semiparametric and nonparametric testing for long memory: A Monte Carlo study, Empirical Economics 22 (1997), pp. 247-271.
• [24] T. P. Hettmansperger and J. D. Naranjo, Some research directions in rank-based inference, in: Directions in Robust Statistics and Diagnostics, Part I , W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 113-120.
• [25] T. P. Hettmansperger and S. J. Sheather, A cautionary note on the method of least median squares, Amer. Statist. 46 (1992), pp. 79-83.
• [26] P. J. Huber, Robust estimation of a location parameter, Ann. Math. Statist. 35 (1964), pp. 73-101.
• [27] P. J. Huber, Robust Statistics, Wiley, New York 1981.
• [28] P. J. Huber, Between robustness and diagnostics, in: Directions in Robust Statistics ad Diagnostics, Part I, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 121-131.
• [29] G. G. Judge, W. E. Griffiths, R. C. Hill, H. Lütkepohl and T. Ch. Lee, The Theory and Practice of Econometrics, Wiley, New York 1985.
• [30] J. Jurečková, Consistency of M-estimators in linear model generated by non-monotone and discontinuous ψ-functions, Probab. Math. Statist. 10 (1988), pp. 1-10.
• [31] J. Jurečková and M. Malý, The asymptotics for studentized k-step M-estimators of location, Sequential Anal. 14 (1995), pp. 229-245.
• [32] J. Jurečková and S. Portnoy, Asymptotics for one-step M-estimators in regression with application to combining efficiency and high breakdown point, Commun. Statist. A 16 (1988), pp. 2187-2199.
• [33] I. Jurečková and P. K Sen, Uniform second order asymptotic linearity of M-statistics in linear models, Statist. Decisions 7 (1989), pp. 263-276.
• [34] J. Jurečková and P. K. Sen, Regression rank scores scale statistics and studentization in linear models, in: Proceedings of the Fifth Prague Symposium on Asymptotic Statistics, Physica-Verlag, 1993, pp. 111-121.
• [35] J. Jurečková and A. H. Welsh, Asymptotic relations between L- and M-estimators in the linear model, Ann. Inst. Statist. Math. 42 (1990), pp. 671-698.
• [36] J. Kmenta, Elements of Econometrics, Macmillan Publishing Company, New York 1986.
• [37] R. Koenker and G. Bassett, Regression quantiles, Econometrica 46 (1978), pp. 33-50.
• [38] T. S. Kuhn, Structure of Scientific Revolution, University of Chicago Press, Chicago, Phoenix Broks 159 (1965).
• [39] A. J. Lawrence, Local and deletion influence, in: Directions in Robust Statistics and Diagnostics, Part I , W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 141-158.
• [40] C. C. Macduffee, The Theory of Matrices, Chelsea Publishing Company, 1946.
• [41] G. S. Maddala, Introduction to Econometrics, Macmillan Press, New York 1988.
• [42] M. Markatou and X. He, Bounded influence and high breakdown point testing procedures in linear models, J. Amer. Statist. Assoc. 89 (1994), pp. 543-549.
• [43] M. Markatou, W. A. Stahel and E. M. Ronchetti, Robust M-type testing procedures for linear models, in: Directions in Robust Statistics and Diagnostics, Part I, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 201-220.
• [44] R. A. Maronna and V. J. Yohai, The breakdown point of simultaneous general M-estimates of regression and scale, J. Amer. Statist. Assoc. 86 (1981), pp. 699-704.
• [45] R. D. Martin, V. J. Yohai and R. H. Zamar, Min-max bias robust regression, Ann. Statist. 17 (1989), pp. 1608-1630.
• [46] J. W. McKean, S. J. Sheather and T. P. Hettmansperger, Regression diagnostics for rank-based method. II, in: Directions in Robust Statistics and Diagnostics, Part II, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 21-32.
• [47] G. E. Mizon, A simple message for autocorrelation correctors: Don't, J. Econometrics 69 (1995), pp. 267-288.
• [48] J. von Neumann, Distribution of the ratio of the mean-square successive difference to the variance, Ann. Math. Statist. 12 (1941), pp. 367-395.
• [49] A. Ö. Önder and A. Zaman, A test of normality based on robust regression residuals, in: Developments in Robust Statistics, R. Dutter, P. Filzmoser, U. Gather and P. J. Rousseeuw (Eds.), Physica-Verlag, Springer-Verlag Company, 2003, pp. 296-306.
• [50] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York-London 1970.
• [51] A. Peters and P. Sibbertsen, Tests on fractional cointegration, in: Developments in Robust Statistics, R. Dutter, P. Filzmoser, U. Gather and P. J. Rousseeuw (Eds.), Physica-Verlag, Springer-Verlag Company, 2003, pp. 307-316.
• [52] E. J. G. Pitman, The 'closest' estimates of statistical parameters, Proc. Cambridge Phil. Soc. 33 (1937), pp. 212-222.
• [53] D. Pollard, Asymptotics for least absolute deviation regression estimator, Econometric Theory 7 (1991), pp. 186-199.
• [54] K. R. Popper, Objective Knowledge, Oxford Clarendon Press, 1972.
• [55] S. Portnoy, Tightness of the sequence of empiric c.d.f. processes defined from regression fractiles, in: Robust and Nonlinear Time-Series Analysis, J. Franke, W. Härdle and D. Martin (Eds.), Springer, New York 1983, pp. 231-246.
• [56] S. Portnoy, Regression quantile diagnostics for multiple outliers, in: Diections in Robust Statistics and Diagnostics, Part II, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 145-158.
• [57] I. Prigogine and I. Stengers, La Nouvelle Alliance, SCIENTIA, 1977, issues 5-12.
• [58] I. Prigogine and I. Stengers, Order out of Chaos. Man's New Dialog with Nature, Bantam Books, New York 1984.
• [59] R. C. Rao and L. C. Zhao, On the consistency of M-estimate in linear model obtained through an estimating equation, Statist. Probab. Lett. 14 (1992), pp. 79-84.
• [60] P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier Detection, Wiley, New York 1987.
• [61] A. M. Rubio and J. Á. Višek, Asymptotic representation of constrained M-estimators, in: Transactions of the Twelfth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, Prague, 1994 (1993), pp. 207-210.
• [62] A. M. Rubio and J. Á. Višek, A note on asymptotic linearity of M-statistics in nonlinear models, Kybernetika 32 (1996), pp. 353-374.
• [63] R. Schall and T. T. Dunne, Diagnostics for regression - ARMA time series, in: Directions in Robust Statistics and Diagnostics, Part II, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 223-242.
• [64] W. Stahel and S. Weisberg (Eds.), Directions in Robust Statistics and Diagnostics, Parts I and II, Springer, New York 1991.
• [65] J. Štĕpán, Teorie Pravdĕpodobnosti (Theory Probability - in Czech), Academia, Praha 1987.
• [66] C. Stone, Adaptive maximum likelihood estimators of a location parameter, Ann. Statist. 3 (1975), pp. 267-284.
• [67] J. W. Tukey, Graphical displays for alternate regression fits, in: Directions in Robust Statistics and Diagnostics, Part II, W. Stahel and S. Weisberg (Eds.), Springer, New York 1991, pp. 145-158.
• [68] C. M. Urzua, On the correct use of omnibus tests for normality, Econom. Lett. 53 (1996), pp. 247-251.
• [69] J. Á. Višek, Adaptive estimation in linear regression model, Kybernetika 28 (1991), Part I., Consistency, pp. 26-36, Part II. Asymptotic normality, pp. 100-119.
• [70] J. Á. Višek, Stability of regression model estimates with respect to subsamples, Computat. Statist. 7 (1992), pp. 183-203.
• [71] J. Á. Višek, A cautionary note on the method of Least Median of Squares reconsidered, in: Transactions of the Twelfth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, Academia Prague-Kluwer Academic Press, Dordrecht 1994, pp. 254-259.
• [72] J. Á. Višek, On high breakdown point estimation, Computat. Statist 11 (1996a), pp. 137-146.
• [73] J. Á. Višek, On the heuristics of statistical results, in: Proceedings of 'PROBASTAT'’94, Tatra Mountains Publ. 7, Bratislava 1996b, pp. 349-357.
• [74] J. Á. Višek, Sensitivity analysis on M-estimates, Ann. Inst. Statist. Math. 48 (1996c) pp. 469-495.
• [75] J. Á. Višek, Contamination level and sensitivity of robust tests, in: Handbook of Statistics, Vol. 15, G. S. Maddala and C. R. Rao (Eds.), Elsevier Science B.V., Amsterdam 1997, pp. 633-642.
• [76] J. Á. Višek, Robust instruments, in: Robust '98, J. Antoch and G. Dohnal (Eds.), published by the Union of Czechoslovak Mathematicians and Physicists, 1998a, pp. 195-224.
• [77] J. Á. Višek, Robust specification test, in: Proceedings of Prague Stochastics '98, M. Hušková, P. Lachout and J. Á. Višek (Eds.), published by the Union of Czechoslovak Mathematicians and Physicists 1998b, pp. 581-586.
• [78] J. Á. Višek, The least trimmed squares - random carriers, Bull. Czech Econometric Soc. 10 (1999a), pp. 1-30.
• [79] J. Á. Višek, Robust estimation of regression model, Bull. Czech Econometric Soc. 9 (1999b), pp. 57-79.
• [80] J. Á. Višek, The robust regression and the experiences from its application on estimation of parameters in a dual economy, in: Proceedings of the Conference "Macromodels '99" organized by Wrocław University, 4-6 December, 1999, in Rydzyna, Poland, 1999c, pp. 424-445.
• [81] J. Á. Višek, On the diversity of estimates, Computat. Statist. Data Anal. 34 (2000a), pp. 67-89.
• [82] J. Á. Višek, Robust instrumental variables and specification test, in: PRASTAN 2000, Proceedings of conference "Mathematical Statistics and Numerical Mathematics and Their Applications, 2000b, pp. 133-164.
• [83] J. Á. Višek, Over- and underfitting the M-estimates, Bull. Czech Econometric Soc. 7 (2000c), pp. 53-83.
• [84] J. Á. Višek, Overfitting the least trimmed squares, in: PRASTAN 2001, Proceedings of conference "Mathematical Statistics and Numerical Mathematics and Their Applications", 2001a, pp. 160-166.
• [85] J. Á. Višek, Regression with high breakdown point, in: ROBUST2000, organized by the Union of the Czech Mathematicians and Physicists and the Czech Statistical Society, 2001b, pp. 324-356.
• [86] J. Á. Višek, Sensitivity analysis of M-estimates of nonlinear regression model: Influence of datu subsets, Ann. Inst. Statist. Math. 54 (2) (2002a), pp. 261-290.
• [87] J. Á. Višek, White test for the least weighted squares, in: COMPSTAT 2002, Berlin, Proceedings of the Conference CompStat 2002 - Short Communications and Poster (CD), S. Klinke, P. Ahrend and L. Richter (Eds.), 2002b.
• [88] J. Á. Višek, Development of the Czech export in nineties, in: Consolidation of Governing and Business in the Czech Republic and in EU, 2003, MatFyz Press, 2003, pp. 193-220.
• [89] R. E. Welsh, Influence function and regression diagnostic, in: Modern Data Analysis, R. L. Launer and A. F. Sigel (Eds.), Academic Press, New York 1982.
• [90] G. Willem, G. Pison, P. J. Rousseeuw and S. van Aelst, A robust Hotelling test, in: Developments in Robust Statistics, R. Dutter, P. Filzmoser, U. Gather and P. J. Rousseeuw (Eds.), Physica-Verlag, Springer-Verlag Company, 2003, pp. 417-431.
• [91] V. J. Yohai and R. A. Maronna, Asymptotic behaviour of M-estimators for the linear model, Ann. Statist. 7 (1979), pp. 248-268.
• [92] K. Zvára, Regresní analýza (Regression Analysis - in Czech), Academia, Praha 1989.