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1. http://www.fsv.cuni.cz Charles University Founded 1348

2. Kočovce Kočovce PRASTAN 2004 PRASTAN 2004 17. - 21. 5. 2004 17. - 21. 5. 2004 THE LEAST WEIGHTED SQUARES UNDER HETEROSCEDASTICITY THE LEAST WEIGHTED SQUARES UNDER HETEROSCEDASTICITY Jan Ámos Víšek Jan Ámos Víšek http://samba.fsv.cuni.cz/~visek/kocovce http://samba.fsv.cuni.cz/~visek/kocovce visek@mbox.fsv.cuni.cz visek@mbox.fsv.cuni.cz Institute of Economic Studies Faculty of Social Sciences Charles University Prague Institute of Information Theory Institute of Economic Studies Faculty of Social Sciences Charles University Prague and Automation Academy of Sciences Prague Institute of Information Theory and Automation Academy of Sciences Prague

3. Schedule of today talk ● Are data frequently heteroscedastic ? ● Is it worthwhile to take it into account ? ● Recalling White’s estimation of covariance matrix of the estimates of regression coefficients under heteroscedasticity ● Recalling Cragg’s improvment of the estimates of regression coefficients under heteroscedasticity ● Recalling the least weighted squares ● Introducing the estimated least weighted squares

4. Brief introduction of notation (This is not assumption but recalling what the heteroscedasticity is - - to be sure that all of us can follow next steps of talk. The assumptions will be given later.)

5. Can we meet with the heteroscedasticity frequently ? ● Data in question represent the aggregates over some regions. ● Explanatory variables are measured with random errors. ● Models with randomly varying coefficients. ● ARCH models. ● Probit, logit or counting models. ● Limited and censored response variable. ● Error component (random effects) model. Heteroscedasticity is assumed by the character (or type) of model.

6. Can we meet with heteroscedasticity frequently ? continued ● Expenditure of households. ● Demands for electricity. ● Wages of employed married women. ● Technical analysis of capital markets. ● Models of export, import and FDI ( for industries ). Heteroscedasticity was not assumed but “empirically found” for given data.

7. Is it worthwhile to take seriously heteroscedasticity ? Let’s look e. g. for a model of the export from given country.

8. Ignoring heteroscedasticity, we arrive at: B means backshift

9. Other characteristics of model White het. test = 244.066 [.000]

10. Significance of explanatory variables when White’s estimator of covariance matrix of regression coefficients was employed.

11. Reducing model according to effective significance

12. Other characteristics of model White het. test = 116.659 [.000]

13. Recalling White’s ideas - assumptions White, H. (1980): A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica, 48, 817 - 838. ● , - independently (non-identically) distributed r.v.’s , - absolutely continuous d. f.’s ● , ●

14. Recalling White’s ideas - assumptions continued ● , , , for large T , ● for large T . ● Remark. No assumption on the type of distribution already in the sense of Generalized Method of Moments.

15. Recalling White’s results

16. continued Recalling White’s results

17. continued Recalling White’s results

18. continued Recalling White’s results

19. Recalling Cragg’s results Cragg, J. G. (1983): More efficient estimation in the presence of heteroscedasticity of unknown form. Econometrica, 51, 751 - 763. We should use has generally T(T+1)/2 elements

20. continued Recalling Cragg’s results Even if rows are independent has T unknown elements, namely We put up with

21. continued Recalling Cragg’s results

22. continued Recalling Cragg’s results An improvement if Should be positive definite. Nevertheless, is still unknown

23. continued Recalling Cragg’s results Asymptotic variance Estimated asymptotic variance

24. Recalling Cragg’s results Example – simulations Model Heteroscedasticity given by T=25 1000 repetitions Columns of matrix P

25. LS 1.000 1.011 0.764 1.000 0.980 0.701 0.409 0.478 0.400 0.590 0.742 0.442 0.278 0.337 0.286 0.471 0.629 0.309 0.254 0.331 0.266 0.445 0.626 0.270 0.247 0.346 0.247 0.437 0.661 0.230 continued Recalling Cragg’s results Example Example – simulations Asymptotic Actual Estimated Asymptotic Actual Estimated j=1 j=1,2 j=1,2,3 j=1,2,3,4 Asymptotic Actual = simulated Estimated

26. Robust regression Requirements on a ( robust ) estimator ● Unbiasedness ● Consistency ● Asymptotic normality ● Reasonably high efficiency ● Scale- and regression-equivariance ● Quite low gross-error sensitivity ● Low local shift sensitivity ● Preferably finite rejection point

27. Requirements on a ( robust ) estimator continued ● Controlable breakdown point ● Available diagnostics, sensitivity studies and accompanying procedures ● Existence of an implementation of the algorithm with acceptable complexity and reliability of evaluation ● An efficient and acceptable heuristics Víšek, J.Á. (2000): A new paradigm of point estimation. Proc. of Data Analysis 2000/II, Modern Statistical Methods - Modeling, Regression, Classification and Data Mining, 95 - 230.

28. The least weighted squares Víšek, J.Á. (2000): Regression withhigh breakdown point. ROBUST 2000, 324 – 356. non-increasing, absolutely continuous

29. Accommodating Cragg’s idea for robust regression Recalling Cragg’s idea

30. Accomodying Cragg’s idea for robust regression Recalling classical weighted least squares

31. The least weighted squares & Cragg’s idea The first step

32. The least weighted squares & Cragg’s idea continued The second step &

33. THANKS for ATTENTION