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Kočovce PRASTAN 2004 Institute of Information Theory and Automation Academy of Sciences Prague and Automation Academy of Sciences Prague Institute of Information Theory Institute of Economic Studies Faculty of Social Sciences Charles University Prague Institute of Economic Studies Faculty of Social Sciences Charles University Prague Jan Ámos Víšek THE LEAST WEIGHTED SQUARES UNDER HETEROSCEDASTICITY Kočovce PRASTAN 2004 THE LEAST WEIGHTED SQUARES UNDER HETEROSCEDASTICITY

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

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.)

● 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. Can we meet with the heteroscedasticity frequently ? ● Error component (random effects) model. Heteroscedasticity is assumed by the character (or type) of model.

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

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

Ignoring heteroscedasticity, we arrive at: B means backshift

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

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

Reducing model according to effective significance

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

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

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

Recalling White’s results

continued

Recalling White’s results continued

Recalling White’s results continued

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

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

Recalling Cragg’s results continued

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

Recalling Cragg’s results continued Asymptotic variance Estimated asymptotic variance

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

continued Example Recalling Cragg’s results Example – simulations Asymptotic Estimated Actual = simulated LS j=1,2,3,4 j=1 j=1,2 j=1,2,3 Asymptotic Actual Estimated

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

● 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, continued Requirements on a ( robust ) estimator

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

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

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

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

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

THANKS for ATTENTION