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Kočovce17. - 21. 5. 2004 PRASTAN 2004

Institute of Information Theory and Automation

Academy of Sciences Prague

and AutomationAcademy of Sciences

Prague

Institute of Information Theory Institute of Economic Studies Faculty of Social Sciences

Charles UniversityPrague

Institute of Economic Studies Faculty of Social Sciences

Charles UniversityPrague

Jan Ámos VíšekJan Ámos Víšek

visek@mbox.fsv.cuni.czvisek@mbox.fsv.cuni.cz

http://samba.fsv.cuni.cz/~visek/kocovce

THE LEAST WEIGHTED SQUARES UNDER HETEROSCEDASTICITY

http://samba.fsv.cuni.cz/~visek/kocovce

Kočovce17. - 21. 5. 2004 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:

Estim. StandardVariable Coeff. Error t-stat. P-valueC 9.643 5.921 1.629 [.104]

log(BX) 0.827 0.033 25.18 [.000]

log(PE) -0.164 0.06 -2.724 [.007]

log(BPE) 0.2 0.062 3.24 [.001]

log(VA) 0.337 0.077 4.365 [.000]

log(BVA) -0.228 0.079 -2.899 [.004]

log(K/L) -0.625 0.159 -3.937 [.000]

log(BK/BL) 0.518 0.157 3.29 [.001]

log(DE/VA) 0.296 0.122 2.419 [.016]

log(BDE/BVA) -0.292 0.119 -2.456 [.015]

log(FDI) 0.147 0.056 2.629 [.009]

log(BFDI) -0.151 0.056 -2.717 [.007]

log(GDPeu) 1.126 0.629 1.789 [.045]

log(BGDPeu) -1.966 0.623 -3.155 [.002]

B means backshift

Mean of dep. var. = 11.115 Durbin-Watson = 1.98 [<.779]

Std. dev. of dep. var. = 1.697 White het. test = 244.066 [.000]

Sum of squared residuals

= 150.997 Jarque-Bera test = 372.887 [.000]Variance of residuals = 0.519 Ramsey's RESET2 = 8.614 [.004]

Std. error of regression = 0.72 F (zero slopes) = 107.422 [.000]

R-squared = 0.828 Schwarz B.I.C. = 365.603Adjusted R-squared = 0.82 Log likelihood = -325.56LM het. test = 19.964

Other characteristics of model

White het. test = 244.066 [.000]

Estim. StandardVariable Coeff. Error t-stat. P-valueC 9.643 4.128 2.336 [.020]

log(BX) 0.827 0.046 18.141 [.000]

log(PE) -0.164 0.107 -1.53 [.127]

log(BPE) 0.2 0.107 1.876 [.062]

log(VA) 0.337 0.203 1.661 [.098]

log(BVA) -0.228 0.192 -1.191 [.235]

log(K/L) -0.625 0.257 -2.435 [.016]

log(BK/BL) 0.518 0.301 1.717 [.087]

log(DE/VA) 0.296 0.292 1.014 [.312]

log(BDE/BVA) -0.292 0.282 -1.034 [.302]

log(FDI) 0.147 0.141 1.039 [.300]

log(BFDI) -0.151 0.123 -1.223 [.222]

log(GDPeu) 1.126 1.097 1.027 [.305]

log(BGDPeu) -1.966 0.995 -1.976 [.049]

Significance of explanatory variables when White’sestimator of covariance matrix of regression coefficients

was employed.

Estim. StandardVariable Coeff. Error t-stat. P-valuelog(BX) 0.804 0.05 16.125 [.000]log(VA) 0.149 0.039 3.784 [.000]log(K/L) -0.214 0.063 -3.38 [.001]log(GDPeu) 1.896 0.782 2.425 [.016]log(BGDPeu) -2.538 0.778 -3.261 [.001]

Reducing model according to effective significance

Mean of dep. var. = 11.115 Durbin-Watson = 1.914 [<.344]

Std. dev. of dep. var. = 1.697 White het. test = 116.659 [.000]

Sum of squared residuals

= 171.003 Jarque-Bera test = 449.795 [.000]

Variance of residuals = 0.572 Ramsey's RESET2 = 3.568 [.060]

Std. error of regression = 0.756 F (zero slopes) = 246.404 [.000]

R-squared = 0.805 Schwarz B.I.C. = 361.696

Adjusted R-squared = 0.801 Log likelihood = -344.53

LM het. test = 17.876

White het. test = 116.659 [.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, 817 - 838.

●

●

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

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, 751 - 763.

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 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 j=1,2,3,4

j=1

j=1,2

j=1,2,3

Asymptotic Actual Estimated 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, 95 - 230.

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 A

TTENTION