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http://www.fsv.cuni.cz. Charles University. Founded 1348. 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. - PowerPoint PPT PresentationTRANSCRIPT
Founded 1348Charles University
http://www.fsv.cuni.cz
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
[email protected]@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