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Copyright 1996 Lawrence C. Marsh Var-Cov Matrix y t
β 1 + β 2 x t2 + β 3 x t3 + e t var(b 1 ) cov(b 1 ,b
) cov(b 1 ,b 3 ) cov(b 1 ,b 2 ,b 3 ) = cov(b
1 ,b 2 ) var(b 2 ) cov(b 2 ,b 3 ) cov(b 1 ,b
) cov(b 2 ,b 3 ) var(b 3 )
1 , b
2 , and b
3 have covariance matrix: 7.17
36 Copyright 1996 Lawrence C. Marsh Normal
y t = β 1 + β 2 x 2t + β 3 x 3t +. . .+ β K x Kt
+ e t y t ~N (
β 1 + β 2 x 2t + β 3 x 3t +. . .+ β K x Kt ), σ
2 e t ~ N(0, σ
2 ) This implies and is implied by: b k
β k , var(b k ) z = ~ N(0,1) for k = 1,2,...,K b k
−
β k var(b
k ) Since b k is a linear function of the y t ’s: 7.18 Copyright 1996 Lawrence C. Marsh Student-t b k
−
β k var(b
k ) ^ t = = b k −
β k se(b k ) Since generally the population variance of b k , var(b k ) , is unknown , we estimate it with which uses σ
2 instead of σ
. var(b
k ) ^ ^ t has a Student-t distribution with df=( T − K ). 7.19
Copyright 1996 Lawrence C. Marsh Interval Estimation b k
k se(b
k ) P − t c ≤
≤ t c = 1 − α
t c is critical value for (T-K) degrees of freedom such that P( t
≥
t c ) = α /2. P b
− t c se(b
k )
≤
β k
≤ b k + t c se(b k )
= 1
− α Interval endpoints: b k
− t c se(b k ) , b k + t c se(b k ) 7.20 Copyright 1996 Lawrence C. Marsh Hypothesis Testing and Nonsample Information Chapter 8 Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein. 8.1
Copyright 1996 Lawrence C. Marsh 1. Student-t Tests 3. F-Tests 4. ANOVA Table 5. Nonsample Information 6. Collinearity 7. Prediction Chapter 8: Overview 8.2
Copyright 1996 Lawrence C. Marsh Student - t Test y t
β 1 + β 2 X t2 + β 3 X t3
+ β 4 X t4 + e
t Student-t tests can be used to test any linear combination of the regression coefficients: H 0 : β 2 + β 3 + β 4 = 1 H 0 : β 1 = 0 H 0 : 3 β 2 − 7 β 3 = 21 H 0 : β 2
−
β 3
≤ 5 Every such t-test has exactly T − K degrees of freedom where K=#coefficients estimated(including the intercept). 8.3
37 Copyright 1996 Lawrence C. Marsh One Tail Test y t
β 1 + β 2 X t2 + β 3 X t3
+ β 4 X t4 + e
t H 0 : β 3 ≤ 0 H 1 : β 3 > 0 b 3 se( b 3 ) t = ~ t
(
−
) t c 0 df = T − K = T − 4 α (1 − α) 8.4
Copyright 1996 Lawrence C. Marsh Two Tail Test y t
β 1 + β 2 X t2 + β 3 X t3
+ β 4 X t4 + e
t H 0 : β 2 = 0 H 1 : β 2 ≠ 0 b 2 se( b 2 ) t = ~ t
(
−
) t c 0 df = T − K = T − 4 α/2 (1 − α) -t c α/2 8.5
Copyright 1996 Lawrence C. Marsh Goodness - of - Fit 0 ≤
2
≤ 1 Coefficient of Determination SST R
= = Σ (y t − y) 2
T ^ SSR Σ (y t − y) 2 t = 1 T 8.6
Copyright 1996 Lawrence C. Marsh Adjusted R-Squared Adjusted Coefficient of Determination Original: Adjusted: SST/
(T − 1) R 2 = 1 − SSE/
(T − K) SST = 1
− SSE
R 2 = SST SSR
8.7 Copyright 1996 Lawrence C. Marsh Computer Output Table 8.2 Summary of Least Squares Results Variable Coefficient Std Error t-value p-value constant 104.79 6.48 16.17 0.000 price − 6.642 3.191 − 2.081 0.042 advertising 2.984 0.167 17.868 0.000 b 2 se( b 2 ) t =
= − 6.642 3.191 − 2.081 = 8.8
Copyright 1996 Lawrence C. Marsh Reporting Your Results y t
104.79
− 6.642
X t2
+ 2.984
X t3 ^ (6.48) (3.191) (0.167) (s.e.) y t = 104.79
−
6.642 X t2 + 2.984
X t3 ^ (16.17) (-2.081) (17.868) (t) Reporting t-statistics: Reporting standard errors: 8.9
38 Copyright 1996 Lawrence C. Marsh Single Restriction F-Test y t
β 1 + β 2 X t2 + β 3 X t3
+ β 4 X t4 + e
t H 0 : β 2 = 0 H 1 : β 2 ≠ 0 df d = T − K = 49 df n = J = 1 (SSE R
− SSE
U )/J SSE U
T − K ) F =
(1964.758 − 1805.168)/1 1805.168/(52 − 3) = = 4.33
By definition this is the t-statistic squared: t =
− 2.081
F = t 2 = 4.33 8.10
Copyright 1996 Lawrence C. Marsh Multiple Restriction F-Test y t
β 1 + β 2 X t2 + β 3 X t3
+ β 4 X t4 + e
t H 0 : β 2 = 0, β 4 = 0 H 1 : H 0 not true df d = T − K = 49 df n = J = 2 (SSE R
− SSE
U )/J SSE U
T − K ) F =
First run the restricted regression by dropping X t2
and X t4 to get SSE R . Next run unrestricted regression to get SSE U . 8.11 Copyright 1996 Lawrence C. Marsh F-Tests
(SSE R
− SSE
U )/J SSE U
T − K ) F =
F-Tests of this type are always right-tailed , even for left-sided or two-sided hypotheses, because any deviation from the null will make the F value bigger (move rightward). 0 F
α (1 − α)
f( F ) F 8.12
Copyright 1996 Lawrence C. Marsh F-Test of Entire Equation y t
β 1 + β 2 X t2 + β 3 X t3 + e t H 0 :
β 2 = β 3 = 0 H 1 : H 0 not true df d
T − K = 49 df n = J = 2 (SSE R
− SSE
U )/J SSE U
T − K ) F =
(13581.35 − 1805.168)/2 1805.168/(52 − 3) = = 159.828 We ignore β 1 .
F c
3.187 α = 0.05 Reject
H 0 ! 8.13 Copyright 1996 Lawrence C. Marsh ANOVA Table Table 8.3 Analysis of Variance Table Sum of Mean Source DF Squares Square F-Value Explained 2 11776.18 5888.09 158.828 Unexplained 49 1805.168 36.84 Total 51 13581.35 p-value : 0.0001 SST R
= = SSR = 0.867
11776.18 13581.35
8.14 Copyright 1996 Lawrence C. Marsh Nonsample Information ln(y t
β 1 + β 2 ln(X t2 )
+ β 3 ln(X t3 ) + β 4 ln(X
t4 )
+ e t A certain production process is known to be Cobb-Douglas with constant returns to scale. β 2 + β 3 + β 4 = 1 where
β 4
= (1 −
β 2 − β 3 ) ln(y
t /X t4 ) = β 1 + β 2 ln(X t2 /X t4 ) + β 3 ln(X
t3 /X t4 ) + e
t y t = β 1 + β 2 X t2
+ β 3 X t3
+ β 4 X t4
+ e t * * * * Run least squares on the transformed model. Interpret coefficients same as in original model. 8.15
39 Copyright 1996 Lawrence C. Marsh Collinear Variables The term “independent variable” means an explanatory variable is independent of of the error term, but not necessarily independent of other explanatory variables. Since economists typically have no control over the implicit “experimental design”, explanatory variables tend to move together which often makes sorting out their separate influences rather problematic. 8.16
Copyright 1996 Lawrence C. Marsh Effects of Collinearity 1. no least squares output when collinearity is exact. 2. large standard errors and wide confidence intervals. 3. insignificant t-values even with high R
and a
significant F-value. 4. estimates sensitive to deletion or addition of a few observations or “insignificant” variables. 5. good “within-sample”(same proportions) but poor “out-of-sample”(different proportions) prediction. A high degree of collinearity will produce: 8.17
Identifying Collinearity Evidence of high collinearity include: 1. a high pairwise correlation between two explanatory variables. 2. a high R-squared when regressing one explanatory variable at a time on each of the remaining explanatory variables. 3. a statistically significant F-value when the t-values are statistically insignificant. 4. an R-squared that doesn’t fall by much when dropping any of the explanatory variables. 8.18
Mitigating Collinearity Since high collinearity is not a violation of any least squares assumption, but rather a lack of adequate information in the sample: 1. collect more data with better information. 2. impose economic restrictions as appropriate. 3. impose statistical restrictions when justified. 4. if all else fails at least point out that the poor model performance might be due to the collinearity problem (or it might not). 8.19
Copyright 1996 Lawrence C. Marsh Prediction Given a set of values for the explanatory variables, (1 X 02 X
03 ), the best linear unbiased predictor of y is given by: y t = β 1 + β 2 X t2
+ β 3 X t3 + e t This predictor is unbiased in the sense that the average value of the forecast error is zero . y 0 = b
1 + b
2 X 02 + b 3 X 03 ^ 8.20 Copyright 1996 Lawrence C. Marsh Extensions of the Multiple Regression Model Chapter 9 Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein. 9.1
40 Copyright 1996 Lawrence C. Marsh Topics for This Chapter 1. Intercept Dummy Variables 2. Slope Dummy Variables 3. Different Intercepts & Slopes 4. Testing Qualitative Effects 5. Are Two Regressions Equal? 6. Interaction Effects 7. Dummy Dependent Variables 9.2
Copyright 1996 Lawrence C. Marsh Intercept Dummy Variables Dummy variables are binary (0,1) D t = 1 if red car, D
t = 0 otherwise. y t
β 1 + β 2 X t + β 3 D t + e t y t =
speed of car in miles per hour X t = age of car in years Police: red cars travel faster . H
: β 3 = 0 H 1 : β 3 > 0 9.3
Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 X t
+ β 3 D t + e t red cars: y t
β 1 + β 3 ) + β 2 x t + e
t
other cars: y t = β 1 + β 2 X t + e
t
y t X t miles per
hour age in years 0 β
+ β 3 β 1 β 2 β 2 red cars
other cars 9.4
Copyright 1996 Lawrence C. Marsh Slope Dummy Variables y t
β 1 + β 2 X t + β 3 D t X t + e t y t = β 1 + ( β 2 + β 3 )X t
+ e t
y t = β 1 + β 2 X t + e
t
y t X t value of porfolio years 0 β 2 +
β 3 β 1 β 2 stocks bonds
Stock portfolio: D t = 1 Bond portfolio: D t = 0
β 1 = initial investment 9.5
Different Intercepts & Slopes y t
β 1 + β 2 X t + β 3 D t
+ β 4 D t X t + e t y t = (
β 1 + β 3 ) + ( β 2 + β 4 )X t + e
t y t = β 1 + β 2 X t
+ e t
y t X t harvest
weight of
corn rainfall
β 2 + β 4 β 1 β 2 “miracle” regular
“miracle” seed: D t = 1 regular seed: D t = 0
β 1 + β 3 9.6 Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 X t + β 3 D t
+ e t β 2 β 1 + β 3 β 2 β 1 y t X t Men
Women 0 y t =
β 1 + β 2 X t + e
t For
men :
D t
= 1. For
women :
D t = 0. years of experience y t = ( β 1 + β 3 ) + β 2 X t + e t wage
rate H 0 :
β 3 = 0 H 1
β
> 0
. . Testing for discrimination in starting wage 9.7
41 Copyright 1996 Lawrence C. Marsh y t = β 1 + β 5 X t + β 6 D t X t + e t β 5 β 5 + β 6 β 1 y t X t Men Women
0 y
t =
β 1 + ( β 5 + β 6 )X t + e t
y t = β 1 + β 5 X t + e
t For men D t = 1.
For women D t = 0. Men and women have the same starting wage, β 1
increase at different rates (diff.= β 6 ).
β 6 > 0 means that men’s wage rates are increasing faster than women's wage rates. years of experience wage rate
9.8 Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 X t + β 3
D t + β 4
D t X t + e
t β 1 + β 3 β 1 β 2 β 2 + β 4 y t X t Men
Women 0
y t = ( β 1 + β 3 ) + ( β 2 + β 4 ) X t + e
t y
t =
β 1 + β 2 X t + e
t Women are given a higher starting wage, β 1
, while men get the lower starting wage, β 1
β 3 , ( β 3 < 0 ). But, men get a faster rate of increase in their wages, β 2 + β 4 , which is higher than the rate of increase for women, β 2 , (since β 4 > 0 ). years of experience An Ineffective Affirmative Action Plan women are started at a higher wage. Note: ( β
< 0 ) wage rate
9.9 Copyright 1996 Lawrence C. Marsh Testing Qualitative Effects 1. Test for differences in intercept . 2. Test for differences in slope . 3. Test for differences in both intercept and
slope . 9.10 Copyright 1996 Lawrence C. Marsh H 0 : β 3 ≤ 0 vs . Η 1 :
β 3
> 0 H 0 : β 4 ≤ 0 vs . Η 1 :
β 4
> 0 Y t = β
1 +β
2 X t + β 3 D t + β
4 D t X t b − 0 3 Est.Var b
3 ˜ t n − 4 b
− 0 4 Est.Var b
4 ˜ t n − 4 men: D t = 1 ; women: D t = 0
Testing for discrimination in starting wage. Testing for discrimination in wage increases. intercept slope
+ e t 9.11 Copyright 1996 Lawrence C. Marsh Testing
: H
o :
β 3
=
β 4 = 0
H 1 : otherwise and
SSE R = ( y t − b 1 − b 2 X t ) 2 t = 1 T ∑ SSE U = ( y t − b 1 − b 2 X t − b 3 D t − b 4 D t X t ) 2 t = 1 T ∑ ( SSE R −
U ) /
2 SSE U / (
T − 4 ) ∼
T − 4 2 intercept and slope 9.12
Are Two Regressions Equal? y t
β 1 + β 2 X t +
β 3
D t + β 4
D t X t + e
t variations of “The Chow Test” I. Assuming equal variances (pooling): men: D
= 1 ; women: D t = 0
H o : β 3 = β 4 = 0 vs. H 1 : otherwise y t = wage rate This model assumes equal wage rate variance. X t = years of experience 9.13
42 Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 X t + e t II. Allowing for unequal variances: y tm
δ 1 + δ 2 X tm + e
tm y tw = γ 1 + γ 2 X tw + e tw Everyone: Men only: Women only: SSE R
β 1 , β 2 . Allowing men and women to be different. SSE
m SSE
w where SSE U = SSE
m +
SSE w F = (SSE R − SSE
U )/J
SSE U /(T − K) J = # restrictions K=unrestricted coefs. (running three regressions) J = 2 K = 4 9.14
Copyright 1996 Lawrence C. Marsh Interaction Variables 1. Interaction Dummies 2. Polynomial Terms (special case of continuous interaction) 3. Interaction Among Continuous Variables 9.15
1. Interaction Dummies y t
β 1 + β 2 X t +
β 3 M t + β 4 B t + e
t For
men : M t
= 1. For women :
M t = 0. For
black :
B t
= 1. For nonblack
: B
t
= 0. No Interaction: wage gap assumed the same: y t = β 1 + β 2 X t + β 3 M t + β 4 B t + β 5 M t B t
+ e t Interaction: wage gap depends on race: Wage Gap between Men and Women y t = wage rate; X t = experience 9.16 Copyright 1996 Lawrence C. Marsh 2. Polynomial Terms y t
β 1 + β 2 X t + β 3
X 2 t + β 4 X 3 t + e
t Linear in parameters but nonlinear in variables: y t
t = age
Polynomial Regression y t X
t People retire at different ages or not at all. 90 20 30 40 50 60 80 70 9.17 Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 X
t + β 3 X 2 t
+ β 4 X 3 t + e t y t = income; X t = age Polynomial Regression Rate income is changing as we age: ∂ y
∂ X t = β 2 + 2 β 3 X
t
+ 3 β 4 X 2 t Slope changes as X
t changes. 9.18
Copyright 1996 Lawrence C. Marsh 3. Continuous Interaction y t
β 1 + β 2 Z t + β 3 B t + β 4 Z t B t
+ e t Exam grade = f(sleep: Z t
B t ) Sleep and study time do not act independently. More study time will be more effective when combined with more sleep and less effective when combined with less sleep. 9.19
43 Copyright 1996 Lawrence C. Marsh Your mind sorts things out while you sleep (when you have things to sort out.) y t
β 1 + β 2 Z t + β 3 B t + β 4 Z t B t
+ e t Exam grade = f(sleep: Z t
B t ) ∂ y t ∂ B t = β 2 + β 4 Z t Your studying is more effective with more sleep. ∂ y
∂ Z t = β 2 + β 4 B t continuous interaction 9.20 Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 Z t + β 3 B t + β 4 Z t B t
+ e t Exam grade = f(sleep: Z t , study time: B t ) If Z t + B t = 24 hours, then B t = (24 −
Z t ) y t = β 1 + β 2 Z t + β 3 (24
−
Z t )
+ β 4 Z t (24 −
Z t )
+ e t y t = (
β 1 + 24
β 3 ) + (
β 2 −β 3 + 24 β 4 ) Z t
−
β 4 Z 2 t
+ e t y t = δ 1 + δ 2 Z t +
δ 3 Z 2 t
+ e t Sleep needed to maximize your exam grade: ∂ y t ∂ Z t = δ 2 + 2 δ 3 Z t
= 0 where
δ 2 > 0 and
δ 3 < 0 − δ
2 2δ 3 Z t
= 9.21
Copyright 1996 Lawrence C. Marsh 1. Linear Probability Model 2. Probit Model 3. Logit Model Dummy Dependent Variables 9.22
Copyright 1996 Lawrence C. Marsh Linear Probability Model y i
β 1 + β 2 X i2 +
β 3
X i3
+ β 4 X i4 + e i X i2 = total hours of work each week 1 quits job 0 does not quit y i
= X i3 = weekly paycheck X i4
= hourly pay (X i3
divided by X i2 ) 9.23
Copyright 1996 Lawrence C. Marsh X i2 y i = β 1 + β 2 X i2 +
β 3
X i3
+ β 4 X i4 + e i y t = 1 0 y t = total hours of work each week y i = b 1 + b 2 X i2 + b 3
X i3
+ b 4 X i4 ^ y i ^ Read predicted values of y i off the regression line : Linear Probability Model 9.24 Copyright 1996 Lawrence C. Marsh 1. Probability estimates are sometimes less than zero or greater than one. 2. Heteroskedasticity is present in that the model generates a nonconstant error variance. Linear Probability Model Problems with Linear Probability Model: 9.25
44 Copyright 1996 Lawrence C. Marsh Probit Model z i
β 1 + β 2 X i2 + . . . 2
π f(z i ) = e − 0.5z
i 2 1 F(z i ) = P[ Z ≤ z i ] = ∫ e − 0.5u
2 du 2 π 1 Normal probability density function: Normal cumulative probability function: z i
latent variable, z i : 9.26
Copyright 1996 Lawrence C. Marsh p i = P[ Z ≤ β
1 +
β 2 X i2 ] = F(
β 1 + β 2 X i2 ) Since z i = β 1 + β 2 X i2 + . . . , we can
substitute in to get :
Probit Model X i2 total hours of work each week y t = 1 0 y t = 9.27 Copyright 1996 Lawrence C. Marsh Logit Model p i
= 1 1 +
e − ( β 1 +
β 2 X i2
+ . . .) Define p i : For
2 > 0
, p i
will approach 1 as X i2 + ∞
p i
is the probability of quitting the job. For
β 2 > 0 , p
i will approach 0 as X i2 − ∞ 9.28
Copyright 1996 Lawrence C. Marsh
Logit Model X i2 total hours of work each week y t = 1 0 y t = p i
= 1 1 +
e − ( β 1 +
β 2 X i2
+ . . .) p i is the probability of quitting the job. 9.29
Maximum Likelihood Maximum likelihood estimation (MLE) is used to estimate Probit and Logit functions. The small sample properties of MLE are not known, but in large samples MLE is normally distributed, and it is consistent and asymptotically efficient . 9.30
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein. 10.1
45 Copyright 1996 Lawrence C. Marsh The Nature of Heteroskedasticity Heteroskedasticity is a systematic pattern in the errors where the variances of the errors are not constant. Ordinary least squares assumes that all observations are equally reliable . For efficiency (accurate estimation/prediction) reweight observations to ensure equal error variance. 10.2
y t = β 1 + β 2 x t
+ e t Regression Model E(e t ) = 0 var(e t ) = σ
2 zero mean: homoskedasticity: nonautocorrelation: cov(e
t , e
s ) =
0 t
≠
s heteroskedasticity: var(e
t ) =
σ t
2 10.3
Copyright 1996 Lawrence C. Marsh Homoskedastic pattern of errors x t
t . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . income
consumption 10.4
Copyright 1996 Lawrence C. Marsh . . x t x 1 x 2 y t f(y t ) The Homoskedastic Case . . x 3 x 4 income consumption 10.5
Copyright 1996 Lawrence C. Marsh Heteroskedastic pattern of errors x t
t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . income
consumption 10.6
Copyright 1996 Lawrence C. Marsh . x
t x 1 x 2 y t f(y
t ) consumption x 3
. The Heteroskedastic Case income rich people poor people 10.7
46 Copyright 1996 Lawrence C. Marsh Properties of Least Squares 1. Least squares still
and
unbiased . 2. Least squares not efficient . 3. Usual formulas give incorrect standard errors for least squares. 4. Confidence intervals and hypothesis tests based on usual standard errors are
. 10.8 Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 x t
+ e t heteroskedasticity: var(e t ) = σ t
2 incorrect formula for least squares variance: var(b 2
σ 2 Σ ( x t − x
) 2 correct formula for least squares variance: var(b 2
Σ σ t 2 ( x t − x
) 2 [Σ (
x t − x
) 2 ] 2 10.9 Copyright 1996 Lawrence C. Marsh Hal White’s Standard Errors White’s estimator of the least squares variance: est.var(b 2 ) =
Σ e t 2 ( x t − x
) 2 [Σ (
x t − x
) 2 ] 2 ^ In large samples White’s standard error (square root of estimated variance) is a correct / accurate / consistent measure. 10.10
Two Types of Heteroskedasticity 1.
Heteroskedasticity. ( continuous function(of x t , for example)) 2. Download 0.54 Mb. Do'stlaringiz bilan baham: 
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