第9章 异方差问题检验与修正

293543 9528.2 18415.4
The scatter graph between R&D expenditure and Sales
15000
10000
5000
0
0
100000
200000
300000
sales (million dollars)
Why Worry About Heteroskedasticity?
heteroskedasticity, cont.
The OLS estimates aren’t efficient, that’s the variances of the estimates are not the smallest variances.
If the standard errors are biased, we can not use the usual t statistics or F statistics for drawing inferences. That is, the t test and F test and the confidence interval based on these test don’t work.
Patterns of heteroskedasticity
Y X
homoskedasticity
Y
Decreasing with X
Y




var(u|X)=s2 (homoskedasticity)
If this is not true, that is if the variance of u is different for different values of the X’s, then the errors are heteroskedastic
ui
2



b1
The consequences of heteroskedasticity, cont.
The R2 and adj-R2 are unaffected by heteroskedasticity.
R2 ESS 1 RSS
TSS
TSS
R
2

1

RSS n TSS n
var(ui|Xi)=si2(heteroskedasticity)
Example of homoskedasticity
f(Y|X)
.
Y
.
. E(Y|X) = b0 + b1X
homoskedasticity
X1
X2
X3
Example of Heteroskedasticity
f(Y|X)
.
There is a shortcut to do the residual plot test when there are more than 1 independent variables. That is, we plot the residual with the fitted value Ŷ, because Ŷ is just the linear combination of all Xs.
OLS estimates are still unbiased and consistent The R2 and adj-R2 are unaffected by
heteroskedasticity The standard errors of the estimates are biased.
What’s heteroskedasticity?
What is Heteroskedasticity?
Recall the assumption of homoskedasticity implied that conditional on the explanatory variables, the variance of the unobserved error, u, was constant
metal
21896.2 258.4 2828.1 electronics 116141.3 6107.5 8787.3
house
26408.3 494.7 225.9 chemistry 122315.7 4454.1 16438.8
manufacture 32405.6 1083 3751.9 polymer
The standard errors of the estimates are biased if we have heteroskedasticity
bˆ1 b1+ XXiiXXu2i ,
var bˆ1

varb1

+XXiiXXu2i
第9章 异方差：检验与修正
Heteroskedasticity: test and correction
Contents
What’s heteroskedasticity? Why worry about heteroskedasticity? How to test the heteroskedasticity? Corrections for heteroskedasticity?
The OLS estimates aren’t efficient. Then, the t test and F test and the confidence interval don’t work.
How to test the heteroskedasticity?
Residual plot
The consequences of
heteroskedasticity
OLS estimates are still unbiased and consistent,
even if we do not assume homoskedasticity.
take the simple regression as an example
k 1 1
Because RSS and TSS are not affected by heteroskedasticity, our R2 and adj-R2 are also not
affected by heteroskedasticity.
The consequences of heteroskedasticity, cont.
XiXXi 2Xva2r2ui

Becauseofheteroskedasticity,then varui si2, whicharenotconstant,therefore,
var bˆ1
Xi
X
s 2 2 i
sales
rdexp profit
packing
6375.3 62.5 185.1 nurse
80552.8 6620.1 13869.9
nonbank
11626.4 92.9 1569.5 space
95294 3918.6 4487.8
service
14655.1 178.3 274.8 consumption 101314.1 1595.3 10278.9
In a word, when there exists heteroskedasticity, we can not use t test and F test as usual. Or else, we’ll get the misleading result.
Summary of the consequences of heteroskedasticity
Residual plot, cont.

e2








X a) homoskedasticity
e2





X b)
e2
e2
e2


.
22
However, OLSestimateofthevarianceofbˆ1
is
Xi X
s2
.
2
Xi X
So, inthiscase, OLSestimatesofthevariancesofthepartial coefficientsarebiased.
The consequences of
X








X

X
c)
d)
e)
Residual plot, cont.
If there are more than one independent variables, we should plot the residual squared with all the independent variables, separately.
Y= b0 + b1 X +u We know the OLS estimator of b1 is
bˆ1

Xi X Yi Xi X2

b1
+
Xi X ui Xi X 2

E
bˆ1

E
b1
+

Xi X Xi X

X Increasing with X

Y
X Complicated heteroskedasticity
The relation between R&D expenditure and Sales