第9章 异方差问题检验与修正

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
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
even if we do not assume homoskedasticity.
➢ take the simple regression as an example
➢ Y= b0 + b1 X +u ➢ We know the OLS estimator of b1 is
bˆ1
Xi X Yi Xi X2
Patterns of heteroskedasticity
Y X
homoskedasticity
Y
Decreasing with X
Y
X Increasing with X
Y
X Complicated heteroskedasticity
The relation between R&D expenditure and Sales
manufacture 32405.6 1083 3751.9 polymer
141649.9 3163.8 9761.4
leisure
35107.7 1620.6 2884.1 computer
175025.8 13210.7 19774.5
paper
40295.4 421.7 4645.7 fuel
Consider a cross-section study of family income and expenditures. It seems plausible to expect that low income individuals would spend at a rather steady rate, while the spending patterns of high income families would be relatively volatile.
14655.1 178.3 274.8 consumption 101314.1 1595.3 10278.9
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
ind
sales rdexp profit ind
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
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?
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)
.
.பைடு நூலகம்
E(Y|X) = b0 + b1X
.
X1
X2
X3
X
Examples
Generally, cross-section data more easily induce heteroskedasticity because of different characteristics of different individuals.
300000
sales (million dollars)
Why Worry About Heteroskedasticity?
The consequences of
heteroskedasticity
OLS estimates are still unbiased and consistent,
If we examine sales of a cross section of firms in one industry, error terms associated with very large firms might have larger variances than those error terms associated with smaller firms; sales of larger firms might be more volatile than sales of smaller firms.
230614.5 1703.8 22626.6
food
70761.6 509.2 5036.4 auto
293543 9528.2 18415.4
The scatter graph between R&D expenditure and Sales
15000
10000
5000
0
0
100000
200000