Heteroskedasticity

Hill et al Chapter 11

Predicting food expenditure

• Are we likely to be better at predicting food expenditure at:– low incomes;– high incomes?

1 2y x

The nature of heteroskedasticity

ˆty = 40.768+0.1283 tx

Violation of assumption MR. 32var( ) var( )t ty e

2var( ) var( )t t ty e

Consequences of Heteroskedasticity

• The least squares estimator is still a linear and unbiased estimator, but it is no longer best. It is no longer B.L.U.E.

• The standard errors usually computed for the least squares estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading.

White’s estimator of the standard error in the presence of hetero.

2 2

2 22var( )

t t

t

x xb

x x

2 2

2 22

ˆˆvar( )

t t

t

x x eb

x x

ˆty = 40.768 + 0.1283 xt

(23.704) (0.0382) (White) (22.139) (0.0305) (incorrect)

White: 2 2se( )cb t b = 0.1283 2.024(0.0382) = [0.051, 0.206]

Incorrect: 2 2se( )cb t b = 0.1283 2.024(0.0305) = [0.067, 0.190]

Proportional Hetero.

1 2t t ty x e

20 var

cov( , ) 0t t t

i j

E e e

e e i j

2 2var t t te x

Transforming the model to make it homoskedastic

* tt

t

yy

x *

1

1t

t

xx

*2

tt

t

xx

x

* tt

t

ee

x

1 1 2 2t t t ty x x e

2 21 1var( ) var var( )t

t t tt tt

ee e x

x xx

Comparing the estimates from OLS and GLS

ˆty =31.924+0.1410 tx

(17.986) (0.0270)

ˆty =40.768 +0.1283 xt

(23.704) (0.0382)

GLS

OLS and White

Detecting Hetero.

• Residual plots.– Simple regression– Multiple regression, plot against:

• each explanatory variable• time• fitted values

• Goldfield and Quandt test

The Goldfield and Quandt Test

• Split the sample in two (according to expected pattern of hetero.)• Compute variances for both samples.• Compute GQ stat:

• Reject null of equal variances if:

2 21 2ˆ ˆGQ

cGQ F

Example of GQ test

2 20 : tH 2 2

1 : t tH x

2285.93.35

682.46GQ

3.35 2.22,cGQ F

A sample with a heteroskedastic partition

Quantity = f (Price, Technology, Weather)

1 2 3t t tq p t e

21

22

0

var 1, ,13

var 14, ,26

t

t

t

E e

e t

e t

Testing the Variance Assumption

2 20 1 2:H

2 21 2 1:H

2122

ˆ 641.6411.11

ˆ 57.76GQ

1 2 13T T

3K

10,1011.11 2.98,GQ F

GLS through transformation

21 2 3 1

21 2 3 2

var 1, ,13

var 14, ,26

t t t t

t t t t

q p t e e t

q p t e e t

1 2 31 1 1 1 1

1 2 32 2 2 2 2

11, ,13

114, ,26

t t t

t t t

q p t et

q p t et

21

2 21 1 1

22

2 22 2 2

1var var 1 1, ,13

1var var 1 14, ,26

tt

tt

ee t

ee t

Implementation of GLSEstimate 2 for each sub-sample by OLS

21̂ = 641.64 2

2̂ = 57.76

ˆtq =138.1+21.72pt+3.283t

(12.7) (8.81) (0.812)