AAEC 4302
ADVANCED
STATISTICAL METHODS IN
AGRICULTURAL RESEARCH
Chapter 15.1
Heteroscedasticity
Heteroscedasticity
• Assumptions of a normal regression model:
– The disturbances are independent random variables
– The standard deviations of all disturbances are equal:
σ(ui) = σu for all i
• Heteroscedasticity occurs when the error term
(and thus the dependent variable Y) does not
have a constant variance across observations:
σ(ui) = σi
Heteroscedasticity
Heteroscedasticity
Heteroscedasticity
• The OLS parameter estimators are still
unbiased, but OLS standard errors are
incorrect
• Also, the OLS parameter estimators are no
longer the most efficient (i.e. minimum
variance), even if the error term is normally
distributed
Heteroscedasticity
• SAVINGS=-1.062+0.295INCOME
(0.851)
(0.075)
R2=0.137 [1.233]
[0.152]
Original standard error (t*=3.94)
Revised standard error (t*=1.94)
Heteroscedasticity
Detection
Examine the residuals
σ(ui) = σXi
•
•
•
•
White Test
Breusch-Pagan Test
Park Test
Glejser Test
Heteroscedasticity
White test
Auxiliary regression
H0: σ(ui) = σu - no heteroscedasticity
H1: σ(ui) = σi
H0 is rejected when nR2 is large
Heteroscedasticity
Estimation
Respecify the original model in such a way that resulting
disturbances are homoscedastic
 Yi 
1
ui
    o    1 
Xi
 Xi 
 Xi 
Heteroscedasticity


SAVINGSi
1
 0.228
  0.197
INCOMEi
 INCOMEi 
(0.222)
(0.044)
SAVINGSi = 0.228 + 0.197 INCOMEi
(0.222)
(0.044)
• GLS (Generalized Least Squares)
– Also known as weighted least squares