Heteroskedasticity Flashcards
What is homoskedasticity
The variance of the disturbance term is constant. If not then MLR 5 is broken.
Causes of heteroskedasticity
Size issues - when x is big, disturbance term is potentially bigger e.g child weight-height.
Is a more common problem when using aggregated data - e.g industries in countries.
Functional forms can be incorrectly specified. If the disturbance term is large take logs to reduce the impact.
Consequences of heteroskedasticity
Does not bias the coefficients
However estimates could be more efficient
SE are wrong so T or F test is invalid
Ways to test for heteroskedasticity
Goldfeld-Quandt test
Breusch-Pagan test
White test
Goldfeld-Quandt test
tests whether ssr is bigger as x gets bigger
1) order data in increasing size of x
2) Omit the middle quarter
3) Compare size off ssr for top and bottom quarters
4) compute F statistic
GQ test: what if there’s more than one explanatory variable ?
Guess at which one is most likely to be heteroskedastic
Breusch-Pagan test
1) (yi − yˆi)^2 are squared residuals
2) Regress squared residuals against the x variables
That means (yi − yˆi)^2 becomes the new dependant variable.
3) Find the R^2 for the new regression
4) Test statistic is nR^2
5) X^2 distribution (chi squared) with df = number of explanatory variables
White test
1) run your OLS equation, obtain the residuals and square them (yi − yˆi)^
2) Regress these against ALL of the explanatory variables, their squares and their cross-products.
Take the R^2 from this equation and the test statistic and use it to work out the test statistic using nR^2
chi squared distribution with df = number of explanatory variables
Which test to use?
If confident of the source use GQ
If unsure use one of the others
If large sample, use white
Why is OLS insufficient in the presence of heteroskedasticity?
It gives equal weight to all observations
How can we solve heteroskedasticity if we know sigma
Need to use weighted least squares, giving more weight to accurate observations. Use σi
if we can.
Variance of ui
σ^2
variance of ui/xi
1/x^2 x var (ui)
What is the new model after dividing by sigma
y’ = β0h + β1x’ + u’
What to do if we don’t know σi
We will know it is proportional to some measurement variable - let’s call it zi. Do the same method but divide through by zi
