How can we detect the presence of heteroscedasticity?
-prior knowledge, we can sometimes expect it
-Plot X&Y, see pattern
-Plot ui2 against Yi^
What are the consequences of undetected heteroscedasticity?
-estimators still linear & unbiased, but not minimum variance
-Coefficient variation estimates are all wrong, so tests all invalid
What is the park test?
-Run the regression of the log of the square residuals against the log of the previous regressors eg: lnYi = B1 + B2 ln Xi + vi
-F test: is there a relationship between residuals & variables
What is a problem with the park test?
-Not clear if it's regression satisfies OLS assumptions -> vi may itself be heteroscedastic
What is the Glesjer test?
1. Run second regression F test of absolute errors against combinations of offending regressors
2. |u^i| ~ sqr(Xi) or Xi2 or 1 / Xi or 1 / sqr(Xi)
- Still has OLD assumption problems, particularly heteroscedasticity & zero mean of error
- More suitable for large samples
What is the Goldfeld-Quandt test?
-Assume that variance is positively related to a regressor -Order observations by that regressor
-Remove c central observations to form two groups
-Create F stat from the RSS of the two groups
What is the White test?
-Run second regression of square residuals against all combinations of regressors crossmultiplied, then all polynomials of the regressors (normally just square).
-Under null nR^2 has a chi distribution for large samples, where df = number of regressors, not including constant in the second model. -not reliant on normality assumption
What are some problems with the White test?
A rejected white test can mean heteroscedasticity or a specification error. Sometimes a white test without crossproducts is considered a test purely for heteroscedasticity, while the test with the crossproducts is a test for both.
How can heteroscedasticity be treated?
-Whites heteroscedasticity corected / robust / sandwich errors
-Transform model based on suspected heteroscedasticity pattern
-Estimate variance then run GLS
-Use log transform
How can GLS be used to correct heteroscedasticity?
-In GLS, you minimise u2i*, where u2i* = u2i/variancei
-is Blue - But we have to know SE(i) for all i (not likely)
What is the White approach for fixing heteroscedasticity?
-Gives new standard errors from matrix technique
-Not as good because estimators remain the same
-SE can be higher or lower
-Only valid for large samples
What is the transformation method for correcting heteroscedasticity?
-Assume some pattern, say Variance(i) = variance*Xi
-We can then divide the entire model by sqr(Xi) to make the whole thing homoscedastic (the square root of regressor in the model)
-But what regressor to use for transformation??