How can we detect the presence of heteroscedasticity?

-prior knowledge, we can sometimes expect it

-Plot X&Y, see pattern

-Plot u_{i}^{2} against Yi^

-Park test

-Glesjer test

-Goldelf-Quandt test

-White test

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 = B_{1} + B_{2} ln X_{i} + v_{i}

-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 Xi^{2} 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

-Assumes normality

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?

-GLS

-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 u^{2}i*, where u^{2}i* = u^{2}_{i}/variance_{i}

-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??