Topic 10: Heteroscedasticity

Question 1

How can we detect the presence of heteroscedasticity?

Answer 1

• prior knowledge, we can sometimes expect it
• Plot X&Y, see pattern
• Plot u_i² against Yi^
• Park test
• Glesjer test
• Goldelf-Quandt test
• White test

Question 2

What are the consequences of undetected heteroscedasticity?

Answer 2

• estimators still linear & unbiased, but not minimum variance
• Coefficient variation estimates are all wrong, so tests all invalid

Question 3

What is the park test?

Answer 3

• Run the regression of the log of the square residuals against the log of the previous regressors eg: lnYi = B₁ + B₂ ln X_i + v_i
• F test: is there a relationship between residuals & variables

Question 4

What is a problem with the park test?

Answer 4

-Not clear if it’s regression satisfies OLS assumptions -> vi may itself be heteroscedastic

Question 5

What is the Glesjer test?

Answer 5

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

Question 6

What is the Goldfeld-Quandt test?

Answer 6

• 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

Question 7

What is the White test?

Answer 7

• 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

Question 8

What are some problems with the White test?

Answer 8

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.

Question 9

How can heteroscedasticity be treated?

Answer 9

• GLS
• Whites heteroscedasticity corected / robust / sandwich errors
• Transform model based on suspected heteroscedasticity pattern
• Estimate variance then run GLS
• Use log transform

Question 10

How can GLS be used to correct heteroscedasticity?

Answer 10

• In GLS, you minimise u²i*, where u²i* = u²_i/variance_i
• is Blue - But we have to know SE(i) for all i (not likely)

Question 11

What is the White approach for fixing heteroscedasticity?

Answer 11

• 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

Question 12

What is the transformation method for correcting heteroscedasticity?

Answer 12

• 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??