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Lecture 11


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LECTURE 11: HETEROSKEDASTICITY HETEROSKEDASTICITY Homoskedasticity: Var( ε i ) = σ 2 Heteroskedasticity: Var( ε i ) = σ i 2 What happens if we violate this assumption of the CLRM? OLS still unbiased but not minimum variance (i.e. not BLUE) Our estimates of the variance will be biased Because of this, our usual hypothesis testing routine is unreliable. Why is OLS inefficient? 2 i β , β ε ˆ min 1 0 With OLS, we weight each 2 i ε ˆ equally, whether it comes from a population with a large variance or with a small variance. Ideally, we would like to give more weight to observations coming from populations with smaller variances, as this would enable us to estimate the PRF more accurately. What do we do? WEIGHTED LEAST SQUARES DETECTION OF HETEROSKEDASTICITY No sure-fire method; generally, we don’t know true σ 2 i and often only have a few observations for each X.
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1. Priors More likely in cross-sectional data Firms (example, firm size and investment) Individuals (earning and savings, education and earnings) 2. Graphical examination of residuals Plot i ε ˆ or 2 i ε ˆ against X to see if there is a pattern Plot i ε ˆ or 2 i ε ˆ against i Y ˆ if there are multiple X’s 3. More rigorous tests There are a number of tests that involve regressing forms of the residual on X’s. One test: WHITE’S GENERAL HETEROSKEDASTICITY TEST a) Y i = β 0 + β 1 X 1i + β 2 X 2i + ε i b) Estimate with OLS, obtain residuals i ε ˆ c) Run the following auxiliary regression: i i 2 i 1 5 2 i 2 4 2 i 1 3 i 2 2 i 1 1 0 2 i v
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