ECON20003 · Quantitative Methods 2
Heteroskedasticity & Dummy Independent Variables
Heteroskedasticity & Dummy Independent Variables covers two practical extensions of regression. Heteroskedasticity — error variance that changes across observations — leaves OLS coefficients unbiased but makes the standard errors wrong, which invalidates the t and F tests; you detect it with a residual-vs-fitted funnel or the Breusch-Pagan/White tests and remedy it with robust standard errors, weighted least squares, or re-specification. Dummy (indicator) variables let categorical information into the model: a 0/1 regressor shifts the intercept, c categories need c − 1 dummies (avoid the dummy-variable trap), and interaction terms shift the slope.
What this chapter covers
- 01Heteroskedasticity: OLS unbiased but inefficient, SEs wrong ⇒ invalid t/F
- 02Detection: residual-vs-fitted funnel; Breusch-Pagan / White (small p ⇒ heteroskedastic)
- 03Remedies: White robust SEs, weighted least squares, re-specify
- 04Dummy variables: 0/1 regressor shifts the intercept
- 05c categories need c − 1 dummies — the dummy-variable trap
- 06Interaction terms shift the slope; coefficient = mean-Y difference vs base category
Breusch-Pagan test and interpreting a dummy coefficient
- 1 markState the BP hypotheses: H₀: homoskedasticity (constant error variance) versus H₁: heteroskedasticity.
- 2 marksDecision rule: reject H₀ if the BP p-value is below α. Here p = 0.008 < 0.05, so reject H₀.
- 1 markConclude on the variance: there is significant evidence of heteroskedasticity, so the ordinary OLS standard errors are unreliable and the t/F tests are invalid as reported.
- 1 markState the remedy: re-run the model with White (heteroskedasticity-robust) standard errors, or use weighted least squares, before doing any inference.
- 2 marksInterpret the dummy in a log-dependent model: a coefficient of 0.12 means a renovated house sells for approximately 12% more than an otherwise-identical non-renovated house, holding the other predictors constant.
Key terms
- Heteroskedasticity
- Non-constant error variance across observations. OLS stays unbiased but becomes inefficient and its standard errors are wrong, so the usual t and F tests are invalid until corrected.
- Breusch-Pagan / White test
- Tests for heteroskedasticity with H₀ of constant variance. A small p-value rejects homoskedasticity, signalling you should switch to robust standard errors or weighted least squares.
- Dummy variable
- A 0/1 indicator that encodes a category. Its coefficient is the mean difference in Y relative to the omitted base category, holding everything else fixed; it shifts the regression intercept.
- Dummy-variable trap
- Including a dummy for every category (plus an intercept) creates perfect collinearity. Avoid it by using c − 1 dummies for c categories, leaving one as the base for comparison.
Heteroskedasticity & Dummy Independent Variables FAQ
If heteroskedasticity is present, are my coefficient estimates wrong?
No — the OLS coefficients stay unbiased. What breaks is the standard errors, which become incorrect and make the t and F tests unreliable. The standard fix is to report White (heteroskedasticity-robust) standard errors, or to use weighted least squares; you do not need to discard the model.
Why use c − 1 dummies for c categories?
Including a dummy for every category alongside the intercept makes them perfectly collinear (the dummies sum to 1, like the intercept), so the model cannot be estimated. Dropping one category as the base avoids this trap, and every dummy coefficient is then read as a difference relative to that base.
Exam move
Practise reading the BP/White p-value off an R diagnostic and stating the consequence (SEs wrong, switch to robust) in one line. For dummies, rehearse interpreting a coefficient as a difference from the base category — and watch for the log-dependent case where it becomes an approximate percentage.
