You fit a regression to your trading desk's daily P&L against three risk factors. The R-squared is 0.62 and every coefficient looks significant. A month later, your hedges drift, the standard errors flip sign on a re-estimate, and your boss asks why the model lied. Diagnostics catch what the t-stats hide.
Ordinary least squares (OLS) delivers the smallest-variance unbiased linear estimator under five conditions: linearity in parameters, zero conditional mean errors, homoskedasticity, no autocorrelation, and no perfect collinearity.
When these hold, OLS is BLUE. Drop any one and a different estimator may dominate, or the standard errors become invalid. The exam tests which assumption breaks each variation and what the consequence is. Bias in beta, bias in SEs, or loss of efficiency.
KEY: Heteroskedasticity and autocorrelation do NOT bias the coefficient estimates. They bias the standard errors. The point estimates of beta stay correct on average; the inference around them is wrong.
Heteroskedasticity is when error variance changes with the regressors. Picture residuals from a regression of P&L on position size. Small positions produce small residuals, large positions produce large residuals.
Common mistakes
- Treating heteroskedasticity as a coefficient problem. Beta estimates are unbiased under heteroskedasticity. Re-running OLS does nothing because the OLS coefficients are already correct on average. Trap: a question shows a fanning residual plot and asks for the next step. Choice C says "re-estimate using only the homoskedastic subsample" is wrong.
- Confusing perfect with imperfect multicollinearity. Perfect multicollinearity prevents OLS from running at all (matrix not invertible). Imperfect just inflates variance. A question that says "the dummy for industry contains all four sectors and the model has an intercept" is describing perfect collinearity (the dummy variable trap), and software drops one.
- Believing high R-squared validates the model. R-squared measures in-sample fit. A regression with omitted variable bias can have R-squared of 0.95 and still produce wrong coefficients. Trap: a question shows R-squared = 0.92 and asks if the coefficients are reliable. The answer depends on whether all relevant variables are included, not on R-squared.
Bottom line
- OLS is BLUE (Best Linear Unbiased Estimator) only when the Gauss-Markov assumptions hold: linearity, zero-mean errors, homoskedasticity, no autocorrelation, no perfect collinearity.
- Heteroskedasticity and autocorrelation leave the coefficient estimates unbiased but ruin the standard errors; fix heteroskedasticity with White-robust SEs, not by re-estimating beta.
- Multicollinearity does not bias coefficients but inflates their variance; the signature is a significant joint F-stat, weak individual t-stats, and VIFs above 10.
- Omitted variable bias appears only when the omitted variable correlates with both Y AND an included regressor; either condition failing means no bias.
Exam shortcut
When a regression diagnostic question gives you a residual pattern and asks "what's the issue?", match the visual: fanning out → heteroskedasticity (robust SEs); curvature → omitted nonlinear term; serial pattern in residuals → autocorrelation (Newey-West SEs). When the question gives high R-squared, high F-stat, but weak individual t-stats, the answer is multicollinearity, check VIFs. Memory aid: "Hetero hurts SEs not betas. Multi hurts variance not bias.
The full lesson (about 2,813 words, 19 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
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