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Summarise the assumptions on which estimating OLS parameters depends.

FRM® Part I · Quantitative Analysis · Linear Regression · FRM1.QA.7.c

The idea

Estimating OLS parameters and drawing inference from them rests on a small set of assumptions. The model is linear in its parameters. The error has zero conditional mean given X, which is called exogeneity, so X tells you nothing about the error's average. An omitted variable correlated with X breaks this and biases the slope. Separately, the observations (X, Y) are independent and identically distributed draws, which is a sampling assumption and not the same as exogeneity. Large outliers are unlikely, meaning the variables have finite fourth moments. X must vary across observations, otherwise the slope cannot be computed. For the classical results on efficiency and simple standard errors, the error variance should also be constant, which is homoskedasticity. Normal errors are needed only for exact small-sample t tests. The usual mistake is thinking heteroskedasticity biases the slope. It does not. Zero conditional mean is what protects against bias.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which assumption is the key condition that makes the OLS slope estimator unbiased?

  1. The error term is normally distributedWhy not: Normality is not needed for unbiasedness; it only matters for exact small-sample tests.
  2. The error has zero conditional mean given the explanatory variable - correct
  3. The explanatory variable has a perfectly constant variance over timeWhy not: Confuses the assumption on the error variance with a constant variance of X; X must vary, not be constant.
  4. The dependent variable is measured in the same units as the explanatory variableWhy not: Units are irrelevant to unbiasedness; rescaling Y only rescales the coefficients.

Unbiasedness needs E(error | X) = 0, which means nothing in the error is systematically related to X. Normality and homoskedasticity are not needed for it.

A harder one

A risk manager regresses firm credit spreads on a leverage ratio, omitting a measure of asset volatility. Asset volatility is positively correlated with leverage and also raises spreads. Which assumption fails and what is the consequence?

  1. Zero conditional mean of the error fails, so the estimated leverage coefficient is biased - correct
  2. Homoskedasticity fails, so the leverage coefficient is biased but the standard errors are fineWhy not: Heteroskedasticity does not bias the slope and affects the standard errors; the problem here is the omitted variable.
  3. Normality of the error fails, so the leverage coefficient is inconsistent only in small samplesWhy not: Normality is not the issue; the bias arises from the correlation of the error with leverage and persists in large samples.
  4. No assumption fails, because omitted variables only reduce R²Why not: Omitting a relevant variable correlated with X puts its effect in the error, which then correlates with X.

Asset volatility is in the error and is correlated with leverage, so E(error | leverage) is not zero. The slope picks up part of volatility's effect, and the bias does not vanish as the sample grows.

The hardest tier

A risk manager's regression of fund returns on a factor includes one observation from a month when a hedge fund collapsed. That single point has a return 15 standard deviations from the rest and moves the estimated slope substantially. Which OLS assumption does this situation most directly challenge?

  1. That large outliers are unlikely (finite fourth moments) - correct
  2. That the error has zero conditional mean given the factorWhy not: A single outlier does not by itself make the error mean related to X; the assumption at risk is about extreme values.
  3. That the model is linear in its parametersWhy not: The functional form is untouched; the problem is the influence of one extreme observation.
  4. That the factor has some variation across observationsWhy not: The factor varies; the problem is a single extreme data point dominating the estimate.

OLS squares residuals, so one extreme observation can dominate the fit. The assumption that large outliers are unlikely (finite fourth moments) supports the normal approximation for estimates and inference.

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