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Set out what a simple linear regression assumes

CFA® Level I · Quantitative Methods · Applications of Simple Linear Regression in Finance · QM.8.a

The idea

Simple linear regression fits a straight line relating one independent variable to one dependent variable, and it rests on four assumptions: the relationship is linear, the errors have zero conditional mean given the independent variable (they are uncorrelated with it), the errors have constant variance across observations, and the errors are normally distributed. Violations have names and consequences — heteroskedasticity is non-constant error variance, serial correlation is dependence between successive errors — and both can leave the fitted coefficients looking more precise than they are, which makes significance tests unreliable rather than merely imperfect.

A practice question

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

What does a simple linear regression estimate?

  1. The correlation between two variablesWhy not: Related but different: correlation is one unitless number, while a regression fits an intercept and a slope that carry the units of the variables.
  2. A straight-line relationship between one independent and one dependent variable — correct
  3. The variance of the dependent variableWhy not: That is a property of the variable on its own, computable before any regression is run. The regression explains part of that variance rather than measuring it.

The model fits an intercept and a slope describing how the dependent variable moves with the independent one. Correlation is related but is a single unitless number rather than a fitted line.

A harder one

What is serial correlation?

  1. Correlation between the independent and dependent variablesWhy not: That is the relationship the regression exists to measure, and it is wanted rather than a problem.
  2. Correlation between two independent variablesWhy not: That is multicollinearity, which needs at least two independent variables and so cannot arise in a simple regression.
  3. Dependence between successive errors — correct

It is common in time series data, where one period's surprise carries into the next.

The hardest tier

Why does serial correlation in the residuals matter even when the coefficient estimates remain unbiased?

  1. It makes the slope estimate meaninglessWhy not: The slope remains unbiased, which the question grants. What breaks is the standard error attached to it.
  2. It changes the sign of the relationshipWhy not: The estimate is unaffected, sign included. Only its apparent precision is distorted.
  3. It understates standard errors, so relationships look more significant than they are — correct

Correlated errors carry less independent information than their count suggests, so the usual formula overstates precision. Analysts then reject nulls they should not, which is how spurious relationships get published.

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