What are the key assumptions of multiple regression and how do I detect violations?

Multiple regression is a CFA Level I favorite, and the exam almost always tests whether you can identify assumption violations from given data. Here are the key assumptions and how to spot trouble:

The Core Assumptions

1. Linearity — The relationship between each independent variable and the dependent variable is linear.
2. Homoscedasticity — Error terms have constant variance across all levels of the independent variables.
3. No Serial Correlation — Error terms are independent of each other (especially relevant for time-series data).
4. No Multicollinearity — Independent variables are not highly correlated with each other.
5. Normality of Errors — Error terms are normally distributed (needed for valid hypothesis tests).

How to Detect Violations in a CFA Vignette

Violation	Detection Method	Red Flag in Data
Heteroscedasticity	Breusch-Pagan test	Residuals fan out as fitted values increase
Serial Correlation	Durbin-Watson statistic	DW far from 2.0 (close to 0 or 4)
Multicollinearity	Variance Inflation Factor (VIF)	VIF > 5 or 10; high R² but insignificant coefficients

Worked Example: An analyst at Cornerstone Research regresses quarterly fund returns (Y) on interest rate changes (X₁) and credit spread changes (X₂). Results:

• R² = 0.81
• t-stat for X₁ = 1.12 (insignificant)
• t-stat for X₂ = 0.97 (insignificant)
• Correlation between X₁ and X₂ = 0.93

The red flag: High R² with insignificant individual coefficients and high correlation between regressors — classic multicollinearity. The model explains returns well overall but can't disentangle the individual effects.

Fix: Drop one variable, combine them into a composite, or use principal component analysis.

Exam Strategy: When you see a regression output in a vignette, immediately check:

1. Is R² high but individual t-stats low? → Multicollinearity
2. Is the Durbin-Watson far from 2? → Serial correlation
3. Do residuals show a pattern? → Heteroscedasticity or non-linearity

Practice identifying these patterns with our CFA Level I question bank.