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CFA® Level II · Quantitative Methods · free lesson
Basics of Multiple Regression and Underlying Assumptions
Multiple regression is the workhorse of quantitative investment research: it attributes fund returns to factors, explains why valuation multiples and spreads differ, and produces forecasts. This module covers what the tool can answer, how to specify a model and read each coefficient as a partial effect, and which assumptions must hold, with residual plots as the first line of defence. Later Level II modules on model fit, misspecification and time series all build on this foundation.
What you will learn
1. Recognise the kinds of investment questions that a regression with several explanatory variables can answer.
- Multiple regression explains one dependent variable with two or more independent variables at the same time.
- Its three main uses are identifying relationships, forecasting and testing existing theories.
- In a factor regression of a fund's excess returns, the slopes measure style exposures and the intercept is alpha.
- A continuous dependent variable suits linear regression; a binary outcome calls for logistic regression.
- Specify from economic reasoning, estimate, diagnose the residuals, then interpret and use the model.
2. Specify a multiple regression model and interpret each slope coefficient as a partial effect, holding the other variables constant.
- A model with k independent variables estimates k + 1 parameters: one intercept and k partial slopes.
- Each slope is the expected change in Y for a one-unit change in that X, holding the other X variables constant.
- When independent variables are correlated, simple-regression slopes and partial slopes differ.
- Apply coefficients in the units used to estimate them, and watch the signs of both coefficient and input.
- The intercept is the prediction when every X equals zero, not an average value; it cancels when comparing two cases.
3. State the assumptions behind ordinary least squares estimation and use residual plots to spot likely violations.
- The five assumptions: linearity, homoskedasticity, independent errors, normal errors, and independent variables with no exact linear relationship.
- An exact linear relationship among the independent variables makes OLS estimation impossible.
- Curved residual patterns suggest nonlinearity; a fan shape suggests heteroskedasticity.
- Runs of same-signed residuals over time suggest serial correlation; Q-Q plots that bend at the ends suggest fat tails.
- Plots flag likely violations; formal tests confirm them.
Module at a glance
- Multiple regression explains a continuous dependent variable with several independent variables; binary outcomes need logistic regression.
- Its uses are identifying relationships, forecasting and testing theories; in factor regressions the intercept is alpha.
- Each slope is a partial effect: the change in Y for a one-unit change in that X with the other X variables held constant.
- Correlated independent variables make simple and partial slopes differ.
- Apply coefficients in their estimation units, mind the signs, and remember the intercept is the prediction when every X is zero.
- OLS assumptions: linearity, homoskedasticity, independent errors, normal errors and no exact linear relationship among the X variables.
- Residual plots flag problems: curves (nonlinearity), fans (heteroskedasticity), runs over time (serial correlation), bent Q-Q tails (non-normality).
This free module has 3 lessons, 14 practice questions with full explanations.