BUSS1020 · Quantitative Business Analysis
Multiple Linear Regression
Multiple Linear Regression (Week 12, Berenson Ch 14.1–14.4, 14.6) generalises the line to several predictors, Ŷ = b₀ + b₁X₁ + … + bₖXₖ, where each slope is the effect of its predictor holding the others constant. You assess overall fit with the F-test and adjusted r², test individual coefficients with t-tests, encode categories with dummy variables, and check the model's assumptions through residual diagnostics. Reading and interpreting Excel regression output is the practical skill that ties the unit together.
What this chapter covers
- 01Multiple regression model and 'holding others constant'
- 02Coefficient of multiple determination r²
- 03Adjusted r² and why it penalises extra predictors
- 04Overall F-test: F = MSR/MSE, H₀ all slopes = 0
- 05Individual t-tests on each coefficient
- 06Dummy variables for categorical predictors
- 07Residual diagnostics and the LINE assumptions
- 08Reading Excel regression output (coefficients, p-values, R Square)
Interpret a multiple regression with a dummy variable
- 2 marks(a) AdSpend coefficient 1.8: holding StaffHours and Metro constant, each extra $1,000 of ad spend is associated with about $1,800 more weekly revenue.
- 2 marks(a) Metro coefficient 6: holding other variables constant, metro branches average about $6,000 more weekly revenue than regional branches (the reference category, Metro = 0).
- 1 mark(b) Substitute AdSpend = 10, StaffHours = 35, Metro = 1: Ŷ = 25 + 1.8(10) + 1.2(35) + 6(1).
- 2 marks(b) Compute: 25 + 18 + 42 + 6 = 91, i.e. predicted revenue of $91,000.
- 1 mark(c) Adjusted r² is below r² because it penalises the model for the number of predictors; r² can only rise when variables are added, whereas adjusted r² rises only if a new predictor improves fit beyond chance.
Key terms
- Multiple regression coefficient
- The estimated change in Y for a one-unit increase in that predictor while holding all other predictors constant — a partial effect, not a total one.
- Adjusted r²
- A version of r² that penalises the addition of predictors, r²_adj = 1 − (1−r²)(n−1)/(n−k−1); it only rises if a new predictor improves fit beyond what chance would give.
- Overall F-test
- A test of whether the model as a whole explains significant variation, F = MSR/MSE, with H₀ that all slope coefficients equal zero.
- Dummy variable
- A 0/1 indicator that lets a categorical predictor enter a regression; its coefficient is the mean difference in Y relative to the omitted reference category.
- Residual diagnostics (LINE)
- Checks of the regression assumptions — Linearity, Independence of errors, Normality of residuals, and Equal variance (homoscedasticity); a funnel-shaped residual plot signals unequal variance.
Multiple Linear Regression FAQ
Why prefer adjusted r² over r² for comparing models?
Because plain r² never decreases when you add a predictor, even a useless one, it always favours bigger models. Adjusted r² penalises extra predictors, so it only improves when a new variable genuinely adds explanatory power — making it the fairer measure when comparing models with different numbers of predictors.
What's the difference between the F-test and the individual t-tests?
The F-test asks whether the model as a WHOLE is useful (are all slopes zero?), while each t-test asks whether one SPECIFIC predictor contributes, holding the others constant. It is possible for the overall F-test to be significant while some individual coefficients are not.
How do I read a residual plot to check assumptions?
Plot residuals against fitted values: a random, patternless band around zero supports the linearity and equal-variance assumptions, while a funnel shape signals heteroscedasticity (unequal variance) and a curved pattern signals a missing nonlinear term. Durbin–Watson checks for autocorrelation of errors.
Exam move
The exam reward here is interpretation, so practise reading a full Excel regression output: locate the coefficients, standard errors, t-stats, p-values, R Square, adjusted R Square and the ANOVA F. Lock in the phrase 'holding others constant' for every coefficient, and rehearse dummy-variable interpretation against a named reference category. Be able to explain in words why adjusted r² differs from r², what the F-test versus individual t-tests answer, and what a residual plot reveals about the LINE assumptions. As the capstone topic appearing in both Part A and Part B, this rewards clean, well-worded answers over heavy arithmetic.
