Multiple linear regression
Multiple linear regression models one outcome variable as a linear function of two or more explanatory variables simultaneously.
Also known asMLRM · multiple regression
FrameworkOrdinary least squares (OLS)
See it move
The infographic is a tree diagram showing monthly earnings Y as the root, decomposed through addition into four branches: the intercept β₀, the term β₁ × hours worked carrying the hours effect, the term β₂ × seniority carrying the seniority effect, and the error term ε. Each slope coefficient represents one regressor's isolated effect on Y while all other regressors in the model are held constant. The branching structure makes visible how multiple regression partitions the outcome simultaneously across several explanatory variables.
The formula
Variables
- Outcome (dependent) variable
- Intercept — predicted Y when all regressors equal zero
- Slope on regressor Xⱼ — effect of a one-unit increase in Xⱼ, holding all other regressors constant
- Explanatory (independent) variable j, for j = 1 to k
- Error term — variation in Y not explained by the regressors
OLS selects β̂₀, β̂₁, …, β̂ₖ to minimise the sum of squared residuals across all n observations.
Check yourself
A regression yields: Salary (€) = 15,000 + 1,200 × Years_experience + 800 × Has_MBA, where Has_MBA = 1 for MBA holders. A student claims: 'The coefficient 1,200 measures the total salary gain from one extra year of experience, including any indirect benefit from experienced workers being more likely to hold an MBA.' What is wrong with this claim?