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What Is Multiple Linear Regression (MLR)?
I'm here to explain multiple linear regression, or MLR, which you might also hear called multiple regression. It's a statistical technique where you use several explanatory variables to predict the outcome of a response variable. The point is to model the linear relationship between those independent variables and the dependent one. Basically, it's just an extension of ordinary least-squares regression, but now you're dealing with more than one explanatory variable.
Key Takeaways
- Multiple linear regression (MLR) is a statistical technique that uses several explanatory variables to predict the outcome of a response variable.
- Multiple regression is an extension of linear (OLS) regression that uses just one explanatory variable.
- MLR is used extensively in econometrics and financial inference.
- Multiple regressions are used to make forecasts, explain relationships between financial variables, and test existing theories.
Formula and Calculation of Multiple Linear Regression (MLR)
The formula for MLR looks like this: y_i = β_0 + β_1 x_{i1} + β_2 x_{i2} + ... + β_p x_{ip} + ε, where for i = n observations, y_i is the dependent variable, x_i are the explanatory variables, β_0 is the y-intercept or constant term, β_p are the slope coefficients for each explanatory variable, and ε is the model's error term, also known as the residuals. You don't need to calculate this by hand; statistical software handles the least-squares estimates for you.
What Multiple Linear Regression (MLR) Can Tell You
Simple linear regression lets you predict one variable based on another, but only with two continuous variables: an independent and a dependent. MLR takes it further by including several explanatory variables. The model relies on assumptions: there's a linear relationship between dependents and independents, independents aren't highly correlated, observations are independent and random, and residuals are normally distributed with mean zero and constant variance.
Remember, MLR assumes linearity between variables, no high multicollinearity, and constant residual variance. The coefficient of determination, R-squared, shows how much variation in the outcome is explained by the independents. R-squared ranges from 0 to 1 and always increases with more predictors, even irrelevant ones, so don't rely on it alone to choose variables.
When you interpret MLR results, beta coefficients hold while keeping other variables constant. You can see the output as an equation or in a table.
Example of How to Use Multiple Linear Regression (MLR)
Let's say you want to know how the market affects ExxonMobil's (XOM) stock price. In simple regression, you'd use the S&P 500 as the independent variable and XOM price as dependent. But reality involves more: oil prices, interest rates, oil futures. That's where MLR comes in—it examines how multiple independents relate to one dependent.
In this setup, y_i is XOM price, x_i1 is interest rates, x_i2 is oil price, x_i3 is S&P 500 value, x_i4 is oil futures price. B0 is the intercept, and B1, B2, etc., measure changes in XOM price per unit change in each x_i. Software computes these.
Running this through software might show, holding others constant, XOM price rises 7.8% with a 1% oil price increase and falls 1.5% with a 1% interest rate rise. R-squared at 86.5% means most variation is explained by these variables. MLR can be nonlinear too, but we're focusing on linear here.
The model predicts based on multiples but includes residuals for slight differences from actual data.
Linear vs. Multiple Regression
Ordinary least-squares regression looks at a dependent variable's response to changes in explanatory variables, but usually more than one matters. That's why you use multiple regression—it handles several independents, assuming linearity and no major correlations between them. It can be linear or nonlinear.
Explain Like I'm 5
Multiple linear regression helps you figure out how different factors affect something you want to predict. Take an oil stock's price: instead of just the market, consider oil prices, interest rates, and indexes together. Each factor influences the price, and MLR calculates how much. It assumes a straight-line relationship and that factors aren't too intertwined, giving you a clear view of drivers and impacts. You'll see it in finance, economics, and beyond for predictions.
Frequently Asked Questions
What makes a multiple regression multiple? It considers more than one explanatory variable's effect on an outcome, holding others constant.
Why use multiple over simple OLS? Dependents are rarely explained by one variable alone, so multiple handles that, assuming no major correlations.
Can you do it by hand? Probably not—it's complex with many variables or data; use software or Excel functions.
What does linear mean here? The model fits a straight line minimizing variances related to the dependent variable.
How is it used in finance? Models like Fama-French Three-Factor expand on CAPM by adding factors like size and value risk for better performance analysis.
The Bottom Line
MLR predicts a variable's outcome using two or more explanatory variables. If one suffices, use simple regression; otherwise, MLR is key. For example, a company's stock price depends on multiple factors—MLR models that to predict it accurately.
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