What is Multiple Linear Regression (MLR)?
Multiple linear regression (MLR), also known simply as multiple regression, is a sophisticated statistical technique used to predict the outcome of a response variable by analyzing several explanatory variables. This method extends ordinary least-squares (OLS) regression, accommodating more than one explanatory variable.
Key Takeaways
- Multiple linear regression (MLR), also referred to as multiple regression, leverages several explanatory variables for prediction.
- It extends from linear (OLS) regression, which utilizes only a single explanatory variable.
- MLR proves invaluable in econometrics and financial inference.
Inspiring Formula and Calculation of Multiple Linear Regression
The multiple linear regression model follows this equation:
[ y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + … + \beta_p x_{ip} + \epsilon ]
Where for each observation (i):
- ( y_i ) = dependent variable
- ( x_i ) = explanatory variables
- ( \beta_0 ) = y-intercept (constant term)
- ( \beta_p ) = slope coefficients for each explanatory variable
- ( \epsilon ) = model’s error term (residuals)
What Multiple Linear Regression Can Tell You
Simple linear regression models the relationship between one independent and one dependent variable. Multiple linear regression models extend this, enabling predictions based on numerous explanatory variables.
Key assumptions of the multiple regression model include:
- A linear relationship exists between dependent and independent variables.
- Independent variables aren’t excessively correlated.
- Observations are selected independently and randomly.
- Residuals follow a normal distribution with a mean of 0 and constant variance (( \sigma^2 )).
The coefficient of determination (R-squared) gauges how much of the outcome’s variation stems from independent variables. While R-squared rises as more predictors are added, it doesn’t always specify the essential predictors. R-squared falls between 0 and 1, with 1 signifying no prediction errors.
Example of How to Use Multiple Linear Regression
Imagine an analyst wants to assess how market movements impact ExxonMobil (XOM)’s price. While a simple regression might use the S&P 500 index, multiple factors influence XOM’s price — such as oil prices, interest rates, and oil futures. Thus, a multiple regression analysis becomes necessary.
For our model:
- Dependent variable (y~i~): XOM’s price
- Explanatory variables (x~i1~, x~i2~, x~i3~, x~i4~): Interest rates, oil prices, S&P 500 index, oil futures prices
- Coefficients (B~0~, B~1~, B~2~, etc.): Measure individual variable influences
Upon modeling XOM price using statistical software, the output may unveil relationships like a 7.8% price increase of XOM with a 1% oil price rise and a 1.5% decrease with a 1% interest rate rise. This shows 86.5% of XOM’s price variation explained by these variables.
Inspired by Linear and Multiple Regression Differences
OLS regression evaluates how a change in an independent variable affects a dependent one. Still, multiple variables usually influence outcomes, necessitating multiple regression, linear or nonlinear.
Multiple regression requires:
- A linear dependent-independent variable relationship
- Independence among explanatory variables
Amazing Factors in Multiple Regressions
Why choose multiple regressions? Because outcomes rarely hinge on a single factor. Multiple regressions reveal how several variables jointly affect the dependent variable, providing a comprehensive view.
Choose Your Path: Manual or Software-Based Regression?
Performing multiple regression manually is daunting. With growing complexity (variables/data), specialized software like Excel simplifies computations.
Understand Linear Equation in Multiple Regressions
Multiple linear regression computes the line of best fit, minimizing variable variances related to a dependent variable. Unlike linear regression, non-linear types also exist, like logistic or quadratic regression.
Financial Mastery Through Multiple Regression
Models, especially factor models, evaluate multi-variable relationships impacting outcomes. The Fama and French Three-Factor Model deepens market risk evaluation in the capital asset pricing model (CAPM), enhancing analysis with additional risk factors. This empowers better asset management performance assessments.
Related Terms: ordinary least squares, linear regression, coefficient of determination, predictive modeling.
References
- Boston University Medical Campus-School of Public Health. “Multiple Linear Regression”.
