Linear Regression: Coefficients, R-Squared & MSE

Ordinary Least Squares (OLS) Linear Regression is the foundation of predictive modeling in business analytics. Whether forecasting quarterly corporate revenue, estimating customer lifetime value (LTV), or quantifying price elasticity, linear regression provides an interpretable, transparent mathematical model that executive leadership can trust.

1. The Mathematical Formulation

Linear regression models the relationship between a continuous dependent target variable $Y$ and one or more independent predictor features $X$:

$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_p X_p + \epsilon$$

Where:

• $\beta_0$ is the intercept (the expected baseline value of $Y$ when all $X=0$).
• $\beta_j$ represents the slope coefficient (the expected change in $Y$ for every 1-unit increase in $X_j$, holding all other variables constant).
• $\epsilon$ is the random Gaussian error term.

2. Training a Revenue Prediction Model in Scikit-Learn

3. Interpreting Regression Coefficients

In business analytics, explaining the coefficients is far more important than just printing predictions:

Example Interpretation:

If meta_ads_spend has a coefficient of 4.25, it implies: "For every additional $1.00 invested in Meta Advertising, quarterly revenue increases by $4.25, assuming AdWords and influencer spend remain constant."

4. Key Performance Metrics