Understanding Simple and Multivariate Linear Regression Models

1 Simple Linear Regression Model

Simple (univariate) linear regression, also called simple linear regression (SLR), is the most basic yet widely used regression model. Its regression equation is:

To estimate the optimal parameters β0 and β1 from a sample, the least squares method is typically employed, aiming to minimize the sum of squared residuals :

Without derivation, the least‑squares estimates are given by:

(Formulas omitted in the source.)

2 Multivariate Linear Regression Model

When a regression analysis involves two or more independent variables, it is called multiple regression . In practice, many phenomena are influenced by several factors, and using a combination of multiple predictors yields more effective and realistic predictions than a single predictor.

For example, household consumption depends not only on disposable income but also on wealth, price level, interest rates, etc., leading to a model with several explanatory variables, known as a multivariate linear regression model .

Parameter estimation for the multivariate regression model, like the simple case, is performed by minimizing the sum of squared errors using the least‑squares method.

References

ThomsonRen github https://github.com/ThomsonRen/mathmodels