MLRs

Multiple linear regression (MLR) is a statistical technique used to model the relationship between a scalar dependent variable and two or more independent variables. The standard form is y = β0 + β1 x1 + β2 x2 + ... + βk xk + ε, where y is the response, x1..xk are predictors, β are coefficients, and ε is the error term. The model is typically estimated by ordinary least squares (OLS), which chooses coefficients that minimize the sum of squared residuals.

Coefficients represent partial effects of predictors on y, holding other variables constant. Predictions for new observations are obtained by plugging in predictor values. Model fit is commonly assessed with R-squared, which measures the proportion of variance explained, and adjusted R-squared that accounts for the number of predictors. An F-statistic tests whether at least one predictor is associated with the response.

Assumptions accompany MLR: linearity of relationships, independence of errors, homoscedasticity (constant error variance), and normality of errors for inference. No perfect multicollinearity among predictors is expected. Violations require remedies such as transforming the response or predictors, adding or removing variables, or using robust or alternative estimation methods. Multicollinearity inflates standard errors and can make coefficient estimates unstable; variance inflation factor (VIF) is a common diagnostic.

Extensions of MLR include incorporating interaction terms and polynomial terms to capture nonlinear relationships, as well as dummy coding for categorical predictors. Regularized variants, such as ridge and lasso, address issues of multicollinearity and model complexity, particularly in high-dimensional settings. MLR remains a foundational tool in econometrics, social sciences, biology, and engineering for understanding and predicting outcomes based on multiple predictors.