In multiple regression, what is the purpose of selecting a particular combination of regression coefficients?

In multiple regression, what is the purpose of selecting a particular combination of regression coefficients?

• A. Maximize the amount of variance in the dependent variable explained by the predictors
• B. Minimize the regression slope
• C. Randomly assign coefficients
• D. Maximize the residual variance

Answer: (A)

In multiple regression, you choose the regression coefficients to make the model’s predictions match the observed data as closely as possible. This is achieved with least squares, which minimizes the sum of squared residuals (the differences between observed Y and predicted Ŷ). When these residuals are minimized, more of the variability in the dependent variable is explained by the predictors, increasing the model’s explanatory power (often summarized by R-squared). So the goal is to maximize the portion of variance in Y that the predictors account for, i.e., maximize explained variance and minimize residual variance. The slopes and intercept arise from the data but are not the objective themselves, and randomly assigning coefficients or maximizing residual variance would worsen the fit.

In multiple regression, you choose the regression coefficients to make the model’s predictions match the observed data as closely as possible. This is achieved with least squares, which minimizes the sum of squared residuals (the differences between observed Y and predicted Ŷ). When these residuals are minimized, more of the variability in the dependent variable is explained by the predictors, increasing the model’s explanatory power (often summarized by R-squared). So the goal is to maximize the portion of variance in Y that the predictors account for, i.e., maximize explained variance and minimize residual variance. The slopes and intercept arise from the data but are not the objective themselves, and randomly assigning coefficients or maximizing residual variance would worsen the fit.