Question

Briefly explain why a large value of \(r^{2}\) is desirable in a regression setting.

Short answer

A high value of \(r^{2}\) is desirable in a regression setting because it denotes that the model is effectively capturing the variance in the dependent variable. With a large \(r^{2}\), a significant portion of the variation in the output variable is explained by the input variables, indicating a good fit of the data, thus facilitating more accurate predictions.

Step-by-step solution

Understand the meaning of \(r^{2}\)

In the context of a regression model, \(r^{2}\), also known as the coefficient of determination, measures the fraction of the total variation in the dependent variable that is captured by the model. It takes on values between 0 and 1.

Interpretation of \(r^{2}\) values

A higher \(r^{2}\) implies that a larger proportion of the variance in the dependent variable is explained by the model. In other words, a higher \(r^{2}\) suggests that the model fits the data better. An \(r^{2}\) of 1 indicates a perfect fit, meaning the model explains all variation in the dependent variable, while an \(r^{2}\) close to 0 indicates that the model explains very little of the variation.

Desirability of a high \(r^{2}\)

Given the interpretation of \(r^{2}\), it's clear that a larger \(r^{2}\) is desirable in a regression setting because it signifies that the model is doing a better job of explaining the variation in the dependent variable. A high \(r^{2}\) value can indicate a good fit for the model, making predictions more accurate and the model more useful.