A source of constant discussion among applied econometricians is the degree to which measures of the overall fit of an estimated equation also measure the quality of that regression. To date, we have introduced something like four different measures of overall fit, but the two most used are $\mathrm { R } ^ { 2 }$ and $\overline { \mathrm { R } } ^ { 2 }$ .
(a) Carefully distinguish between $\mathrm { R } ^ { 2 }$ and $\overline { \mathrm { R } } ^ { 2 }$ .
(b) Of the two, which do you recommend typically using? Why?
(c) What drawbacks are there to the use of the measure you chose (as your answer to part (b) above) as the primary determinant of the overall quality of a regression?

Question

Quizplus · Accepted Answer

The Answer is (a) The main difference between

R^2

and

\overline{R}^2

lies in how they account for the number of independent variables in the regression model.

R^2

measures the proportion of the variance in the dependent variable that is explained by the independent variables in the model. However, it does not account for the number of independent variables, which means that adding more variables to the model will always increase

R^2

, even if those variables are not truly adding explanatory power. On the other hand,

\overline{R}^2

adjusts for the number of independent variables in the model, providing a more accurate measure of the overall fit.

(b) Typically, it is recommended to use

\overline{R}^2

as the primary measure of overall fit in a regression model. This is because

\overline{R}^2

takes into account the number of independent variables in the model, providing a more accurate assessment of the model's quality. By adjusting for the degrees of freedom,

\overline{R}^2

helps to prevent overfitting and provides a more reliable indication of the true explanatory power of the model.

(c) However, there are drawbacks to using

\overline{R}^2

as the primary determinant of the overall quality of a regression. One drawback is that it may penalize models with a large number of independent variables, even if those variables are truly adding explanatory power. Additionally,

\overline{R}^2

may not capture the full complexity of the relationship between the independent and dependent variables, as it is based on the assumption of a linear relationship. Therefore, while

\overline{R}^2

is a more robust measure of overall fit compared to

R^2

, it should be used in conjunction with other diagnostic tools and considerations to fully assess the quality of a regression model.

A Source of Constant Discussion Among Applied Econometricians Is the Degree