f. distinguish between and interpret the R Square and adjusted R Square in multiple regression.
R square
R square measures the fraction of the total variation in the dependent variable that is explained by the independent variable. This was discussed in Reading 11.
The formula for R square as defined in Reading 11 is:
= 1 minus Unexplained variation
Total variation
(NB – Note in previous blog how the regression tables are used to obtain “Unexplained Variation & Total Variation”)
The problem with R square is that one can increase R square simply by including many additional independent variables that explain even a slight amount of the previously unexplained variation, even if the amount they explain is not statistically significant.
Solution!
Excel Statistics (and I gather other statistically packages) calculate an Adjusted R square.
Adjusted R square is adjusted for degrees of freedom, and does not automatically increases because other variables are added.
The output from a statistic package would look like this:
Excel Statistics (and I gather other statistically packages) calculate an Adjusted R square.
Adjusted R square is adjusted for degrees of freedom, and does not automatically increases because other variables are added.
The output from a statistic package would look like this:
If I understand the LOS correctly (as well as the examples, L2 candidates are not required to calculate adjusted R square
Interpret Adjusted R square
If we use adjusted R square to compare regression models, it is important that the dependent variable be defined the same way in both models and that the sample sizes used to estimate the models are the same.
For example, it makes a difference for the value of R square if the dependent variable is the share price or the ln(share price), even if the independent variables are identical.
Furthermore, we should be aware that a high adjusted R square does not necessarily indicate that the regression is well specified in the sense of including the correct set of variables.
Interpret Adjusted R square
If we use adjusted R square to compare regression models, it is important that the dependent variable be defined the same way in both models and that the sample sizes used to estimate the models are the same.
For example, it makes a difference for the value of R square if the dependent variable is the share price or the ln(share price), even if the independent variables are identical.
Furthermore, we should be aware that a high adjusted R square does not necessarily indicate that the regression is well specified in the sense of including the correct set of variables.
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