Understanding Coefficient of Determination ($R^2$) Properties

The Coefficient of Determination, denoted as $R^2$, is a statistical measure used in regression analysis. It represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.

$R^2$ Relationship with Regressors

Let's analyze the statements concerning the number of regressors:

• Statement A: $R^2$ is an increasing function of the number of regressors. This is generally true. $R^2$ never decreases when you add more independent variables to a regression model. It either increases or stays the same. For simplicity in exam contexts, it's often treated as increasing.
• Statement B: $R^2$ is a decreasing function of the number of regressors. This is incorrect, as $R^2$ cannot decrease as variables are added.
• Statement C: $R^2$ is unaffected by increasing or decreasing the number of regressors. This is incorrect because $R^2$ typically changes (increases or stays the same) when the number of regressors changes.

$R^2$ Value and Sign

Now, let's consider the value of $R^2$:

• Statement D: $R^2$ is always positive. In standard OLS regression with an intercept term, $R^2$ is always non-negative ($R^2 \ge 0$). It measures the goodness of fit compared to a simple mean model. While $R^2$ can be negative in certain specific statistical models (e.g., without an intercept) or if the model fits worse than the baseline, in the context of typical regression problems and exam questions assuming OLS with an intercept, it's considered non-negative. The phrasing "always positive" might be used loosely here to mean non-negative in the standard context.

Final Conclusion

Based on the analysis:

• Statement A holds true (non-decreasing effect of adding regressors).
• Statement D holds true within the common understanding of $R^2$ in OLS regression (non-negative).

Therefore, the correct options are A and D.