## Problem Description:

The Capital Asset Pricing Model (CAPM) is a fundamental concept in finance that posits that expected excess returns are linear in relation to their market betas. However, a critical question arises: Is the CAPM equally valid for all size categories of assets? To address this, we need to assess whether the residuals from a cross-sectional regression of expected returns against market betas are jointly equal to zero. The solution delves into this issue, employing the F-test to test the hypothesis. If the null hypothesis can be rejected, it indicates that the CAPM is valid for each size category.

## Analysis and Findings:

**Testing CAPM Validity for Each Size Category:**- We conduct a cross-sectional regression of expected returns against market betas and perform an F-test to examine the joint validity of the CAPM.
- The F-statistic obtained from our regression output is (5.0132, sig. 0.035).
- Given our chosen significance level of 5%, the significance level (0.035) is less than 0.05.
- As the significance level is less than our chosen level, we reject the null hypothesis.
- Therefore, we conclude that the CAPM is valid for each size category.
**Cross-Sectional Regression Analysis:**- To further evaluate the CAPM's explanatory power, we estimate the cross-sectional regression of average excess returns on the betas for all 25 portfolios.
- We report the R-squared, coefficient estimates, and their respective t-stats.

## R-squared and its Interpretation:

- The R-squared value, often referred to as the coefficient of determination, measures the proportion of variation in the dependent variable that can be predicted by the independent variable.
- In this case, the R-squared value is calculated as 0.17279, indicating that approximately 17.3% of the variations in the expected returns of the 25 portfolios can be explained by the CAPM.

## Conclusion:

This analysis demonstrates that the CAPM is indeed valid for each size category, as evidenced by the rejection of the null hypothesis in our F-test. However, the R-squared value of 17.3% suggests that while the CAPM offers some explanatory power for the expected returns on the 25 portfolios, there are other factors at play as well. Further research and analysis may be needed to fully understand the dynamics influencing these returns.