Estimating beta for the cost of equity

In short

• The cost of equity (required return on equity) discounts FCFE directly and is the equity part of the WACC that discounts FCFF.
• At one point in time and in one country, $r_f$ and the MRP are the same for every stock, so in the CAPM only beta makes costs of equity differ. Over time all inputs move.
• Beta is estimated by regression: on excess returns $(r_i - r_f)$ on $(r_m - r_f)$, or more simply with the market model on raw returns; both give similar slopes.
• Adjusted beta = $w_1$ × raw beta + $w_2$ × 1.0, commonly $\tfrac{2}{3}$ and $\tfrac{1}{3}$: it pulls the estimate towards 1.0 because betas mean-revert.
• Practical choices: which index is the market, how noisy the estimate is (low $R^2$, high standard error), and the horizon and frequency of returns (about 100 weekly returns over two years is a common default).