Exercise: Stock Beta via OLS — Numerical Example

Prerequisites: Linear Regression Derivation

Problem

You have two years of daily returns ($T = 504$ days) for a stock $s$ and an index $m$. The true data-generating process is $r_{s,t} = 0.0001 + 1.3\cdot r_{m,t} + \epsilon_t$, with $\epsilon_t \sim \mathcal{N}(0, 0.01^2)$ (stock-specific noise) and $r_{m,t} \sim \mathcal{N}(0.0005, 0.012^2)$ (market returns).

1. Simulate the data with a seeded RNG.

2. Estimate $\hat\alpha$ and $\hat\beta$ using the closed-form OLS formulas from Exercise 1. Compare to the true values.

3. Compute the standard error $\text{SE}(\hat\beta) = \sigma/\sqrt{\sum(r_{m,t} - \bar r_m)^2}$ (using the true $\sigma = 0.01$, or use residual $\hat\sigma$ for a real-world estimate). Report a 95% confidence interval for $\hat\beta$.

4. Interpret. Is your estimate statistically significantly different from the reference "market beta" of $1.0$? From the "true" beta of $1.3$? What is the minimum sample size $T$ you would need to distinguish a beta of $1.3$ from $1.0$ at 5% significance?

Hint