Solution: Stock Beta via OLS — Numerical Example

Exercise: Stock Beta via OLS: Numerical Example

Parts 1–3

import numpy as np

rng = np.random.default_rng(0)
T = 504
r_m = 0.0005 + 0.012 * rng.standard_normal(T)
eps = 0.01 * rng.standard_normal(T)
r_s = 0.0001 + 1.3 * r_m + eps

# OLS estimates
m_bar, s_bar = r_m.mean(), r_s.mean()
beta_hat = np.sum((r_m - m_bar)*(r_s - s_bar)) / np.sum((r_m - m_bar)**2)
alpha_hat = s_bar - beta_hat * m_bar

print(f"alpha_hat = {alpha_hat:.5f} (true: 0.0001)")
print(f"beta_hat  = {beta_hat:.3f} (true: 1.3)")
# alpha_hat = 0.00014 (true: 0.0001)
# beta_hat  = 1.322 (true: 1.3)

# residual noise estimate
residuals = r_s - alpha_hat - beta_hat * r_m
sigma_hat = np.sqrt(np.sum(residuals**2) / (T - 2))
print(f"sigma_hat = {sigma_hat:.5f} (true: 0.01)")
# sigma_hat = 0.00989 (true: 0.01)

# SE of beta_hat
SE_beta = sigma_hat / np.sqrt(np.sum((r_m - m_bar)**2))
print(f"SE(beta) = {SE_beta:.3f}")
# SE(beta) = 0.037

# 95% CI
CI = (beta_hat - 1.96 * SE_beta, beta_hat + 1.96 * SE_beta)
print(f"95% CI = [{CI[0]:.3f}, {CI[1]:.3f}]")
# 95% CI = [1.249, 1.394]

$\hat\beta \approx 1.32$ with 95% CI $[1.25, 1.39]$ .

Part 4

Beta vs 1.0: CI $[1.25, 1.39]$ does not contain $1.0$ . We reject the null "beta = 1" at 5% significance. The stock has a statistically significant "high beta" character.
Beta vs 1.3: CI contains $1.3$ . We do not reject the null "beta = 1.3"; the estimate is consistent with the true value.

Minimum $T$ to distinguish $\beta = 1.3$ from $\beta = 1.0$ at 5%. We need

|1.3 - 1.0|/\text{SE}(\hat\beta) \ge 1.96

, i.e.

\text{SE}(\hat\beta) \le 0.153

. Using

\text{SE}(\hat\beta) = \sigma/(\sigma_m\sqrt T)

with

\sigma = 0.01

\sigma_m = 0.012

T \ge \left(\frac{\sigma}{\sigma_m\cdot 0.153}\right)^2 = \left(\frac{0.01}{0.012\cdot 0.153}\right)^2 \approx 30.

So roughly $T \ge 30$ trading days suffice to distinguish $\beta = 1.3$ from $\beta = 1.0$ at 5% significance — about 6 weeks of data. In practice this number is often much higher due to autocorrelation in returns (Newey-West correction inflates SE) and regime changes.

Takeaways

OLS delivers consistent, unbiased estimates under i.i.d. gaussian noise — with standard errors matching the closed-form formula.
SE of beta shrinks as $1/\sqrt T$ and as $1/\sigma_m$ . More data and higher market volatility both tighten the estimate. This is why regressions on calm periods are noisier than on volatile ones.
Beta estimates have non-trivial uncertainty. A 2-year estimate with SE $\approx 0.04$ means "beta could plausibly be anywhere from 1.25 to 1.4." This matters in risk budgeting; don't treat $\hat\beta$ as exact.
Real-world beta estimation is harder. Autocorrelated returns, time-varying beta, outliers, and non-gaussian tails all break the Gauss-Markov assumptions. Robust regression, rolling windows, and Kalman filtering are the standard fixes.