How to Use an F-Test to Compare Stock Return Variances: A Step-by-Step Guide

Null and Alternative Hypotheses for One‑ and Two‑Tailed Tests

The null hypothesis (H₀) states that the population variances are equal, while the alternative hypothesis (H₁) differs depending on whether a one‑tailed or two‑tailed test is performed.

Note: Always place the larger sample variance in the numerator of the F‑statistic, ensuring the statistic is ≥ 1; consequently, only the right‑hand rejection region needs to be considered, regardless of test direction.

Example: Comparing IBM and HP Monthly Return Variances

We examine 36 months of monthly returns for IBM and HP (2004‑200? data). The observed standard deviations are σ̂₁ = … and σ̂₂ = … (values omitted). Using a significance level α = …, we test whether the variances differ.

(1) Hypotheses : H₀: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂² (two‑tailed).

(2) F‑test : Compute the F‑statistic F = s₁² / s₂² (larger variance on top).

(3) Calculate the statistic using the sample variances.

(4) Critical value : Obtain the critical F value from the F‑distribution table with appropriate degrees of freedom.

(5) Since the computed F does not fall in the rejection region, we fail to reject H₀.

(6) Conclusion : The standard deviations of IBM and HP monthly returns are not significantly different.

Reference

Zhu Shunquan, *Economic and Financial Data Analysis and Its Python Application*.