Can a Stock’s Volatility Remain Unchanged? Applying the Chi‑Square Test for Variance

It discusses testing whether a single population variance equals (or is greater/less than) a given constant using the chi‑square test, presenting the null and alternative hypotheses for both two‑tailed and one‑tailed cases.

The chi‑square statistic is calculated as \(\chi^2 = (n-1)\frac{s^2}{\sigma_0^2}\), where \(s^2\) is the sample variance and \(\sigma_0^2\) the hypothesized variance.

Example 1: A stock’s historical monthly return standard deviation before 2003 is \(\sigma_0\). Using the 36 months of returns from 2004‑2006, we test at a chosen significance level whether the current standard deviation still equals \(\sigma_0\).

Solution:

(1) State the null hypothesis \(H_0:\sigma^2 = \sigma_0^2\) and the alternative hypothesis \(H_1:\sigma^2 \neq \sigma_0^2\) (or one‑sided as needed).

(2) Apply the chi‑square test using the formula above.

(3) Determine the critical chi‑square values from the table for the chosen significance level \(\alpha\). For a two‑tailed test with \(\alpha\) and degrees of freedom \(df = n-1 = 35\), each tail has area \(\alpha/2\); the critical values are \(\chi^2_{\alpha/2,35}\) and \(\chi^2_{1-\alpha/2,35}\).

(4) Compare the computed chi‑square statistic with the critical values. If it falls outside the interval defined by the critical values, reject \(H_0\); otherwise, do not reject.

(5) In this example the statistic does not lie in the rejection region, so we cannot reject the null hypothesis.

(6) Conclusion: The stock’s standard deviation is not significantly different from the historical value; there is no evidence that its volatility has changed.

Reference:

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