How do you use the chi-squared test to test a hypothesis about variance?

Question

AcadiFi · Accepted Answer

The chi-squared test for variance is a natural extension of hypothesis testing to the second moment of a distribution. While z-tests and t-tests focus on the mean, the chi-squared test focuses on dispersion — critically important in risk management.

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## Setup

**Hypotheses (two-tailed example):**
- H0: sigma^2 = sigma_0^2 (variance equals hypothesized value)
- H1: sigma^2 != sigma_0^2

**Test Statistic:**

chi^2 = (n - 1) x s^2 / sigma_0^2

Where:
- n = sample size
- s^2 = sample variance
- sigma_0^2 = hypothesized population variance

Under H0, this statistic follows a chi-squared distribution with (n - 1) degrees of freedom.

## Worked Example

Crestview Asset Management claims that the daily volatility (standard deviation) of their equity fund is 1.2%. A risk auditor collects 31 trading days and computes a sample standard deviation of 1.55%. Test at the 5% significance level whether the true variance exceeds the claimed variance.

**Hypotheses (one-tailed, upper):**
- H0: sigma^2 <= (0.012)^2 = 0.000144
- H1: sigma^2 > 0.000144

**Test statistic:**
chi^2 = (31 - 1) x (0.0155)^2 / (0.012)^2
chi^2 = 30 x 0.00024025 / 0.000144
chi^2 = 0.007208 / 0.000144 = **50.05**

**Critical value:** chi^2(0.05, 30) = 43.77

Since 50.05 > 43.77, we **reject H0**. The evidence suggests the fund's true variance is significantly greater than what Crestview claims.

## Key Properties of the Chi-Squared Distribution

1. **Always non-negative** — variance cannot be negative
2. **Right-skewed** — becomes more symmetric as df increases
3. **Not symmetric** — so two-tailed tests require both upper and lower critical values from the table
4. **Mean = df, Variance = 2 x df**

**FRM exam tip:** The chi-squared test appears in questions about validating VaR model assumptions (is the variance what the model predicts?) and in backtesting contexts. Always state degrees of freedom explicitly.

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