How do you detect heteroskedasticity in a linear regression, and why does it matter for FRM?

Question

AcadiFi · Accepted Answer

Heteroskedasticity is the condition where the variance of regression residuals is not constant across observations. In risk management, this is extremely common — volatility clustering means the variance of returns changes over time, directly violating the OLS assumption of homoskedasticity.

## Why It Matters

If heteroskedasticity is present but ignored:
1. **Coefficient estimates are still unbiased** (OLS is BLUE only under homoskedasticity)
2. **Standard errors are WRONG** — typically underestimated, making t-statistics too large
3. **Hypothesis tests become unreliable** — you may conclude a variable is significant when it isn't
4. **Confidence intervals are too narrow**

## Detection Method 1: Breusch-Pagan Test

**Procedure:**
1. Run the original regression: Y = beta_0 + beta_1 X + epsilon
2. Compute squared residuals: e_i^2
3. Regress e_i^2 on the original independent variables: e^2 = gamma_0 + gamma_1 X + u
4. Test statistic: n x R^2 from the auxiliary regression, which follows chi^2(k) where k = number of regressors

## Detection Method 2: White Test

The White test is more general — it also captures non-linear forms of heteroskedasticity.

**Procedure:**
1. Run original regression, collect residuals
2. Regress e^2 on X, X^2, and cross-products of all independent variables
3. Test statistic: n x R^2 from this auxiliary regression, chi^2(p) where p = number of regressors in auxiliary

## Example

Hargrove Analytics regresses monthly hedge fund returns (Y) on market returns (X) using 120 months of data.

Original regression: R_fund = 0.3% + 0.85 x R_market, R^2 = 0.42

**Breusch-Pagan auxiliary regression:**
e^2 = 0.0004 + 0.0031 x R_market, R^2_aux = 0.074

BP statistic = 120 x 0.074 = **8.88**
Critical value: chi^2(0.05, 1) = 3.84

Since 8.88 > 3.84, **reject homoskedasticity**. The residuals are larger when market returns are large (volatility clustering).

```mermaid
flowchart TD
    A["Run OLS Regression"] --> B["Collect Residuals e_i"]
    B --> C["Square Residuals e_i^2"]
    C --> D{"Choose Test"}
    D -->|"Breusch-Pagan"| E["Regress e^2 on X"]
    D -->|"White"| F["Regress e^2 on X, X^2, cross-products"]
    E --> G["Test: n x R^2 ~ chi^2(k)"]
    F --> G
    G --> H{"Significant?"}
    H -->|"Yes"| I["Heteroskedasticity Present
Use robust SEs or WLS"]
    H -->|"No"| J["Homoskedasticity OK
OLS SEs valid"]
```

## Remedies

- **White's robust standard errors** (heteroskedasticity-consistent SEs) — most common fix
- **Weighted Least Squares** (WLS) — if you know the form of heteroskedasticity
- **GARCH models** — if the heteroskedasticity has a time-series structure

For more econometrics practice, explore our FRM Part I question bank.

How do you detect heteroskedasticity in a linear regression, and why does it matter for FRM?

Why It Matters

Detection Method 1: Breusch-Pagan Test

Detection Method 2: White Test

Example

Remedies

Master Part I with our FRM Course

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