Building a Volatility Model

Procedure

A systematic four-step approach to building ARCH/GARCH models for asset return series.

Overview

Building a volatility model consists of four sequential steps, moving from the mean equation to the volatility specification and finally to model validation.

Step 1: Specify the Mean Equation

Test for serial dependence in the return data and build an econometric model to remove any linear dependence.

Actions:

• Examine ACF/PACF of returns $\{r_t\}$
• Fit an appropriate model (ARMA/ARIMA) if needed: $$r_t={\mu }_t+a_t$$
• ${\mu }_t$ can be modeled with regression or ARMA specification
• The goal is to obtain residuals $\{a_t\}$ that are serially uncorrelated

Check: Ljung-Box test on $\{a_t\}$ should not reject white noise

Step 2: Test for ARCH Effects

Use the squared residuals from the mean equation to test for ARCH effects.

Actions:

• Compute squared residuals $\{a_t^2\}$
• Apply tests for ARCH effects:
  • Ljung-Box test on $\{a_t^2\}$
  • LM test (Lagrange Multiplier)

Decision:

• If tests are significant → proceed to Step 3
• If not significant → no ARCH effects present, standard model suffices

Step 3: Specify a Volatility Model

If ARCH effects are significant, specify and estimate a volatility model.

Actions:

• Use PACF of $\{a_t^2\}$ to determine ARCH order $m$
• Consider GARCH for parsimony (often GARCH(1,1) suffices)
• Perform joint estimation of mean and volatility equations
• Use Maximum Likelihood Estimation

Typical specification:

$$r_t=\mu +a_t,a_t={\sigma }_t{ϵ}_t,{\sigma }_t^2={\alpha }_0+{\alpha }_1{a}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2$$

Step 4: Check the Fitted Model

Carefully check the fitted model and refine if necessary.

Actions:

• Compute standardized residuals: ${\tilde{a}}_t={\hat{a}}_t/{\hat{\sigma }}_t$
• Apply diagnostic checks
• If inadequacies found, return to Step 3 and modify specification

Flowchart

flowchart TD
    A[(Data: r_t)] --> B[[Step 1: Mean Equation]]
    B --> C[/Residuals: a_t/]
    C --> D{Step 2: ARCH Effects?}
    D -->|No| E((Done))
    D -->|Yes| F[[Step 3: Specify ARCH/GARCH]]
    F --> G[[Step 4: Check Model]]
    G --> H{Model Adequate?}
    H -->|Yes| I((Done))
    H -->|No| F
    
    style A fill:#e1f5fe
    style B fill:#fff9c4
    style F fill:#fff9c4
    style G fill:#fff9c4
    style C fill:#e8f5e9
    style E fill:#e8f5e9
    style I fill:#e8f5e9
    style D fill:#ffe0b2
    style H fill:#ffe0b2

Related

• Testing for ARCH Effects
• GARCH
• GARCH