Model Checking for ARCH/GARCH

Procedure

Diagnostic checking of fitted ARCH/GARCH models using standardized residuals and goodness-of-fit measures.

Standardized Residuals

The foundation of model diagnostics is the standardized residual:

$${\tilde{a}}_t=\frac{{\hat{a}}_t}{{\hat{\sigma }}_t}$$

If the model is correct, $\{{\tilde{a}}_t\}$ should behave like the assumed innovation process $\{{ϵ}_t\}$ (typically i.i.d. $N(0,1)$).

Checking the Mean Equation

Test: Ljung-Box statistic on standardized residuals $\{{\tilde{a}}_t\}$

• $H_0:$ No autocorrelation in residuals
• If rejected → mean equation inadequate, need different ARMA specification

Checking the Volatility Equation

Test: Ljung-Box statistic on squared standardized residuals $\{{\tilde{a}}_t^2\}$

• $H_0:$ No remaining ARCH effects
• If rejected → volatility model inadequate, need higher order or different specification

Checking Distributional Assumption

Normality Tests:

• QQ-plot: Visual check of normality
• Shapiro-Wilk test: Formal test for normality
• Jarque-Bera test: Tests skewness and kurtosis jointly

If normality is rejected, consider:

• Student-t distribution
• Skewed distributions
• Generalized Error Distribution (GED)

Goodness-of-Fit Measures

Sum of Squared Residuals (SSR)

Since ${ϵ}_t=a_t/{\sigma }_t$:

$$SSR=\sum _{t=1}^T\frac{{\hat{a}}_t^2}{{\hat{\sigma }}_t^2}$$

Log-Likelihood Value

From the fitted model:

$$2\ell =\sum _{t=1}^T\left[\ln ({\hat{\sigma }}_t^2)+\frac{{\hat{a}}_t^2}{{\hat{\sigma }}_t^2}\right]-T\ln (2\pi )$$

Information Criteria

Use for model comparison (lower is better):

Criterion	Formula	Penalty
AIC	$-2\ell +2n$	$2n$
BIC	$-2\ell +n\ln (T)$	$n\ln (T)$

where $n$ = number of parameters, $T$ = sample size.

Model Selection

BIC has a stronger penalty for complexity than AIC, generally favoring more parsimonious models.

Summary Checklist

Component	Test	Target
Mean equation	Ljung-Box on ${\tilde{a}}_t$	$p>0.05$
Volatility equation	Ljung-Box on ${\tilde{a}}_t^2$	$p>0.05$
Distribution	QQ-plot, Shapiro-Wilk	Resemble assumed dist.
Overall fit	AIC, BIC	Compare across models

Related

• Standardized Residuals
• Building a Volatility Model
• Maximum Likelihood Estimation