Standardized Residuals

The standardized residuals in ARCH/GARCH models are used for diagnostic checking and represent the “shocks” normalized by the estimated volatility.

Definition

Given the residuals ${\hat{a}}_t$ from the mean equation and the estimated conditional standard deviation ${\hat{\sigma }}_t$, the standardized residual is:

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

Properties

If the model is correctly specified, the standardized residuals $\{{\tilde{a}}_t\}$ should:

1. Mimic the behavior of ${ϵ}_t$ (the true innovation process)
2. Be serially uncorrelated (no autocorrelation in levels)
3. Have constant variance of 1 (homoscedastic)
4. Follow the assumed distribution (typically normal or Student-t)

Diagnostic Checks

Standardized residuals are used for model checking:

Check	Statistic	Purpose
Mean equation	Ljung-Box on $\{{\tilde{a}}_t\}$	Test for remaining autocorrelation
Volatility equation	Ljung-Box on $\{{\tilde{a}}_t^2\}$	Test for remaining ARCH effects
Normality	QQ-plot, Shapiro-Wilk	Test distributional assumption

NOTE

If the Ljung-Box test on $\{{\tilde{a}}_t^2\}$ shows significant autocorrelation, the volatility model is inadequate and needs refinement.

Related

• GARCH
• ARCH(m) Model
• GARCH(m,s) Model