ARMA vs ARCH/GARCH Relationship

About

The fundamental insight connecting linear time series models (ARMA) to volatility models (ARCH/GARCH) through the lens of squared series.

The Key Analogy

Linear Model	Squared Series	Volatility Model
AR(p) process	$a_t^2$ follows AR	ARCH(p)
ARMA(p,q) process	$a_t^2$ follows ARMA	GARCH(p,q)

ARCH as AR Process

An ARCH(m) model can be viewed as an AR(m) process applied to the squared series $a_t^2$:

$$a_t^2={\alpha }_0+{\alpha }_1{a}_{t-1}^2+...+{\alpha }_m{a}_{t-m}^2+{\eta }_t$$

where ${\eta }_t=a_t^2-{\sigma }_t^2$ is a martingale difference sequence.

GARCH as ARMA Process

A GARCH(m,s) model can be written as an ARMA(max(m,s), s) process for $a_t^2$:

$$a_t^2={\alpha }_0+\sum _{i=1}^{\max (m,s)}({\alpha }_i+{\beta }_i)a_{t-i}^2+{\eta }_t-\sum _{j=1}^s{\beta }_j{\eta }_{t-j}$$

This representation explains why:

• GARCH is more parsimonious than high-order ARCH (just like ARMA vs AR)
• The stationarity condition for GARCH mirrors that of ARMA: $\sum ({\alpha }_i+{\beta }_i)<1$

Why This Matters

This relationship means we can use familiar tools from ARMA modeling:

• PACF of $a_t^2$ → determine ARCH order
• ACF/PACF of $a_t^2$ → identify GARCH structure
• Stationarity conditions → ensure finite unconditional variance

Related

• ARCH(m) Model
• GARCH(m,s) Model
• AR(p) Process Model
• ARMA(p,q) Process Model