GARCH(m,s) Model

A Generalized Autoregressive Conditionally Heteroscedastic model extends ARCH by including lagged conditional variances, providing a more parsimonious representation of volatility dynamics.

Definition

Let $\{a_t\}$ be shocks from a mean equation. A GARCH(m,s) model is:

$$a_t={\sigma }_t{ϵ}_t,{\sigma }_t^2={\alpha }_0+\sum _{i=1}^m{\alpha }_i{a}_{t-i}^2+\sum _{j=1}^s{\beta }_j{\sigma }_{t-j}^2$$

where:

• ${ϵ}_t\sim \text{i.i.d.}(0,1)$
• ${\alpha }_0>0$, ${\alpha }_i\ge 0$ (at least one ${\alpha }_i>0$), ${\beta }_j\ge 0$
• ${\sum }_{i=1}^{\max (m,s)}({\alpha }_i+{\beta }_i)<1$ (stationarity condition)

GARCH(1,1) as Special Case

The most widely used specification:

$${\sigma }_t^2={\alpha }_0+{\alpha }_1{a}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2$$

This simple 3-parameter model often outperforms high-order ARCH models.

Relationship to ARCH

When ${\beta }_j=0$ for all $j$:

$$\text{GARCH}(m,s)\Rightarrow \text{ARCH}(m)$$

Unconditional Variance

For a stationary GARCH(m,s) process:

$$E(a_t^2)=\frac{{\alpha }_0}{1-{\sum }_{i=1}^{\max (m,s)}({\alpha }_i+{\beta }_i)}$$

For GARCH(1,1):

$$E(a_t^2)=\frac{{\alpha }_0}{1-{\alpha }_1-{\beta }_1},{\alpha }_1+{\beta }_1<1$$

ARMA Representation

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

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

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

Kurtosis (Heavy Tails)

For GARCH(1,1) with Gaussian innovations (when $1-({\alpha }_1+{\beta }_1{)}^2-2{\alpha }_1^2>0$):

$$\text{Kurtosis}=\frac{3[1-({\alpha }_1+{\beta }_1{)}^2]}{1-({\alpha }_1+{\beta }_1{)}^2-2{\alpha }_1^2}>3$$

Volatility Persistence

The parameters have distinct roles:

Parameter	Role	Interpretation
${\alpha }_1$	Response to news	Larger ${\alpha }_1$ = bigger response to new information ($a_t\to {\sigma }_{t+1}^2$)
${\beta }_1$	Persistence	Larger ${\beta }_1$ = more persistence in conditional variance (${\sigma }_t^2\to {\sigma }_{t+1}^2$)

Related

• ARCH(m) Model
• Principle of Parsimony
• GARCH Relationship