ARCH(m) Model

An Autoregressive Conditionally Heteroscedastic model of order $m$ captures time-varying volatility by making the conditional variance a function of past squared shocks.

Definition

Let $\{a_t\}$ be a series of shocks (residuals) from a mean equation. An ARCH(m) model is:

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

where:

• ${ϵ}_t\sim \text{i.i.d.}(0,1)$ (white noise with unit variance)
• ${\alpha }_0>0$ and ${\alpha }_i\ge 0$ for $i=1,...,m$ (ensures positive variance)

ARCH(1) as Special Case

When $m=1$:

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

A large value of $a_{t-1}^2$ leads to a large conditional variance at time $t$, creating volatility clustering.

Unconditional Mean

$$E(a_t)=E({\sigma }_t{ϵ}_t)=E({\sigma }_t)E({ϵ}_t)=0$$

The unconditional mean of an ARCH process is zero.

Unconditional Variance

Assuming stationarity (${\sum }_{i=1}^m{\alpha }_i<1$):

$$\text{Var}(a_t)=E(a_t^2)=\frac{{\alpha }_0}{1-{\sum }_{i=1}^m{\alpha }_i}$$

For ARCH(1) specifically:

$$\text{Var}(a_t)=\frac{{\alpha }_0}{1-{\alpha }_1},0\le {\alpha }_1<1$$

Kurtosis (Heavy Tails)

For ARCH(1) with Gaussian innovations, the kurtosis is:

$$\text{Kurtosis}=\frac{E(a_t^4)}{[E(a_t^2){]}^2}=\frac{3(1-{\alpha }_1^2)}{1-3{\alpha }_1^2}>3$$

This requires $0\le {\alpha }_1^2<\frac{1}{3}$ for the fourth moment to exist.

Interpretation

The excess kurtosis is positive, meaning the tail distribution of $a_t$ is heavier than normal—a key stylized fact of financial returns.

Serial Correlation Properties

• $a_t$ is serially uncorrelated: $\text{Cov}(a_t,a_{t-k})=0$ for $k\ne 0$
• But $a_t$ is dependent: $a_t^2$ follows an AR(m) process

Related

• GARCH(m,s) Model
• GARCH Relationship
• Conditional Variance