Autoregressive conditional heteroskedasticity

Builds on Time series Conditional variance Heteroscedasticity Independent and identically distributed random variables

Definition

Let $q \ge 1$ be an integer. A real time series $(\varepsilon_t)_{t \in \Z}$ is an ARCH( $q$ ) process if it satisfies the following conditions.

$\varepsilon_t = \sigma_t z_t$ for all $t$ , where $(z_t)_{t \in \Z}$ are independent and identically distributed random variables with mean $0$ and variance $1$ .
$\sigma_t^2 = \alpha_0 + \sum_{i=1}^{q} \alpha_i \varepsilon_{t-i}^2$ for all $t$ , where $\alpha_0 > 0$ and $\alpha_1, \dots, \alpha_q \ge 0$ are constants and $\sigma_t \ge 0$ .
$z_t$ is independent of $(\varepsilon_s)_{s < t}$ for all $t$ .

Then $\sigma_t^2$ is the conditional variance of $\varepsilon_t$ given the past $(\varepsilon_s)_{s < t}$ .

Remarks

The GARCH( $p$ , $q$ ) model replaces clause 2 by $\sigma_t^2 = \alpha_0 + \sum_{i=1}^{q} \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^{p} \beta_j \sigma_{t-j}^2$ with $\beta_j \ge 0$ .

Example: take $q = 1$ , $\alpha_0 = 1$ and $\alpha_1 = 1/2$ . If the process is weakly stationary with variance $v$ , then taking expectations in clause 2 gives $v = 1 + v/2$ , so $v = 2$ . In general a weakly stationary ARCH( $q$ ) process exists if and only if $\alpha_1 + \cdots + \alpha_q < 1$ , and then its variance is $\alpha_0 / (1 - \alpha_1 - \cdots - \alpha_q)$ .

Non-example: the rule $\sigma_t^2 = 1 - \frac12 \varepsilon_{t-1}^2$ fails clause 2 because $\alpha_1 < 0$ . It gives a negative "variance" as soon as $\abs{\varepsilon_{t-1}} > \sqrt{2}$ .

An ARCH process is uncorrelated in time, but it is not independent: the squares $\varepsilon_t^2$ follow an autoregressive pattern.

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Conditional variance concept
Heteroscedasticity concept
Independent and identically distributed random variables concept
Time series concept
Autoregressive conditional heteroskedasticity definition

Tag	`P3YM2DDGX9`
URI	`m2://P3YM2DDGX9`
Kind	definition
Writing	W0 stub
MSC	62M10, 91G70, 62P20
Defines	Autoregressive conditional heteroskedasticity
Uses	Time series, Conditional variance, Heteroscedasticity, Independent and identically distributed random variables

Autoregressive conditional heteroskedasticity

Definition

Remarks

Reading path

Further reading

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