GARCH Model

If an autoregressive moving average model (ARMAⁱ model) is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCHⁱ, Bollerslev(1986)) model.

$x_t = \mu + a_t$

$\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i a_{t-i}^2}+\sum_{j=1}^q{\beta_j \sigma_{t-j}^2}$

$a_t = \sigma_t \times \epsilon_t$

$\epsilon_t \sim P_{\nu}(0,1)$

Where:

$x_t$ is the time series value at time t.
$\mu$ is the mean of GARCH model.
$a_t$ is the model's residual at time t.
$\sigma_t$ is the conditional standard deviation (i.e. volatility) at time t.
$p$ is the order of the ARCH component model.
$\alpha_o,\alpha_1,\alpha_2,...,\alpha_p$ are the parameters of the the ARCH component model.
$q$ is the order of the GARCH component model.
$\beta_1,\beta_2,...,\beta_q$ are the parameters of the the GARCH component model.
$\left[\epsilon_t\right]$ are the standardized residuals:

$\left[\epsilon_t\right] \sim i.i.d$

$E\left[\epsilon_t\right]=0$

$\mathit{VAR}\left[\epsilon_t\right]=1$
is the probability distribution function for . Currently, the following distributions are supported:
1. Normal distribution
  $P_{\nu} = N(0,1)$ .
2. Student's t-distribution
  $P_{\nu} = t_{\nu}(0,1)$
  
  $\nu \succ 4$
3. Generalized error distribution (GEDⁱ)
  $P_{\nu} = \mathit{GED}_{\nu}(0,1)$
  
  $\nu \succ 1$

Remarks

Clustering: a large $a_{t-1}^2$ or $\sigma_{t-1}^2$ gives rise to a large $\sigma_t^2$ . This means a large $a_{t-1}^2$ tends to be followed by another large $a_{t}^2$ , generating, the well-known behavior, of volatility clustering in financial time series.
Fat-tails: The tail distribution of a GARCH(p,q) process is heavier than that of a normal distribution.
Mean-reversion: GARCH provides a simple parametric function that can be used to describe the volatility evolution. The model converge to the unconditional variance of $a_t$ :

$\sigma_{\infty}^2 \rightarrow V_L=\frac{\alpha_o}{1-\sum_{i=1}^{max(p,q)}\left(\alpha_i+\beta_i\right)}$

References

Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740

GARCH Model

Remarks

References

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