Generalized Autoregressive Conditional Heteroskedasticity (GARCH) 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:
  $$\left[\epsilon_t\right] \sim i.i.d$$
  $$E\left[\epsilon_t\right]=0$$
  $$\mathit{VAR}\left[\epsilon_t\right]=1$$
• $P_{\nu}$ is the probability distribution function for $\epsilon_t$. 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 (GED)
     $$P_{\nu} = \mathit{GED}_{\nu}(0,1) $$
     $$\nu \succ 1 $$

• 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 $a_t$
• In financial markets, the negative returns have higher influence on volatility levels than positive ones do. GARCH model does not have different treatment based on shock/error term direction.
• GARCH(p,q) model with normal-distributed innovation has p+q+2 estimated parameters
• GARCH(p,q) model with GED or student's t-distributed innovation has p+q+3 estimated parameters

Related Links

• Wikipedia - Autoregressive conditional heteroskedasticity

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