If an autoregressive moving average model (ARMA model) is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH, 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:

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

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

$$\mathit{VAR}\left[\epsilon_t\right]=1$$ - $P_{\nu}$ is the probability distribution function for $\epsilon_t$. Currently, the following distributions are supported:

- Normal distribution

$$P_{\nu} = N(0,1) $$. - Student's t-distribution

$$P_{\nu} = t_{\nu}(0,1) $$

$$\nu \succ 4 $$ - Generalized error distribution (GED)

$$P_{\nu} = \mathit{GED}_{\nu}(0,1) $$

$$\nu \succ 1 $$

- Normal distribution

## 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 $a_t$- In fincnail 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

## 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

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