# Generalized Autoregressive Conditional Heteroskedasticity (GARCH) Model

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 the 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 ARCH component model.
• $q$ is the order of the GARCH component model.
• $\beta_1,\beta_2,...,\beta_q$ are the parameters of 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): $$P_{\nu} = \mathit{GED}_{\nu}(0,1)$$ $$\nu \succ 1$$

## 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 converges to the unconditional variance of $a_t$.
• In financial markets, negative returns have a higher influence on volatility levels than positive ones do. The GARCH model does not have different treatments 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.