# GARCH Modeling

Models with generalized autoregressive conditional heteroskedasticity (GARCH) model

The generalized autoregressive conditional heteroskedasticity (GARCH) model is basically an autoregressive moving average model (ARMA model), but for the error variance.

$$x_t = \mu + a_t$$ $$\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i a_{t-i}^2}+\sum_{j=1}^q{\beta_j \sigma_{t-j}^2}$$ $$a_t = \sigma_t \times \epsilon_t$$ $$\epsilon_t \sim P_{\nu}(0,1)$$ Where:

• $x_t$ is the time series value at time t.
• $\mu$ is the mean of GARCH in Excel 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)

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

$\nu \succ 1$

## GARCH Modeling Function ARCH/GARCH Modeling

The ARCH/GARCH modeling functionality automates the ARMA model construction steps: guessi...