GARCH Analysis

$x_t = \mu + a_t$ $\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i a_{t-i}^2}+\sum_{j=1}^q{\beta_j \sigma_{t-j}^2}$ $a_t = \sigma_t \times \epsilon_t$ $\epsilon_t \sim P_{\nu}(0,1)$ 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) $P_{\nu} = \mathit{GED}_{\nu}(0,1)$ $\nu \succ 1$
• 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 in Excel (p,q) process is heavier than that of a normal distribution.
• Mean-reversion: GARCH in Excel provides a simple parametric function that can be used to describe the volatility evolution. The model converge to the unconditional variance of $$a_t$$: $\sigma_{\infty}^2 \rightarrow V_L=\frac{\alpha_o}{1-\sum_{i=1}^{max(p,q)}\left(\alpha_i+\beta_i\right)}$