\[x_t = \mu + a_t\]
\[\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i a_{t-i}^2}+\sum_{j=1}^q{\beta_j \sigma_{t-j}^2}\]
\[a_t = \sigma_t \times \epsilon_t\]
\[\epsilon_t \sim P_{\nu}(0,1)\]
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:
- Normal distribution \(P_{\nu} = N(0,1)\).
- Student's t-distribution \(P_{\nu} = t_{\nu}(0,1)\) \(\nu \succ 4\)
- Generalized error distribution (GED) \[P_{\nu} = \mathit{GED}_{\nu}(0,1)\] \[\nu \succ 1\]
- 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 \(a_t\): \[\sigma_{\infty}^2 \rightarrow V_L=\frac{\alpha_o}{1-\sum_{i=1}^{max(p,q)}\left(\alpha_i+\beta_i\right)}\]
