In finance, the return of a security may depend on its volatility (risk). To model such phenomena, the GARCH-in-mean (GARCH-M) model adds a heteroskedasticity term into the mean equation. It has the specification: \[x_t = \mu + \lambda \sigma_t + a_t\] \[\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i a_{t-i}^2}+\sum_{j=1}^q{\beta_j \sigma_{t-j}^2}\] \[a_t = \sigma_t \times \epsilon_t\] \[\epsilon_t \sim P_{\nu}(0,1)\]
- Where:
- \(x_t\) is the time series value at time t.
- \(\mu\) is the mean of GARCH model.
- \(\lambda\) is the volatility coefficient (risk premium) for the mean.
- \(a_t\) 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) \[P_{\nu} = \mathit{GED}_{\nu}(0,1)\] \[\nu \succ 1\]
- Remarks
-
- A positive risk-premium (i.e. \(\lambda\)) indicates that data series is positively related to its volatility.
- Furthermore, the GARCH-M model implies that there are serial correlations in the data series itself which were introduced by those in the volatility \(\sigma_t^2\) process.
- The mere existence of risk-premium is, therefore, another reason that some historical stocks returns exhibit serial correlations.
