GARCH-in Mean (GARCH-M) Model

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:

1. The GARCH-M(p,q) model is written as:
   $$x_t = \mu + \lambda \sigma_t + a_t$$
   $$\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i a_{t-i}^2}+\sum_{j=1}^q{\beta_j \sigma_{t-j}^2}$$
   $$ a_t = \sigma_t \times \epsilon_t$$
   $$ \epsilon_t \sim P_{\nu}(0,1)$$
   Where:
   
   • $x_t$ is the time series value at time t.
   • $\mu$ is the mean of the GARCH model.
   • $\lambda$ is the volatility coefficient 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 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
        $$P_{\nu} = t_{\nu}(0,1) $$
        $$\nu \gt 4 $$
     3. Generalized Error Distribution (GED)
        $$P_{\nu} = \mathit{GED}_{\nu}(0,1) $$
        $$\nu \gt 1 $$

Remarks