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

• 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:
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) $P_{\nu} = \mathit{GED}_{\nu}(0,1)$ $\nu \succ 1$
Remarks
1. A positive risk-premium (i.e. $$\lambda$$) indicates that data series is positively related to its volatility.
2. 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.
3. The mere existence of risk-premium is, therefore, another reason that some historical stocks returns exhibit serial correlations.