# GARCH-M Modeling

Model the conditional volatility as a GARCH process, but adds a heteroskedasticity term into the mean equation.

## GARCH-M with NumXL

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) $$P_{\nu} = \mathit{GED}_{\nu}(0,1)$$ $$\nu \succ 1$$

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