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:

- 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:

- Normal Distribution

$$P_{\nu} = N(0,1) $$ - Student's t-Distribution

$$P_{\nu} = t_{\nu}(0,1) $$

$$\nu \gt 4 $$ - Generalized Error Distribution (GED)

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

$$\nu \gt 1 $$

- Normal Distribution

## Remarks

- GARCH-M(p,q) model with normal-distributed innovation has p+q+3 estimated parameters
- GARCH-M(p,q) model with GED or student's t-distributed innovation has p+q+4 estimated parameters
- 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.

## References

- Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740

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