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

## 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 (λ) indicates that the 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 stock returns exhibit serial correlations.

## Related Links

## References

- James Douglas Hamilton; Time Series Analysis, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
- Tsay, Ruey S.; Analysis of Financial Time Series, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.

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