GARCH-M Model

In finance, the return of a security may depend on its volatility (risk). To model such phenomena, the GARCHⁱ-in-mean (GARCH-Mⁱ) 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 . 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 (GEDⁱ)
  
  $P_{\nu} = \mathit{GED}_{\nu}(0,1)$
  
  $\nu \succ 1$

Remarks

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.

GARCH-M Model

Remarks

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