GARCH-in Mean (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:

  1. 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:
      1. Normal Distribution
        $$P_{\nu} = N(0,1) $$
      2. Student's t-Distribution
        $$P_{\nu} = t_{\nu}(0,1) $$
        $$\nu \gt 4 $$
      3. 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 (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

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