6.7 Extensions of ARCH Models

The ARCH model has been extended to allow the conditional variance to be a determinant of the mean, and under the name ARCH-M was introduced by (,,). This is an attractive form in financial applications, since it is natural to suppose that the expected return on an asset is proportional to the expected risk of the asset. The general model used in financial series is an ARMA process in the mean and innovations with time dependent conditional heteroskedasticity are represented by an ARCH process. Its expression is given by

$\displaystyle y_{t}$ $\displaystyle =$ $\displaystyle \mu + \delta\sigma_t^2+ \sum_{i=1}^p\phi_iy_{t-i}+
\sum_{i=1}^q\theta_{i}u_{t-i}+u_{t}\cr u_{t}$ (6.30)

Note that the greatest complexity introduced in this model comes from evaluating the derivatives of innovations and the conditional variance with respect to the parameters in a non obvious way and consequently, the explicit expression in the first order likelihood conditions are not available in a such simple form. Moreover, the information matrix is not block diagonal between the parameters of the mean and variance, so the estimation procedure must be carried out simultaneously (for more details, see McCurdy and Morgan (1988)).

The conditions in a GARCH process to have a finite unconditional variance are often unsatisfied in a high frequently sampled financial series. Engel and Bollerslev (1986) introduced a new variety of ARCH models, in which $ \alpha_1+\beta_1=1$, in the formulation of GARCH(1,1) given in (6.28), named Integrated GARCH model (IGARCH). It is easy to see in this model, that

$\displaystyle E[\sigma^2_{t+k}\vert \sigma^2_t] = \sigma^2_t + \alpha_0 k
$

very similar to the conditional mean specification of a random walk, however this model is strictly stationary unlike a random walk. Consequently, the IGARCH model has some characteristics of integrated processes. At this stage, could be interesting to go further in understanding the persistence effect. In a few words, if the shocks (imputs) in the conditional variance persist indefinitely, the process is said to be persistent in variance. It is clear from (6.6.1), that the effect of a shock persists faraway but not very much only when $ \alpha_1+\beta_1<1$. That is to say, the persistence disappears in terms of the past of the process , i.e. in an unconditional sense(for more details, see section 5.5 in Gouriéroux ; 1997). Many empirical researchers in financial markets have found evidence that bad news and good news have different behaviour in the models, revealing an asymmetric behaviour in stock prices, negative surprises give to increase the volatility more than positive surprise, this effect is described as the leverage effect. For this reason,Nelson (1991) proposes a new model, in which the conditional variance is

$\displaystyle \log \sigma_t^2 = \alpha_0 + \sum_{j=1}^q \alpha_j \log
\sigma_{t...
...heta\varepsilon_{t-j} +(
\gamma \vert\varepsilon_{t-j}\vert- (2/\pi)^{1/2} )\}
$

where the $ \gamma$ parameter allows this asymmetric effect.