By Christian Gouriéroux (auth.)

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"Gourieroux bargains a pleasant stability of thought and alertness during this e-book on ARCH modeling in finance…The publication is definitely written and has broad references. Its concentrate on finance will attract monetary engineers and monetary danger managers."

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**Example text**

It is an ARMA[Max(p, q), p] representation for the process £2 but with an error term U that does not necessarily have a constant variance. We emphasize the fact that the GARCH terminology is similar to that of the ARMA, except that p denotes the order of the moving average part and not necessarily of the autoregressive polynomial. iii) ARMA-GARCH Models (Weiss 1986) Finally, the GARCH modelling may be applied to the innovation process instead of the initial process. We may simultaneously introduce several additional effects either in the conditional mean or in the conditional variance.

5 Exercises 27 3. The ergodicity condition is supposed to be satisfied, and the stationary solution of the threshold autoregressive equation is retained. Specify the form of the mean and covariances of the process Z. Show that the sequence of covariances satisfies a linear recursive equation of a given order and, therefore, Z admits a weak ARMA representation. Find this representation. 4. Follow the same approach to find a weak ARMA representation for the Y process. 3 Univariate ARCH Models The aim of this chapter is to describe the major specifications with conditional heteroscedasticity found in the literature.

Aq )(1 d - \11)-1 (aI, ... , aq )' ~ 0, and deduce that the marginal distribution of Y1 has fatter tails than the normal distribution [see Milhoj (1985) and Diebold (1986)]. 5. a2 ~ 0. at 1. Verify that the fourth order moment exists if a2 + 3a; + 3ai + 3a;a2 - 3a1 < 1. 2. Show that the autocorrelations of yl2 are such that (2) PI al I (2) 2 2 (2) (2) (2) = - - . P2 = --(a2+ a l -a2), Ph =aIPh_1 +a2Ph_2' 1 - a2 1 - a2 3. Taking into account the various constraints on al. a2. determine the variation domain of p~2) and p~2) and compare this domain with the one usually obtained for an AR(2) process [see Bollerslev (1988)].