By Christian Gouriéroux (auth.)

ISBN-10: 1461218608

ISBN-13: 9781461218609

ISBN-10: 1461273145

ISBN-13: 9781461273141

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"Gourieroux deals a pleasant stability of concept and alertness during this ebook on ARCH modeling in finance…The publication is definitely written and has large references. Its concentrate on finance will entice monetary engineers and fiscal threat managers."

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**Extra resources for ARCH Models and Financial Applications**

**Example text**

The basic idea is to increase the order of the autoregressive polynomial. 11) i-I where U = (u t ) is a martingale difference sequence. 2 General Properties of ARCH Processes and depends on the past through the q most recent values of 35 £;. ii) GARCH (p, q) Models (Bollerslev 1986) The ARCH model is based on an autoregressive representation of the conditional variance. One may also in a usual way add a moving average part. The GARCH processes (generalized autoregressive conditionally heteroscedastic) are thus obtained.

Two types of additional regressors may be distinguished: a) Some are known functions of the lagged values of the observed process. For instance, we could test the null hypothesis of a weak linear AR(l) process Y, = pY,_1 +£" or of the presence of a threshold effect: This test can be performed using the classical Student t -statistic for the parameter a. b) In other cases, the auxiliary regressors are known functions of Y'-I, Y'-2, ... £'-2, ... For instance, if Y is strictly stationary, we know that the series may be suitably approximated by its Volterra's expansion.

I-I The estimator is then defined as a solution to the maximization problem: max log L(y; 8). (J 44 4. Estimation and Tests It is known that in general the properties of this estimator, denoted OT and called pseudo maximum likelihood estimator (PML), both depend on the true underlying distribution and on the one used to compute the likelihood function (here the normal distribution). However, under standard regularity conditions (White 1981; Gourieroux et al. 1984; Gallant 1987; Gourieroux and Monfort 1989), this estimator is consistent even if the underlying distribution is not conditionally normal; that is, this property does not depend on the distribution that is used to build the likelihood function.

### ARCH Models and Financial Applications by Christian Gouriéroux (auth.)

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