Download e-book for kindle: ARCH Models and Financial Applications by Christian Gouriéroux (auth.)

By Christian Gouriéroux (auth.)

ISBN-10: 1461218608

ISBN-13: 9781461218609

ISBN-10: 1461273145

ISBN-13: 9781461273141

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Sample text

In order to study the second order stationarity, it is then sufficient to consider the variance, and to show that it is asymptotically time independent. 19. A process e satisfying a GARCH(p,q) model with positive coefficients c ~ 0, ai ~ 0, i = I, ... ,q, (3j ~ 0, j = I, ... , p, is asymptotically second order stationary if q a(I)+{3(I) = p Lai + L{3j < 1. j-I i-I Proof. q) e;=c+ L p (ai+{3i)e;_i+ Ur -L{3jUr-j. q) Ee; = c+ L (a; + (3i )E(e;_;). q) 1- L (ai + (3i)L i , i-I are strictly outside the unit circle, the sequence Ee; converges and the process is asymptotically second order stationary.

Q) 1- L (ai + (3i)L i , i-I are strictly outside the unit circle, the sequence Ee; converges and the process is asymptotically second order stationary. (**) This condition implies that a(l) + (3(I) < 1. Indeed, if a(l) + (3(I) were greater than orequalto 1, we would have l-a(O)- (3(0) > and l-a(I)- (3(I) ~ 0; there would then be a root of the characteristic polynomial that would be real and lie between and I. ° ° 38 3. Univariate ARCH Models (***) Conversely, let us assume that a(1) + JS(1) < 1.

F. 1 ah l (f3) LT __1_ aml(a) t I_I 2 h:/ (f3) [YI - aa UI ' ml(a)f ahl (f3) h;-(f3) af3 2 2 ~ h l (f3) ~(UI - 1). 1, and from their expressions one derives the matrix J by J = E [_1_ o h,(O) _1_ am,(O) am,(O) + ahl(O) ahl(O)] . 8) The second matrix 1 = Eo [ill~: I, il~: II] depends on the conditional third order moment of Y, 48 4. 9) are direct extensions of the one derived in the LLd. case. One can easily check that the two matrices I and J coincide when the true underlying distribution is normal since K,(O) = 3, and M3,(O) = O.

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


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