I. Consider the ARCH(1) model at=t(2+0.5(at-1)2)1/2, where 1, 2 ... are independent N(0,1) random variables. Note that
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Question:
I. Consider the ARCH(1) model at=εt(2+0.5(at-1)2)1/2, where ε1, ε2 ... are independent N(0,1) random variables. Note that at is stationary.
1. Prove that the conditional expected value E(at|at-1,...) and the unconditional expected value E(at) of at are equal to 0.
2. Find the conditional variance Var(at|at-1,...) and the unconditional variance Var(at) of at.
3. Show that Cov(at, at-h)=0 for any h=1,2,...
II. Consider the stationary AR(1)/ARCH(1) model for the log-return rt given by rt=1+0.1rt-1+at, with at defined as above in I.
1. Find the unconditional expected value E(rt) of rt.
2. Find the unconditional variance Var(rt) of rt.
Related Book For
Probability and Random Processes With Applications to Signal Processing and Communications
ISBN: 978-0123869814
2nd edition
Authors: Scott Miller, Donald Childers
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