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 at is stationary. 1. Prove that the conditional

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.