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Question 2. Let Yo, Y1, ... denote the observable price series of a stock. Define the log-daily returns by Xt = log Yt for t = 1, 2, .... We model {X,} by the stationary ARCH(1) series of order 1 of the form Xt = Otft. 1. Define a 1-st order autoregressive conditional heteroscedastic model ARCH(1) for {X} by stating the form of of and appropriate assumptions on {}. [3] 2. Given the filtration Ft-1, for t 2 1, show that . the conditional mean of Xt is E(X(Ft-1) = 0 and . the conditional variance of Xt is E(X? | Ft-1) = of. [5] 3. Show that X, and Xt_1 are uncorrelated for all t 2 2. [3] 4. Using the expression of the conditional variance, obtain an expression for the uncondi- tional variance of Xt, namely Var(Xt). [3] 5. State the difference between an ARCH(1) and a GARCH(1,1) model. [2]