A. Suppose that E(ut ut - 1, ut - 2, c) = 0, that var1ut
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A. Suppose that E(ut ut - 1, ut - 2, c) = 0, that var1ut ut - 1, ut - 2, c2 follows the ARCH(1) model s2 t = a0 + a1
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A. Suppose that E(ut ut - 1, ut - 2, c) = 0, that var1ut ut - 1, ut - 2, c2 follows the ARCH(1) model s2 t = a0 + a1u2 t - 1, and that the process for ut is stationary. Show that var1ut2 = a0> 11 - a12. (Hint: Use the law of iterated expectations E(u2 t) = E[E(u2 t ut - 1)].) b. Extend the result in (a) to the ARCH(p) model. c. Show that gp i = 1ai 6 1 for a stationary ARCH(p) model. d. Extend the result in (a) to the GARCH(1,1) model. e. Show that a1 + f1 6 1 for a stationary GARCH(1,1) model.
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