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Unformatted text preview: 2. Consider the following ARCH(1) model: y t = β + ε t ε t = r t ( γ ) u t where u t ∼ IN(0,1) and (1) r t ( γ ) 2 = γ + γ 1 (y t-1- β ) 2 . (a) Describe the features of data that this model is designed to capture. Why is the variance of u t assumed to be 1? (b) Construct the likelihood function of the model in (1). How would you estimate the parameters? (c) Suppose we generalize the model so that the mean of u t is α and the scale of u t is σ : y t = β + h t ( φ )( α + σ u t ) h t ( φ ) 2 = [1 + φ 1 (y t-1- β ) 2 ] . (2) What is φ 1 in terms of γ and γ 1 ? Suppose that α ≠ 0. Does the presence of β in (1) account for non-zero α ? Carefully explain your answer. (d) Given your answer to (c), if α ≠ 0, is the estimator of ( β , γ ) from (1) consistent? If not, propose a test for consistency....
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- Fall '08
- Economics, Estimation theory, Yt, Bias of an estimator, stationary equivalent characterizations