psx245b

# psx245b - 2 Consider the following ARCH(1 model y t = β ε...

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University of California D. Steigerwald Department of Economics Economics 245B Problem Set X 1. Suppose you have two univariate processes, {y t } and {x t }. (a) Are I(0) and stationary equivalent characterizations of a process? Why or why not? (b) Formally define the concept of cointegration. How would you determine if y t and x t are cointegrated? (c) Suppose you find that y t and x t are cointegrated. Characterize the properties of the ordinary least squares (OLS) estimator of the cointegrating relation. Suppose that y t and x t are not cointegrated, what are the characteristics of the OLS estimator? (d) Suppose y t and x t are both I(1) and are cointegrated as y t = α 0 + α 1 x t + η t . (1) Carefully explain why y t = β 0 + β 1 x t + β 2 (y t-1 - α 0 - α 1 x t-1 ) + ε t (2) is called an error correction model. Suppose in addition to (1) that x t = u t . If u t and η t are correlated, how will this affect the properties of parameter estimators of (2) and how can the specification be modified to enhance its desirability? If α 0 is erroneously excluded from (2), how will this affect the estimators of the parameters in (2)?

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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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psx245b - 2 Consider the following ARCH(1 model y t = β ε...

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