psiv245b - Write down the sample periodogram estimator Show...

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University of California D. Steigerwald Department of Economics Economics 245B Problem Set IV 1. Given a time series y t , t = 1,. ..,T, where T is an even number, consider its finite Fourier representation given by the cyclical trend model (1) y a T T t K n = + = - 0 1 1 2 [ α K cos( λ K t) + β K sin( λ K t)] + (-1) t / T α n + ε t where ε t IN(0, σ 2 ), and λ K = 2 π K T and K = 1,. ..,T/2 . a. Suppose that we also ran the following regression of (2) y t = 2 1 T α * cos( λ 1 t) + η t . Would the estimate of α 1 from equation (1) equal the estimate of α 1 * from equation (2)? Please explain your answer. b. If we added additional frequency terms to equation (1) would your answer to part (a) change? Please be sure to justify your answer. c. How could one test that a component of a specific frequency belongs in the model? 2a. Consider estimating the non-normalized sample spectral density function.
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Unformatted text preview: Write down the sample periodogram estimator. Show that this estimator is asymptotically unbiased. Is it possible to have an estimator that is asymptotically unbiased but not consistent? Explain. 2b. How could the sample periodogram estimator be used to estimate the covariance matrix? Can we separately identify each of the elements in the matrix? Describe any potential problems with the sample periodogram estimator and how to construct a new estimator to overcome them. 3. Give the theoretical autocovariance and autocorrelation functions for the following two MA processes y t = (1 - .5L) ε t y t = (1 + 0.5L + 0.4L 2 ) ε t where ε t ∼ iid(0,1)...
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