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psvi245b - Be explicit in setting up the problem and...

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University of California D. Steigerwald Department of Economics Economics 245B Problem Set VI 1. Consider the problem of selecting the correct order of a moving average process with a sample of T observations. If the true order is q*, and the autocovariance function at lag τ is denoted V( τ ) we introduce the following test statistic for order q: ( 29 ( 29 S V i V j g i j q n ij = = + $ $ $ , 1 where g = VV is the (T-1) × (T-1) autocovariance matrix and $ g ij is the (i,j) element of the inverse of the sample analog of g. a. Describe intuitively what the value of S should be for all q q*. What issues arise in the selection n? b. How does this test statistic relate to the likelihood function (does it choose the value of q that maximizes the likelihood function)? 2. Given y t = ε t + θε t-1 t = 1,...,T, where ε t is iid, E[ ε t ] = 0, E[ ε t 2 ] = σ 2 , and θ < 1 . a. How would you estimate θ using the Kalman filter?
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Unformatted text preview: Be explicit in setting up the problem and carefully explain the steps. b. Is there measurement error in this problem? If so, where? c. What are the advantages to using the Kalman filter? d. How does the Kalman filter compare to exact maximum likelihood estimation? e. Once you have an estimate of θ , how would you forecast y t+n ? 3. Consider the VAR(1) model given by: Y t = Φ Y t-1 + ε t t = 1,. ..,T ε t ∼ Multivariate independent Normal (Q, Ω ) . a. Describe in detail, including the likelihood function, how you would estimate Φ using Approximate MLE. You may assume y 1 ~ IN(0 , Σ ) where vec I vec N ( ) ( ) ( ). Σ Φ Φ Ω =-⊗-2 1 b. Describe in detail how you would estimate Φ using Exact MLE. c. Discuss any other “useful” approach to estimating Φ ....
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