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psi245b - 2 σ a Under what conditions is Y t a weakly...

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University of California D. Steigerwald Department of Economics Economics 245B Problem Set I 1. Consider the time-series model t t t U X Y + = b t = 1,… ,n 1 1 - + = t t t V V U a t V ~ ( 29 , IN 2 0 s a. Under what conditions is t Y stationary? b. If we assume that Y t is stationary and 0 0 = V , describe carefully how to obtain conditional maximum likelihood estimators of b and q using the Gauss-Newton algorithm. How would you obtain consistent initial values of $ $ β θ and ? c. Construct a test of the hypothesis a 1 = 0 using the Lagrange multiplier principle. 2. Two dynamic models are (1) ( 29 ( 29 α β L Y L X U t t t = + (2) ( 29 Y L L X U t t t = + β α ( ) where, in each case, α (L) and β (L) are polynomials in the lag operator of order p and q , respectively, X t is an exogenously generated time series, and U t ( 29 IN 0 2 , σ
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Unformatted text preview: 2 , σ . a. Under what conditions is Y t a weakly stationary random variable in each of the above models? b. Give an example of how each type of model might arise in practice. c. For the case p = 1 : (i) Describe and justify a large sample estimation method for model (1), and consider the consequences of applying it if (2) is the true model. (ii) Describe how you would estimate (2) in large samples, and outline the consequences if (1) is the true model. d. Given sample observations Y t ,X t ,t = 1,. ..,n, and a forecast of X n + 1 , denoted X n F + 1 , what are the optimal forecasts of Y n+1 from models (1) and (2) with p = q = 1?...
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