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psii245b

psii245b - b Derive the asymptotic covariance matrix for 29...

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University of California D. Steigerwald Department of Economics Economics 245B Problem Set II 1. Consider the linear regression model with a lagged dependent variable Y t = β 0 + β 1 X t + β 2 Y t-1 + U t , for t = 1,. ..,T. Now assume that U 0 and Y 0 are known and let U t = ρ U t-1 + V t , where E[V t ] = 0 and E[ V t 2 ] = σ 2 . We know that the OLS estimator of β 2 is inconsistent if the errors exhibit serial dependence. Show that this is the case by determining what T -1 Σ Y t-1 U t converges to in probability 2. For the model in question 1, additionally assume that V t is normally distributed. Derive the information matrix for the parameter vector ( β 0 , β 1 , β 2 , ρ , σ 2 ) . 3. Consider the process Y Y U t t t = + - φ 1 where φ< 1,U t ( 29 IN t T 0 12, 2 , , , ..., . σ = a. Write both the full and approximate log-likelihood functions for such a process.
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Unformatted text preview: b. Derive the asymptotic covariance matrix for ( 29 \$ \$ , \$ ′= θ φσ 2 . Hint: Since the difference between the exact and conditional log-likelihood functions is negligible in large samples, use the conditional log-likelihood function. c. Based on (a) above, derive the equation that you would solve to obtain an estimate of φ in each case. How would you estimate φ based on Yule-Walker equations? What is the relationship between the three methods? Hint: You may wish to concentrate out σ 2 to form the concentrated log-likelihood function....
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