psviiii245b

psviiii245b - E[ t ] = 0 for all t, x t is I(0), and y t is...

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University of California D. Steigerwald Department of Economics Economics 245B Problem Set IX 1. Given y t T t t t = + = - ε θε 1 1 , ,..., , where { ε t } i.i.d., E t ε = 0, E t ε σ 2 2 = , θ < 1. a. How would you estimate θ using the Kalman filter? 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 + ? 2. Let y t follow an ARCH (1) process f(y t y t-1 ) f(0, σ t 2 ) , where σ δ δ t t y 2 0 1 1 2 = + - represents the conditional variance of y t . If we assume that the initial value, y 0 , has the unconditional distribution f(y 0 ) N(0, 2 ) derive the unconditional variance of y t .
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3. Consider the regression model y t = β 0 + β 1 x t + ε t where { ε t } t=1,. ..,T is a sequence of independent normal random variables.
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Unformatted text preview: E[ t ] = 0 for all t, x t is I(0), and y t is I(0). a. Suppose the variance of t equals t 2 . What difficulties characterize statistical hypothesis tests based on OLS estimators of = ( , 1 ) ? b. How would you test for heteroskedasticity of the form t 2 = 2 z t , where z t are known? Show that the variance of the OLS estimator is higher than the variance of the generalized least squares (GLS) estimator obtained after the appropriate transformation. Are the GLS estimators unbiased? c. Describe the pattern of data associated with heteroskedasticity of the form 2 = 0 + 1 t-1 2 . How would you estimate the parameters ( , , 1 )? d. How would you test for heteroskedasticity of the form t t 2 1 1 2 = +-?...
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psviiii245b - E[ t ] = 0 for all t, x t is I(0), and y t is...

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