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# hw6 - σ 2 w(a Find the initial least squares estimator of...

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The homework has to be stapled. PSTAT 174/274. Homework #6 Name: Due: 12/06/06 at the beginning of the class Score: /10 Credit: People you worked with: Sources consulted(Reference): 1. Consider the steady model and denote the signal to noise ratio σ 2 w 2 n by c . X t = μ t + n t μ t = μ t - 1 + W t (a) (2 points) Show that the first order autocorrelation coefficient of (1 - B ) X t is - 1 / (2 + c ) and that higher order autocorrelations are all zero. (b) (1 point) For the ARIMA(0,1,1) model (1 - B ) X t = W t + θW t - 1 , show that the first order autocorrelation coefficient of (1 - B ) X t is θ/ (1 + θ 2 ) and that higher order autocorrelations are all zero. (c) (1 point) These two models have equivalent autocorrelation properties when θ/ (1 + θ 2 ) = - 1 / (2 + c ). Find the invertible solution with | θ | < 1. 2. (2 point each) Consider the following special linear growth model. X t = μ t + n t (1) μ t = μ t - 1 + β t - 1 (2) β t = β t - 1 + w t , (3) where n t , w t are independent normal with zero means and respective variance
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Unformatted text preview: σ 2 w . (a) Find the initial least squares estimator of the state vector at time t = 2, in terms of the observations x 1 and x 2 . (b) If σ 2 w = 0, so that we have ordinary linear regression with constant coeﬃceints and a third observation x 3 becomes available, apply the Kalman ﬁlter to show that the estimator of the state vector at time t = 3 is given by [ˆ μ 3 , ˆ β 3 ] = ± 5 6 x 3 + 1 3 x 2 ± 1 6 x 1 , 1 2 ( x 3 ± x 1 ) ² . 3. (1 point each) Consider AR(2) process, X t = φ 1 X t-1 + φ 2 X t-2 + W t , W t ∼ N (0 ,σ 2 ) . (a) Find a state space representation based on the state vector θ t = ( X t ,X t-1 ). (b) Find a state space representation based on the state vector θ t = ( X t , ˆ X t ), where ˆ X t is the optimal one step ahead predictor at time t ....
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