ECON217_HW_ARMA - 7. Find the moving average...

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ECON217_HW_ARMA 1. If a time series {X t } is covariance stationary, what do we know about E(X t ) and COV(X t , X t-k ) for t = 1, . .., T and k = 0, 1, 2, . .? 2. If {X t } is a white noise process, what do we know about E(X t ), and COV(X t , X t-k ) for for t = 1, . .., T and k = 0, 1, 2, . .? 3. Define and compare the autocorrelation function and the partial autocorrelation function of a stationary time series. 4. Suppose Y t follows Y t = φ Y t-1 + ε t ; ε t ~ WN (0 , σ 2 ) . a. State the assumption(s) on φ that will make {Y t } stationary. b. Assuming {Y t } is stationary. Find the autocorrelation function and the partial autocorrelation function. 5. Suppose Y t follows Y t = ε t + θ ε t-1 ; ε t ~ WN (0, σ 2 ). a. State the assumption(s) that will make {Y t } stationary. b. Find the autocorrelation function of {Y t }. c. Write down the partial autocorrelation function of {Y t }. 6. Consider a time series record {X 1 , . .., X T }. Discuss how you would specific a time series model using the Box-Jenkins three-step approach and the information criterion approach.
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Unformatted text preview: 7. Find the moving average representation, the impulse response, and the forecast of each of the following processes: a) (1-φ L)Y t = ε t , b) (1-L)Y t = α + t , c) Y t = (1+ θ L) t , and d) Y t = α + (1+ L) t . 8. Consider the second-order autoregressive process y t = a + a 2 y t-2 + t ,where ⏐ a 2 ⏐ < 1. a. Find: i. E t-2 y t ii. E t-1 y t iii. E t y t +2 iv. Cov( y t , y t-1 ) v. Cov( y t , y t-2 ) vi. the partial autocorrelations 11 and 22 b. Find the impulse response function. Given y t-2 , trace out the effects on an t shock on the { y t } sequence. c. Determine the forecast function: E t y t + s . The forecast error ) ( s e t is the difference between y t + s and E t y t + s . Derive the correlogram of the { ) ( s e t } sequence. [Hint: Find E t ) ( s e t , Var [ ] ) ( s e t , and [ ] ) ( ) ( j s e s e E t t t − for j = 0 to s ]. 9. Enders, chapter 2, question 11....
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This note was uploaded on 09/29/2010 for the course ECON Econometri taught by Professor Fairlie during the Winter '09 term at University of California, Santa Cruz.

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