Time Series Analysis Assignment 1 Submission date: 8pm 15th October 2018. In all the questions, {εt} is a white noise with Var(εt) = σ2. Exercise 1. Consider the following ARIMA (p, d, q) time series:...

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Time Series Analysis Assignment 1 Submission date: 8pm 15th October 2018. In all the questions, {εt} is a white noise with Var(εt) = σ2. Exercise 1. Consider the following ARIMA (p, d, q) time series: Yt = 1/2Yt−1 + 1/2Yt−2 − 1/3εt−1 + εt. 1) Identify p, d and q. [5] 2) Is this series stationary? How do you know? Describe how the auto- correlation plot at various lags h would look for this series. [5] Exercise 2. Consider an MA(1) time series Yt = εt − θεt−1. 1) Show that the autocorrelation at h = 1 is given by ρ1 = −θ 1+θ2 . [5] 2) Using the above result, show that when θ is replaced by 1θ , the au- tocorrelation function for an MA(1) process does not change for any value of h. [5] Exercise 3. Sketch without proof the ACF and PACF for the following time series: 1) AR(1) with φ = 0.5. [5] 2) AR(1) with φ = −0.5. [5] 3) MA(1) with θ = 0.5. [5] 4) MA(1) with θ = −0.5. [5] Exercise 4. The �gures below show time series plots, ACF plots, and PACF plots for two time series Xt and Yt. Also shown are plots for the di�erenced series ∆Yt. 1) Propose simple ARIMA(p,d,q) models for both series, justifying your model choice with reference to the �gures. [5] 2) Write equations for your proposed models in the form of: Xt = . . . and Yt = . . . [5] 1 Time Series x Time X t 0 100 200 300 400 500 − 4 − 2 0 2 4 0 5 10 15 20 25 0. 0 0. 2 0. 4 0. 6 ACF(x) Lag A C F 0 5 10 15 20 25 − 0. 2 0. 0 0. 2 0. 4 0. 6 Lag P ar tia l A C F PACF(x) Time Series y Time Y t 0 100 200 300 400 500 − 40 − 20 0 20 40 60 0 5 10 15 20 25 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 ACF(y) Lag A C F 0 5 10 15 20 25 − 0. 4 0. 0 0. 4 0. 8 Lag P ar tia l A C F PACF(y) Time Series ∆y Time ∆Y t 0 100 200 300 400 500 − 4 − 2 0 2 4 0 5 10 15 20 25 0. 0 0. 2 0. 4 0. 6 ACF(∆y) Lag A C F 0 5 10 15 20 25 0. 0 0. 2 0. 4 0. 6 Lag P ar tia l A C F PACF(∆y) 2