Homework 6 Question 1 Consider the following model: Xt = Xt−1 +Wt − λWt−1, (1) where Wt is a white noise process with zero expectation and innovation variance σ2, |λ| < 1 and X0 = 0. 1. Classify this model as an ARIMA model: determine the order and the parameters. 2. Based on the model above we have thatWt = Xt−Xt−1+λWt−1 and henceWt−1 = Xt−1−Xt−2+λWt−2 (and so on). Using the latter information, use a recursion, based on expression (1), to show that Xt = ∞∑ i=1 (1− λ)λi−1Xt−i +Wt. 3. Based on this result, we can build the forecast: Xtt+1 = ∞∑ i=1 (1− λ)λi−1Xt+1−i. Show that this is equivalent to: Xtt+1 = (1− λ)Xt + λXt−1t , where Xt−1t = ∑∞ i=1(1− λ)λi−1Xt−i and comment on this result 1. Question 2 Consider the model Yt = ΦYt−3 + et − θet−1, where et has variance σ2. 1. Identify Yt as a certain SARIMA(p, d, q)× (P,D,Q)s model. That is, specify each of p, d, q, P,D,Q, s. You may assume that |Φ| < 1. 2. Find the variance of Yt. 3. What are the forecasts for Yt+1 and Yt+4? 4. What are the error variances for your forecasts above? 5. If σ2 = 1, Φ = .7, and θ = −.5, find 95% limits for your forecasts above. You may assume that et are normally distributed. Also, the four most recent yt values are 0.13, -0.50, 0.38 and 1.53. The four most recent et values are 0.08, -0.60, 0.75, 0.95. Question 3 Consider a simulated time series of length T = 105 whose ACF is presented in the figure below. Using the figure below: 1. Propose a reasonable model for this time series. Justify your answer. 2. Propose a rough estimate of the model’s parameters. Justify your answer. 1When using this approach to forecast (and restricting 0 < λ < 1), this approach is called the Exponentially Weighted Moving Average (EWMA) which is a popular and easy-to-use forecasting technique. For this approach, (1− λ) is called the “smoothing parameter” where larger values of λ lead to smoother forecasts. 1 Lags AC F x ACF plot 0 10 20 30 40 50 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 # Question 4 Show that a ARCH(p) model for Wt is a AR(p) model for W 2t . 2
