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