STAT 457/554 A01 (Spring 2021)
Assignment 5: Due 8:30am, April 6
STAT 457: Questions 1 - 4
STAT 554: Questions 1, 3, 4, 5
Submit your R code and plots for Questions 1, 3 and 4.
Upload your solution including R code and plots as one pdf file on the
Brightspace.
1. (5 marks) Let St represent the monthly sales data in “sales” (n = 150). Fit
an ARIMA model to St. Discuss your model fitting in a step-by-step fashion,
presenting your
(a) initial examination of the data,
(b) transformations, if necessary,
(c) identification of ARIMA model,
(d) parameter estimation,
(e) residual diagnostics and model choice.
2. (5 marks) Consider an invertible MA(1) model, xt = wt + θwt−1, where wt is a
white noise series with mean 0 and variance σ2w.
(a) Show that |ρx(1)| ≤ 0.5.
(b) Consider the sample autocorrelation function ρˆx(1) based on n observations
from the model. Is it possible that |ρˆx(1)| > 0.5? Why?
3. (5 marks) Use R to construct the following plots and comment the features in the
ACF and PACF.
(a) Plot the ACF and PACF of the seasonal ARIMA(1, 0, 0)7 with Φ = 0.75.
(b) Plot the ACF and PACF of the seasonal ARIMA(0, 0, 1)7 with Θ = 0.75.
(c) Plot the ACF and PACF of the seasonal ARIMA(1, 0, 1)7 with Φ = 0.75 and
Θ = 0.75.
4. (5 marks) Fit a seasonal ARIMA model to the unemployment data in
“UnempRate”.
(a) Present your model choice and the residual diagnostics.
(b) Use the estimated model to forecast the next 12 months. Present the plot
with the forecasts and the 95% prediction intervals for each of the 12 forecasts.
5. (5 marks) A first-order autoregressive model is generated from the white noise
series wt using the generating equations xt = −0.5xt−1 +wt, where wt is a white
noise series with mean 0 and variance σ2w.
(a) Find the power spectrum function fx(ω) of xt.
(b) Plot fx(ω) vs ω and comment on the power and frequencies in xt.
(c) Let yt = 3xt + 20. Find the power spectrum function fy(ω) of yt.





































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