STAT463 Homework #4
due Monday, 2/11
1. (data from Shumway & Stoffer, 2e) The data in “varve.dat” contains sedimentary deposit (sand and silt from melting glaciers) thicknesses for n = 634 consecutive years in Mas- sachusetts. It can be read into R with y=ts(scan(file.choose())).
(a) Plot the data and comment on violations of stationarity.
(b) Argue that the (natural) log of the data stabilizes the variance over time. Is the trans- formed data now stationary? Why or why not?
(c) Define the differences dt = log yt − log yt−1. Plot, and argue in favor of stationary now. Hint: use the R function diff(log(y)).
(d) Plot and interpret the estimated ACF (autocorrelation function) for the differenced data. What significant spike(s) do you see?
(e) Recall the moving average model dt = et − θet−1, where et are independent with mean 0 and variance σ2. Find its autocorrelation function ρk = Cor(dt,dt−k).
(f) Comparing ρk with the estimated ACF in (d) above, comment on whether this model is reasonable for dt.
2. Load the deere2 data set that is included in the TSA package. (use data(deere2) to make the data available). The data contains deviations from a specified target, in units of length, for an industrial process at Deere&Co.
(a) Display the time series plot for these data, and comment on the appearance. Is a stationary model reasonable?
(b) Plot the sample ACF and sample PACF, and select, with justification, tentative orders for an ARMA model for this time series.
3. For each of the following, find the values of p, d, and q such that Yt ∼ ARIMA(p, d, q), and state whether Yt is stationary and whether ∇dYt is stationary.
(a) Yt = Yt−1 − 0.25Yt−2 + et − 0.1et−1
(b) Yt = 0.5Yt−1 − 0.5Yt−2 + et − 0.5et−1 + 0.25et−2
(c) Yt = 0.9Yt−1 + 0.09Yt−2 + et