MSDS596 Homework 9 (Due in class on Nov 14) Fall 2017
Notes. The lowest grade among all twelve homework will be dropped, so NO late submission will be
accepted. All homework assignment must be written on standard 8.5 by 11 paper and stapled together.
Computer generated output without detailed explanations and remarks will not receive any credit. You may
type out your answers, but make sure to use different fonts to distinguish your own words with computer
output. Only hard copies are accepted, except under special circumstances. For the simulation and data
analysis problems, keep the code you develop as you may be asked to present your work later.
1 (20 pts). Let U1 and U2 be independent random variables with zero means and Var(U1) = Var(U2) = σ2.
Consider the time series
xt = U1 sin(2πωt) + U2 cos(2πωt),
where ω ∈ [0, 1) is a fixed constant. Show that this series is weakly stationary with autocovariance function
γh = σ2cos(2πωh).
Find the autocorrelation function as well.
2 (30 pts). Let wt, t ∈ Z be a normal white noise (i.e. they are iid normal) with variance 1, and consider
the time series
xt = wtwt−1, yt = x2t.
(a) Find the mean, autocovariance, and autocorrelation functions of xt.
(b) Simulate xt of length T = 500. Give the time series plot, and the sample autocorrelations plot.
Comment.
(c) Perform the Ljung-Box test on the simulated series xt, using m = 1, 2, 3, 4, 5, 6.
(d) Find the mean, autocovariance, and autocorrelation functions of yt.
(e) Simulate yt of length T = 500. Give the time series plot, and the sample autocorrelations plot.
Comment.
(f) Perform the Ljung-Box test on the simulated series yt, using m = 1, 2, 3, 4, 5, 6.
3 (50 pts). Tourism is one of the largest economic components of Hawaii (another is the Pearl Harbor
Navy Base). The data hawaii-new.dat contains monthly record of the number of tourists visited
Hawaii from January, 1970 to December, 1995. The first column shows the year-month. The second column
is the total. The third and fourth columns show the number of west-bound (mainly from US and
Canada) and east bound (mainly from Japan and Australia) visitors. Perform the following analysis. [Use
read.table("hawaii-new.dat") to load the data to R.]
(a) Draw time series plots of the three series on the same graph. Comment on what you observe (trend,
seasonality, variance, possible outliers, relationship between the three series and others).
(b) Perform a log transformation of the total series. Draw a time series plot and comment on it.
(c) If you are to use a polynomial trend model for the log transformed total series, which order of the
polynomial (e.g. linear, quadratic, cubic etc) will you use? Your decision should be based on sound
statistical reasoning and formal testing or variable selection procedure. Fit the trend model, plot the
fitted line with the log-transformed time series and plot the de-trended series. What do you think
about the results?
(d) Fit a trend-seasonal model to the log transformed total series. Plot the fitted values with the log
transformed data and plot the de-trend-de-seasoned series. Comment on the estimated coefficients
of the seasonal factors.
(e) Use the trend-seasonal model in (d) to predict the total number of tourists (in log) who will visit
Hawaii each month in 1996, assuming the noises in the trend-seasonal model are i.i.d. Plot your
predictions (in dash lines) with the last three years of the original data.
(f) For the log transformed total series, determine the difference(s) needed to make the series stationary
(by look). Show your working process.