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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.

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