OLLSCOIL NA hEIREANN, CORCAIGH THE NATIONAL
OLLSCOIL NA hEIREANN, CORCAIGH
THE NATIONAL UNIVERSITY OF IRELAND, CORK
COLAISTE NA hOLLSCOILE, CORCAIGH
UNIVERSITY COLLEGE, CORK
ST4064 - Time Series Analysis
2014-2015
Problem Set 2
Due date: 20th March 2015
Instructions: Answer all questions. Show all relevant steps.
Question 1
The R dataset nottem contains average air temperatures recorded in degrees Fahrenheit at
Nottingham Castle over 20 years, from January 1920 to December 1939. Load this dataset in
the R environment, using x = nottem. Our objective is to compare several methods for dealing
with the seasonality in this time series.
(a) Check, numerically rather than visually, whether there is a linear trend in the time series.
(b) Apply the method of seasonal means to estimate the average seasonal variations in the
series. Note that you should center the series first by removing its mean. Provide the
values you obtain for the seasonal means.
(c) Compute the sum of the seasonal means obtained in (b) and explain this result.
(d) Apply the Holt-Winters method to the time series, with a multiplicative seasonal component and report the final sum of squared errors achieved for that fit.
(e) Apply the Holt-Winters method to the time series, with an additive seasonal component.
Report the final sum of squared errors achieved for that fit. How do the two Holt-Winter
fits compare?
(f) Fit a SARIMA(1,0,0)×(2,1,0)12 to the time series x. Report on the coefficient estimates
and associated standard errors you obtain.
(g) Compute and report on the sum of squared residuals for the fit obtained in (f). Compare
it to the value obtained in (d) and indicate which model you would rather use, based on
this quantity.
(h) When fitting an AR(1) to the series x adjusted for mean and seasonal means (computed
in (b)), we obtain a sum of squared errors of 1154.706, which makes it a better model in
terms of minimum MSE than the models considered in (d), (e) and (f). Describe another
advantage of using the seasonal means over the other methods.
Question 2
(a) Consider the following non-stationary process
Xt = 0.7Xt−1 + 0.3Xt−2 + εt − 0.5εt−1
2
where {ε}∞
t=0 denotes a sequence of i.i.d. random variables with distribution N (0, σ ).
(i) Transform Xt into a stationary process Yt , and check for stationarity and invertibility
of the differenced process.
(ii) Classify Xt as an ARIMA(p, d, q) process.
(iii) The most recently observed values in the original series (Xt ) are
x57 = 22.7
x58 = 21.1
with estimated residual eˆ58 = 2.18. Provide estimates x
ˆ58 (1) and x
ˆ58 (2).
(b) Recall that for a generic ARMA(1, 1) model of the form Wt = αWt−1 + εt + βεt−1 , the
ACF coefficients are given by
ρ1 =
(1 + αβ)(α + β)
,
1 + 2αβ + β 2
ρ2 = αρ1
Apply the method of moments to estimate parameters α and β of an ARMA(1,1) process
for which the sample ACF coefficients were found to be
r1 = −0.59,
r2 = 0.18
Question 3
The file series4064 PS2.csv contains a time series with a trend, and from which we wish to
forecast future values. Load this dataset in the R environment, using the following command:
x = ts(read.csv("series4064_PS2.csv",header=T), start=1, freq=1)
(a) Detrend the time series x using least squares; report the values obtained for that trend
and plot the detrended series y, its SACF and SPACF.
(b) Fit AR(2), MA(2), and ARMA(2,2) models to the series y. Decide whether or not you
should include an intercept parameter in your fit, and implement accordingly. Briefly
discuss on the relative fits of the three models.
(c) Examine the independence of the model residuals using an appropriate statistical test.
Interpret the output of this test.
(d) Obtain a 10-step ahead forecast for the MA(2) model and provide the standard deviation
of the forecast values.
(e) Plot the MA(2) forecast for both y and x (in separate plots), including confidence bands
for both these forecasts.
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