LINKÖPINGS UNIVERSITET
Institutionen för datavetenskap
Statistik, CL, ANd
732G06 TIME SERIES ANALYSIS
Fall semester 2008
Assignment
Assignment week 41: Seasonal data and ARIMA modelling
In this assignment you should apply regular and seasonal differencing to achieve a
stationary time series and then find out what the best ARIMA-model could be for the
(original) times series.
Further you should try to model residuals from a time series regression.
The assignment should be submitted by the end of week 44.
G. Forecasting by using Seasonal ARMA-models
Data set: The Cars and Motorcycles data set
Preparatory differentiations
Study the time series of monthly (new-)registrations of private cars during 1980-1998,
which can be found in the file 'vehicles.txt'. Examine how different kinds of
differentiation (regular and seasonal) have impact on the sample auto-correlation function
(SACF) and possibly also the sample partial auto-correlation function (SPACF). The
select an alternative for differentiation that gives auto-correlations that might be
described by an ARMA- or a Seasonal-ARMA-model.
Preliminary model selection by using SACF and SPACF.
Use the SACF and the SPACF to get some clues about whether you should use an
ARMA-model or a Seasonal ARMA-model to describe the series after the selected
method of differentiation has been performed. In addition try to figure out if you should
first fit AR-models or MA-models.
Estimation of parameters and calculation of Mean Square Error
Estimate the parameters in a number of different ARMA- or Seasonal ARMA-models
and compare the calculated Mean Square Error of the different models. Note that it is not
always possible to fit a certain model to the available data and that the program can be
interrupted, if the model used is not feasible. Select a model that is characterised by a
relative small number of parameters (a parsimonious model) and a low value of the
calculated Mean Square Error.
Model validation
Use Ljung-Box’s test on the selected model to judge upon its fit to the data. Further,
display the SACF for the residuals in a graph and plot the residuals in time order. Did you
manage to find any satisfactory prediction model? Did you gain on using Seasonal
ARMA-models compared with using models with P=Q=0?
Specify the model
Write down the formula for the entire ARIMA model, including differentiation,
autoregressive and moving average terms. Using this model calculate by hand the
predicted value for the next time point (October 1998).
H. Time series regression with autocorrelated errors
Data set: Hjalmaren month
Use the data in file 'Hjalmarenmonth.txt' and fit a time series regression including a term
for trend (time) and seasonal dummies to describe the seasonal effects. Save the residuals
from this model and determine
(i) if the residuals are independent random errors or if they must be modelled
(ii) identify the model that describes the residuals best, if such a model is necessary