Analysis of Time Series
For AS90641
Part 2
Extra for Experts
September 2005
Created by Polly Stuart
1
Contents
• This resource is designed to suggest
some ways students could meet the
requirements of AS 90641.
• It shows some common practices in
New Zealand schools and suggests
other simplified statistical methods.
• The suggested methods do not
necessarily reflect practices of Statistics
New Zealand.
2
Aims
• This presentation (and the next) takes
you through some extra types of
analysis you could try for time series
data.
• It also makes suggestions for writing
your report
• You will need to open the spreadsheet:
Example sales.xls
• Choose the worksheet labeled
Clothing.
3
Beginnings
• You have already learned a basic
analysis of a time series and how to
isolate some components.
• We are now going to do a more
complex analysis.
• Before doing any analysis you need to:
– Graph the raw data
– Identify the components of the data
– Decide on the best method of
analysis.
4
Look at :
the trend
the seasonal component
the irregular component
C l o t hi ng and so f t g o o d s sal es
550
$(million)
500
450
400
350
300
2500
M ar
M ar
M ar
M ar
M ar
M ar
M ar
M ar
M ar
M ar
M ar
M ar
M ar
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
5
Step 1: Using Indexes
Indexes show how prices have changed over time.
They show the percentage increase in prices since
a base period. The index for the base period is
usually 1000.
An index of 1150 shows that prices have increased
15 percent since the base period.
You can use indexes to ‘deflate’ time series data
which contains dollar values.
Statistics New Zealand indexes include:
Consumers Price Index
Labour Cost Index
Food Price Index
Farm Expenses Price Index
6
Consumers Price Index
• The Consumers Price Index (CPI)
measures the change in prices of a
specific basket of goods and services in
New Zealand.
• For retail sales of clothing this is an
appropriate index to use as clothing is
included in the ‘basket’ of goods priced.
• Open the CPI worksheet and copy the
series into the next column of the clothing
worksheet.
Look at the CPI data. Which is the base period? How
do you know?
7
If the value of sales from clothing shops are
increasing over time there several possible
reasons:
• Prices have increased because of inflation
• The number of people in the population is growing
so there are more possible customers needing
clothes
• Sales are actually increasing because people are
buying more clothing
• Something else?
To help find out if total sales are increasing
because of inflation we can turn the sales into
constant 1999 dollars using the value of the CPI
for each year.
8
Constant dollars
The present base period for the Consumers
Price Index (CPI) is 1999.
Assume that the CPI now is 1150.
In 1999, $100 could buy the same amount as:
1150
 100  $115
1000
can buy now
Now, $100 can buy the same amount as:
1000
100  $86.96
1150
could buy in 1999
9
Calculate your deflated value
Use this
formula to
calculate
the value in
constant
1999
dollars.
We will
use
constant
1999
dollars
for the
rest of
this
exercise.
10
Step 2: Deciding on an appropriate
model
• Some data follows an additive model
where:
Data value = trend + seasonal +
irregular
• Other data follows a multiplicative
model where:
Data value = trend x seasonal x
irregular
11
Additive
When the size of the
seasonal
component stays
about the same as
the trend changes,
then an additive
method is usually
best.
Series for which an additive series is
appropriate
250
200
150
100
50
0
Mar 1991
Mar 1992
Mar 1993
Mar 1994
Original series
Trend series
12
Multiplicative
When the size of
the seasonal
component
increases as the
trend increases,
then a
multiplicative
method may be
better.
Series for which a multiplicative model is appropriate
300
250
200
150
100
50
0
Mar 1991
Mar 1992
Mar 1993
Mar 1994
Original series
Trend series
13
Look again at the graph below
• Which model seems more suitable?
In the previous PowerPoint we used an additive
model and we will do this also for this data
(An example of using a multiplicative model is
given at the end of the third presentation).
C l o t hi ng a nd s o f t g o o d s r e t a i l t r a d e
550
500
450
400
350
300
250
$million
0Mar
1991
Mar
Mar
Mar
Mar
Mar
Mar
Mar
Mar
1992
1993
1994
1995
1996
1997
1998
1999 2000
Mar
Mar
Mar
Mar
2001 2002 2003
14
Step 3: Analyse the data
• Do the spreadsheet analysis as far as
calculating the seasonally adjusted
data.
• Use the constant dollar values for your
analysis.
15
Your spreadsheet should look like this:
16
Step 4: Describe and justify your
model for the trend
• Try some different models for the
moving average.
• Decide which one will give a sensible
forecast.
17
Trend
Describe what you can see.
y = -0.0864x + 381.6
Clothing and softgoods sales
$(m illion)
500
Clothing
1999
dollars
Estimated
trend
450
400
350
Linear
(Estimated
trend)
300
2500
Mar
1991
Mar
1993
Mar
1995
Mar
1997
Mar
1999
Mar
2001
Mar
2003
Does this linear trend model look sensible?
18
• Many trends cannot be modelled by a single
straight line
• A quadratic model may be tempting…
y = 0.1097x 2 - 5.572x + 431.66
Clothing and softgoods sales
$(m illion)
500
Clothing
1999
dollars
Estimated
trend
450
400
350
Poly.
(Estimated
trend)
300
2500
Mar
1991
Mar
1993
Mar
1995
Mar
1997
Mar
1999
Mar
2001
Mar
2003
But is it realistic?
19
• Remember the shape of a parabola.
• Do you think that sales (in constant dollars)
are going to grow at that rate?
y = 0.1097x 2 - 5.572x + 431.66
Clothing and softgoods sales
$(m illion)
600
550
500
450
400
350
300
2500
Mar
1991
Clothing
1999
dollars
Estimated
trend
Poly.
(Estimated
trend)
Mar
1993
Mar
1995
Mar
1997
Mar
1999
Mar
2001
Mar
2003
20
• An option is to use a linear model over the
trend at the end of the series.
• This is likely to give the most realistic forecast.
Clothing and softgoods sales from 1998
y = 4.3368x + 335.87
$(million)
500
450
400
350
300
2500
Mar
1998
Clothing
1999
dollars
Estimated
trend
Mar
1999
Mar
2000
Mar
2001
Mar
2002
Mar
2003
Linear
(Estimated
trend)
21
Step 5: Describing the seasonal
component
• A graph can help you to see the patterns more
clearly.
22
Seasonal sales patterns
$(m illion)
50
0
Mar 1991
Mar 1995
Mar 1999
Mar 2003
-50
Describe the patterns you can see.
You can also identify amounts easily from the
graph.
23
Step 6: Analysing the irregular
component
• There is always random variation in a
time series, the irregular component.
• When a very unusual event happens it
may cause a spike in the data, called an
outlier.
• This can distort the trend and seasonal
component values.
• The larger the spike the more distortion.
• It is useful to calculate the irregular
component and look for outliers.
24
Subtract the values in the ‘Seasonal’
column from the ‘Seasonal and Irregular’
column. A graph is often useful.
25
Outliers
Highlight the date and irregular columns for
the graph.
Irregular Com ponent
$ m i l l i on 19 9 9
15
10
5
0
M ar 1991
M ar 1995
M ar 1999
M ar 2003
-5
-10
Both the pattern of the irregular component
and any extreme values are worth
commenting on.
26
This is not the end!
Continue the analysis and
write a report on retail
clothing sales.
Some ideas are given in the
next presentation,
Reporting.
27