September 2005 Created by Polly Stuart 1

Analysis of Time Series

For AS90641

Part 2 Extra for Experts

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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.

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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.

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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.

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Look at : the trend

the seasonal component

the irregular component

Clothing and softgoods sales

250

300

350

400

450

500

550

Mar

1991

Mar

1992

Mar

1993

Mar

1994

Mar

1995

Mar

1996

Mar

1997

Mar

1998

Mar

1999

Mar

2000

Mar

2001

Mar

2002

Mar

2003

0

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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

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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?

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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.

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Constant dollars

115$1001000

1150

The present base period for the Consumers Price Index (CPI) is 1999.

Assume that the CPI now is 1150.

Now, $100 can buy the same amount as:

96.86$1001150

1000

can buy now

could buy in 1999

In 1999, $100 could buy the same amount as:

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We will use constant 1999 dollars for the rest of this exercise.

Use this formula to calculate the value in constant 1999 dollars.

Calculate your deflated value

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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

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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

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50

100

150

200

250

Mar 1991 Mar 1992 Mar 1993 Mar 1994

Original series

Trend series

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When the size of the seasonal component increases as the trend increases, then a multiplicative method may be better.

Multiplicative

Series for which a multiplicative model is appropriate

0

50

100

150

200

250

300

Mar 1991 Mar 1992 Mar 1993 Mar 1994

Original series

Trend series

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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).

Clothing and softgoods retail trade

250300350400450500550

Mar

1991

Mar

1992

Mar

1993

Mar

1994

Mar

1995

Mar

1996

Mar

1997

Mar

1998

Mar

1999

Mar

2000

Mar

2001

Mar

2002

Mar

2003

0

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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.

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Your spreadsheet should look like this:

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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.

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Trend

Does this linear trend model look sensible?

Describe what you can see.

Clothing and softgoods salesy = -0.0864x + 381.6

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300

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500

Mar1991

Mar1993

Mar1995

Mar1997

Mar1999

Mar2001

Mar2003

$(million)

Clothing1999dollarsEstimatedtrend

Linear(Estimatedtrend)

0

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• Many trends cannot be modelled by a single straight line

• A quadratic model may be tempting…

But is it realistic?

Clothing and softgoods salesy = 0.1097x2 - 5.572x + 431.66

250

300

350

400

450

500

Mar1991

Mar1993

Mar1995

Mar1997

Mar1999

Mar2001

Mar2003

$(million)

Clothing1999dollarsEstimatedtrend

Poly.(Estimatedtrend)

0

20

• Remember the shape of a parabola.• Do you think that sales (in constant dollars)

are going to grow at that rate?

Clothing and softgoods salesy = 0.1097x2 - 5.572x + 431.66

250300350400450500550600

Mar1991

Mar1993

Mar1995

Mar1997

Mar1999

Mar2001

Mar2003

$(million)

Clothing1999dollarsEstimatedtrend

Poly.(Estimatedtrend)

0

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• 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 1998y = 4.3368x + 335.87

250300350

400450500

Mar1998

Mar1999

Mar2000

Mar2001

Mar2002

Mar2003

$(million)Clothing1999dollars

Estimatedtrend

Linear(Estimatedtrend)

0

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Step 5: Describing the seasonal component• A graph can help you to see the patterns more

clearly.

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Describe the patterns you can see.

You can also identify amounts easily from the graph.

Seasonal sales patterns

-50

0

50

Mar 1991 Mar 1995 Mar 1999 Mar 2003

$(million)

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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.

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Subtract the values in the ‘Seasonal’ column from the ‘Seasonal and Irregular’ column. A graph is often useful.

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Outliers

Both the pattern of the irregular component and any extreme values are worth commenting on.

Irregular Component

-10

-5

0

5

10

15

Mar 1991 Mar 1995 Mar 1999 Mar 2003

$million 1999

Highlight the date and irregular columns for the graph.

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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.