1. Analysis of Time Series
For AS90641
Part 2
Extra for Experts
Achievement with Excellence
Analyse time series data and prepare a report on the analysis.
The analysis may be based on data that cannot be modelled by a single straight linear trend over its entire range.
The analysis should include at least some of:
choosing and justifying a model for the analysis
seasonally adjusting data and interpreting the results in context
comparing with related time series data
using formal methods of analysis
(e.g. least squares regression lines).
The report should include discussion of at least some of:
relevance and usefulness of forecast
discussion of features of the time series data
potential sources of bias
improvements to the model
limitations of the analysis.
September 2005
Created by Polly Stuart

2. Contents
This resource is designed to suggest some ways students could meet the requirements of AS
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.
More information about techniques used at Statistics New Zealand is available in the information section on Schools Corner.
It is important to be aware that Statistics New Zealand does not use time series analysis for forecasting. In addition, many of the other methods of seasonal adjustment shown here are very much simplified.

3. 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.
The analyses shown are suggestions for students to try. They enable a more in depth analysis of the data and, hopefully, give opportunities for comments at Excellence level.
This presentation goes through some calculations and graphs that might be useful. The next talks about possible ways of using these in a report.

4. 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.
Students need to be encouraged to look at the data before they rush into analysis. For a more complex analysis they need to plan in advance what methods they will use.

5. the seasonal component the irregular component
Look at : the trend
the seasonal component
the irregular component
A reminder of what each point is composed of.

6. 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
Deflation is the opposite of inflation.
It is a way of removing the effects of inflation so that you can compare the purchasing power of the dollars.
This is explained further in slide 9.
Development and use of indexes is mentioned in the explanatory notes for Excellence.
Deflation is a technique used at Statistics New Zealand. Here is a link to an example:
This is the document which shows the analysis of the series used in this activity (among others).

7. 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.
The composition of an index is explained in the next few slides.
Look at the CPI data. Which is the base period? How do you know?

8. 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.
1999 is used as this is the current base period for the CPI.

9. 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:
can buy now
Now, $100 can buy the same amount as:
If we say something is worth $100 today, that is different from saying that 10 years ago. That is because the $100 has less purchasing power today than it did then.
Using constant dollars shows the relative cost of an item. So, for example, a television probably costs less now than 10 years ago in real terms. Constant dollars can help us estimate those changes.
could buy in 1999

10. Calculate your deflated value
We will use constant 1999 dollars for the rest of this exercise.
Use this formula to calculate the value in constant 1999 dollars.
The rest of the analysis uses the constant dollar values, rather than the actual dollar values.

11. 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
While the focus for the standard is on additive models it is useful for students to consider the other possibility of multiplicative and to be able to comment on the decision they make.
This is explained in more detail in the next 2 slides and, if they wish, students can learn how to do multiplicative models at the end of the third PowerPoint.

12. Additive
When the size of the seasonal component stays about the same as the trend changes, then an additive method is usually best.
Actual series are never this clear cut.

13. Multiplicative
When the size of the seasonal component increases as the trend increases, then a multiplicative method may be better.
Its not always really clear from looking which model is best for the data. In practice the data is often analysed both ways and the one that seems to work best is used.

14. 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).
The irregular component makes it difficult to be sure what is happening, but the data looks more like an additive series.

15. 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.
This process has been covered in the first slide show and should be familiar to students. The only difference is that the constant dollar values calculated earlier are used rather than the actual raw data.

16. Your spreadsheet should look like this:
Students can check that they have correctly remembered the process.

17. 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.
It is important to play around and try different options for the regression model. This should include using only part of the data. In the next few slides some of these are examined in more detail.

18. Trend Describe what you can see.
This model will give a very poor forecast. Students need to be able to recognise this.
Fitting straight lines to time series is regarded as dubious by some statisticians. In addition, the moving average trend has a lot of information which is lost by focus on a regression line.
Does this linear trend model look sensible?

19. Many trends cannot be modelled by a single straight line
A quadratic model may be tempting…
Quadratics are usually a bad idea and this is explored further in the next slide.
But is it realistic?

20. Remember the shape of a parabola.
Do you think that sales (in constant dollars) are going to grow at that rate?
A polynomial model soon begins to increase more quickly that the data. Forecasts based on these are generally too high.
This idea is explored again in the ‘Marriage activity’ which is linked to from the webpage.

21. This is likely to give the most realistic forecast.
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.
For the purposes of forecasting the shorter series modeled with a linear model will generally give a better result. The concern with the validity of fitting a regression line remains.
The pitfalls of this are also looked at in the ‘Marriage activity’.

22. Step 5: Describing the seasonal component
A graph can help you to see the patterns more clearly.
Looking at results in graphs is a good habit for students. It usually clarifies patterns in the data.

23. Describe the patterns you can see.
You can also identify amounts easily from the graph.
Students need to look at the graphs they have done in Excel so they can identify points exactly.

24. 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.
Calculating the irregular component is shown on the next slide.

25. Subtract the values in the ‘Seasonal’ column from the ‘Seasonal and Irregular’ column. A graph is often useful.
Once again, showing the irregular on a graph is useful. This is done on the next slide.

26. Outliers Highlight the date and irregular columns for the graph.
The 2 or 3 highest and lowest irregular values are worth noting. The overall pattern of larger irregular in the middle and smaller at the ends is also worth noting. An example is given in the next PowerPoint presentation, which shows writing a report.
Both the pattern of the irregular component and any extreme values are worth commenting on.

27. 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.