1. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Chapter 9: Projecting the Future Please note: there are 3 Excel handouts that go with this presentation (Timeseries1, Timeseries2 and Timeseries3)Timeseries1Timeseries2 Timeseries3

2. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Objectives Be able to work with data collected over time Use linear regression where appropriate Model the trend and seasonal variation Make predictions using an appropriate model

3. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Much of the data of interest to us is collected over time. We are likely to be interested in the actual values, the trend and the predictable variations. Examples include: Employment and unemployment Sales (particularly seasonal variation) Energy use Travel patterns

4. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Using a linear (regression) model It might be the case, that our data of interest could be described by a straight line. Suppose we have the following sales figures for the last 5 years. Year12345 Sales180168172164158

5. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage We typically work with years of 1, 2, 3 and so on rather than 2000, 2001, 2002 just because it can make the number easier It is always useful to graph the data and comment on the relationship.

6. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage It can be seen from the graph that sales have fallen steadily over the last 5 years and can reasonably be described by a straight line:

7. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Using Excel, we can get the following values: correlation-0.9150 intercept 182.8 gradient (slope)-4.8 The correlation is close to -1 suggesting a strong negative relationship

8. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The equation of the line is y = 182.8 - 4.8x If we want to forecast the coming year (using this methodology), let x = 6 y = 182.8 – 4.8 x 6 = 154.0 This assumes the continuation of the observed linear relationship.

9. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Modelling a Time Series We typically expect to see the following over time: Trend (T) – this is the general movement in the data over time Seasonal factors (S) – predictable variations Residual factors (R) – the variations we cannot explain

10. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage 1. The Additive model Y = T + S + R 2. The Multiplicative model Y = T x S x R We will consider the use of two models:

11. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The Additive Model The elements of interest are added together The average value of the residual is assumed to be 0 The variations around the trend are predictable and roughly of the same order of magnitude

12. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Quarter Year 1Year 2Year 3 y * * * * * A typical scatter of points for the additive model

13. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Quarter Year 1Year 2Year 3 y * * * * * The pattern becomes clearer as we join points

14. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Quarter Year 1Year 2Year 3 y * * * * * Trend We are interested in (additive) variation around the trend

15. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The Multiplicative Model The elements of interest are multiplied together The effects of the residual are assumed to cancel out and the residual assumed to be 1 The variations around the trend are predictable and proportionate to the trend

16. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Quarter Year 1Year 2Year 3 y * * * * * * * * * * * A typical scatter of points for the multiplicative model

17. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Quarter Year 1Year 2Year 3 y * * * * * * * * * * * The pattern becomes clearer if we join points

18. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Quarter Year 1Year 2Year 3 y * * * * * * * * * * * Trend Note increased spread within the data We are interested in the (proportionate) variation around the trend

19. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Consider an example (in this case the volume consumption of a commodity)

20. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The Determination of Trend Whichever model we use, we do need to find trend. Essentially we smooth out the variations In this case the variations are over the 4 seasons of the year (quarterly) We use the method of moving average

21. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage To find the moving average: We take the first four quarterly values and add We then divide by four to get an average We then move along one place, take the next four values (3 old values + 1 new one) and divide by 4 to get next average Repeat the process to end of data

22. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Finding a 4 quarter average

23. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage However, there is a problem If we are dealing with even numbers, the moving average does not align with the original values We need to add an additional step, where we average pairs – see following table

24. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Finding a 4 quarter average

25. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Handout 1 shows: The use of a spreadsheet to determine the 4 quarter moving total, the 4 quarter moving average and the determination of trend by ‘centring’ Plot of the sales of the commodity and trend

26. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The determination of the seasonal effect and making a prediction using the additive model See handout 2 Handout 2 shows: The determination of trend (as before) The difference between the actual values and the trend values Y-T (using the additive model)

27. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Differences from the trend using Y-T

28. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Handout 2 also shows: A table that finds the average quarterly difference (giving the estimated seasonal effect) The predictions for year 4 using values from the extended trend line and the adjustment for seasonal effect

29. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The determination of the seasonal effect and making a prediction using the multiplicative model See handout 3 Handout 3 shows: The determination of trend (as before) The difference between the actual values and the trend values Y/T (using the multiplicative model)

30. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Differences from the trend using Y/T

31. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Handout 3 also shows: A table that finds the average quarterly difference (giving the estimated seasonal effect) The predictions for year 4 using values from the extended trend line and the adjustment for seasonal effect

32. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage A note of prediction It is necessary to project the trend line forward using regression or some other method (in our examples we have used linear regression) We need to be aware of the assumptions being made by such extension of the trend line e.g. the the present trend is continuing

33. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Having extended the trend line forward we adjust our forecast according to the estimated seasonal effect Different models (additive and multiplicative in this case) are likely to give different answers

34. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage A note on number of periods being used for the calculation of the trend In our example we used a 4 quarter moving average (that we then needed to centre) because the natural variation was over 4 quarters/seasons of the year However, if we were looking at sales over 7 working days, then we would use a 7 point moving average

35. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage If we are using an odd number (5 or 7 working days for example) then the moving average is centred and we don’t need the additional averaging stage (you will see examples in the text book)

36. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Conclusions Much of the data we work with is collected over time e.g sales We can use a number of forecasting models In particular, we need to think about the trend and seasonal variation The answers we get will depend on the quality of data collected and models used