1. 18 - 1 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved.

2. 18 - 2 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Define the four components of a time series 1. Determine a linear trend equation 2. Compute a moving average 3. Compute the trend equation for a nonlinear trend 4. Use trend equations to forecast future time periods and to develop seasonally adjusted forecasts 5. Determine and interpret a set of seasonal indexes 6. Deseasonalize data using a seasonal index 8. When you have completed this chapter, you will be able to: Identify cyclical fluctuations 7. Compute and evaluate forecasts 9.

3. 18 - 3 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. … is a collection of data recorded over a period of time ( data may be recorded weekly, monthly, or quarterly) …is the long run direction of the Time Series …is the fluctuation above and below the trend line …is the pattern in a time series; these patterns tend to repeat themselves from year to year

4. 18 - 4 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Episodic variations … are unpredictable, but can usually be identified, such as a flood or hurricane Residual variations … are random in nature and cannot be identified …is divided into two components: Continued…

5. 18 - 5 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Text Chart 18-1… Excel Secular Trend

6. 18 - 6 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Secular Trend …almost constant Text Chart 18-2 Excel

7. 18 - 7 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Text Chart 18-3 Excel Secular Trend …Increasing Trend

8. 18 - 8 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Text Chart 18-4 Excel Secular Trend …Declining Trend

9. 18 - 9 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Text Chart 18-5 Excel Cyclical Variation

10. 18 - 10 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Figure 18-6 Text Chart 18-6 Excel Seasonal Variation

11. 18 - 11 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Linear Trend The long term trend equation (linear ) Estimated by the least squares equation for time t is: abt b tyyt tt a y n b t n ()() ()            nn nn  22 y

12. 18 - 12 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. C ontinued… The owner of Farley Homes would like a forecast for the next couple of years of new homes that will be constructed in the Edmonton area. Listed below are the sales of new homes constructed in the area for the last 5 years. Year 1997 1998 1999 2000 2001 Sales ($1000) 4.3 5.6 7.8 9.2 9.7

13. 18 - 13 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Year SalestSales*tt 2 19974.31 4.3 1 19985.62 11.2 4 19997.83 23.4 9 20009.24 36.816 20019.75 48.525 Total 36.6 15 124.255 …least squares equation for time t Continued…

14. 18 - 14 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. nswer Year 1997 1998 1999 2000 2001 Sales ($1000) 4.3 5.6 7.8 9.2 9.7 = 1.44 5/)15(55 5/)15(6.362.124 2     / / 2 2     ntt ntytyty b = 3.0 5 15 44.1 5 6.36                    n t b n y a Develop a trend equation using the least squares method by letting 1997 be the time period 1

15. 18 - 15 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. nswer * Five year (1997 – 2001) + 2002 and 2003 The t ime series equation is: = 3.00 + 1.44 t y The forecast for the year 2003 is: = 3.00 + 1.44(7) * = 13.08 y

16. 18 - 16 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. See Using Click on Tools Click on DATA ANALYSIS See…

17. 18 - 17 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Highlight Regression …Click OK See… See Using

18. 18 - 18 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Line Fit Plot Data Regression See Using

19. 18 - 19 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. If the trend is not linear but rather the increases tend to be a constant percent, the y values are converted to logarithms, and a least squares equation is determined using the lns: Non-Linear Trend ln()[ ln()][ ln()]abt  y

20. 18 - 20 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Text Figure 18-11 Excel

21. 18 - 21 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Moving-Average Method … is used to smooth out a time series. This is accomplished by “moving” the arithmetic mean through the time series. …the moving-average is the basic method used in measuring the seasonal fluctuation …to apply the moving-average method to a time series, the data should follow a fairly linear trend and have a definite rhythmic pattern of fluctuations

22. 18 - 22 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Moving-Average Method Using Text Chart 18-9 Excel

23. 18 - 23 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The method most commonly used to compute the typical seasonal pattern is called the Ratio-to-Moving-Average Method …it eliminates the trend, cyclical, and irregular components from the original data (y) …the numbers that result are called the typical seasonal indexes

24. 18 - 24 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. YearWinterSpringSummerFall 19966.74.66.712.7 19976.54.66.513.6 19986.95.06.914.1 19997.05.57.015.0 20007.15.77.114.5 20018.0.6.211.414.9 Listed below are the quarterly sales (in $ millions) of Toys International for the years 1996 through 2001. Determine a quarterly seasonal index using the ratio-to-moving average method. Note … that the fall quarter sales are the largest and the spring sales are the smallest each year

25. 18 - 25 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 1. …determine the moving total for the time series 6. … apply the correction factor Steps 2. …determine the moving average for the time series 3. …the moving averages are then centered 4. …the specific seasonal for each period is then computed by dividing the y values with the centered moving averages 5. … organize the specific seasonals in a table

26. 18 - 26 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Steps Text Chart 18-6 1. …determine the moving total for the time series

27. 18 - 27 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Text Chart 18-13 Excel

28. 18 - 28 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Text Chart 18-14 Excel

29. 18 - 29 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Moving Average The Moving-Average Method

30. 18 - 30 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The resulting series (sales) is called deseasonalized sales or seasonally adjusted sales The reason for deseasonalizing a series (sales) is to remove the seasonal fluctuations so that the trend and cycle can be studied A set of typical indexes is very useful in adjusting a series, sales, for example

31. 18 - 31 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Text Chart 18-17 Excel

32. 18 - 32 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Test your learning … www.mcgrawhill.ca/college/lind Click on… Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s E-Stat data …and much more!

33. 18 - 33 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. This completes Chapter 18