1. ©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin Time Series and Forecasting Chapter 16

2. 2 Goals Define the components of a time series Compute moving average Determine a linear trend equation Compute a trend equation for a nonlinear trend Use a trend equation to forecast future time periods and to develop seasonally adjusted forecasts Determine and interpret a set of seasonal indexes Deseasonalize data using a seasonal index Test for autocorrelation

3. 3 Time Series What is a time series? – a collection of data recorded over a period of time (weekly, monthly, quarterly) – an analysis of history, it can be used to make current decisions and plans based on a long-term forecasting. – Usually assumes the past pattern to continue into the future.

4. 4 Components of a Time Series Secular Trend – the smooth long term direction of a time series Cyclical Variation – the rise and fall of a time series over periods (longer than one year) Seasonal Variation – Patterns of change in a time series within a year which tends to repeat each year Irregular Variation – classified into: Episodic – unpredictable but identifiable Residual – (or chance fluctuation) unidentifiable

5. 5 Secular Trend – Home Depot Example

6. 6 Secular Trend – EMS Calls Example

7. 7 Secular Trend – Manufactured Home Shipments in the U.S.

8. 8 Cyclical Variation – Sample Chart

9. 9 Seasonal Variation – Sample Chart

10. 10 The Moving Average Method Useful in smoothing time series to see its trend. How? By “moving” the arithmetic mean values through the time series. – Average out the “cyclical” and “irregular” variations, with only “secular trend” remaining. – Applicable when time series follows fairly linear trend that have definite rhythmic pattern

11. 11 Moving Average Method - Example

12. 12 Three-year and Five-Year Moving Averages

13. 13 Linear Trend The long term trend of many business series often approximates a straight line

14. 14 Linear Trend Plot

15. 15 Linear Trend – Using the Least Squares Method Use the least squares method in Simple Linear Regression (Chapter 13) to find the best linear relationship between two variables Code time (t) and use it as the independent variable – E.g. let t be 1 for the first year, 2 for the second, and so on (if data are annual)

16. 16 Year Sales ($ mil.) 20027 200310 20049 200511 200613 The sales of Jensen Foods, a small grocery chain, since 2002 are: Linear Trend – Using the Least Squares Method: An Example Yeart Sales ($ mil.) 200217 2003210 200439 2005411 2006513

17. 17 Linear Trend – Using the Least Squares Method: An Example Using Excel

18. 18 Nonlinear Trends A linear trend equation is used when the data are increasing (or decreasing) by equal amounts A nonlinear trend equation is used when the data are increasing (or decreasing) by increasing amounts over time When data increase (or decrease) by equal percents or proportions plot will show curvilinear pattern

19. 19 Log Trend Equation – Gulf Shores Importers Example Top graph is plot of the original data Bottom graph is the log base 10 of the original data which now is linear (Excel function: =log(x) or log(x,10) Using Data Analysis in Excel, generate the linear equation Regression output shown in next slide

20. 20 Log Trend Equation – Gulf Shores Importers Example

21. 21 Log Trend Equation – Gulf Shores Importers Example

22. 22 Seasonal Variation One of the components of a time series Seasonal variations are fluctuations that coincide with certain seasons and are repeated year after year Understanding seasonal fluctuations help plan for sufficient goods and materials on hand to meet varying seasonal demand Analysis of seasonal fluctuations over a period of years help in evaluating current sales

23. 23 Seasonal Index A number, usually expressed in percent, that expresses the relative value of a season with respect to the average for the year (100%) Ratio-to-moving-average method – The method most commonly used to compute the typical seasonal pattern – It eliminates the trend (T), cyclical (C), and irregular (I) components from the time series

24. 24 The table below shows the quarterly sales for a toy company for the years 2001 through 2006. The sales are reported in millions of dollars. Determine a quarterly seasonal index using the ratio-to- moving-average method. Seasonal Index – An Example

25. 25 Step (1) – Organize time series data in column form Step (2) Compute the 4-quarter moving totals Step (3) Compute the 4-quarter moving averages Step (4) Compute the centered moving averages by getting the average of two 4-quarter moving averages Step (5) Compute ratio by dividing actual sales by the centered moving averages

26. 26 Seasonal Index – An Example

27. 27 Actual versus Deseasonalized Sales for the Toy Company Deseasonalized (seasonally adjusted) Sales = Sales / Seasonal Index

28. 28 Seasonal Index – An Example Using Excel

29. 29 Actual versus Deseasonalized Sales for Toy company – Time Series Plot – Using Excel

30. 30 Using Deseasonallized Data to Forecast- Toys Sales Example (establish linear trend)

31. 31 Seasonal Index – An Example Using Excel Given the deseasonalized linear equation for Toys International sales as Ŷ=8.109 + 0.0899t, generate the seasonally adjusted forecast for the each of the quarters of 2007 Quartert Ŷ (unadjusted forecast) Seasonal Index Quarterly Forecast (seasonally adjusted forecast) Winter2510.356750.7657.923 Spring2610.446660.5756.007 Summer2710.536571.14112.022 Fall2810.626481.51916.142 Ŷ = 8.109 + 0.0899(28) Ŷ X SI = 10.62648 X 1.519

32. 32 Durbin-Watson Statistic Tests the autocorrelation, which tend to arise in time series data. – Successive error terms are correlated. The Durbin-Watson statistic, d, is computed by first determining the residuals for each observation : e t = (Y t – Ŷ t ) Then compute d using the following equation:

33. 33 Durbin-Watson Test for Autocorrelation – Interpretation of the Statistic Range of d is 0 to 4 d = 2 No autocorrelation d close to 0Positive autocorrelation d beyond 2Negative autocorrelation Hypothesis Test: H 0 : No residual correlation (ρ = 0) H 1 : Positive residual correlation (ρ > 0) Critical values for d are found in Appendix B.10 using Appendix B.10 α - significance level n – sample size K – the number of predictor variables

34. 34 Durbin-Watson Critical Values (  =.05)

35. 35 Durbin-Watson Test for Autocorrelation: An Example The Banner Rock Company developed a special rocker (chair) for senior citizens which it advertises extensively on TV. The CEO of this company is studying the association between his advertising expense (X) and the number of rockers sold over the last 20 months (Y). He collected the following data. He would like to use the model to forecast sales, based on the amount spent on advertising, but is concerned that because he gathered these data over consecutive months that there might be problems of autocorrelation. MonthSales (000)Ad ($millions) 11535.5 21565.5 31535.3 41475.5 51595.4 61605.3 71475.5 81475.7 91525.9 101606.2 111696.3 121765.9 131766.1 141796.2 151846.2 161816.5 171926.7 182056.9 192156.5 202096.4

36. 36 Durbin-Watson Test for Autocorrelation: An Example Step 1: Generate the regression equation

37. 37 Durbin-Watson Test for Autocorrelation: An Example The resulting equation is: Ŷ = - 43.802 + 35.95X The coefficient (r) is 0.828 The coefficient of determination (r 2 ) is 68.5% (note: Excel reports r 2 as a ratio. Multiply by 100 to convert into percent) There is a strong, positive association between sales and advertising Is there potential problem with autocorrelation?

38. 38 ∑(e i -e i-1 ) 2 ∑(e i ) 2 =E4^2 =(E4-F4)^2 =-43.802+35.95*C3 =B3-D3 =E3 Durbin-Watson Test for Autocorrelation: An Example

39. 39 Hypothesis Test: H 0 : No residual correlation (ρ = 0) H 1 : Positive residual correlation (ρ > 0) Critical values for d given α=0.5, n=20, k=1 found in Appendix B.10 d l =1.20 d u =1.41 d l =1.20 d u =1.41 Reject H 0 Positive Autocorrelation Inconclusive Fail to reject H 0 No Autocorrelation Durbin-Watson Test for Autocorrelation: An Example

40. 40 END OF CHAPTER 16