1. Time Series and Forecasting Chapter 16 McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Learning Objectives LO1 Define the components of a time series LO2 Compute moving average LO3 Determine a linear trend equation LO4 Use a trend equation for a nonlinear trend LO5 Use a trend equation to compute forecasts LO6 Determine and interpret a set of seasonal indexes LO7 Determine and interpret a set of seasonal indexes LO8 Deseasonalize data using a seasonal index LO9 Calculate seasonally adjusted forecasts LO10 Test for autocorrelation 16-2

3. TIME SERIES is a collection of data recorded over a period of time (weekly, monthly, quarterly), an analysis of history, that can be used by management to make current decisions and plans based on long-term forecasting. It usually assumes past pattern to continue into the future Time Series and its Components Components of a Time Series 1. Secular Trend – the smooth long term direction of a time series 2. Cyclical Variation – the rise and fall of a time series over periods longer than one year 3. Seasonal Variation – Patterns of change in a time series within a year which tends to repeat each year 4. Irregular Variation – classified into: Episodic – unpredictable but identifiable Residual – also called chance fluctuation and unidentifiable LO1 Define the components of a time series. 16-3

4. Secular Trend – Examples LO1 16-4

5. Cyclical Variation – Sample Chart 1991199620012006 2011 LO1 16-5

6. Seasonal Variation – Sample Chart LO1 16-6

7. Irregular variation Behavior of time series other than trend cycle or seasonal Subdivided into:  Episodic  Residual Very difficult to predict LO1 16-7

8. The Moving Average Method Useful in smoothing time series to see its trend Basic method used in measuring seasonal fluctuation Applicable when time series follows fairly linear trend that have definite rhythmic pattern LO2 Compute a moving average. 16-8

9. Moving Average Method - Example LO2 16-9

10. 3-year and 5-Year Moving Averages LO2 16-10

11. Weighted Moving Average A simple moving average assigns the same weight to each observation in averaging Weighted moving average assigns different weights to each observation Most recent observation receives the most weight, and the weight decreases for older data values In either case, the sum of the weights = 1 LO2 16-11

12. Weighted Moving Average - Example Cedar Fair operates seven amusement parks and five separately gated water parks. Its combined attendance (in thousands) for the last 17 years is given in the following table. A partner asks you to study the trend in attendance. Compute a three- year moving average and a three-year weighted moving average with weights of 0.2, 0.3, and 0.5 for successive years. LO2 16-12

13. Weighted Moving Average - Example LO2 16-13

14. Weighed Moving Average – An Example LO2 16-14

15. Linear Trend The long term trend of many business series often approximates a straight line LO3 Determine a linear trend equation. 16-15

16. Linear Trend Plot LO3 16-16

17. 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 2 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) LO4 Use a linear trend equation to compute forecasts. 16-17

18. Year Sales ($ mil.) 20057 200610 20079 200811 200913 Yeart Sales ($ mil.) 200517 2006210 200739 2008411 2009513 The sales of Jensen Foods, a small grocery chain located in southwest Texas, since 2005 are: Linear Trend – Using the Least Squares Method: An Example LO4 16-18

19. Linear Trend – Using the Least Squares Method: An Example Using Excel LO4 16-19

20. 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 LO5 Compute a trend equation for a nonlinear trend. 16-20

21. Log Trend Equation – Gulf Shores Importers Example Graph on right 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 LO5 16-21

22. Log Trend Equation – Gulf Shores Importers Example LO5 16-22

23. Log Trend Equation – Gulf Shores Importers Example LO5 16-23

24. 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 LO6 Determine and interpret a set of seasonal indexes. 16-24

25. 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 LO6 16-25

26. The table below shows the quarterly sales for Toys International 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 LO6 16-26

27. 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 LO6 16-27

28. Seasonal Index – An Example LO6 16-28

29. Adjusted Seasonal Indexes LO6 16-29

30. Actual versus Deseasonalized Sales for Toys International Deseasonalized Sales = Sales / Seasonal Index LO7 Deseasonalize data using a seasonal index. 16-30

31. Actual versus Deseasonalized Sales for Toys International – Time Series Plot using Minitab LO7 16-31

32. Seasonally Adjusted Forecast Given the deseasonalized linear equation for Toys International sales as Ŷ=8.109 + 0.0899t, generate the seasonally adjusted forecast for each of the quarters of 2010 Ŷ = 8.10 + 0.0899(28) Ŷ X SI = 10.62648 X 1.519 LO8 Calculate seasonally adjusted forecasts. 16-32

33. Durbin-Watson Statistic Tests the autocorrelation among the residuals 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: LO9 Test for autocorrelation. 16-33

34. 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 LO9 16-34

35. Durbin-Watson Critical Values (  =.05) LO9 16-35

36. Durbin-Watson Test for Autocorrelation: An Example The Banner Rock Company manufactures and markets its own rocking chair. The company developed special rocker for senior citizens which it advertises extensively on TV. Banner’s market for the special chair is the Carolinas, Florida and Arizona, areas where there are many senior citizens and retired people The president of Banner Rocker 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 LO9 16-36

37. Durbin-Watson Test for Autocorrelation: An Example Step 1: Generate the regression equation LO9 16-37

38. 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? LO9 16-38

39. Durbin-Watson Test for Autocorrelation: An Example ∑(e i -e i-1 ) 2 ∑(e i ) 2 =E4^2 =(E4-F4)^2 =-43.802+35.95*C3 =B3-D3 =E3 LO9 16-39

40. Durbin-Watson Test for Autocorrelation: An Example Hypothesis Test: H 0 : No residual correlation (ρ = 0) H 1 : Positive residual correlation (ρ > 0) Critical values for d given α=0.05, 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 LO9 16-40