1. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 15 Analyzing and Forecasting Time-Series Data

2. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-2 Chapter Goals After completing this chapter, you should be able to: Develop and implement basic forecasting models Identify the components present in a time series Compute and interpret basic index numbers Use smoothing-based forecasting models, including single and double exponential smoothing Apply trend-based forecasting models, including linear trend, nonlinear trend, and seasonally adjusted trend

3. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-3 The Importance of Forecasting Governments forecast unemployment, interest rates, and expected revenues from income taxes for policy purposes Marketing executives forecast demand, sales, and consumer preferences for strategic planning College administrators forecast enrollments to plan for facilities and for faculty recruitment Retail stores forecast demand to control inventory levels, hire employees and provide training

4. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-4 Time-Series Data Numerical data obtained at regular time intervals The time intervals can be annually, quarterly, daily, hourly, etc. Example: Year:1999 2000 2001 2002 2003 Sales: 75.3 74.2 78.5 79.7 80.2

5. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-5 Time Series Plot the vertical axis measures the variable of interest the horizontal axis corresponds to the time periods A time-series plot is a two-dimensional plot of time series data

6. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-6 Time-Series Components Time-Series Cyclical Component Random Component Trend Component Seasonal Component

7. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-7 Upward trend Trend Component Long-run increase or decrease over time (overall upward or downward movement) Data taken over a long period of time Sales Time

8. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-8 Downward linear trend Trend Component Trend can be upward or downward Trend can be linear or non-linear Sales Time Upward nonlinear trend Sales Time (continued)

9. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-9 Seasonal Component Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly Sales Time (Quarterly) Winter Spring Summer Fall

10. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-10 Cyclical Component Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to trough Sales 1 Cycle Year

11. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-11 Random Component Unpredictable, random, “residual” fluctuations Due to random variations of Nature Accidents or unusual events “Noise” in the time series

12. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-12 Index Numbers Index numbers allow relative comparisons over time Index numbers are reported relative to a Base Period Index Base period index = 100 by definition Used for an individual item or measurement

13. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-13 Index Numbers Simple Index number formula: (continued) where I t = index number at time period t y t = value of the time series at time t y 0 = value of the time series in the base period

14. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-14 Index Numbers: Example Company orders from 1995 to 2003: YearNumber of Orders Index (base year = 2000) 199527285.0 199628890.0 199729592.2 199831197.2 1999322100.6 2000320100.0 2001348108.8 2002366114.4 2003384120.0 Base Year:

15. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-15 Orders in 1996 were 90% of base year orders Orders in 2000 were 100% of base year orders (by definition, since 2000 is the base year) Orders in 2003 were 120% of base year orders Index Numbers: Interpretation

16. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-16 Aggregate Price Indexes An aggregate index is used to measure the rate of change from a base period for a group of items Aggregate Price Indexes Unweighted aggregate price index Weighted aggregate price indexes Paasche IndexLaspeyres Index

17. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-17 Unweighted Aggregate Price Index Unweighted aggregate price index formula: where I t = unweighted aggregate price index at time t  p t = sum of the prices for the group of items at time t  p 0 = sum of the prices for the group of items in the base period

18. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-18 Combined expenses in 2004 were 18.8% higher in 2004 than in 2001 Automobile Expenses: Monthly Amounts ($): YearLease paymentFuelRepair Total Index (2001=100) 20012604540345100.0 20022806040380110.1 20033055545405117.4 200431050 410118.8 Unweighted Aggregate Price Index Example

19. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-19 Weighted Aggregate Price Indexes Paasche index q t = weighting percentage at q 0 = weighting percentage at time t base period p t = price in time period t p 0 = price in the base period Laspeyres index

20. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-20 Commonly Used Index Numbers Consumer Price Index Producer Price Index Stock Market Indexes Dow Jones Industrial Average S&P 500 Index NASDAQ Index

21. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-21 Deflating a Time Series Observed values can be adjusted to base year equivalent Allows uniform comparison over time Deflation formula: where = adjusted time series value at time t y t = value of the time series at time t I t = index (such as CPI) at time t

22. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-22 Deflating a Time Series: Example Which movie made more money (in real terms)? Year Movie Title Total Gross $ 1939 Gone With the Wind 199 1977Star Wars461 1997Titanic601 (Total Gross $ = Total domestic gross ticket receipts in $millions)

23. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-23 Deflating a Time Series: Example GWTW made about twice as much as Star Wars, and about 4 times as much as Titanic when measured in equivalent dollars (continued) Year Movie Title Total Gross CPI (base year = 1984) Gross adjusted to 1984 dollars 1939 Gone With the Wind 19913.91431.7 1977Star Wars46160.6760.7 1997Titanic601160.5374.5

24. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-24 Trend-Based Forecasting Estimate a trend line using regression analysis Year Time Period (t) Sales (y) 1999 2000 2001 2002 2003 2004 123456123456 20 40 30 50 70 65 Use time (t) as the independent variable:

25. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-25 Trend-Based Forecasting The linear trend model is: Year Time Period (t) Sales (y) 1999 2000 2001 2002 2003 2004 123456123456 20 40 30 50 70 65 (continued)

26. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-26 Trend-Based Forecasting Forecast for time period 7: Year Time Period (t) Sales (y) 1999 2000 2001 2002 2003 2004 2005 12345671234567 20 40 30 50 70 65 ?? (continued)

27. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-27 Comparing Forecast Values to Actual Data The forecast error or residual is the difference between the actual value in time t and the forecast value in time t: Error in time t:

28. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-28 Two common Measures of Fit Measures of fit are used to gauge how well the forecasts match the actual values MSE (mean squared error) Average squared difference between y t and F t MAD (mean absolute deviation) Average absolute value of difference between y t and F t Less sensitive to extreme values

29. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-29 MSE vs. MAD Mean Square Error where: y t = Actual value at time t F t = Predicted value at time t n = Number of time periods Mean Absolute Deviation

30. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-30 Autocorrelation Autocorrelation is correlation of the error terms (residuals) over time (continued) Violates the regression assumption that residuals are random and independent Here, residuals show a cyclic pattern, not random

31. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-31 Testing for Autocorrelation The Durbin-Watson Statistic is used to test for autocorrelation H 0 : ρ = 0 (residuals are not correlated) H A : ρ ≠ 0 (autocorrelation is present) Durbin-Watson test statistic:

32. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-32 Testing for Positive Autocorrelation  Calculate the Durbin-Watson test statistic = d (The Durbin-Watson Statistic can be found using PHStat or Minitab) Decision rule: reject H 0 if d < d L H 0 : ρ = 0 (positive autocorrelation does not exist) H A : ρ > 0 (positive autocorrelation is present) 0dUdU 2dLdL Reject H 0 Do not reject H 0  Find the values d L and d U from the Durbin-Watson table (for sample size n and number of independent variables p) Inconclusive

33. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-33 Example with n = 25: Durbin-Watson Calculations Sum of Squared Difference of Residuals3296.18 Sum of Squared Residuals3279.98 Durbin-Watson Statistic1.00494 Testing for Positive Autocorrelation (continued) Excel/PHStat output:

34. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-34 Here, n = 25 and there is one independent variable Using the Durbin-Watson table, d L = 1.29 and d U = 1.45 d = 1.00494 < d L = 1.29, so reject H 0 and conclude that significant positive autocorrelation exists Therefore the linear model is not the appropriate model to forecast sales Testing for Positive Autocorrelation (continued) Decision: reject H 0 since d = 1.00494 < d L 0 d U =1.45 2 d L =1.29 Reject H 0 Do not reject H 0 Inconclusive

35. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-35 Nonlinear Trend Forecasting A nonlinear regression model can be used when the time series exhibits a nonlinear trend One form of a nonlinear model: Compare R 2 and s ε to that of linear model to see if this is an improvement Can try other functional forms to get best fit

36. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-36 Multiplicative Time-Series Model Used primarily for forecasting Allows consideration of seasonal variation Observed value in time series is the product of components whereT t = Trend value at time t S t = Seasonal value at time t C t = Cyclical value at time t I t = Irregular (random) value at time t

37. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-37 Finding Seasonal Indexes Ratio-to-moving average method: Begin by removing the seasonal and irregular components (S t and I t ), leaving the trend and cyclical components (T t and C t ) To do this, we need moving averages Moving Average: averages of consecutive time series values

38. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-38 Moving Averages Used for smoothing Series of arithmetic means over time Result dependent upon choice of L (length of period for computing means) To smooth out seasonal variation, L should be equal to the number of seasons For quarterly data, L = 4 For monthly data, L = 12

39. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-39 Moving Averages Example: Four-quarter moving average First average: Second average: etc… (continued)

40. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-40 Seasonal Data QuarterSales 1 2 3 4 5 6 7 8 9 10 11 etc… 23 40 25 27 32 48 33 37 50 40 etc… … …

41. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-41 Calculating Moving Averages Each moving average is for a consecutive block of 4 quarters QuarterSales 123 240 325 427 532 648 733 837 9 1050 1140 Average Period 4-Quarter Moving Average 2.528.75 3.531.00 4.533.00 5.535.00 6.537.50 7.538.75 8.539.25 9.541.00 etc…

42. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-42 Centered Moving Averages Average periods of 2.5 or 3.5 don’t match the original quarters, so we average two consecutive moving averages to get centered moving averages Average Period 4-Quarter Moving Average 2.528.75 3.531.00 4.533.00 5.535.00 6.537.50 7.538.75 8.539.25 9.541.00 Centered Period Centered Moving Average 329.88 432.00 534.00 636.25 738.13 839.00 940.13 etc…

43. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-43 Calculating the Ratio-to-Moving Average Now estimate the S t x I t value Divide the actual sales value by the centered moving average for that quarter Ratio-to-Moving Average formula:

44. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-44 Calculating Seasonal Indexes QuarterSales Centered Moving Average Ratio-to- Moving Average 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 50 40 … 29.88 32.00 34.00 36.25 38.13 39.00 40.13 etc… … 0.837 0.844 0.941 1.324 0.865 0.949 0.922 etc… …

45. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-45 Calculating Seasonal Indexes QuarterSales Centered Moving Average Ratio-to- Moving Average 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 50 40 … 29.88 32.00 34.00 36.25 38.13 39.00 40.13 etc… … 0.837 0.844 0.941 1.324 0.865 0.949 0.922 etc… … Average all of the Fall values to get Fall’s seasonal index Fall Do the same for the other three seasons to get the other seasonal indexes (continued)

46. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-46 Interpreting Seasonal Indexes Suppose we get these seasonal indexes: Season Seasonal Index Spring0.825 Summer1.310 Fall0.920 Winter0.945  = 4.000 -- four seasons, so must sum to 4 Spring sales average 82.5% of the annual average sales Summer sales are 31.0% higher than the annual average sales etc… Interpretation:

47. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-47 Deseasonalizing The data is deseasonalized by dividing the observed value by its seasonal index This smooths the data by removing seasonal variation

48. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-48 Deseasonalizing QuarterSales Seasonal Index Deseasonalized Sales 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 50 40 0.825 1.310 0.920 0.945 0.825 1.310 0.920 0.945 0.825 1.310 0.920 … 27.88 30.53 27.17 28.57 38.79 36.64 35.87 39.15 44.85 38.17 43.48 … etc… (continued)

49. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-49 Unseasonalized vs. Seasonalized

50. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-50 Forecasting Using Smoothing Methods Exponential Smoothing Methods Single Exponential Smoothing Double Exponential Smoothing

51. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-51 Single Exponential Smoothing A weighted moving average Weights decline exponentially Most recent observation weighted most Used for smoothing and short term forecasting

52. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-52 Single Exponential Smoothing The weighting factor is  Subjectively chosen Range from 0 to 1 Smaller  gives more smoothing, larger  gives less smoothing The weight is: Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 for forecasting (continued)

53. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-53 Exponential Smoothing Model Single exponential smoothing model where: F t+1 = forecast value for period t + 1 y t = actual value for period t F t = forecast value for period t  = alpha (smoothing constant) or:

54. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-54 Exponential Smoothing Example Suppose we use weight  =.2 Quarter (t) Sales (y t ) Forecast from prior period Forecast for next period (F t+1 ) 1 2 3 4 5 6 7 8 9 10 etc… 23 40 25 27 32 48 33 37 50 etc… NA 23 26.4 26.12 26.296 27.437 31.549 31.840 32.872 33.697 etc… 23 (.2)(40)+(.8)(23)=26.4 (.2)(25)+(.8)(26.4)=26.12 (.2)(27)+(.8)(26.12)=26.296 (.2)(32)+(.8)(26.296)=27.437 (.2)(48)+(.8)(27.437)=31.549 (.2)(48)+(.8)(31.549)=31.840 (.2)(33)+(.8)(31.840)=32.872 (.2)(37)+(.8)(32.872)=33.697 (.2)(50)+(.8)(33.697)=36.958 etc… F 1 = y 1 since no prior information exists

55. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-55 Sales vs. Smoothed Sales Seasonal fluctuations have been smoothed NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only.2

56. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-56 Double Exponential Smoothing Double exponential smoothing is sometimes called exponential smoothing with trend If trend exists, single exponential smoothing may need adjustment Add a second smoothing constant to account for trend

57. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-57 Double Exponential Smoothing Model where: y t = actual value in time t  = constant-process smoothing constant  = trend-smoothing constant C t = smoothed constant-process value for period t T t = smoothed trend value for period t F t+1 = forecast value for period t + 1 t = current time period

58. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-58 Double Exponential Smoothing Double exponential smoothing is generally done by computer Use larger smoothing constants  and β when less smoothing is desired Use smaller smoothing constants  and β when more smoothing is desired

59. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-59 Exponential Smoothing in Excel Use tools / data analysis / exponential smoothing The “damping factor” is (1 -  )

60. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 15-60 Chapter Summary Discussed the importance of forecasting Addressed component factors present in the time-series model Computed and interpreted index numbers Described least square trend fitting and forecasting linear and nonlinear models Performed smoothing of data series moving averages single and double exponential smoothing