1. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-1 Chapter 15 Time Series Forecasting and Index Numbers Statistics for Managers Using Microsoft ® Excel 4 th Edition

2. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-2 Chapter Goals After completing this chapter, you should be able to: identify the components present in a time series develop and implement basic forecasting models apply trend-based forecasting models, including linear and nonlinear trend models compute and interpret basic index numbers

3. Statistics for Managers Using Microsoft Excel, 4e © 2004 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. Statistics for Managers Using Microsoft Excel, 4e © 2004 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. Statistics for Managers Using Microsoft Excel, 4e © 2004 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. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-6 Time-Series Components Time-Series Cyclical Component Irregular Component Trend Component Seasonal Component

7. Statistics for Managers Using Microsoft Excel, 4e © 2004 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. Statistics for Managers Using Microsoft Excel, 4e © 2004 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. Statistics for Managers Using Microsoft Excel, 4e © 2004 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. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-10 Cyclical Component Long-term wave-like patterns  5-7 year cycles Regularly occur but may vary in length Often measured peak to peak or trough to trough Sales 1 Cycle Year

11. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-11 Irregular Component Unpredictable, random, “residual” fluctuations Due to random variations of Nature Accidents or unusual events Non-repeating

12. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-12 Multiplicative Time-Series Model for Annual Data Used primarily for economic forecasting Observed value in time series is the product of components whereT i = Trend value at year i C i = Cyclical value at year i

13. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-13 Multiplicative Time-Series Model with a Seasonal Component Used primarily for economic forecasting Allows consideration of seasonal variation whereT i = Trend value at time i S i = Seasonal value at time i C i = Cyclical value at time i

14. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-14 Multiplicative Time-Series Model for Business Applications Used primarily for business forecasting Allows consideration of seasonal variation whereT i = Trend value at time i S i = Seasonal value at time i

15. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-15 Smoothing the Annual Time Series Moving averages are not used by forecasting software as they are to simplistic. Almost all software for inventory control use exponential smoothing

16. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-16 Exponential Smoothing A weighted moving average Weights decline exponentially Most recent observation weighted most Used for smoothing and short term forecasting (often one period into the future)

17. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-17 Exponential Smoothing The weight (smoothing coefficient) is W Subjectively determined or calculated from history Range from 0 to 1 Smaller W gives more smoothing, larger W gives quicker response (continued)

18. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-18 Exponential Smoothing Model Exponential smoothing model where: E i = exponentially smoothed value for period I E i-1 = exponentially smoothed value already computed for period i - 1 Y i = observed value in period i W = weight (smoothing coefficient), 0 < W < 1 For i = 2, 3, 4, …

19. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-19 Exponential Smoothing Example Suppose we use weight W =.2 Time Period (i) Sales (Y i ) Forecast from prior period (E i-1 ) Exponentially Smoothed Value for this period (E i ) 1 2 3 4 5 6 7 8 9 10 etc… 23 40 25 27 32 48 33 37 50 etc… -- 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… E 1 = Y 1 since no prior information exists

20. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-20 Sales vs. Smoothed Sales 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. A model with trend is needed.

21. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-21 Forecasting Time Period i + 1 The smoothed value in the current period (i) is used as the forecast value for next period (i + 1) : The computations presented represent the basic exponential smoothing model. Software packages use a model with trend and seasonal adjustments.

22. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-22 Exponential Smoothing in Excel Use tools / data analysis / exponential smoothing The “damping factor” is (1 - W)

23. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-23 Trend-Based Forecasting Estimate a trend line using simple linear regression analysis Year Coded (X) Sales (Y) 1999 2000 2001 2002 2003 2004 012345012345 20 40 30 50 70 65 Use Coded (X) as the independent variable Sales as the dependent

24. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-24 Trend-Based Forecasting The linear trend model is: YearCoded (X) Sales (Y) 1999 2000 2001 2002 2003 2004 012345012345 20 40 30 50 70 65 (continued)

25. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-25 Trend-Based Forecasting Forecast for time period 6: Year Time Period (X) Sales (y) 1999 2000 2001 2002 2003 2004 2005 01234560123456 20 40 30 50 70 65 ?? (continued)

26. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-26 Nonlinear Trend Forecasting A nonlinear regression model should be used when the time series exhibits a nonlinear trend The Quadratic form is one type of a nonlinear model: The quadratic model was presented in chapter 14 the only difference is the independent variable is time Compare r 2 and standard error to that of linear model to see if this is an improvement Can try other functional forms to get best fit

27. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-27 Quadratic Model Worksheet Year, X i YiYi Coded X i Coded X 2 i 19992000 20004011 20013024 20025039 200370416 200465525 Run a multiple regression with Y i, as dependent and X i, X 2 i as independent

28. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-28 The Quadratic Trend Model Excel Output

29. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-29 Exponential Trend Model Estimated exponential forecasting equation: Interpretation: is the estimated annual compound growth rate (in %)

30. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-30 Exponential Trend Worksheet Year, X i YiYi Coded X i Log Y i 19992001.3010 20004011.6021 20013021.4771 20025031.6990 20037041.8451 20046551.8129 Run a simple linear regression with Log Y as dependent and Coded X as independent.

31. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-31 The Exponential Trend Model or Excel Output of Values in logs Year Coded X Sales (Y) 1999 0 20 2000 1 40 2001 2 30 2002 3 50 2003 4 70 2004 5 65 antilog=10 x

32. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-32 Trend-Based Forecasting Forecast for time period 6: Year Time Period (X) Sales (y) 1999 2000 2001 2002 2003 2004 2005 01234560123456 20 40 30 50 70 65 ?? (continued)

33. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-33 The Least Squares Trend Models in PHStat Use PHStat | Simple Linear Regression for linear trend and exponential trend models and Use PHStat | Multiple Regression for quadratic trend model Use Tools | Data Analysis… | Regression for either simple or multiple regression

34. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-34 Autoregressive Modeling Used for forecasting Takes advantage of autocorrelation 1st order - correlation between consecutive values 2nd order - correlation between values 2 periods apart p th order Autoregressive models: Random Error

35. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-35 Autoregressive Model: Example Year Units 97 4 98 3 99 2 00 3 01 2 02 2 03 4 04 6 The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last eight years. Develop the second order Autoregressive model.

36. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-36 Autoregressive Model: Example Solution Year Y i Y i-1 Y i-2 97 4 -- -- 98 3 4 -- 99 2 3 4 00 3 2 3 01 2 3 2 02 2 2 3 03 4 2 2 04 6 4 2 Excel Output  Develop the 2nd order table  Use Excel to estimate a regression model Dependent Variable

37. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-37 Autoregressive Model Example: Forecasting Use the second order model to forecast number of units for 2005:

38. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-38 Autoregressive Modeling Steps 1. Choose p (note that df = n – 2p – 1) 2. Form a series of “lagged predictor” variables Y i-1, Y i-2, …,Y i-p 3. Use Excel to run regression model using all p variables 4. Test significance of A p If null hypothesis rejected, this model is selected If null hypothesis not rejected, decrease p by 1 and repeat

39. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-39 Selecting A Forecasting Model Look at a Scatter Chart Develop appropriate models Perform a residual analysis Look for pattern or direction Look at the adjusted r 2 or Measure residual error using MAD Use simplest model Principle of parsimony

40. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-40 Residual Analysis Random errors Trend not accounted for Cyclical effects not accounted for Seasonal effects not accounted for T T T T ee e e 00 0 0

41. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-41 Measuring Errors Choose the model that gives the smallest measuring errors Mean Absolute Deviation (MAD) Not sensitive to extreme observations

42. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-42 Principal of Parsimony Suppose two or more models provide a good fit for the data Select the simplest model Simplest model types: Least-squares linear Least-squares quadratic 1st order autoregressive More complex types: 2nd and 3rd order autoregressive Least-squares exponential

43. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-43 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

44. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-44 Simple Price Index Simple Price Index: where I i = index number for year i P i = price for year i P base = price for the base year

45. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-45 Index Numbers: Example Airplane ticket prices from 1995 to 2003: YearPrice Index (base year = 2000) 199527285.0 199628890.0 199729592.2 199831197.2 1999322100.6 2000320100.0 2001348108.8 2002366114.4 2003384120.0 Base Year:

46. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-46 Prices in 1996 were 90% of base year prices Prices in 2000 were 100% of base year prices (by definition, since 2000 is the base year) Prices in 2003 were 120% of base year prices Index Numbers: Interpretation

47. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-47 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

48. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-48 Unweighted Aggregate Price Index Unweighted aggregate price index formula: = unweighted price index at time t = sum of the prices for the group of items at time t = sum of the prices for the group of items in time period 0 i = item t = time period n = total number of items

49. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-49 Unweighted total expenses 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

50. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-50 Weighted Aggregate Price Indexes Paasche index : weights based on : weights based on current period 0 quantities period quantities = price in time period t = price in period 0 Laspeyres index

51. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-51 Common Price Indexes Consumer Price Index (CPI) Producer Price Index (PPI) Stock Market Indexes Dow Jones Industrial Average S&P 500 Index NASDAQ Index

52. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-52 Pitfalls in Time-Series Analysis Assuming the mechanism that governs the time series behavior in the past will still hold in the future Using mechanical extrapolation of the trend to forecast the future without considering personal judgments, business experiences, changing technologies, and habits, etc.

53. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 15-53 Chapter Summary Discussed the importance of forecasting Addressed component factors of the time-series model Performed Exponential smoothing of a data series Described least square trend fitting and forecasting Linear, quadratic and exponential models Addressed autoregressive models Described procedure for choosing appropriate models Discussed pitfalls concerning time-series analysis Computed and interpreted basic index numbers