1. © 2000 Prentice-Hall, Inc. Chap. 11- 1 The Least Squares Linear Trend Model Year Coded X Sales 95 0 2 96 1 5 97 2 2 98 3 2 99 4 7 00 5 6

2. © 2000 Prentice-Hall, Inc. Chap. 11- 2 The Least Squares Linear Trend Model (Continued) Excel Output Projected to year 2001

3. © 2000 Prentice-Hall, Inc. Chap. 11- 3 Year Coded X Sales 95 0 2 96 1 5 97 2 2 98 3 2 99 4 7 00 5 6 The Quadratic Trend Model

4. © 2000 Prentice-Hall, Inc. Chap. 11- 4 The Quadratic Trend Model (Continued) Excel Output

5. © 2000 Prentice-Hall, Inc. Chap. 11- 5 The Exponential Trend Model or Excel Output of Values in logs Year Coded Sales 94 0 2 95 1 5 96 2 2 97 3 2 98 4 7 99 5 6

6. © 2000 Prentice-Hall, Inc. Chap. 11- 6 Model Selection Using Differences Use a Linear Trend Model if the First Differences Are More or Less Constant Use a Quadratic Trend Model if the Second Differences Are More or Less Constant

7. © 2000 Prentice-Hall, Inc. Chap. 11- 7 Model Selection Using Differences Use an Exponential Trend Model if the Percentage Differences Are More or Less Constant (continued)

8. © 2000 Prentice-Hall, Inc. Chap. 11- 8 Autoregressive Modeling Used for forecasting Takes advantage of autocorrelation  1st order - correlation between consecutive values  2nd order - correlation between values 2 periods apart Autoregressive model for pth order: Random Error

9. © 2000 Prentice-Hall, Inc. Chap. 11- 9 Autoregressive Model: Example The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last 8 years. Develop the 2nd order Autoregressive model. Year Units 93 4 94 3 95 2 96 3 97 2 98 2 99 4 00 6

10. © 2000 Prentice-Hall, Inc. Chap. 11- 10 Autoregressive Model: Example Solution Year Y i Y i-1 Y i-2 93 4 --- --- 94 3 4 --- 95 2 3 4 96 3 2 3 97 2 3 2 98 2 2 3 99 4 2 2 00 6 4 2 Excel Output Develop the 2nd order table Use Excel to run a regression model

11. © 2000 Prentice-Hall, Inc. Chap. 11- 11 Autoregressive Model Example: Forecasting Use the 2nd order model to forecast number of units for 2001:

12. © 2000 Prentice-Hall, Inc. Chap. 11- 12 Autoregressive Modeling Steps 1. Choose p: note that df = n - p - 1 2. Form a series of “lag 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

13. © 2000 Prentice-Hall, Inc. Chap. 11- 13 Selecting A Forecasting Model Perform A residual analysis  Look for pattern or direction Measure sum of square error - SSE (residual errors) Measure residual error using MAD Use simplest model  Principle of parsimony

14. © 2000 Prentice-Hall, Inc. Chap. 11- 14 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 00

15. © 2000 Prentice-Hall, Inc. Chap. 11- 15 Measuring Errors Choose a model that gives the smallest measuring errors Sum square error (SSE)  Sensitive to outliers

16. © 2000 Prentice-Hall, Inc. Chap. 11- 16 Measuring Errors Mean absolute deviation (MAD)  Not sensitive to extreme observations (continued)

17. © 2000 Prentice-Hall, Inc. Chap. 11- 17 Principal of Parsimony Suppose 2 or more models provide good fit for data Select the simplest model  Simplest model types: Least-squares linear Least-square quadratic 1st order autoregressive  More complex types: 2nd and 3rd order autoregressive Least-squares exponential

18. © 2000 Prentice-Hall, Inc. Chap. 11- 18 Forecasting With Seasonal Data Use categorical predictor variables with least- square trending fitting Exponential model with quarterly data:  The b i provides the multiplier for the ith quarter relative to the 4th quarter  Q i = 1 if ith quarter and 0 if not  X j = the coded variable denoting the time period

19. © 2000 Prentice-Hall, Inc. Chap. 11- 19 Forecasting With Quarterly Data: Example Quarter 1994 1995 1996 1997 Standards and Poor’s Composite Stock Price Index: Excel Output Appears to be an excellent fit. r 2 is.98

20. © 2000 Prentice-Hall, Inc. Chap. 11- 20 Quarterly Data: Example Excel Output Regression Equation for the first quarter:

21. © 2000 Prentice-Hall, Inc. Chap. 11- 21 Chapter Summary Discussed the importance of forecasting Addressed component factors of the time- series model Performed smoothing of data series  Moving averages  Exponential smoothing

22. © 2000 Prentice-Hall, Inc. Chap. 11- 22 Chapter Summary Described least square trend fitting and forecasting  Linear, quadratic and exponential models Addressed autoregressive models Described procedure for choosing appropriate models Discussed seasonal data (use of dummy variables) (continued)