1. ©2003 Thomson/South-Western 1 Chapter 17 – Quantitative Business Forecasting Slides prepared by Jeff Heyl, Lincoln University ©2003 South-Western/Thomson Learning™ Introduction to Business Statistics, 6e Kvanli, Pavur, Keeling

2. ©2003 Thomson/South-Western 2 Quantitative Forecasting  Regression Models  Time Series Models

3. ©2003 Thomson/South-Western 3 Sales for Clayton Corp. 400 400 – 300 300 – 200 200 – 100 100 – Forecast  350 Sales (thousands of units) t (time) Forecast period (t = 16) |||||||||||||||| 1988199119962003 2002 (t = 1) (t = 15) Data Figure 17.1

4. ©2003 Thomson/South-Western 4 Procedure for Forecasting with Time Series Data Identification of variable of interest Identification of different forecasting methodologies Estimation of model Calculation of forecast accuracy and final model selection Generation of forecasts Reexamination of forecasting accuracy at a later time Reexamination of present model or possible consideration of alternate forecasting models Model selection and forecasting Model review Figure 17.2

5. ©2003 Thomson/South-Western 5 Naive Predictor Figure 17.3

6. ©2003 Thomson/South-Western 6 Naive Predictor Figure 17.4

7. ©2003 Thomson/South-Western 7 Time Series Containing Trend and Seasonality 120 120 – 100 100 – 80 80 – 60 60 – 40 40 – 20 20 – Sales (millions of dollars) Actual data (y 1 ) Deseasonalized data (d t ) |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 t 1998199920002001 Figure 17.5

8. ©2003 Thomson/South-Western 8 Time Series Trend and Seasonality  Calculate the deseasonalized data from the original time series  Construct a least squares line through the deseasonalized data  Calculate the forecast for the time period T+1

9. ©2003 Thomson/South-Western 9 Video-Comp Example 110 110 – 100 100 – 90 90 – 80 80 – 70 70 – 60 60 – 50 50 – 40 40 – 30 30 – 20 20 – 10 10 – |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 t = 1 t = 16 2002 |11|111 |22|222 d t = 19.372 + 5.037t ^ dtdtdtdt d 18 ^ d 17 ^ t Deseasonalized data (d t ) Deseasonalized data (millions of dollars) Figure 17.6

10. ©2003 Thomson/South-Western 10 Excel Solution Figure 17.7

11. ©2003 Thomson/South-Western 11 Exponential Smoothing This technique uses all the preceding observations to determine a smoothed value for a particular time period S t = smoothed value for time period, t = Ay t + (1 - A)S t-1 t = 2, 3, 4,...

12. ©2003 Thomson/South-Western 12 Exponential Smoothing YtYtYtYt t No trend Figure 17.8

13. ©2003 Thomson/South-Western 13 Jefferson Civic Center 198915.05.05.05.0 199028.05.36.57.7 199132.14.984.32.66 199247.15.195.76.66 199354.85.155.254.99 199462.04.843.622.30 199577.85.135.717.25 199685.05.125.365.23 1997914.16.029.7313.21 19981013.06.7211.3613.02 19991113.57.3912.4313.45 20001214.28.0713.3214.12 20011314.08.6713.6614.01 YeartY t S t (A =.1)S t (A =.5)S t (A =.9) Table 17.1

14. ©2003 Thomson/South-Western 14 Attendance Example 15 15 – 14 14 – 13 13 – 12 12 – 11 11 – 10 10 – 9 9 – 8 8 – 7 7 – 6 6 – 5 5 – 4 4 – 3 3 – 2 2 – 1 1 – |1989 1991 1993 1995 1997 1999 2001 t Average attendance (thousands) S t (A =.1) S t (A =.5) S t (A =.9) Actual data (y t ) Figure 17.9

15. ©2003 Thomson/South-Western 15 Forecasting Using Simple Exponential Smoothing Figure 17.10

16. ©2003 Thomson/South-Western 16 Forecasting Using Simple Exponential Smoothing Figure 17.11 15 15 – 14 14 – 13 13 – 12 12 – 11 11 – 10 10 – 9 9 – 8 8 – 7 7 – 6 6 – 5 5 – 4 4 – 3 3 – 2 2 – 1 1 – |1989 1991 1993 1995 1997 1999 2001 t Average attendance (thousands) y t = S t-1 ytytytyt ^

17. ©2003 Thomson/South-Western 17 Linear Exponential Smoothing S t = Ay t + (1 - A)(S t - 1 + b t - 1 ) t = 2, 3, 4,... Smoothing Observations b t = B(S t - S t - 1 ) + (1 - B)b t - 1 t = 2, 3, 4,... Smoothing Trend y t + 1 = S t + b t t = 1, 2, 3,... ^ y t + m = S t + mb t t = 1, 2, 3,... ^

18. ©2003 Thomson/South-Western 18 Linear Exponential Smoothing Procedure 2 Use the first five years to estimate the initial trend Procedure 1 Let b 1 = 0 provided you have a large number of years, this procedure provides an adequate initial estimate for the trend Procedures for Summarizing the Results

19. ©2003 Thomson/South-Western 19 Summary for Linear Exponential Smoothing Figure 17.12

20. ©2003 Thomson/South-Western 20 Predicted Values YtYtYtYt t 500 500 – 400 400 – 300 300 – 200 200 – 100 100 – |11|111 |22|222 |44|444 |55|555 |66|666 |77|777 |88|888 |99|999 |10 |33|333 11 12 14 15 16 17 18 19 20 13 Actual (y t ) Procedure 1 (y t ) ^ Procedure 2 (y t ) ^ Figure 17.13

21. ©2003 Thomson/South-Western 21 Exponential Smoothing for Trend and Seasonality Winters’ Method S t = A + (1 - A)(S t - 1 + b t - 1 ) F t = B + (1 - B)F t - 1 b t = C(S t - S t - 1 ) + (1 - C)b t - 1 t= L + 1, L + 2, L + 3,... y t F t - L ytytStStytytStSt

22. ©2003 Thomson/South-Western 22 Forecasting Using Linear and Seasonal Exponential Smoothing Procedure 1: 3.Set the initial smoothed value for quarter 4 (S 0 ) equal to the actual value for quarter 4 (t + 1) 2.Set the initial trend estimate (b 0 ) equal to 0 1.Set the initial seasonal factors equal to 1 y t + m = (S t + mb t ) F t + m - L ^

23. ©2003 Thomson/South-Western 23 Forecasting Using Linear and Seasonal Exponential Smoothing Procedure 2: 3.The initial smoothed value for quarter 4, S 0, is the forecast value for each of the 4 quarters in year t + 1 2. Deseasonalize the data for the first two years and calculate the least squares line through these deseasonalized values, d t 1.Use the first two years of data to determine the seasonal indexes

24. ©2003 Thomson/South-Western 24 Jackson City Example Figure 17.14

25. ©2003 Thomson/South-Western 25 Jackson City Example Figure 17.15 YtYtYtYt t 500 500 – 400 400 – 300 300 – 200 200 – 100 100 – |11|111 |22|222 |44|444 |55|555 |66|666 |77|777 |88|888 |99|999 |10 |33|333 11 12 14 15 16 17 18 19 20 13 Actual (y t ) Procedure 1 (y t ) ^ Procedure 2 (y t ) ^

26. ©2003 Thomson/South-Western 26 Choosing the Appropriate Forecasting Procedure Exponential smoothing procedures are excellent for short-term forecasts, whereas the component decomposition is useful for medium- and long- range forecasting  Short term forecast: one to three months  Medium-range forecast: four months to two years  Long-range forecast: two or more years Length of the forecast

27. ©2003 Thomson/South-Western 27 Comparing Predicted and Observed Values There is no consensus among statisticians as to which measure is preferable MAD (mean absolute deviation) = ∑|e t | n MAPE (mean absolute percentage error) = ∑et2∑et2nn∑et2∑et2nnn MSE (mean square error) = ∑ etetytytetetytytn

28. ©2003 Thomson/South-Western 28 Comparison of Procedures Method 1 MAD = 12/3 = 4.0 MSE = 48/3 = 16.0 MAPE =.295/3 =.098 Method 2 MAD = 11/3 = 3.67 MSE = 57/3 = 19.0 MAPE =.260/3 =.087 Table 17.6 (abbreviated)