1. 4 - 1© 2011 Pearson Education, Inc. publishing as Prentice Hall 4 4 Forecasting

2. 15-2 ■Time Frame in Forecasting ■Patterns in Past Data ■Forecasting Methods ■Forecast Accuracy Chapter Topics Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3. 4 - 3© 2011 Pearson Education, Inc. publishing as Prentice Hall What is Forecasting?  Process of predicting a future event  Underlying basis of all business decisions  Production  Inventory  Personnel  Facilities ??

4. 4 - 4© 2011 Pearson Education, Inc. publishing as Prentice Hall The Realities!  Forecasts are seldom perfect.  Should include expected value and measure of error.  Most techniques assume an underlying stability in the system  Long-term forecasts are less accurate than short-term forecasts. Too long term forecasts are useless  Product family and aggregated forecasts are more accurate than individual product forecasts

5. 4 - 5© 2011 Pearson Education, Inc. publishing as Prentice Hall  Short-range forecast  Up to 1 year, generally less than 3 months  Purchasing, job scheduling, workforce levels, job assignments, production levels  Medium-range forecast  3 months to 3 years  Aggregate production planning, budgeting  Long-range forecast  3 + years  New product planning, facility location, research and development Forecasting Time Horizons

6. 4 - 6© 2011 Pearson Education, Inc. publishing as Prentice Hall Forecasting Approaches  Used when little data exist  New products  New technology  Involves intuition, experience  e.g., forecasting sales on Internet Qualitative Methods

7. 4 - 7© 2011 Pearson Education, Inc. publishing as Prentice Hall Quantitative Approaches  Used when situation is ‘stable’ and historical data exist  Existing products  Current technology  Involves mathematical techniques  e.g., forecasting sales of LCD televisions

8. 4 - 8© 2011 Pearson Education, Inc. publishing as Prentice Hall Quantitative Approaches 1.Naive approach 2.Moving averages 3.Exponential smoothing 4.Trend projection 5.Linear regression time-series models associative model

9. 4 - 9© 2011 Pearson Education, Inc. publishing as Prentice Hall  Set of evenly spaced numerical data  Obtained by observing response variable at regular time periods  Forecast based only on past values, no other variables important  Assumes that factors influencing past and present will continue influence in future Time Series Forecasting

10. 4 - 10© 2011 Pearson Education, Inc. publishing as Prentice Hall Trend Seasonal Cyclical Random Time Series Components

11. 15-11 (a) Trend; (b) Cycle; (c) Seasonal; (d) Trend w/Season Patterns in Past Data Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

12. 4 - 12© 2011 Pearson Education, Inc. publishing as Prentice Hall Components of Demand Demand for product or service |||| 1234 Time (years) Average demand over 4 years Trend component Actual demand line Random variation Figure 4.1 Seasonal peaks

13. 4 - 13© 2011 Pearson Education, Inc. publishing as Prentice Hall  Persistent, overall upward or downward pattern  Changes due to population, technology, age, culture, etc.  Typically several years duration Trend Component

14. 4 - 14© 2011 Pearson Education, Inc. publishing as Prentice Hall  Regular pattern of up and down fluctuations  Due to weather, customs, etc.  Occurs within a single year Seasonal Component Number of PeriodLengthSeasons WeekDay7 MonthWeek4-4.5 MonthDay28-31 YearQuarter4 YearMonth12 YearWeek52

15. 4 - 15© 2011 Pearson Education, Inc. publishing as Prentice Hall  Repeating up and down movements  Affected by business cycle, political, and economic factors  Multiple years duration Cyclical Component 05101520

16. 4 - 16© 2011 Pearson Education, Inc. publishing as Prentice Hall  Erratic, unsystematic, ‘residual’ fluctuations  Due to random variation or unforeseen events  Short duration and nonrepeating Random Component MTWTFMTWTF

17. 4 - 17© 2011 Pearson Education, Inc. publishing as Prentice Hall Naive Approach  Assumes demand in next period is the same as demand in most recent period  e.g., If January sales were 68, then February sales will be 68  Sometimes cost effective and efficient  Can be good starting point

18. 4 - 18© 2011 Pearson Education, Inc. publishing as Prentice Hall  MA is a series of arithmetic means  Used if little or no trend  Used often for smoothing  Provides overall impression of data over time Moving Average Method Moving average = ∑ demand in previous n periods n

19. 4 - 19© 2011 Pearson Education, Inc. publishing as Prentice Hall January10 February12 March13 April16 May19 June23 July26 Actual3-Month MonthShed SalesMoving Average (12 + 13 + 16)/3 = 13 2 / 3 (13 + 16 + 19)/3 = 16 (16 + 19 + 23)/3 = 19 1 / 3 Moving Average Example 101213 101213 (10 + 12 + 13)/3 = 11 2 / 3

20. 4 - 20© 2011 Pearson Education, Inc. publishing as Prentice Hall Graph of Moving Average ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Shed Sales 30 – 28 – 26 – 24 – 22 – 20 – 18 – 16 – 14 – 12 – 10 – Actual Sales Moving Average Forecast

21. 4 - 21© 2011 Pearson Education, Inc. publishing as Prentice Hall  Used when some trend might be present  Older data usually less important  Weights based on experience and intuition Weighted Moving Average Weighted moving average = ∑ (weight for period n) x (demand in period n) ∑ weights

22. 4 - 22© 2011 Pearson Education, Inc. publishing as Prentice Hall January10 February12 March13 April16 May19 June23 July26 Actual3-Month Weighted MonthShed SalesMoving Average [(3 x 16) + (2 x 13) + (12)]/6 = 14 1 / 3 [(3 x 19) + (2 x 16) + (13)]/6 = 17 [(3 x 23) + (2 x 19) + (16)]/6 = 20 1 / 2 Weighted Moving Average 101213 131210 [(3 x 13) + (2 x 12) + (10)]/6 = 12 1 / 6 Weights AppliedPeriod 3 3Last month 2 2Two months ago 1 1Three months ago 6Sum of weights

23. 4 - 23© 2011 Pearson Education, Inc. publishing as Prentice Hall  Increasing n smooths the forecast but makes it less sensitive to changes  Do not forecast trends well  Require extensive historical data Potential Problems With Moving Average

24. 4 - 24© 2011 Pearson Education, Inc. publishing as Prentice Hall Moving Average And Weighted Moving Average 30 – 25 – 20 – 15 – 10 – 5 – Sales demand ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Actual sales Moving average Weighted moving average Figure 4.2

25. 4 - 25© 2011 Pearson Education, Inc. publishing as Prentice Hall  Form of weighted moving average  Weights decline exponentially  Most recent data weighted most  Requires smoothing constant (  )  Ranges from 0 to 1  Subjectively chosen  Involves little record keeping of past data Exponential Smoothing

26. 4 - 26© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing New forecast =Last period’s forecast +  (Last period’s actual demand – Last period’s forecast) F t = F t – 1 +  (A t – 1 - F t – 1 ) whereF t =new forecast F t – 1 =previous forecast  =smoothing (or weighting) constant (0 ≤  ≤ 1)

27. 4 - 27© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand(t-1) = 142 Ford Mustangs Actual demand (t-1)= 153 Smoothing constant  =.20

28. 4 - 28© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand (t-1)= 142 Ford Mustangs Actual demand = (t-1)153 Smoothing constant  =.20 New forecast (t)= 142 +.2(153 – 142)

29. 4 - 29© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= 142 +.2(153 – 142) = 142 + 2.2 = 144.2 ≈ 144 cars

30. 4 - 30© 2011 Pearson Education, Inc. publishing as Prentice Hall Impact of Different  225 – 200 – 175 – 150 – |||||||||123456789|||||||||123456789 Quarter Demand  =.1 Actual demand  =.5

31. 4 - 31© 2011 Pearson Education, Inc. publishing as Prentice Hall Impact of Different  225 – 200 – 175 – 150 – |||||||||123456789|||||||||123456789 Quarter Demand  =.1 Actual demand  =.5  Chose high values of  when underlying average is likely to change  Choose low values of  when underlying average is stable

32. 4 - 32© 2011 Pearson Education, Inc. publishing as Prentice Hall Choosing  The objective is to obtain the most accurate forecast no matter the technique We generally do this by selecting the model that gives us the lowest forecast error Forecast error= Actual demand - Forecast value = A t - F t

33. 4 - 33© 2011 Pearson Education, Inc. publishing as Prentice Hall Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |Actual - Forecast| n Mean Squared Error (MSE) MSE = ∑ (Forecast Errors) 2 n

34. 4 - 34© 2011 Pearson Education, Inc. publishing as Prentice Hall Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = ∑ 100|Actual i - Forecast i |/Actual i n i = 1

35. 4 - 35 Example 4, page 146 During the past 8 quarters, the Port of Baltimore has unloaded large quantities of grain from ships. The Port’s Operations Manager wants to test the forecasting method exponential smoothing to see how well the this method works in predicting tonnage unloaded. He guesses that the forecast of grain unloaded in the first quarter was 175 tons. © 2011 Pearson Education, Inc. publishing as Prentice Hall

36. 4 - 36© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62

37. 4 - 37© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD = ∑ |deviations| n = 82.45/8 = 10.31 For  =.10 = 98.62/8 = 12.33 For  =.50

38. 4 - 38© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 = 1,526.54/8 = 190.82 For  =.10 = 1,561.91/8 = 195.24 For  =.50 MSE = ∑ (forecast errors) 2 n

39. 4 - 39© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 MSE190.82195.24 = 44.75/8 = 5.59% For  =.10 = 54.05/8 = 6.76% For  =.50 MAPE = ∑ 100|deviation i |/actual i n i = 1

40. 4 - 40© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 MSE190.82195.24 MAPE5.59%6.76%

41. 4 - 41© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified to respond to trend Forecast including (FIT t ) = trendExponentially smoothed (F t ) +smoothed(T t ) forecasttrend

42. 4 - 42© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment F t =  (A t - 1 ) + (1 -  )(F t - 1 + T t - 1 ) T t =  (F t - F t - 1 ) + (1 -  )T t - 1 Step 1: Compute F t Step 2: Compute T t Step 3: Calculate the forecast FIT t = F t + T t

43. 4 - 43 EXAMPLE 7, page 149 A Portland manufacturer wants to forecast the demand for a pollution- control equipment. Past data shows that there is an increasing trend. The company assumes the initial forecast for month 1 was 11 units and the trend over that period was 2 units. α = 0.2 β =0.4 © 2011 Pearson Education, Inc. publishing as Prentice Hall

44. 4 - 44© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 217 320 419 524 621 731 828 936 10 Table 4.1

45. 4 - 45© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 217 320 419 524 621 731 828 936 10 Table 4.1 F 2 =  A 1 + (1 -  )(F 1 + T 1 ) F 2 = (.2)(12) + (1 -.2)(11 + 2) = 2.4 + 10.4 = 12.8 units Step 1: Forecast for Month 2

46. 4 - 46© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.80 320 419 524 621 731 828 936 10 Table 4.1 T 2 =  (F 2 - F 1 ) + (1 -  )T 1 T 2 = (.4)(12.8 - 11) + (1 -.4)(2) =.72 + 1.2 = 1.92 units Step 2: Trend for Month 2

47. 4 - 47© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.801.92 320 419 524 621 731 828 936 10 Table 4.1 FIT 2 = F 2 + T 2 FIT 2 = 12.8 + 1.92 = 14.72 units Step 3: Calculate FIT for Month 2

48. 4 - 48© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.801.9214.72 320 419 524 621 731 828 936 10 Table 4.1 15.182.1017.28 17.822.3220.14 19.912.2322.14 22.512.3824.89 24.112.0726.18 27.142.4529.59 29.282.3231.60 32.482.6835.16

49. 4 - 49© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Figure 4.3 |||||||||123456789|||||||||123456789 Time (month) Product demand 35 – 30 – 25 – 20 – 15 – 10 – 5 – 0 – Actual demand (A t ) Forecast including trend (FIT t ) with  =.2 and  =.4

50. 4 - 50© 2011 Pearson Education, Inc. publishing as Prentice Hall Trend Projections Fitting a trend line to historical data points to project into the medium to long-range Linear trends can be found using the least squares technique y = a + bx ^ where y= computed value of the variable to be predicted (dependent variable) a= y-axis intercept b= slope of the regression line x= the independent variable ^

51. 4 - 51© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Method Time period Values of Dependent Variable Figure 4.4 Deviation 1 (error) Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y-value) Trend line, y = a + bx ^

52. 4 - 52© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Method Time period Values of Dependent Variable Figure 4.4 Deviation 1 (error) Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y-value) Trend line, y = a + bx ^ Least squares method minimizes the sum of the squared errors (deviations)

53. 4 - 53© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Method Equations to calculate the regression variables b =  xy - nxy  x 2 - nx 2 y = a + bx ^ a = y - bx

54. 4 - 54© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Example b = = = 10.54 ∑xy - nxy ∑x 2 - nx 2 3,063 - (7)(4)(98.86) 140 - (7)(4 2 ) a = y - bx = 98.86 - 10.54(4) = 56.70 TimeElectrical Power YearPeriod (x)Demand (megawatt)x 2 xy 2006174174 20072794158 20083809240 200949016360 2010510525525 2011614236852 2012712249854 ∑x = 28∑y = 692∑x 2 = 140∑xy = 3,063 x = 4y = 98.86

55. 4 - 55© 2011 Pearson Education, Inc. publishing as Prentice Hall b = = = 10.54 ∑xy - nxy ∑x 2 - nx 2 3,063 - (7)(4)(98.86) 140 - (7)(4 2 ) a = y - bx = 98.86 - 10.54(4) = 56.70 TimeElectrical Power YearPeriod (x)Demandx 2 xy 2003174174 20042794158 20053809240 200649016360 2007510525525 2008614236852 2009712249854 ∑x = 28∑y = 692∑x 2 = 140∑xy = 3,063 x = 4y = 98.86 Least Squares Example The trend line is y = 56.70 + 10.54x ^

56. 4 - 56© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Example ||||||||| 200620072008200920102011201220132014 160 – 150 – 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – 60 – 50 – Year Power demand Trend line, y = 56.70 + 10.54x ^

57. 4 - 57© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Requirements 1.We always plot the data to insure a linear relationship 2.We do not predict time periods far beyond the database 3.Deviations around the least squares line are assumed to be random and normally distributed.

58. 4 - 58© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Variations In Data The multiplicative seasonal model can adjust trend data for seasonal variations in demand (jet skis, snow mobiles)

59. 4 - 59© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Variations In Data 1.Find average historical demand for each season 2.Compute the average demand over all seasons 3.Compute a seasonal index for each season 4.Estimate next year’s total demand 5.Divide this estimate of total demand by the number of seasons, then multiply it by the seasonal index for that season Steps in the process:

60. 4 - 60© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan80851059094 Feb7085858094 Mar8093828594 Apr909511510094 May11312513112394 Jun11011512011594 Jul10010211310594 Aug8810211010094 Sept8590959094 Oct7778858094 Nov7572838094 Dec8278808094 DemandAverageAverage Seasonal Month2012201320142012-2014MonthlyIndex

61. 4 - 61© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan80851059094 Feb7085858094 Mar8093828594 Apr909511510094 May11312513112394 Jun11011512011594 Jul10010211310594 Aug8810211010094 Sept8590959094 Oct7778858094 Nov7572838094 Dec8278808094 DemandAverageAverage Seasonal Month2012201320142012-2014MonthlyIndex 0.957 Seasonal index = Average 2012-2014 monthly demand Average monthly demand = 90/94 =.957

62. 4 - 62© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan808510590940.957 Feb70858580940.851 Mar80938285940.904 Apr9095115100941.064 May113125131123941.309 Jun110115120115941.223 Jul100102113105941.117 Aug88102110100941.064 Sept85909590940.957 Oct77788580940.851 Nov75728380940.851 Dec82788080940.851 DemandAverageAverage Seasonal Month2012201320142012-2014MonthlyIndex

63. 4 - 63© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan808510590940.957 Feb70858580940.851 Mar80938285940.904 Apr9095115100941.064 May113125131123941.309 Jun110115120115941.223 Jul100102113105941.117 Aug88102110100941.064 Sept85909590940.957 Oct77788580940.851 Nov75728380940.851 Dec82788080940.851 DemandAverageAverage Seasonal Month2012201320142012-2014 MonthlyIndex Expected annual demand = 1,200 Janx.957 = 96 1,200 12 Febx.851 = 85 1,200 12 Forecast for 2015

64. 4 - 64© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Time Demand 2015 Forecast 2014 Demand 2013 Demand 2012 Demand

65. 4 - 65© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Used when changes in one or more independent variables can be used to predict the changes in the dependent variable Most common technique is linear regression analysis We apply this technique just as we did in the time series example

66. 4 - 66© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Forecasting an outcome based on predictor variables using the least squares technique y = a + bx ^ where y= computed value of the variable to be predicted (dependent variable) a= y-axis intercept b= slope of the regression line x= the independent variable though to predict the value of the dependent variable ^

67. 4 - 67© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Example SalesArea Payroll ($ millions), y($ billions), x 2.01 3.03 2.54 2.02 2.01 3.57 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Sales Area payroll

68. 4 - 68© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Example Sales, y Payroll, xx 2 xy 2.0112.0 3.0399.0 2.541610.0 2.0244.0 2.0112.0 3.574924.5 ∑y = 15.0∑x = 18∑x 2 = 80∑xy = 51.5 x = ∑x/6 = 18/6 = 3 y = ∑y/6 = 15/6 = 2.5 b = = =.25 ∑xy - nxy ∑x 2 - nx 2 51.5 - (6)(3)(2.5) 80 - (6)(3 2 ) a = y - bx = 2.5 - (.25)(3) = 1.75

69. 4 - 69© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Example y = 1.75 +.25x ^ Sales = 1.75 +.25(payroll) If payroll next year is estimated to be $6 billion, then: Sales = 1.75 +.25(6) Sales = $3,250,000 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Nodel’s sales Area payroll 3.25

70. 4 - 70© 2011 Pearson Education, Inc. publishing as Prentice Hall Standard Error of the Estimate  A forecast is just a point estimate of a future value  This point is actually the mean of a probability distribution Figure 4.9 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Nodel’s sales Area payroll 3.25

71. 4 - 71© 2011 Pearson Education, Inc. publishing as Prentice Hall Standard Error of the Estimate We use the standard error to set up prediction intervals around the point estimate S y,x = ∑y 2 - a∑y - b∑xy n - 2

72. 4 - 72© 2011 Pearson Education, Inc. publishing as Prentice Hall Standard Error of the Estimate 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Nodel’s sales Area payroll 3.25 S y,x = = ∑y 2 - a∑y - b∑xy n - 2 39.5 - 1.75(15) -.25(51.5) 6 - 2 S y,x =.306 The standard error of the estimate is $306,000 in sales

73. 4 - 73 Prediction Intervals The ranges or limits of a forecast are estimated by: Upper limit = Y + t*Sy,x Lower limit = Y - t*Sy,x where: Y = forecasted value (point estimate) t = number of standard deviations from the mean of the distribution to provide a given probability of exceeding the limits through chance syx = standard error of the forecast © 2011 Pearson Education, Inc. publishing as Prentice Hall

74. 4 - 74 Prediction Intervals Estimate the confidence interval for annual sales of next year (remember payroll next year is estimated to be $6 billion) so that the probability that the actual sales exceeding these limits will be approximately equal to 0.05. Sales = 1.75 +.25(6) Sales = $3,250,000 © 2011 Pearson Education, Inc. publishing as Prentice Hall

75. 4 - 75 Prediction Intervals Step 1: Compute the standard error of the forecasts, syx. Step 2: Determine the appropriate value for t. n = 6, so degrees of freedom = n – 2 = 4 level of significance = =.05 The Student’s t Distribution Table shows: (t α/2=0.025) ) =2.78 © 2011 Pearson Education, Inc. publishing as Prentice Hall

76. 4 - 76 Prediction Intervals Step 3: Compute upper and lower limits. Upper limit = $3,250,000 + 2.78 * $306,000 =$4,100,680 Lower limit = $3,250,000 - 2.78 * $306,000 = $2,399,320 We are 95% confident the actual sales for next year will be between $2,399,320 and $4,100,680. © 2011 Pearson Education, Inc. publishing as Prentice Hall

77. 4 - 77© 2011 Pearson Education, Inc. publishing as Prentice Hall  How strong is the linear relationship between the variables?  Correlation does not necessarily imply causality!  Coefficient of correlation, r, measures degree of association  Values range from -1 to +1 Correlation

78. 4 - 78© 2011 Pearson Education, Inc. publishing as Prentice Hall Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ]

79. 4 - 79© 2011 Pearson Education, Inc. publishing as Prentice Hall Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ] y x (a)Perfect positive correlation: r = +1 y x (b)Positive correlation: 0 < r < 1 y x (c)No correlation: r = 0 y x (d)Perfect negative correlation: r = -1

80. 4 - 80© 2011 Pearson Education, Inc. publishing as Prentice Hall  Coefficient of Determination, r 2, measures the percent of change in y predicted by the change in x  Values range from 0 to 1  Easy to interpret Correlation For the Nodel Construction example: r =.901 r 2 =.81

81. 4 - 81© 2011 Pearson Education, Inc. publishing as Prentice Hall Multiple Regression Analysis If more than one independent variable is to be used in the model, linear regression can be extended to multiple regression to accommodate several independent variables y = a + b 1 x 1 + b 2 x 2 … ^ Computationally, this is quite complex and generally done on the computer

82. 4 - 82© 2011 Pearson Education, Inc. publishing as Prentice Hall Multiple Regression Analysis y = 1.80 +.30x 1 - 5.0x 2 ^ In the Nodel example, including interest rates in the model gives the new equation: An improved correlation coefficient of r =.96 means this model does a better job of predicting the change in construction sales Sales = 1.80 +.30(6) - 5.0(.12) = 3.00 Sales = $3,000,000