Business Forecasting

1. Business Forecasting Chapter 4 Data Collection and Analysis in Forecasting

2. Chapter Topics • Preliminary Adjustments to Data • Data Transformation • Patterns in Time Series Data • The Classical Decomposition Method

3. Preliminary Data Adjustments • Trading Day Adjustments • Price Change Adjustments • Population Change Adjustments

4. Trading Day Adjustments

5. Trading Day Adjustment

6. Trading Day Adjustments

7. Price Change Adjustments

8. Price Change Adjustments • Having computed the price index, we are now able to deflate the sales revenue with the weighted price index in the following way:

9. Price Change Adjustments • To see the impact of separating the effect of price level changes, we graph the price of computers in constant and current dollars.

10. Population Change Adjustments Disposable Personal Income and Per Capita Income for theU.S. 1990 and 2005 Disposable Income Population Per Capita Disposable Year Billions of Dollars (in Millions) Income ($)

11. Data Transformation • Most appropriate remedial measure for variance heterogeneity. • Original data are converted into a new scale, resulting in a new data set that is expected to satisfy the condition of homogeneity of variance. • Several transformation techniques are available.

12. Data Transformation • Linear Transformation: • An important assumption in using the regression model for forecasting is that the pattern of observation is linear. • Obviously, there are many situations in which this is not a valid assumption. • For example, if we were forecasting monthly sales and it was believed that those sales varied according to the season of the year, then the assumption of linearity would not hold.

13. Linear Transformation • A forecasting equation may be of the form: • The above could easily be transformed into a linear form for estimation purposes:

14. Logarithmic Transformation

15. Operating Revenue Log of Operating Revenue Actual Log 10,000 10.00 8,000 6,000 4,000 2,000 1.00 0 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Time Logarithmic Transformation Figure 4.2 Actual and Logarithmically Transformed Operating Revenue for Southwest Airlines

16. Square Root Transformation

17. Operating Revenue Square Root 10,000 120.00 100.00 8,000 80.00 6,000 Square Root 60.00 Operating Revenue 4,000 40.00 2,000 20.00 0 0.00 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Time Square Root Transformation(Scaled Square Root Data)

18. Square Root Transformation(Unscaled Square Root Data)

19. Classical Time Series Model • Secular Trend (T ) • Seasonal Variation (S ) • Cyclical Variation (C ) • Random or Irregular Variation

20. Trend • Linear Trend • Non-linear Trend

21. Trend • Computing the Linear Trend • The Freehand Method • The Semi-average Method • The Least Squares Method

22. Freehand Method

23. Freehand Method Since a linear trend by this method is simply an approximation of a straight line equation, we have to determine the intercept and the slope of the line. Based on our data, we have:

24. Freehand Method

25. Freehand Method • Now we can use this equation to make a forecast of the trend. For example, the forecast for 2006 would be:

26. Freehand Method • Based on your understanding, what are the pitfalls of using the freehand method? • Simple method but not objective. • Why not objective?

27. The Semi-average Method • Simple but objective method in fitting a trend line. • Divides the data into two equal parts and computes the average for each part. • The computed averages for each part provide two points on a straight line. • The slope of the line is computed by taking the difference between the averages and dividing it by half of the total number of observations.

28. The Semi-average Method Fitting a Straight Line by the Semi-Average Method to Income from the Export of Durable Goods, 1996–2005 Year Income Semi-total Semi-average Coded Time 1996 394.9 −2 1997 466.2 −1 1998481.2 2,415.1 483.02 0 1999 503.6 1 2000 569.2 2 2001 522.2 3 2002 491.2 4 2003499.8 2,679 535.8 5 2004 556.1 6 2005 609.7

29. The Semi-average Method We see that the intercept of the line is: 483.02 The slope is: The fitted equation is:

30. The Semi-average Method • For the year 2005, the forecast revenue from export of durable goods is:

31. The Least Squares Method • Provides the best method of fitting a trend. • The intercept and the slope are computed as follows:

32. The Least Squares Method • Using the data from the previous example, we have:

33. The Least Squares Method • The fitted trend line equation is: x = 0 in 2000 ½ 1 x = ½ year Y = Billions ofDollars • Note: Since x is measured in a half year, we have to multiply it by two to get the full year.

34. The Least Squares Method • To compare the two methods, we note: • Semi-average: • Least squares:

35. Nonlinear Trend • In many business and economic environments we observe that the time series does not follow a constant rate of increase or decrease, but follows an increasing or decreasing pattern. • Whenever there is dramatic change in production technology, we expect the trend line not to follow a constant linear pattern.

36. Nonlinear Trend • A polynomial function best exemplifies business conditions. • A second-degree parabola provides a good historical description of an increase or decrease per time period.

37. Nonlinear Trend • To solve for the constants a, b, and c in the previous equation, we use the following simultaneous equations:

38. Nonlinear Trend World Carbon Emissions from Fossil Fuel Burning 1982–1994 Year Million tonnes X Y x xY 1982 4,960 −6 −29,760 178,560 36 1,296 1983 4,947 −5 −24,735 123,675 25 625 1984 5,109 −4 −20,436 81,744 16 256 1985 5,282 −3 −15,846 47,538 9 81 1986 5,464 −2 −10,928 21,856 4 16 1987 5,584 −1 −5,584 5,584 1 1 1988 5,801 0 0 0 0 0 1989 5,912 1 5,912 5,921 1 1 1990 5,941 2 11,882 23,764 4 16 1991 6,026 3 18,078 54,234 9 81 1992 5,910 4 23,640 94,560 16 256 1993 5,893 5 29,465 147,325 25 625 1994 5,925 6 35,550 213,300 36 1,296 72,754 0 17,238 998,061 182 4,550

39. Nonlinear Trend • The data from the table is used to compute the following: x = 0 in 1988 1x = one year Y = million tonnes

40. Logarithmic Trend • When we wish to fit a trend line to percentage rates of change, we use the logarithmic trend line. • This is more prevalent when dealing with economic growth in an environment. • The logarithmic trend equation is:

41. Logarithmic Trend • The least squares trend is computed as:

42. Logarithmic Trend Example Chinese Exports Year ($100 Million) log Yxx log Y 1990 620.9 2.793 −15 −41.89 225 1991 719.1 2.857 −13 −37.13 169 1992 849.4 2.929 −11 −32.22 121 1993 917.4 2.963 −9 −26.66 81 1994 1,210.1 3.083 −7 −21.57 49 1995 1,487.8 3.173 −5 −15.86 25 1996 1,510.5 3.179 −3 −9.53 9 1997 1,827.9 3.262 −1 −3.26 1 1998 1,837.1 3.264 1 3.26 1 1999 1,949.3 3.290 3 9.86 9 2000 2,492.0 3.397 5 16.98 25 2001 2,661.0 3.425 7 23.97 49 2002 3,256.0 3.513 9 31.61 81 2003 4,382.28 3.642 11 40.05 121 2004 5,933.2 3.773 13 49.05 169 2005 7,619.5 3.882 15 58.22 225 52.42 0.00 44.891360.0

43. Logarithmic Trend Example (continued)

44. Logarithmic Trend Example (continued) • The estimated trend line equation is:

45. Logarithmic Trend(continued) • Check the goodness of fit by substituting two data points such as 1992 and 2003, into the fitted equation. • For 1992, we will have:

46. Logarithmic Trend(continued) • For 2003, we will have:

47. Logarithmic Trend Interpretation of the estimated trend line would be similar to a linear trend. However, before we can interpret the estimated values, we have to convert the log values into actual values of the data points. This is done by taking the antilog.

48. Logarithmic Trend • The results are: And

49. Logarithmic Trend • To determine the rate of change or the slope of the line we have: R = antilog 0.033 = 1.079 Since the rate of change (r) was defined as R −1, then r = 1.079 −1 = 0.079 r = 7.9 percent per half-year Therefore the growth rate is 15.8% or 16% per year.

50. Other Approaches to Trend Line • Two more sophisticated methods of determining whether there is a trend in the data: • Differencing • Autocorrelation (Box–Jenkins Methodology) • Allows the analyst to see whether a linear equation, a second-degree polynomial, or a higher-degree equation should be used to determine a trend.