Business Forecasting

Chapter 4Data Collection and Analysis

in Forecasting

Chapter Topics

Preliminary Adjustments to Data

Data Transformation

Patterns in Time Series Data

The Classical Decomposition Method

Preliminary Data Adjustments

Trading Day Adjustments

Price Change Adjustments

Population Change Adjustments

Trading Day Adjustments

Yr J F M A M J J A S O N D

05

21 20 23 21 22 22 21 23 22 21 22 22

06

22 20 23 20 23 22 21 23 22 22 22 21

07

23 20 22 21 23 21 22 23 20 23 22 21

Trading Day AdjustmentAverage Trading Days for Each Month Month Average Number of Trading Days January 22.00

February 20.00

March 22.67

April 20.67

May 22.67

June 21.67

July 21.33

August 23.00

September 21.33

October 22.00

November 22.00

December 21.33

Trading Day Adjustments

Trading Day Computation for October Trading Day Actual Adjusted Year Trading Days Coefficient Data Data 2005 21 21/22 = 0.954 5,000,000 5,241,090

2006 22 22/22 = 1.000 5,000,000 5,000,000

2007 23 23/22 = 1.045 4,000,000 3,827,751

Price Change Adjustments Compunet Sales Data, 1990–2005 (1) (2) (3) (4) (5) (6) Compunet Sales Computer Software Price Sales in in Current $ Price Index Price Index Index* Constant $ Year (Thousands) 1995=100 1995=100 1995=100 (Thousands) 1990 89.0 58.40 63.81 59.48 149.63

1991 90.0 57.03 71.98 60.02 149.94

1992 95.4 66.57 77.61 68.78 138.71

1993 96.6 72.47 86.19 75.21 128.44

1994 98.9 79.27 91.55 81.73 121.01

1995 99.4 100.00 100.00 100.00 99.40

1996 100.2 110.14 114.61 111.03 90.24

1997 106.2 123.15 144.10 127.34 83.40

1998 107.5 131.92 166.22 138.78 77.46

1999 108.3 145.23 204.56 157.10 68.94

2000 120.0 153.40 236.19 169.96 70.60

2001 105.0 129.20 234.18 150.20 69.91

2002 103.8 116.79 224.66 138.37 75.02

2003 102.1 117.70 229.76 140.11 72.87

2004 98.7 124.51 247.05 149.02 66.23

2005 99.6 128.74 260.05 155.01 64.26

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:

63.1490.48.59

100)0.89( 1990

salesDeflated

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.

Figure 4.1 Computer Sales in Current and Constant Dollars

0 20 40 60 80

100 120 140 160

1986 1991 1996 2001 2006 Time

Sales

Current Dollars Constant Dollars

Population Change Adjustments

Disposable Personal Income and Per Capita Income for the U.S. 1990 and 2005

Disposable Income Population Per Capita DisposableYear Billions of Dollars (in Millions) Income ($)

1990 4285.8 250.2 17,129.50

1991 4464.3 253.5 17,610.65

…… …… ……. ……….

…… …… ……. ……….

1999 6695.0 279.3 23,970.64

2000 7194.0 282.4 25,474.50

2001 7486.8 285.4 26,232.66

2002 7830.1 288.3 27,159.56

2003 8162.5 291.1 28,040.19

2004 8681.6 293.9 29,539.30

2005 9036.1 296.7 30,455.34

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.

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.

Linear Transformation

A forecasting equation may be of the form:

The above could easily be transformed into a linear form for estimation purposes:

u .eY X

uXY logloglog

Logarithmic TransformationSouthwest Airlines Operating Revenue between 1990 and 2006 Operating Revenue Logarithmic Year (Million Dollars) Transformation 1990 1237 3.09

1991 1379 3.14

1992 1803 3.26

…… …… ……

…… …… ……

2003 5937 3.77

2004 6530 3.81

2005 7584 3.88

2006 9086 3.96

Logarithmic Transformation

Time

Ope

rati

ng R

even

ue

0

2,000

4,000

6,000

8,000

10,000

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008

1.00

10.00

Log

of

Ope

rati

ng R

even

ue

Actual Log

Figure 4.2 Actual and Logarithmically Transformed Operating Revenue for Southwest Airlines

Square Root Transformation

Southwest Airlines Operating Revenue for the Years 1990 and 2006 Operating Revenue Square Root Year (Million Dollars) Transformation 1990 1,237 35.17

1991 1,379 37.13

…… …… ……

…… …… ……

2002 5,522 74.31

2003 5,937 77.05

2004 6,530 80.81

2005 7,584 87.09

2006 9,086 95.32

Square Root Transformation(Scaled Square Root Data)

0

2,000

4,000

6,000

8,000

10,000

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008

Time

0.00

20.00

40.00

60.00

80.00

100.00

120.00Operating Revenue Square Root

Square Root

Operating Revenue

Square Root Transformation(Unscaled Square Root Data)

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Actual

Transformed

Classical Time Series Model

Secular Trend (T )

Seasonal Variation (S )

Cyclical Variation (C )

Random or Irregular Variation

Trend Linear Trend

Non-linear Trend

bxaYt

tbat eY

2cTbTaYt

Trend Computing the Linear Trend

The Freehand Method

The Semi-average Method

The Least Squares Method

Freehand MethodU.S. Private Fixed Investment for the Years 1995–2005 Total Private Fixed Investment Year ($ Billion) 1995 1,112.9

1996 1,209.9

1997 1,317.8

1998 1,438.4

1999 1,558.8

2000 1,679.0

2001 1,646.1

2002 1,570.2

2003 1,649.8

2004 1,830.6

2005 2,036.2

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.

bxaYt

xYt 48.612.845ˆ

Based on our data, we have:

Freehand Method

0

200

400

600

800

1000

1200

1400

1600

1 2 3 4 5 6 7 8 9 10

Time

Mil

lion

s of

Dol

lars

Actual

Trend Line

Freehand Method

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

)12(94.839.112,1ˆ tY

DollarsBillion 18.120,22006 Y

Freehand Method

Based on your understanding, what are the pitfalls of using the freehand method?

Simple method but not objective. Why not objective?

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.

The Semi-average MethodFitting 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 −21997 466.2 −11998 481.2 2,415.1 483.02 01999 503.6 12000 569.2 22001 522.2 32002 491.2 42003 499.8 2,679 535.8 52004 556.1 62005 609.7

The Semi-average Method

We see that the intercept of the line is:483.02

The fitted equation is:

The slope is:

56.105

02.48380.535

b

xYt 56.1002.483ˆ

The Semi-average Method

For the year 2005, the forecast revenue from export of durable goods is:

(7) 10.56 483.02ˆ tY

Billion $556.94ˆ tY

The Least Squares Method Provides the best method of fitting a

trend. The intercept and the slope are

computed as follows:

n

Ya

2x

xYb

The Least Squares Method Using the data from the previous

example, we have:

4.50910

1.094,5a

79.7330

3.571,2b

The Least Squares Method The fitted trend line equation is:

7.79x 509.4Yt ˆ

Note: Since x is measured in a half year, we have to multiply it by two to get the full year.

x = 0 in 2000 ½ 1 x = ½ yearY = Billions of Dollars

The Least Squares Method

To compare the two methods, we note:

Least squares:

Semi-average:

xYt 56.1002.483ˆ

15.58x 509.4Yt ˆ

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.

Nonlinear Trend

A polynomial function best exemplifies business conditions.

2ˆttt cxbxaY

A second-degree parabola provides a good historical description of an increase or decrease per time period.

Nonlinear Trend To solve for the constants a, b, and c in

the previous equation, we use the following simultaneous equations:

2xcnaY

422 xcxaYx

2x

xYb

Nonlinear TrendWorld Carbon Emissions from Fossil Fuel Burning 1982–1994Year Million tonnes

1982 4,960 −6 −29,760 178,560 36 1,2961983 4,947 −5 −24,735 123,675 25 6251984 5,109 −4 −20,436 81,744 16 2561985 5,282 −3 −15,846 47,538 9 811986 5,464 −2 −10,928 21,856 4 161987 5,584 −1 −5,584 5,584 1 11988 5,801 0 0 0 0 01989 5,912 1 5,912 5,921 1 11990 5,941 2 11,882 23,764 4 161991 6,026 3 18,078 54,234 9 811992 5,910 4 23,640 94,560 16 2561993 5,893 5 29,465 147,325 25 6251994 5,925 6 35,550 213,300 36 1,296

72,754 0 17,238 998,061 182 4,550

X Y x xY Yx 2 2x 4x

Nonlinear Trend The data from the table is used to

compute the following:

22410719487395 x.x..,Yt x = 0 in 19881x = one yearY = million tonnes

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.

blog xalogYlog t

The logarithmic trend equation is:

Logarithmic Trend

The least squares trend is computed as:

n

Ya

log

log

2

) log ( log

x

Yxb

Logarithmic Trend Example

1990 620.9 2.793 −15 −41.89 2251991 719.1 2.857 −13 −37.13 1691992 849.4 2.929 −11 −32.22 1211993 917.4 2.963 −9 −26.66 811994 1,210.1 3.083 −7 −21.57 491995 1,487.8 3.173 −5 −15.86 251996 1,510.5 3.179 −3 −9.53 91997 1,827.9 3.262 −1 −3.26 11998 1,837.1 3.264 1 3.26 11999 1,949.3 3.290 3 9.86 92000 2,492.0 3.397 5 16.98 252001 2,661.0 3.425 7 23.97 492002 3,256.0 3.513 9 31.61 812003 4,382.28 3.642 11 40.05 1212004 5,933.2 3.773 13 49.05 1692005 7,619.5 3.882 15 58.22 225

52.42 0.00 44.89 1360.0

Chinese ExportsYear ($100 Million) log Y x x log Y

2x

Logarithmic Trend Example (continued)

28.316

42.52 log log

n

Ya

033.01360

89.44) log ( log

2

x

Yxb

Logarithmic Trend Example (continued)

The estimated trend line equation is:

xtY 033.028.3ˆ log

211997 in x 0

year x 211

Logarithmic Trend (continued)

Check the goodness of fit by substituting two data points such as 1992 and 2003, into the fitted equation.

0.033(-11) + 3.28 = Y Log 1992ˆ

For 1992, we will have:

2.917 = Y Log 1992ˆ

Logarithmic Trend (continued)

0.033(11) + 3.28 = ˆ Log 2003Y

For 2003, we will have:

3.643 = ˆ Log 2003Y

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.

Logarithmic Trend

The results are:

917.2 antilog)ˆ(log antilog1992 YY

04.8261992 Y

And

643.3 antilog)ˆ(log antilog2003 YY

42.395,42003 Y

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.

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.

Differencing

First Difference

1 ttt YYY

12 ttt YYY

Second Difference

Differencing Method ExampleFirst and Second Difference of Hypothetical

Data

Yt First Difference Second Difference

20,00022,000 2,00024,300 2,300 30026,900 2,600 30029,800 2,900 30033,000 3,200 300

Seasonal Analysis

Seasonal variation is defined as a predictable and repetitive movement observed around a trend line within a period of 1 year or less.

There are several reasons for measuring seasonal variations. When analyzing the data from a time series, it

is important to be able to know how much of the variation in the data is due to the seasonal factors.

Seasonal Variation (Continued)

We may use seasonal variation patterns in making projections or forecasts of a short-term nature.

By eliminating the seasonal variation from a time series, we may discover the cyclical pattern of the time series.

Seasonal VariationComputation of Ratio of Original Data to Moving Average

Passenger Moving Ratio of Original

Year and Enplanement 12-month Moving Data to Moving

Month (Million) Total Average Average, %

(1) (2) (3) (4) (5) 2001 Jan. 44.107

Feb. 43.177

March 53.055

April 50.792

May 51.120

June 53.471

July 55.803 560.359 46.70 119.50

Aug. 56.405 554.809 46.23 122.00

Sept. 30.546 550.277 45.86 66.61

Oct. 40.291 545.723 45.48 88.60

Seasonal Variation To compute a seasonal index, we do the following:

sum the modified means

Months Year Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. 2001 119.5 122.0 66.6 88.6 90.4 91.5

2002 87.0 87.9 111.4 102.5 104.7 108.6 111.1 110.3 86.2 103.2 94.2 105.7

2003 91.3 86.6 104.7 97.7 101.5 107.4 114.7 110.8 90.3 101.2 94.4 99.1

2004 86.7 89.0 105.8 103.5 102.2 109.0 113.6 108.6 90.0 101.6 96.7 97.6

2005 88.6 86.6 108.0 100.4 104.8 109.0 114.0 Mean 87.4 87.1 106.2 100.2 102.8 108.3 112.9 109.9 88.8 102.0 95.1 96.1

Seasonal Variation Mean Middle Seasonal Month Three Ratios Index Jan. 87.42 87.68 Feb. 87.05 87.31 March 106.19 106.50 April 100.19 100.50 May 102.80 103.11 June 108.32 108.64 July 112.91 113.25 Aug. 109.90 110.23 Sep. 88.81 89.07 Oct. 102.00 102.31 Nov. 95.09 95.38 Dec. 96.07 96.36

Seasonal Variation

If production is full, we expect the index to equal 100 for each month. If not, we have to adjust it by computing a correction factor.

Compute the seasonal index.

002.197.1197

1200Factor Correction

Seasonal VariationPartial Deseasonalized Data for Passenger Enplanement, 2001–2005 Deseasonalized Year and Passengers Seasonal Passenger Month (Million) Index Enplanement (1) (2) (3) [col 2 ÷ col 3] X 100 2001 Jan. 44.107 87.68 50.306 Feb. 43.177 87.31 49.452 March 53.055 106.50 49.814 April 50.792 100.50 50.539 ….. ..……. ……… ……. 2005 Sept. 50.776 89.07 57.005 Oct. 53.971 102.31 52.754 Nov. 52.962 95.38 55.530 Dec. 53.007 96.36 55.008

Cyclical Variation

Similar to seasonal variation except that it occurs every 5 to 10 years.

There is a systematic pattern in the data that mirrors what is happening in the economy.

Movements from a recession to a depression or recession to recovery follow a cycle.

Every time series data has a random component. If there were no random components, we would have perfect prediction of future values. However, this is not the case with real-world conditions.

The cyclical component is measured as a proportion of the trend.

Cyclical VariationCitrus Received by the Cooperative during 1994–2006, and the Estimated Trend Boxes of Citrus Year (in 1,000) Trend X Y Yt

1994 6.5 6.7 1995 7.2 7.1 1996 7.6 7.5 1997 8.4 7.9 1998 8.5 8.3 1999 8.0 8.7 2000 8.6 8.9 2001 8.9 8.7 2002 9.5 9.5 2003 10.2 9.9 2004 10.6 10.3 2005 10.8 10.7 2006 11.0 11.1

Cyclical Variation

0

2

4

6

8

10

12

Bo

xes

(in

th

ou

san

ds)

Time

Boxes of Citrus Trend of Boxes of Citrus

Cyclical Fluctuations around the Trend Line

Cyclical VariationCalculation of Percent of Trend Boxes of Citrus Percent of Year (in 1,000) Trend trend X Y Yt (Y/Yt 100) 1994 6.5 6.7 97.0 1995 7.2 7.1 101.4 1996 7.6 7.5 101.3 1997 8.4 7.9 106.3 1998 8.5 8.3 102.4 1999 8.0 8.7 91.9 2000 8.6 8.9 96.6 2001 8.9 8.7 102.3 2002 9.5 9.5 100.0 2003 10.2 9.9 103.0 2004 10.6 10.3 102.9 2005 10.8 10.7 100.9 2006 11.0 11.1 99.1

Cyclical Variation Example

80

85

90

95

100

105

110

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Time

Per

cent

Tre

nd

Percent Trend Linear (Percent Trend)

Chapter Summary

Preliminary Adjustments to Data: Trading Day Adjustment Price Adjustment Population Adjustment

Data Transformation

Patterns in Time Series Data

The Classical Decomposition Method