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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

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:

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.

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 ($)

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:

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 TransformationFigure 4.2 Actual and Logarithmically Transformed Operating Revenue for Southwest Airlines

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)Classical Time Series Model

- Secular Trend (T )
- Seasonal Variation (S )
- Cyclical Variation (C )
- Random or Irregular Variation

Trend

- Linear Trend
- Non-linear Trend

Trend

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

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:

Freehand Method

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

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 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

The Semi-average Method

We see that the intercept of the line is:

483.02

The slope is:

The fitted equation is:

The Semi-average Method

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

The Least Squares Method

- Provides the best method of fitting a trend.
- The intercept and the slope are computed as follows:

The Least Squares Method

- Using the data from the previous example, we have:

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.

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.

- 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:

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

Nonlinear Trend

- The data from the table is used to compute the following:

x = 0 in 1988

1x = one year

Y = 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.

- The logarithmic trend equation is:

Logarithmic Trend

- The least squares trend is computed as:

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

Logarithmic Trend Example (continued)

- The estimated trend line equation is:

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:

Logarithmic Trend(continued)

- For 2003, we will have:

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

- 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 Method Example

First and Second Difference of Hypothetical Data

Yt First Difference Second Difference

20,000

22,000 2,000

24,300 2,300 300

26,900 2,600 300

29,800 2,900 300

33,000 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 Variation

Computation of Ratio of Original Data to MovingAverage

Seasonal Variation

- To compute a seasonal index, we do the following:
- sum the modified means

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.

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.

Chapter Summary

- Preliminary Adjustments to Data:
- Trading Day Adjustment
- Price Adjustment
- Population Adjustment
- Data Transformation
- Patterns in Time Series Data
- The Classical Decomposition Method

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