1. Chapter Nineteen McGraw-Hill/Irwin
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

2. Time Series and Forecasting
Chapter Nineteen
Time Series and Forecasting
GOALS
When you have completed this chapter, you will be able to:
ONE Define the four components of a time series.
TWO Compute a moving average.
THREE Determine a linear trend equation.
FOUR Compute the trend equation for a nonlinear trend.

3. Time Series and Forecasting
Chapter Nineteen continued
Time Series and Forecasting
GOALS
When you have completed this chapter, you will be able to:
FIVE Use trend equations to forecast future time periods and to develop seasonally adjusted forecasts.
SIX Determine and interpret a set of seasonal indexes.
SEVEN Deseasonalize data using a seasonal index.

4. Components of a Time Series
A Time Series is a collection of data recorded over a period of time. The data may be recorded weekly, monthly, or quarterly.
The Secular Trend
There are four components to a time series:
The Cyclical Variation
The Seasonal Variation
The Irregular Variation
Components of a Time Series

5. The Cyclical Variation is the rise and fall of a time series over periods longer than one year.

6. The Secular Trend is the smooth long run direction of the time series.

7. The Seasonal Variation is the pattern of change in a time series within a year. These patterns tend to repeat themselves from year to year.
Seasonal Variation

8. Components of a Time Series
The Irregular Variation is divided into two components:
Episodic Variations
are unpredictable, but can usually be identified, such as a flood or hurricane.
Residual variations are random in nature and cannot be identified.
Components of a Time Series

9. The long term trend equation (linear) :
Y’ = a + bt
Y’ is the projected value of the Y variable for a selected value of t
a is the Y-intercept, the estimated value of Y when t=0.
b is the slope of the line
t is an value of time that is selected
Linear Trend

10. The owner of Strong Homes would like a forecast for the next couple of years of new homes that will be constructed in the Pittsburgh area. Listed below are the sales of new homes constructed in the area for the last 5 years.
Year
Sales ($1000)
1999
4.3
2000
5.6
2001
7.8
2002
9.2
2003
9.7
Example 1

11. EXCEL
Regression tool
Example 1 Continued

12. EXCEL Chart and trend line function
Example 1 continued

13. The same results can be derived using MINITAB’s
Stat/time series/ trend analysis function.
The time series equation is:
Y’ = t
The forecast for the year 2003 is:
Y’ = (7) = 13.08
Example 1 continued

14. If the trend is not linear but rather the increases tend to be a constant percent, the Y values are converted to logarithms, and a least squares equation is determined using the logs.
Nonlinear Trends

15. Technological advances are so rapid that often initial prices decrease at an exponential rate from month to month. Hi-Tech Company provides the following information for the 12-month period after releasing its latest product.
Example 2

16. Example 2 continued

17. Example 2 continued

18. Example 2 continued

19. Example 2 continued

20. Take antilog to find estimate
Take antilog to find estimate. Thus, the estimated sales price for the 25th period would be:
Price25 = antilog(-.0476* ) =
Example 2 continued

21. The Moving-Average Method
The Moving-Average method is used to smooth out a time series. This is accomplished by “moving” the arithmetic mean through the time series.
To apply the moving-average method to a time series, the data should follow a fairly linear trend and have a definite rhythmic pattern of fluctuations.
The moving-average is the basic method used in measuring the seasonal fluctuation.
The Moving-Average Method

22. The numbers that result are called the Typical Seasonal Indexes.
The method most commonly used to compute the typical seasonal pattern is called the Ratio-to-Moving-Average method.
It eliminates the trend, cyclical, and irregular components from the original data (Y).
The numbers that result are called the Typical Seasonal Indexes.
Seasonal Variation

23. Determining a Seasonal Index
Using an example of sales in a large toy company, let us look at the steps in using the moving average method.
Winter
Spring
Summer
Fall
1998
6.7
4.6
10.0
12.7
1999
6.5
9.8
13.6
2000
6.9
5.0
10.4
14.1
2001
7.0
5.5
10.8
15.0
2002
7.1
5.7
11.1
14.5
2003
8.0
6.2
11.4
14.9
Determining a Seasonal Index

24. Step 1: Determine the moving total for the time series.
Step 2: Determine the moving average for the time series.
Step 3: The moving averages are then centered.
Step 4: The specific seasonal for each period is then computed by dividing the Y values with the centered moving averages.
Step 5: Organize the specific seasonals in a table.
Step 6: Apply the correction factor.
Steps

25. The resulting series (sales) is called deseasonalized sales or seasonally adjusted sales.
The reason for deseasonalizing a series (sales) is to remove the seasonal fluctuations so that the trend and cycle can be studied.
A set of typical indexes is very useful in adjusting a series (sales, for example)
Deseasonalizing Data

26. Deseasonalized Toy Sales
Winter
Spring
Summer
Fall
1998
8.7
8.0
8.3
1999
8.5
8.6
8.9
2000
9.0
9.1
9.2
2001
9.5
9.4
9.8
2002
9.3
9.9
9.7
2003
10.4
10.8
10.0
Example 3

27. Deseasonalized Toy Sales
Example 3 continued