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

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.

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.

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

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

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

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

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

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

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

EXCEL Regression tool Example 1 Continued

EXCEL Chart and trend line function Example 1 continued

The same results can be derived using MINITAB’s Stat/time series/ trend analysis function. The time series equation is: Y’ = 3.00 + 1.44t The forecast for the year 2003 is: Y’ = 3.00 + 1.44(7) = 13.08 Example 1 continued

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

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

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Take antilog to find estimate Take antilog to find estimate. Thus, the estimated sales price for the 25th period would be: Price25 = antilog(-.0476*25+2.9596) = 58.830 Example 2 continued

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

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

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

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

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

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

Deseasonalized Toy Sales Example 3 continued