19- Chapter Nineteen McGraw- © 2005 The McGraw-Hill Companies, Inc., All 19- 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 19- 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 19- A Time Series is a collection of data recorded over a period of time The data may be recorded weekly, monthly, or quarterly There are four components to a time series: The Secular Trend The Cyclical Variation The Seasonal Variation The Irregular Variation Components of a Time Series 19- The Cyclical Variation is the rise and fall of a time series over periods longer than one year Cyclical Variation 19- The Secular Trend is the smooth long run direction of the time series The Secular Trend 19- 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 19- 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 19- The long term trend equation (linear) : Y’ = a + bt o Y’ is the projected value of the Y variable for a selected value of t o a is the Y-intercept, the estimated value of Y when t=0 o b is the slope of the line o t is an value of time that is selected Linear Trend 19- 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 years Year 1999 2000 2001 2002 2003 Sales ($1000) 4.3 5.6 7.8 9.2 9.7 Example 19- 13 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 continued 19- 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 log(Y ') = [log(a )] + [log(b)]t Nonlinear Trends 19- 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 19- 16 Example continued 19- 17 Example continued 19- 18 Example continued 19- 19 Example continued 19- 20 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 continued 19- 21 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 19- 22 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 19- 23 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 4.6 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 19- 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 19- 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 19- 26 Deseasonalized Toy Sales Winter Spring Summer Fall 1998 8.7 8.0 8.7 8.3 1999 8.5 8.0 8.6 8.9 2000 9.0 8.7 9.1 9.2 2001 9.1 9.5 9.4 9.8 2002 9.3 9.9 9.7 9.5 2003 10.4 10.8 10.0 9.8 Example 19- 27 Deseasonalized Toy Sales Example continued ... 11.4 14.9 Determining a Seasonal Index 19- 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... Indexes Seasonal Variation 19- 23 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... period after releasing its latest product Example 19- 16 Example continued 19- 17 Example continued 19- 18 Example continued 19- 19 Example continued 19- 20 Take antilog to find estimate Thus,