8. Trend and seasonal effects - Forecasting with seasonal indexes

One out of every seven transactions at Starbucks is paid for with a Starbucks card: a reloadable, stored-value card. The table and graph that follow show quarterly time-series data on the number of Starbucks card activations. (Source: Starbucks Corporation for data from 2005:4 forward; the first three quarters of 2005 are imputed.)

\f30 25 20 Activations (millions) 15 - 10 - www 5- 0 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 2005 2006 2007 2008 The graph indicates a seasonal pattern in Starbucks card activations. Activations are highest in the fourth quarter of the year and lowest in the second and third quarters of the year.Estimating the regression model: yt = B0 + [ht + a with y = the number of card activations, and t = time period 1, 2, ..., 16 )rovides the sample regression line: 9' = 624+ 0.66t =ill in the ratios yt/SiI for the third quarter in the following table. Then compute the seasonal index for the third quarter, and enter it in the table. (Note: In this case, no adjustment is required to force the average of the seasonal indexes to equal 1.) Fill in the ratios yd}?t for the third quarter in the following table. Then compute the seasonal index for the third quarter, and enter it in the table. (Note: In this case, no adjustment is required to force the average of the seasonal indexes to equal 1.) Quarter Ratio 2005:3 2006:3 2007:3 2008:3 44444 Index The deseasonalized number of Starbucks card activations for 2008:3 is Y million. The forecast of the number of Starbucks card activations for the third quarter of 2009 is v million