1. Example 16.5 Regression-Based Trend Models

2. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b INTEL.XLS n This file contains quarterly sales data for the chip manufacturing firm Intel from the beginning of 1986 through the second quarter of 1996. n Each sales value is expressed in millions of dollars. n Check that an exponential trend fits these sales data fairly well. n Then estimate the relationship and interpret it.

3. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b Time Series Plot of Sales with Exponential Trend Superimposed

4. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b The Time Series Plot of Sales n The time series plot shows that sales are clearly increasing at an increasing rate, which a linear trend would not capture. n The smooth curve of the plot is an exponential trendline, which appears to be an adequate fit. n Alternatively, we can try to “straighten out” the data by taking the log of sales with Excel’s LN function. n The following is a plot of the log data.

5. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b Time Series Plot of Log Sales with Linear Trend Superimposed

6. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b The Time Series Plot of Log Sales n This plot goes together logically with the time series plot of Sales in the sense that if an exponential trendline fits the original data well, then a linear trendline will fit the transformed data well, and vice versa. n Either is evidence of an exponential trend in the sales data.

7. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b Estimating the Exponential Trend n To estimate the exponential trend, we run a regression of the log of sales, LnSales, versus Time. n A portion of the resulting data and output appears on the next slide.

8. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b Data Setup for Regression of Exponential Trend

9. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b Regression Output n The regression output shows that the estimated log of sales is given by Forecasted LnSales = 5.6883 + 0.0657Time n Looking at the coefficient of Time, we can say that Intel’s sales are increasing by approximately 6.6% per quarter during this period. n This translates to an annual percentage increase of about 29%. Perhaps the slight tailing off that we see at the right indicates that Intel can’t keep up this fantastic rate forever.

10. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b Forecasting the Future n The data also shows how to develop forecasts and 95% forecast intervals for future quarters. n The following steps are required. –Fitted values. Calculate the forecasts of historical sales in column E by entering the formula =$J$22*EXP($J$19*A4) in cell E4 and copying down. This is the formula based directly on the equation. –Absolute percentage errors. Calculate the absolute values of the percentage forecast errors in column F by entering the formula =ABS((E4-C4)/C4) in cell F4 and copying down.

11. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b Forecasting the Future -- continued –MAPE and standard deviation of percentage errors. Calculate the mean absolute percentage error in cell J23 with the formula =AVERAGE(F4:F5). Then approximate the standard deviation of the percentage errors, s pe, in cell J24 with the formula =1.25*J23. –Forecasts. For any future period in row 49, calculate a forecast in cell C49 with the formula =$J$22*EXP($J$19*A49) and copy it down for any other future periods. –Forecast intervals. To obtain 95% forecast intervals, enter the formulas =C49*(1-2*$J$24) and =C49*(1+2*$J$24) in cells D49 and E49, and copy them down for any other future periods.

12. 16.116.1 | 16.1a | 16.2 | 16.3 | 16.4 | 16.6 | 16.2a | 16.7 | 16.7a | 16.7b16.1a16.216.316.416.6 16.2a16.7 16.7a16.7b Forecasting the Future -- continued n In summary, we forecast Intel’s future sales by projecting that its current sales will continue to increase by 6.6% per quarter. n However, the wide forecasting intervals indicate the high level of uncertainty that exists for future sales.