Example 13.3 Quarterly Sales at Intel Regression-Based Trend Models

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Objective To estimate Intel’s exponential growth and to see whether it has been maintained during the entire period from 1986 until early 2001.

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b INTEL.XLS n This file contains quarterly sales data for the chip manufacturing firm Intel from the first quarter of 1986 through the first quarter of n Each sales value is expressed in millions of dollars. n Are Intel’s sales growing exponentially through this entire period?

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Solution n We will first estimate and interpret an exponential trend for the years Then we will see how well the projection of this trend into the future fits the data after n The time series plot through 1996 appears on the next slide. n We have used Excel’s Chart/Add Trendline menu item, with the Exponential option, to superimpose an exponential tend line on this plot.

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Time Series Plot of Sales with Exponential Trend Superimposed

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Solution -- continued n The fit is evidently quite good. n Equivalently, the next slide illustrates the time sereis of log sales for this same period, with the linear trend line superimposed. n It fit is equally good.

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Time Series Plot of Log Sales with Linear Trend Superimposed

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Solution -- continued n We can also use StatPro’s regression procedure to estimate this exponential trend as shown on the next slide. n We first add a time variable in column C and make a logarithmic transformation of Sales in column D. n Then we regress Log(Sales) on Time to obtain the regression output. n Note that the two coefficients in cells H18 and H19 are the same as those shown for the linear trend on the previous slide.

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Regression Output for Estimating Exponential Trend

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b What does it all mean? n The estimated equation is Forecasted Sales = e t n The most important constant in this equation is the regression coefficient of Time, b= Expressed as a percentage, this coefficient implies that Intel’s sales were increasing by approximately 6.64% per quarter throughout this 11 year period. n To use this equation for forecasting into the future, we start with the final observation, 6440 in quarter 4 of 1996, and multiply by for as many quarters as we are forecasting ahead.

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Forecasts n Has this exponential growth continued beyond 1996 at Intel? n As you might have guessed, it has not, due to slumping sales in the computer industry and increase competition from other chip manufacturers. n We checked this by creating the Forecast column in the table on the next slide. n This implements are estimate of the equation.

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Creating Forecasts of Sales

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Forecasts -- continued n We then use StatPro’s time series plot procedure to plot the two series Sales and Forecast, shown on the next slide. n It is clear that sales in the forecast period remained rather constant – nowhere near the 6.64% growth they exhibited in the estimation period. n As Intel clearly realizes, nothing that good lasts forever.

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Time Series Plot of Forecasts Superimposed on Sales

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Standard Error of Estimate n The standard error of estimate is found in cell I9 in the Forecast of Sales table shown earlier. n This value, , is in log units, not original dollar units. n Therefore, it is a totally misleading indicator of the forecast errors we might make from the exponential trend equation. n To obtain more meaningful measures, we first obtain the forecasts of sales. Then we easily obtain any of the three forecast error measures discussed earlier. n The results appear on the next slide.

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Measures of Forecast Errors

| 13.1a | 13.2 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Calculations n The squared errors, absolute errors, and absolute percentage errors are first calculated with the formulas =(B4-E4)^2, =ABS(B4-E4), and =G4/B4 in cells F4, G4, and H4, which are then copied down. n The error measures then appear in cells L24, L25, and L26. n The corresponding formulas for RMSE, MAE, and MAPE are straightforward. n Forecasts for the 11-year estimate period were off, on average, by about 7.5%.