Jill Tilly was a recent graduate of a university business school when she took a job with Busby Associates, a large exporter of farm equipment. Busby's president noticed a forecasting course on Jill's resume during the hiring process and decided to start Jill's employment with a forecasting project that had been discussed many times by Busby's top managers.
Busby's president believed there was a strong relationship between the company's export sales and the national figures for exports. The national figures were readily available from government sources, so Jill's project was to forecast a good, representative export variable. If this effort was successful, Busby's president believed the company would have a powerful tool for forecasting its own export sales.
At the time, Jill located the most recent copy of the Survey of Current Business in a local library and recorded the quarterly figures for consumer goods exports in billions of dollars. She believed this variable was a good representative of total national exports. Anticipating the possibility of forecasting using regression analysis, she also recorded values for other variables she thought might possibly correlate well with this dependent variable. She ended up with values for four variables for 14 quarters.
She then computed three additional variables from her dependent variable values: change in Y, percent change in Y , and Y lagged one period. By the time she began to think about various ways to forecast her variable, she had collected the data shown in Table 11-2.
Jill keyed her data into a computer program that performed regression analysis and computed the correlation matrix for her seven variables. After examining this matrix, she chose three regressions with a single predictor variable and six regressions with two predictor variables. She then ran these regressions and chose the one she considered best: It used one predictor ( Y lagged one period) with the following results:
Jill examined the residual autocorrelations and found them to be small. She concluded she did not need to correct for any autocorrelation in her regression. Jill believed that she had found a good predictor variable ( Y lagged one period).
1: Consumer goods, exports, billions of dollars
2: Gross personal savings, billions of dollars
3: National income retail trade, billions of dollars
4: Fixed weight price indexes for national defense purchases, military equipment, 1982 = 100
5: Change in dependent variable from previous period
6: Percent change in dependent variable from previous period
7: Dependent variable lagged one period
Source for variables 1 through 4: Survey of Current Business (U.S. Department of Commerce) 70 (7) (July 1990).
Jill realized that her sample size was rather small: 13 quarters. She returned to the Survey of Current Business to collect more data points and was disappointed to find that during the years in which she was interested the definition of her dependent variable changed, resulting in an inconsistent time series. That is, the series took a jump upward halfway through the period that she was studying.
Jill pointed out this problem to her boss, and it was agreed that total merchandise exports could be used as the dependent variable instead of consumer goods exports. Jill found that this variable remained consistent through several issues of the Survey of Current Business and that several years of data could be collected. She collected the data shown in
Table 11-3, lagged the data one period, and again ran a regression analysis using Y lagged one period as the predictor variable.
This time she again found good statistics in her regression printout, except for the lag 1 residual autocorrelation. That value,.47, was significantly different from zero at the 5% level. Jill was unsure what to do. She tried additional runs, including the period number and the change in Y as additional predictor variables. But after examining the residuals, she was unable to conclude that the autocorrelation had been removed. Jill decided to look at other forecasting techniques to forecast her new dependent variable: total merchandise exports. She used the time series data shown in the Y column of Table 11-3. The time series plot of total merchandise exports for the years 1984 through the second quarter of 1990 is shown in Figure 11-4.
After studying Figure 11-4, Jill decided to use only the last 16 data points in her forecasting effort. She reasoned that beginning with period 9 the series had shown a relatively steady increase, whereas before that period it had exhibited an increase and a decline.
An examination of her data revealed no seasonality, so Jill decided to try two smoothing methods:
Simple exponential smoothing
Holt's linear exponential smoothing, which can accommodate a trend in the data
For simple exponential smoothing, Jill restricted the smoothing constant, a, to lie between 0 and 1 and found the best a to be very close to 1. The Minitab program selected the optimal values for the Holt smoothing constants, a and b. Error measures and the choices for the smoothing constants are shown in Table 11-4. Given the results shown in Table 11-4, Jill decided Holt's linear smoothing offered the most promise and used this procedure to generate forecasts for the next four quarters.
Jill noted that the optimum smoothing constant using simple exponential smoothing was almost 1.00 (.99). Apparently, in order to track through the data in this fashion, the program was basically using each data value to predict the next. This is equivalent to using a simple naive method to forecast-that is, a model that says consecutive differences are random.
FIGURE 11-4 Plot of Quarterly Data Values: Total Merchandise Exports, First Quarter of 1984 to Second Quarter of 1990 ($ billions)
Jill realized that with each passing quarter a new actual value of total merchandise exports would be available and that the forecasts for future periods could be updated.
Using Holt's smoothing method, the forecasts for the next four quarters beyond the end of her data are
Jill then met with her boss to discuss her results. She indicated that she thought she had a good way to forecast the national variable, total merchandise exports, using exponential smoothing with trend adjustments. Her boss asked her to explain this method, which she did. Her next assignment was to use actual data to verify the hunch of Busby's president: that Busby's exports were well correlated with national exports. If she could establish this linkage, Busby would have a good forecasting method for its exports and could use the forecasts to plan future operations.
The optimum smoothing constants used by Holt's linear exponential smoothing were ? = 1.77 and ? =.14. As new data come in over the next few quarters, Jill should probably rerun her data to see if these values change. How often do you think she should do this?