Forecasting - Time-Series Extrapolation
Forecasting - Time-Series Extrapolation
A reasonable approach to forecasting is simply to extrapolate historical data forward. This approach, termed time-series forecasting, can be very effective, especially for short-term forecasting. A time-series is simply data collected over time, such as weekly sales data for three years.
Three factors must be present for time-series forecasting to be appropriate. First, time-series data must exist. Without reliable past data there is nothing to extrapolate. Second, the future must be like the past, at least with respect to forces influencing the time-series. Since past data are being extrapolated forward, any environmental change influencing the time series can make the extrapolation err. In particular, a time-series forecast has little ability to forecast "turning points," where the rate of growth or decline of the time-series will change significantly. Thus, time-series forecasting is best suited in rather stable situations and for short-term forecasting when the assumption that the future will be like the past is acceptable. Third, it must be possible to detect patterns or trends in the past data. If there is a large random element in the data, it might be difficult to detect any useful trends or other patterns.
Time-series forecasting involves several questions. First, what is the nature of the trend? Is it linear, for example? Second, how much historical data should be used? Third, how should seasonal and other cyclical fluctuations be handled?
Trend Projection
An advantage of time-series forecasting is that it is easy to understand conceptually. Historical data simply are plotted and extrapolated forward. The process can be done completely visually, or it can be done by using a regression program. In either case, a plot of historical data usually provides a useful feel for the data that a computer output sometimes lacks.
The simplest trend is the straight line, shown in Figure 22-3(a). The equation is simply:
y = a + bt
where t is time.
The domestic demand for oil products, a key item in energy-consumption forecasting. Suppose that the problem was to forecast the consumption for 1972 and 1973 using data from 1957 to 1970. A straight-line (least-squares) fit to this data would be:
8.1 + .43r
and is shown in Figure 22-4. Notice that the line is above the actual consumption during the 1961 to 1967 period and that its forecast for 1972 and 1973 is extremely low. A better straight-line forecast is obtained if the line is based only on the period from 1963 to 1971. The straight-line (least-squares) fit to the 1963 to 1971 data is:
9.7 + .61t
and also is shown in Figure 22-4. Notice the fit is quite good and that the forecast, although still low, improves.
In a linear or straight-line model, the growth is assumed to be a constant amount per year (0.61 millions of barrels per day for the 1963 to 1971 data). If the market instead grows at a constant percentage rate (such as 5 percent each year), then the appropriate model is the exponential model.Data actually follow the exponential curve very closely through 1973, so closely that a plot would tend to confuse the figure. However, if an exponential curve is fitted to the data from 1957 to 1973 the consumption forecast for 1974 would be 18.5, far
above the 16.7 that actually occurred in 1974. The reason, of course, is that the consumption growth pattern was broken in 1974 by the Arab oil embargo. The example illustrates the inability of a time-series to predict when there will be a dramatic shift in the underlying environment of the forecast. The exponential model, which could provide very accurate predictions for any year prior to 1974, became inappropriate because of the oil embargo. However, because the oil embargo was so unexpected, there was probably no forecasting device that would have predicted it.
An important forecasting task is to forecast the growth, maturity, and decline of new products. Such a pattern is illustrated nicely by Figure 22-5, which shows the sales pattern for black-and-white and color television sets. The decline phase of a product life cycle often follows a downward-sloping exponential curve, as shown in Figure 22-3(c). In Figure 22-5, an exponential curve is fitted to the black-and-white television set sales using only the data from 1966 to 1973.
Still another useful curve, an S-shaped curve, is shown in Figure 22-3(d). It is appropriate when there is an introduction stage, a rapid growth stage, and finally a maturity stage. In Figure 22-5 the black-and-white data from 1946 to 1966 seem to follow this type of curve, as do the color television sales from 1960 to 1973.
Sometimes it is useful to separate initial purchases and replacement purchase demand and forecast each separately. An initial purchase is the
first purchase such as the first purchase of a television set. A replacement purchase is made when the first set breaks down or becomes obsolete. Initial purchases tend to follow a growth-and-decline cycle, as the number of potential customers eventually declines. Replacement or repeat purchases, however, usually grow to a very stable plateau.
Weighting the Data
How much historical data should be used? At first glance, it would seem all available data should be used. However, data that are too old will represent conditions that have changed and therefore actually could tend to detract from the forecast. Recall the superior prediction performance of the Figure 22-4 line based only on the 1963 to 1971 data. At one extreme, only the last data point should be used, since the conditions surrounding it will be the closest to those of the future. There are three intermediate positions that deserve mention: moving average, exponential smoothing, and past turning points.
Moving Average An alternative to using the last data point is to use the moving average of the last n data points. For example, if the task were to forecast the shipments of the monthly volume of mail delivered to a particular mail route, we might use the average of the last 12 months, and each forecast then would be based on that figure.
Exponential Smoothing In exponential smoothing, instead of weighting the last n data points equally, we use exponentially decreasing sets of weights so that the more recent data are weighted more heavily than less recent data. This technique allows all the data to be used, but the more recent data will have more influence on the forecast. There are other, even more general weighting schemes; however, the added conceptual complexity and cost are usually not justified.2
Identifying Past Turning Points If a past turning point, a point in time where there was substantial change in the growth rate caused by an environmental change, can be identified, then a forecast might be based on data since that point in time. The year 1966 represents such a turning point for black-and-white television sales, because, as Figure 22-5 indicates, it was in that year that color television sales became very substantial. Thus, the use of data from 1966 on to form the basis for a forecast of black-and-white television sales is reasonable.
Seasonal and Cyclical Indexes
It is important to distinguish between a trend and a seasonal or cyclical fluctuation. A growth pattern simply may represent a cyclical upturn or even a seasoned fluctuation. Of course, if a 12-month moving average is used, the seasonal effect will be removed. However, it is useful to consider other ways to remove seasonal and cyclical effects that do not restrict one to using 12-month moving averages. Further, in short-term forecasting, the time period involved will be weeks, months, or quarters, and it becomes necessary to forecast the seasonal and cyclical effects as well as the trend;
The solution is to develop indexes that will represent the seasonal and cyclical effect.
Seasonal Index The seasonal index represents the effect of seasonal fluctuations. It can be created in a variety of ways.3 A commonly used index, developed by the Bureau of the Census, is based on the ratio of a given month's sales to the average monthly sales over a 12-month period. The 12 months used are centered at the month in question. If the month were September, for instance, the year would include 5.5 months before September and 5.5 months after. The seasonal index for September might then be the average of this ratio over a period of years.4
Table 22-2 provides an illustration using the quarterly attendance at a museum. The right column shows the actual attendance. The middle column contains the seasonal indexes. If there were no seasonal effects, the first quarter would be expected to have 25 percent of a given year's demand. However, because of seasonal factors, the first quarter, on the average, gets only 80 percent of this amount (or 20 percent of a given year's demand), whereas the third quarter gets 120 percent (or 30 percent of a given year's demand). The seasonally adjusted attendance is in the left column. If the seasonally adjusted attendance for the first quarter, 150, is multiplied by the seasonal index, .80, the actual attendance is:
150 x .80 = 120
Trends can be estimated and forecasts made using seasonally adjusted data. Then the seasonal index can be used to convert the forecast to actual numbers. For example, the seasonally adjusted data in Table 22-2 indicate a decline in attendance. Projecting this decline, the forecasted seasonally adjusted attendance for the first quarter of the next year might be 125. Applying the seasonal index of .80 would mean an actual forecast of 100.
The seasonal index also can be used to interpret past data. For example, actual attendance in the third quarter appeared good and reflected a nice growth trend; but, in fact, after applying the seasonal index, the attendance in the third quarter was disappointing, indicating a decline.
Cyclical Fluctuations Cyclical indexes can be developed in the same manner as seasonal indexes; however, most cycles, like business cycles, do not behave as regularly as seasonal factors, and it can be most difficult even to determine the length of the cycle.
A reasonable approach to forecasting is simply to extrapolate historical data forward. This approach, termed time-series forecasting, can be very effective, especially for short-term forecasting. A time-series is simply data collected over time, such as weekly sales data for three years.
Three factors must be present for time-series forecasting to be appropriate. First, time-series data must exist. Without reliable past data there is nothing to extrapolate. Second, the future must be like the past, at least with respect to forces influencing the time-series. Since past data are being extrapolated forward, any environmental change influencing the time series can make the extrapolation err. In particular, a time-series forecast has little ability to forecast "turning points," where the rate of growth or decline of the time-series will change significantly. Thus, time-series forecasting is best suited in rather stable situations and for short-term forecasting when the assumption that the future will be like the past is acceptable. Third, it must be possible to detect patterns or trends in the past data. If there is a large random element in the data, it might be difficult to detect any useful trends or other patterns.
Time-series forecasting involves several questions. First, what is the nature of the trend? Is it linear, for example? Second, how much historical data should be used? Third, how should seasonal and other cyclical fluctuations be handled?
Trend Projection
An advantage of time-series forecasting is that it is easy to understand conceptually. Historical data simply are plotted and extrapolated forward. The process can be done completely visually, or it can be done by using a regression program. In either case, a plot of historical data usually provides a useful feel for the data that a computer output sometimes lacks.
The simplest trend is the straight line, shown in Figure 22-3(a). The equation is simply:
y = a + bt
where t is time.
The domestic demand for oil products, a key item in energy-consumption forecasting. Suppose that the problem was to forecast the consumption for 1972 and 1973 using data from 1957 to 1970. A straight-line (least-squares) fit to this data would be:
8.1 + .43r
and is shown in Figure 22-4. Notice that the line is above the actual consumption during the 1961 to 1967 period and that its forecast for 1972 and 1973 is extremely low. A better straight-line forecast is obtained if the line is based only on the period from 1963 to 1971. The straight-line (least-squares) fit to the 1963 to 1971 data is:
9.7 + .61t
and also is shown in Figure 22-4. Notice the fit is quite good and that the forecast, although still low, improves.
In a linear or straight-line model, the growth is assumed to be a constant amount per year (0.61 millions of barrels per day for the 1963 to 1971 data). If the market instead grows at a constant percentage rate (such as 5 percent each year), then the appropriate model is the exponential model.Data actually follow the exponential curve very closely through 1973, so closely that a plot would tend to confuse the figure. However, if an exponential curve is fitted to the data from 1957 to 1973 the consumption forecast for 1974 would be 18.5, far
above the 16.7 that actually occurred in 1974. The reason, of course, is that the consumption growth pattern was broken in 1974 by the Arab oil embargo. The example illustrates the inability of a time-series to predict when there will be a dramatic shift in the underlying environment of the forecast. The exponential model, which could provide very accurate predictions for any year prior to 1974, became inappropriate because of the oil embargo. However, because the oil embargo was so unexpected, there was probably no forecasting device that would have predicted it.
An important forecasting task is to forecast the growth, maturity, and decline of new products. Such a pattern is illustrated nicely by Figure 22-5, which shows the sales pattern for black-and-white and color television sets. The decline phase of a product life cycle often follows a downward-sloping exponential curve, as shown in Figure 22-3(c). In Figure 22-5, an exponential curve is fitted to the black-and-white television set sales using only the data from 1966 to 1973.
Still another useful curve, an S-shaped curve, is shown in Figure 22-3(d). It is appropriate when there is an introduction stage, a rapid growth stage, and finally a maturity stage. In Figure 22-5 the black-and-white data from 1946 to 1966 seem to follow this type of curve, as do the color television sales from 1960 to 1973.
Sometimes it is useful to separate initial purchases and replacement purchase demand and forecast each separately. An initial purchase is the
first purchase such as the first purchase of a television set. A replacement purchase is made when the first set breaks down or becomes obsolete. Initial purchases tend to follow a growth-and-decline cycle, as the number of potential customers eventually declines. Replacement or repeat purchases, however, usually grow to a very stable plateau.
Weighting the Data
How much historical data should be used? At first glance, it would seem all available data should be used. However, data that are too old will represent conditions that have changed and therefore actually could tend to detract from the forecast. Recall the superior prediction performance of the Figure 22-4 line based only on the 1963 to 1971 data. At one extreme, only the last data point should be used, since the conditions surrounding it will be the closest to those of the future. There are three intermediate positions that deserve mention: moving average, exponential smoothing, and past turning points.
Moving Average An alternative to using the last data point is to use the moving average of the last n data points. For example, if the task were to forecast the shipments of the monthly volume of mail delivered to a particular mail route, we might use the average of the last 12 months, and each forecast then would be based on that figure.
Exponential Smoothing In exponential smoothing, instead of weighting the last n data points equally, we use exponentially decreasing sets of weights so that the more recent data are weighted more heavily than less recent data. This technique allows all the data to be used, but the more recent data will have more influence on the forecast. There are other, even more general weighting schemes; however, the added conceptual complexity and cost are usually not justified.2
Identifying Past Turning Points If a past turning point, a point in time where there was substantial change in the growth rate caused by an environmental change, can be identified, then a forecast might be based on data since that point in time. The year 1966 represents such a turning point for black-and-white television sales, because, as Figure 22-5 indicates, it was in that year that color television sales became very substantial. Thus, the use of data from 1966 on to form the basis for a forecast of black-and-white television sales is reasonable.
Seasonal and Cyclical Indexes
It is important to distinguish between a trend and a seasonal or cyclical fluctuation. A growth pattern simply may represent a cyclical upturn or even a seasoned fluctuation. Of course, if a 12-month moving average is used, the seasonal effect will be removed. However, it is useful to consider other ways to remove seasonal and cyclical effects that do not restrict one to using 12-month moving averages. Further, in short-term forecasting, the time period involved will be weeks, months, or quarters, and it becomes necessary to forecast the seasonal and cyclical effects as well as the trend;
The solution is to develop indexes that will represent the seasonal and cyclical effect.
Seasonal Index The seasonal index represents the effect of seasonal fluctuations. It can be created in a variety of ways.3 A commonly used index, developed by the Bureau of the Census, is based on the ratio of a given month's sales to the average monthly sales over a 12-month period. The 12 months used are centered at the month in question. If the month were September, for instance, the year would include 5.5 months before September and 5.5 months after. The seasonal index for September might then be the average of this ratio over a period of years.4
Table 22-2 provides an illustration using the quarterly attendance at a museum. The right column shows the actual attendance. The middle column contains the seasonal indexes. If there were no seasonal effects, the first quarter would be expected to have 25 percent of a given year's demand. However, because of seasonal factors, the first quarter, on the average, gets only 80 percent of this amount (or 20 percent of a given year's demand), whereas the third quarter gets 120 percent (or 30 percent of a given year's demand). The seasonally adjusted attendance is in the left column. If the seasonally adjusted attendance for the first quarter, 150, is multiplied by the seasonal index, .80, the actual attendance is:
150 x .80 = 120
Trends can be estimated and forecasts made using seasonally adjusted data. Then the seasonal index can be used to convert the forecast to actual numbers. For example, the seasonally adjusted data in Table 22-2 indicate a decline in attendance. Projecting this decline, the forecasted seasonally adjusted attendance for the first quarter of the next year might be 125. Applying the seasonal index of .80 would mean an actual forecast of 100.
The seasonal index also can be used to interpret past data. For example, actual attendance in the third quarter appeared good and reflected a nice growth trend; but, in fact, after applying the seasonal index, the attendance in the third quarter was disappointing, indicating a decline.
Cyclical Fluctuations Cyclical indexes can be developed in the same manner as seasonal indexes; however, most cycles, like business cycles, do not behave as regularly as seasonal factors, and it can be most difficult even to determine the length of the cycle.
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