Demand Estimation and Forecasting

Chapter 6

Demand Estimation and Forecasting

6.1MARKET RESEARCH APPROACHES

A firm’s sales or demand can be estimated by market research approaches. These refer to consumer surveys, consumer clinics, and market experiments. Consumer surveys involve questioning a sample of consumers about how they would respond to particular changes in the price of the commodity and of related commodities, to changes in their incomes, and to changes in other determinants of demand. Consumer clinics are laboratory experiments in which participants are given a sum of money and asked to spend it in a simulated store to see how they react to changes in the commodity price, commodity packaging, displays, prices of competing commodities, and other factors affecting demand. In market experiments,the researcher changes the commodity price or other determinants of demand under experimental control (such as packaging and the amount and type of promotion) in a particular real-world store or stores and examines consumers’ responses to the changes.

Since many economic decisions (such as businesses’ plans to add to plant and equipment, and consumers intentions of purchasing houses, automobiles, washing machines, refrigerators, and so on) are made well in advance of actual expenditures, surveys and opinion polls on the buying intentions of businesses and consumers can be used to forecast a firm’s sales, or demand. These are often referred to as qualitative forecasts.

EXAMPLE 1.In 1962 University of Florida researchers conducted a market experiment in Grand Rapids, Michigan, to determine the price elasticity and cross-price elasticity of demand for three types of oranges: those from the Indian River district of Florida, those from the interior district of Florida, and those from California. Nine supermarkets participated in the experiment, which involved changing the prices of the three types of oranges each day for 31 consecutive days and recording the quantity sold of each variety. The researchers found that the price elasticity of demand for the three types of oranges was, respectively, –3.07, –3.01, and –2.76. They also found that while the cross-price elasticity of demand was larger than 1 between the two types of Florida oranges, it was less than 0.20 between each type of Florida orange and the California ones. Thus, consumers regarded the two types of Florida oranges as close substitutes, but they did not view the California oranges in that way. This information can be used to forecast the demand for each type of orange, resulting from changes in prices and in other determinants of demand.

EXAMPLE 2.Some of the best-known published surveys that can be used to forecast economic activity in general and in particular sectors of the economy are surveys of business executives’ plant and equipment expenditure plans (published in Business Week and in the Survey of Current Business),surveys of plans for inventory changes and sales expectations (conducted by the Department of Commerce, McGraw-Hill, the National Association of Purchasing Agents, and others), and surveys of consumer expenditure plans (conducted periodically by the Bureau of the Census and the Survey Research Center of the University of Michigan). For more specific forecasts of its own sales, a firm may rely on polls of its own executives and sales force or its own polls of consumer intentions.

6.2TIME-SERIES ANALYSIS

One of the most frequently used methods of forecasting a firm’s sales or demand is time-series analysis. Time-series data are data arranged chronologically by days, weeks, months, quarters, or years. Most economic time series exhibit a secular trend (long-run increases and decreases), cyclical fluctuations (the wavelike movement above and below the trend that appears every several years), seasonal variation (the fluctuations that regularly recur each year because of weather or social customs), and irregular or random influences arising from strikes, natural disasters, wars, or other unique events. While these components are shown separately for the time-series (sales) data in Fig. 6-1, they all operate at the same time in the real world.

The simplest form of time-series analysis is to forecast the past trend by fitting a straight line to the data (on the assumption that the past trend will continue in the future). The linear regression model will take the form of

St = S0 + bt(6-1)

where St is the value of the time series in forecasting for period t, S0 is the estimated value of the time series (the constant of the regression) in the base period (i.e., at time period t = 0), b is the absolute amount of growth per period, and t is the time period in which the time series is to be forecasted (see Example 3).

Sometimes, an exponential trend (showing a constant percentage change rather than a constant amount of change in each period) fits the data better. The exponential trend model can be specified as

St = S0 (1 + g)t(6-2)

where g is the constant percentage growth rate to be estimated. To estimate g, we transform equation (6-2)into natural logarithms and run the following regression:

ln St = ln S0 + t ln (1 + g)(6-2)

Taking into consideration seasonal variation (when present) can significantly improve the trend forecast and can be done by the ratio4o-trend method. To do so, we simply find the average ratio by which the actual value of the time series differs from the corresponding estimated trend value in each period and then multiply the forecasted trend value by this ratio. (See Example 4.) An alternative is to use dummy variables.

Fig. 6-1

EXAMPLE 3.Fitting a trend line to the electricity sales data (consumption in millions of kilowatt-hours) running from the first quarter of 1985(t = 1) to the last quarter of 1988 (t = 16) given in Table 6.1 we get

St = 11.90 + 0.394tR2 = 0.50

(400)

Table 6.1

1986.4 / 1986.3 / 1986.2 / 1986.1 / 1985.4 / 1985.3 / 1985.2 / 1985.1 / Quarter
16 / 13 / 17 / 12 / 14 / 12 / 15 / 11 / Quantity
1988.4 / 1988.3 / 1988.2 / 1988.1 / 1987.4 / 1987.3 / 1987.2 / 1987.1 / Quarter
19 / 16 / 20 / 15 / 17 / 15 / 18 / 14 / Quantity

The regression results indicate that electricity sales in the last quarter of 1984 (i.e., S0) are estimated to be 11.90 million kwh and to increase at an average of 0.394 million kwh per quarter. The trend variable is statistically significant at better than the 1 percent level (from the t statistic below the estimated slope coefficient) and “explains” 50 percent of the variation in electricity consumption. Thus, based on the past trend, we can forecast electricity consumption (in million kwh) in the city to be

S17 = 11.90 + 0.394 (17) = 18.60in the first quarter of 1989

S18 = 11.90 + 0.394 (18) = 18.99in the second quarter of 1989

S19 = 11.90 + 0.394 (19) = 19.39in the third quarter of 1989

S20 = 11.90 + 0.394 (20) = 19.78in the fourth quarter of 1989

Similar results are obtained with an exponential trend (see Problem 6.6).

EXAMPLE 4.By incorporating the strong seasonal variation in the data in Table 6.1 (consumption in the second and fourth quarters of each year is consistently higher than in the first and third quarters) we can significantly improve the above forecast. To do this we first estimate, or forecast, electricity consumption in each quarter from 1985 to 1988 by substituting actual consumption into the above estimated equation. Then we find the ratio of actual to forecasted consumption in each quarter and calculate the average. This is shown in Table 6.2. Finally, we multiply the trend forecasts obtained in Example 3 by the average seasonal factors estimated in Table 6.2 (i.e., 0.887 for the first quarter, 1.165 for the second quarter, and so on) and get the following new forecasts based on both the linear trend and the seasonal adjustment:

S17 = 18.60(0.887) = 16.50in the first quarter of 1989

S18 = 18.99(1.165) = 22.12in the second quarter of 1989

S19 = 19.39(0.097) = 17.59in the third quarter of 1989

S20 = 19.78(1.042) = 20.61in the fourth quarter of 1989

Note that with the inclusion of the seasonal adjustment, the forecasted values for electricity sales seem to closely replicate the past seasonal pattern (i.e., they are higher in the second and fourth quarters than in the first and third quarters). Similar results are obtained by using dummy variables. (See Problem 6.7.)

Table 6.2

Quarter

/

Forecasted

/ Actual / Actual / Forecasted
1985.1 / 12.29 / 11.00 / 0.895
1986.1 / 13.87 / 12.00 / 0.865
1987.1 / 15.45 / 14.00 / 0.906
1988.1 / 17.02 / 15.00 / 0.881
Average = 0.887
1985.2 / 12.69 / 15.00 / 1.182
1986.2 / 14.26 / 17.00 / 1.192
1987.2 / 15.84 / 18.00 / 1.136
1988.2 / 17.42 / 20.00 / 1.148
Average = 1.165
1985.3 / 13.08 / 12.00 / 0.917
1986.3 / 14.66 / 13.00 / 0.887
1987.3 / 16.23 / 15.00 / 0.924
1988.3 / 17.81 / 16.00 / 0.898
Average = 0.907
1985.4 / 13.48 / 14.00 / 1.039
1985.4 / 15.05 / 16.00 / 1.063
1985.4 / 16.63 / 17.00 / 1.022
1985.4 / 18.20 / 19.00 / 1.044
Average = 1.042

6.3SMOOTHING TECHNIQUES

Smoothing techniques can be used to forecast future values of a time series as some average of its past values when the series exhibits little trend or seasonal variation but a great deal of irregular or random variation. The simplest smoothing technique is the moving average.Here the forecasted value of a time series in a given period is equal to the average value of the series in a number of previous periods. For example, with a three-period moving average, the forecasted value of the time series for any period is given by the average value of the series in the previous three periods.

A better smoothing technique is exponential smoothing.Here the forecast for period t + 1 (i.e., Ft + 1) is a weighted average of the actual and forecasted values of the time series in period t. The value of the time series at period t (i.e., At)is assigned a weight (w)between 0 and 1 inclusive, and the forecast for period t (i.e., Ft) is then assigned the weight of 1 – w.The initial value of Ft is usually taken as the mean for the entire observed time series. The greater the value of w,the greater is the weight given to the value of the time series in period t as opposed to previous periods. The value of Ft+ 1is given by

Ft+1 = wAt + (1 – w ) Ft(6-4)

To determine which moving average or weight in exponential smoothing leads to a better forecast, we calculate the root mean square error (RMSE) of each forecast and utilize the forecast with the lowest RMSE. The formula for RMSE is

(6-5)

where  stands for “the sum of,” “i is the index variable,” Atis the actual value of the time series in period t, Ftis the forecasted value, and n is the number of time periods or observations. The forecast difference (i.e., A – F)is squared in order to penalize larger errors proportionately more than smaller ones.

EXAMPLE 5.Table 6.3 shows the calculation of a three-quarter and a five-quarter moving average for a firm’s market share during eight quarters [columns (1) and (2)]. The forecast for the fourth quarter in column (3) is equal to the average of the values for the first three quarters in column (2). For the ninth quarter, the forecast is 19.67 with a three-quarter moving average and 20.6 with a five-quarter moving average.

The RMSEs are, respectively:

Since the three-quarter moving average forecast has the lower RMSE, we prefer it.

Table 6.3

Quarter / Firm’s Actual Market Share (A) / Three-Quarter Moving Average Forecast (F) / A – F / (A – F)2 / Five-Quarter Moving Average Forecast (F) / A – F / (A – F)2
( 1 ) / ( 2 ) / ( 3 ) / ( 4 ) / ( 5 ) / ( 6 ) / ( 7 ) / ( 8 )
1 / 20 / –– / –– / –– / –– / –– / ––
2 / 22 / –– / –– / –– / –– / –– / ––
3 / 23 / –– / –– / –– / –– / –– / ––
4 / 24 / 21.67 / 2.33 / 5.4289 / –– / –– / ––
5 / 20 / 23.00 / –3.00 / 9.0000 / –– / –– / ––
6 / 23 / 22.33 / 0.67 / 0.4489 / 21.8 / 1.2 / 1.44
7 / 19 / 22.33 / –3.33 / 11.0889 / 22.4 / –3.4 / 11.56
8 / 17 / 20.67 / –3.67 / 13.4689 / 21.8 / –4.8 / 23.04
9 / –– / 19.67 / –– / –– / 20.6 / –– / ––
Total / –– / –– / –– / 39.4356 / –– / –– / 36.04

EXAMPLE 6.Table 6.4 shows the calculations used to obtain forecasts for the firm’s market share data in Table 6.3 by exponential smoothing with w = 0.3 and
w = 0.5.We let F1= 21 [the average of the values in column (2)] to get the calculations started. Applying equation (6-4) to find the forecast for F2with w = 0.3 and w = 0.5, we get, respectively:

F2 = 0.3(20) + (1 – 0.3) (21) = 20.7andF2 = 0.5(20) + (1 – 0.5) (21) = 20.5

Forecasts for the other quarters are similarly obtained. The RMSEs are, respectively,

Therefore, we prefer the forecast of F9= 18.9 obtained by exponential smoothing with w = 0.5 to the forecast obtained with w = 0.3, and to the three-quarter and five-quarter moving average forecasts.

Table 6.4

Quarter / Firm’s Actual Market Share (A) / Forecast (F)
with w = 0.3 / A – F / (A – F)2 / Forecast (F)
with w = 0.3 / A – F / (A – F)2
( 1 ) / ( 2 ) / ( 3 ) / ( 4 ) / ( 5 ) / ( 6 ) / ( 7 ) / ( 8 )
1 / 20 / 21.0 / –1.0 / 1.00 / 21.0 / –1.0 / 1.00
2 / 22 / 20.7 / 1.3 / 1.69 / 20.5 / 1.5 / 2.25
3 / 23 / 21.1 / 1.9 / 3.61 / 21.3 / 1.7 / 2.89
4 / 24 / 21.7 / 2.3 / 5.29 / 22.2 / 1.8 / 3.24
5 / 20 / 22.4 / –2.4 / 5.76 / 23.1 / –3.1 / 9.61
6 / 23 / 21.7 / 1.3 / 1.69 / 21.6 / 1.4 / 1.96
7 / 19 / 22.1 / –3.1 / 9.61 / 22.3 / –3.3 / 10.89
8 / 17 / 21.2 / –4.2 / 17.64 / 20.7 / –3.7 / 13.69
9 / –– / 19.9 / –– / –– / 18.9 / –– / ––
Total / –– / –– / –– / 46.29 / –– / –– / 45.53

6.4BAROMETRIC METHODS

Barometric forecasting relies on changes in leading indicators (time series that anticipate or lead changes in general economic activity) to forecast turning points (peaks and troughs) in business cycles. While our interest is primarily in leading indicators, some time series move in step, or coincide, with movements in general economic activity and are therefore called coincident indicators.Still others follow, or lag behind, movements in economic activity and are called lagging indicators (see Problem 6.11). Table 6.5 presents a list of the 12 best leading indicators (out of a much longer list available), together with their mean lead ( – ) time (in months) with respect to the actual cyclical peak or trough.

Table 6.5Short List of 12 Leading Indicators

Indicators

/ Lead in Months
Average work week of production workers, manufacturing / –7.3
Layoff rate, manufacturing / –8.6
New orders, consumer goods and materials, 1972 dollars / –6.7
Vendor performance, companies receiving slower deliveries / –7.3
Index of net business formation / –7.4
Contracts and orders, plant and equipment, 1972 dollars / –5.5
New building permits, private housing units / –10.9
Change in inventories on hand and on order, 1972 dollars / –5.6
Change in sensitive materials prices / –8.8
Stock prices, 500 common stocks / –7.0
Change in total liquid assets-8.5 / –8.5
Money supply (M2), 1972 dollars / –11.8
Composite index of the 12 leading indicators / –8.2

Source: U. S. Department of Commerce, Bureau of Economic Analysis, Handbook of Cyclical Indicators (Washington, D. C.: U. S. Government Printing office, May 1977), pp. 174-191.

Table 6.5 also gives the composite index,a weighted average of the 12 leading indicators, with larger weights assigned to those indicators that do a better job of forecasting. The composite index smooths out random variations and provides more reliable forecasts and fewer wrong signals than individual indicators. Another method of overcoming the difficulty arising when some of the 12 leading indicators move up and some move down is the diffusion index. This gives the percentage of the 12 leading indicators that are moving up. All these indicators are published monthly in Business Conditions Digest by the U.S. Department of Commerce. Three or four successive one-month declines in the composite index and a diffusion index of less than 50 percent are usually a prelude to a recession, (i.e., a slowing down) of economic activity.

Fig. 6-2

EXAMPLE 7.The top panel of Fig. 6-2 shows that the composite index of 12 leading indicators turned down prior to (i.e., led) the recessions of 1969-1970, 1973-1975, 1980, and 1981-1982 (the shaded regions in the figure). The bottom panel shows that the diffusion index for the leading indicators was generally below 50percent in the months preceding recessions.

6.5ECONOMETRIC METHODS

The most common method of estimating and forecasting firm’s sales or demand is regression analysis. The first step is to specify the model, (i.e., to identify the determinants of sales or demand). The second step is to collect the data (over time or across different economic units at a given point in time) for the variables in the model. The third step is to determine the functional form. The simplest is the linear model. Here, the estimated slope coefficients measure the change in sales or demand per unit change in the independent or explanatory variables. An alternative specification is the logarithmic form, in which the estimated coefficients measure percentage changes, or elasticities. Both forms are often estimated, and the one that gives better results is usually reported. Finally, we must evaluate the regression results. That is, we check that the signs of the estimated coefficients conform to theory; conduct t tests on the statistical significance of the estimated coefficients; determine the proportion of the total variation in sales, or demand, that is “explained” by the model (R2); and make sure that the regression results are free of econometric problems. (See Section 4.3.)

To forecast sales, or demand, we simply substitute the projected or forecasted values of the independent or explanatory variables into the estimated equation and then solve the equation. Forecasts obtained in this way are generally superior to forecasts obtained by other methods because regression analysis identifies the determinants of sales or demand and also provides an estimate of the magnitude of the forecast (not just whether sales or demand will rise or fall). Regression analysis can also incorporate trend analysis, seasonal adjustment, and leading indicators. Forecasts of more complex relationships such as gross national product (GNP) or sales of major sectors or industries are usually based on multiple-rather than single-equation models. These range from a few equations to hundreds. (See Problems 6.18 and 6.19.)

EXAMPLE 8.The following regression equation presents the estimated demand for air travel between the United States and Europe from 1965to 1978 (numbers in parentheses are t values):

ln Qt = 2.737 – 1.247 ln Pt + 1.905 ln GNPt = 0.97D = W = 1.83

(– 5.071)( 7.286 )

whereQ1=number of passengers per year traveling between the United States and Europe from 1965to 1978, in thousands

Pt=average yearly air fare between New York and London, adjusted for inflation.

GNPt=U.S. gross national product in each year, adjusted for inflation

Since the regression is run on the natural logs of the variables, the estimated coefficients represent demand elasticities. Thus, EP= –1.247 and EI = 1.905. The very high t values indicate that both estimated coefficients (elasticities) are statistically significant at better than the 1 percent level. Air fares and GNP “explain” 97 percent of the variation in the. number of passengers flying between New York and London. Since both estimated slope coefficients have the correct sign and are statistically significant, there seems to be no multicollinearity problem. The high value of the Durbin-Watson statistic D = W)indicates that there is no autocorrelation problem either.

EXAMPLE 9.Suppose that an airline estimated that 1979 air fares (adjusted for inflation) between New York and London (i.e., Pt + 1)would be $550 and real GNP (i.e., GNPt + 1) would be $1,480. The natural log of 550 (i.e., In 550) is 6.310 and ln 1480 is 7.300. Substituting these values into the regression we get

ln Qt+1 = 2.737 – 1.247 (6.310) + 1.905(7.300) = 8.775

The antilog of 8.775 is 6,470, i.e., a forecast of 6,470,000 passengers for 1979. The accuracy of this forecast depends on the accuracy of the estimated demand coefficients and of the estimated or forecasted values of the independent or explanatory variables in the demand equation.

Glossary

Barometric forecastingThe method of forecasting turning points in business cycles by the use of leading economic indicators.

Coincident indicatorsTime series that move in step, or coincide, with movements in the level of general economic activity.