9781118041581

In 1977, Company A’s common stock sold for $50 per share, the same price as a share

of Company B’s stock. Over the next 35 years, the value of A’s stock increased at an

average rate of 5 percent per year; the value of B’s stock increased by 6 percent per

year on average. Find the 2012 price for each company’s stock. Comment on your

findings.

156 Chapter 4 Estimating and Forecasting Demand

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STATION 5

Forecasting

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In many economic settings, the value of a variable today influences the value

of the same variable tomorrow. Increased sales in one month frequently mean

increased sales in the following month. An elevated rate of inflation in the cur-

rent quarter is likely to spell higher rates in succeeding quarters. To consider

the simplest case, suppose that a firm’s sales in the current period depend on

its sales in the previous period according to

where a and b are coefficients. We can estimate this equation by OLS regres-

sion using last period’s sales (or sales “lagged” one period) as the explanatory

variable. For instance, if the constant term a is positive and b is greater than 1,

then sales grow (more than proportionally) over time. Alternatively, if the coef-

ficient b is smaller than one, sales grow at a decreasing rate.

QtabQt 1 ,

exponential equations give closely similar results. Both curves have about the

same shape and fit the data equally well. (Thus, we have not provided a sepa-

rate graph of the exponential curve.)

When it comes to forecasting, the significant difference is between the lin-

ear and nonlinear specifications. For example, using the linear equation, we

forecast sales for the next year (year 13) to be

The forecasts for quadratic and exponential equations are slightly higher,

213.1 and 212.4, respectively. The gap between the predictions widens as we

forecast further and further into the future. The linear equation predicts

constant additions to sales year after year; the nonlinear equations predict

steeper and steeper sales increases over time. Therefore, the respective fore-

casts for year 16 are 235.8, 244.5, and 250.1; for year 20, they are 270.2,

289.8, and 311.0. Note that, as the time horizon lengthens, the exponential

predictions exceed the quadratic predictions by a greater and greater mar-

gin. As time goes by, one can compare these predictions to actual sales expe-

rience to judge which equation produces the more accurate forecasts on

average.

Q 13 98.2(8.6)(13)210.0.

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