9The elasticity coefficient, in calculus notation, is defined as (dY/Y)/(dX/ X) = [(dY/dX)(X/Y)]. Readers familiar with differential calculus will readily see that f>2 is in fact the elasticity coefficient.
A technical note: The calculus-minded reader will note that d(ln X)/dX = 1/X or d(ln X) = dX/X, that is, for infinitesimally small changes (note the differential operator d) the change in ln X is equal to the relative or proportional change in X. In practice, though, if the change in X is small, this relationship can be written as: change in ln X = relative change in X, where = means approximately. Thus, for small changes,
(lnXt — lnXt-i) = (Xt — Xt-i)/Xt-i = relative change inX
Incidentally, the reader should note these terms, which will occur frequently: (1) absolute change, (2) relative or proportional change, and (3) percentage change, or percent growth rate. Thus, (Xt — Xt—1) represents absolute change, (Xt — Xt—i)/Xt—i = (Xt/Xt—i — 1) is relative or proportional change and [(Xt — Xt—i)/Xt—i]i00 is the percentage change, or the growth rate. Xt and Xt—i are, respectively, the current and previous values of the variable X.
CHAPTER SIX: EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL 177
transformation as shown in Figure 6.3b will then give the estimate of the price elasticity (—f2).
Two special features of the log-linear model may be noted: The model assumes that the elasticity coefficient between Y and X, f2, remains constant throughout (why?), hence the alternative name constant elasticity model.10 In other words, as Figure 6.3b shows, the change in ln Y per unit change in ln X (i.e., the elasticity, f2) remains the same no matter at which ln X we measure the elasticity. Another feature of the model is that although a and f2 are unbiased estimates of a and f2, f 1 (the parameter entering the original model) when estimated as f = antilog (a) is itself a biased estimator. In most practical problems, however, the intercept term is of secondary importance, and one need not worry about obtaining its unbiased estimate.11
In the two-variable model, the simplest way to decide whether the loglinear model fits the data is to plot the scattergram of ln Yi against ln Xi and see if the scatter points lie approximately on a straight line, as in Figure 6.3b.
AN ILLUSTRATIVE EXAMPLE: EXPENDITURE ON DURABLE GOODS IN RELATION TO TOTAL PERSONAL CONSUMPTION EXPENDITURE
Table 6.3 presents data on total personal consumption expenditure (PCEXP), expenditure on durable goods (EXPDUR), expenditure on nondurable goods (EXPNONDUR), and expenditure on services (EXPSERVICES), all measured in 1992 billions of dollars.12
Suppose we wish to find the elasticity of expenditure on durable goods with respect to total personal consumption expenditure. Plotting the log of expenditure on durable goods against the log of total personal consumption expenditure, you will see that the relationship between the two variables is linear. Hence, the doublelog model may be appropriate. The regression results are as follows:
¡rTEXDURt = -9.6971 + 1.9056 ln PCEXt se = (0.4341) (0.0514) (6.5.5)
where * indicates that the p value is extremely small.
As these results show, the elasticity of EXPDUR with respect to PCEX is about 1.90, suggesting that if total personal expenditure goes up by 1 percent, on average, the expenditure on durable goods goes up by about 1.90 percent. Thus, expenditure on durable goods is very responsive to changes in personal consumption expenditure. This is one reason why producers of durable goods keep a keen eye on changes in personal income and personal consumption expenditure. In exercises 6.17 and 6.18, the reader is asked to carry out a similar exercise for nondurable goods expenditure and expenditure on services.
10A constant elasticity model will give a constant total revenue change for a given percentage change in price regardless of the absolute level of price. Readers should contrast this result with the elasticity conditions implied by a simple linear demand function, Yi = f 1 + f 2 Xi + u. However, a simple linear function gives a constant quantity change per unit change in price. Contrast this with what the log-linear model implies for a given dollar change in price.
"Concerning the nature of the bias and what can be done about it, see Arthur S. Goldberger, Topics in Regression Analysis, Macmillan, New York, 1978, p. 120.
12Durable goods include motor vehicles and parts, furniture, and household equipment; nondurable goods include food, clothing, gasoline and oil, fuel oil and coal; and services include housing, electricity and gas, transportation, and medical care.
178 PART ONE: SINGLE-EQUATION REGRESSION MODELS
AN ILLUSTRATIVE EXAMPLE: . . . (Continued) TABLE 6.3
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