182 PART ONE: SINGLE-EQUATION REGRESSION MODELS
Symbolically, we have
where, as usual, A denotes a small change. Equation (6.6.12) can be written, equivalently, as
This equation states that the absolute change in Y ( = AY) is equal to slope times the relative change inX. If the latter is multiplied by 100, then (6.6.13) gives the absolute change in Y for a percentage change in X. Thus, if (AX/X) changes by 0.01 unit (or 1 percent), the absolute change in Y is 0.01(p2); if in an application one finds that p2 = 500, the absolute change in Y is (0.01)(500) = 5.0. Therefore, when regression (6.6.11) is estimated by OLS, do not forget to multiply the value of the estimated slope coefficient by 0.01, or, what amounts to the same thing, divide it by 100. If you do not keep this in mind, your interpretation in an application will be highly misleading.
The practical question is: When is a lin-log model like (6.6.11) useful? An interesting application has been found in the so-called Engel expenditure models, named after the German statistician Ernst Engel, 1821-1896. (See exercise 6.10.) Engel postulated that "the total expenditure that is devoted to food tends to increase in arithmetic progression as total expenditure increases in geometric progression."16
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