Note: EXPSERVICES = expenditure on services, billions of 1992 dollars.
EXPDUR = expenditure on durable goods, billions of 1992 dollars. EXPNONDUR = expenditure on nondurable goods, billions of 1992 dollars.
PCEXP = total personal consumption expenditure, billions of 1992 dollars. Source: Economic Report of the President, 1999, Table B-17, p. 347.
6.6 SEMILOG MODELS: LOG-LIN AND LIN-LOG MODELS
How to Measure the Growth Rate: The Log-Lin Model
Economists, businesspeople, and governments are often interested in finding out the rate of growth of certain economic variables, such as population, GNP, money supply, employment, productivity, and trade deficit.
Suppose we want to find out the growth rate of personal consumption expenditure on services for the data given in Table 6.3. Let Yt denote real expenditure on services at time t and Y0 the initial value of the expenditure on services (i.e., the value at the end of 1992-IV). You may recall the following well-known compound interest formula from your introductory course in economics.
CHAPTER SIX: EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL 179
where r is the compound (i.e., over time) rate of growth of Y. Taking the natural logarithm of (6.6.1), we can write ln Yt = ln Yo + t ln(1 + r) (6.6.2)
Adding the disturbance term to (6.6.5), we obtain13
This model is like any other linear regression model in that the parameters fy1 and fy2 are linear. The only difference is that the regressand is the logarithm of Y and the regressor is "time," which will take values of 1, 2, 3, etc.
Models like (6.6.6) are called semilog models because only one variable (in this case the regressand) appears in the logarithmic form. For descriptive purposes a model in which the regressand is logarithmic will be called a log-lin model. Later we will consider a model in which the regressand is linear but the regressor(s) are logarithmic and call it a lin-log model.
Before we present the regression results, let us examine the properties of model (6.6.5). In this model the slope coefficient measures the constant proportional or relative change in Y for a given absolute change in the value of the regressor (in this case the variable t), that is,14
relative change in regressand
absolute change in regressor
If we multiply the relative change in Y by 100, (6.6.7) will then give the percentage change, or the growth rate, in Y for an absolute change in X, the regressor. That is, 100 times fy2 gives the growth rate in Y; 100 times fy2 is
13We add the error term because the compound interest formula will not hold exactly. Why we add the error after the logarithmic transformation is explained in Sec. 6.8.
14Using differential calculus one can show that fy2 = d(ln Y)/dX = (1/Y)(dY/dX) = (dY/Y)/dX, which is nothing but (6.6.7). For small changes in Y and X this relation may be approximated by
180 PART ONE: SINGLE-EQUATION REGRESSION MODELS
known in the literature as the semielasticity of Y with respect to X. (Question: To get the elasticity, what will we have to do?)
AN ILLUSTRATIVE EXAMPLE: THE RATE OF GROWTH EXPENDITURE ON SERVICES
To illustrate the growth model (6.6.6), consider the data on expenditure on services given in Table 6.3. The regression results are as follows:
InExS, = 7.7890 + 0.00743t se = (0.0023) (0.00017) (6.6.8)
Note: EXS stands for expenditure on services and * denotes that the p value is extremely small.
The interpretation of Eq. (6.6.8) is that over the quarterly period 1993:1 to 1998:3, expenditure on services increased at the (quarterly) rate of 0.743 percent. Roughly, this is equal to an annual growth rate of 2.97 percent. Since 7.7890 = log of EXS at the beginning of the study period, by taking its antilog we obtain 2413.90 (billion dollars) as the beginning value of EXS (i.e., the value at the end of the fourth quarter of 1992). The regression line obtained in Eq. (6.6.8) is sketched in Figure 6.4.
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