Instantaneous versus Compound Rate of Growth. The coefficient of the trend variable in the growth model (6.6.6), h2, gives the instantaneous (at a point in time) rate of growth and not the compound (over a period of time) rate of growth. But the latter can be easily found from (6.6.4) by taking the antilog of the estimated h2 and subtracting 1 from it and multiplying the difference by 100. Thus, for our illustrative example, the estimated slope coefficient is 0.00743. Therefore, [antilog(0.00743) — 1] = 0.00746 or 0.746 percent. Thus, in the illustrative example, the compound rate of growth on expenditure on services was about 0.746 percent per quarter, which is slightly higher than the instantaneous growth rate of 0.743 percent. This is of course due to the compounding effect.
Linear Trend Model. Instead of estimating model (6.6.6), researchers sometimes estimate the following model:
That is, instead of regressing the log of Y on time, they regress Y on time, where Y is the regressand under consideration. Such a model is called a linear trend model and the time variable t is known as the trend variable. If the slope coefficient in (6.6.9) is positive, there is an upward trend in Y, whereas if it is negative, there is a downward trend in Y.
CHAPTER SIX: EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL 181
For the expenditure on services data that we considered earlier, the results of fitting the linear trend model (6.6.9) are as follows:
In contrast to Eq. (6.6.8), the interpretation of Eq. (6.6.10) is as follows: Over the quarterly period 1993-I to 1998-III, on average, expenditure on services increased at the absolute (note: not relative) rate of about 20 billion dollars per quarter. That is, there was an upward trend in the expenditure on services.
The choice between the growth rate model (6.6.8) and the linear trend model (6.6.10) will depend upon whether one is interested in the relative or absolute change in the expenditure on services, although for comparative purposes it is the relative change that is generally more relevant. In passing, observe that we cannot compare the r2 values of models (6.6.8) and (6.6.10) because the regressands in the two models are different. We will show in Chapter 7 how one compares the R2's of models like (6.6.8) and (6.6.10).
Unlike the growth model just discussed, in which we were interested in finding the percent growth in Y for an absolute change in X, suppose we now want to find the absolute change in Y for a percent change in X. A model that can accomplish this purpose can be written as:
For descriptive purposes we call such a model a lin-log model. Let us interpret the slope coefficient fa2.15 As usual, change in Y 2 change in ln X
change in Y relative change in X
The second step follows from the fact that a change in the log of a number is a relative change.
15Again, using differential calculus, we have
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