Refer to exercise 7.16 where we have presented data on the demand for roses in the Detroit metropolitan area for the period 1971 —II to 1975-II. For illustrative purposes, we will consider the demand for roses as a function only of the prices of roses and carnations, leaving out the income variable for the time being. Now we consider the following models:
where Y is the quantity of roses in dozens, X2 is the average wholesale price of roses ($/dozen), and X3 is the average wholesale price of carnations ($/dozen). A priori, a2 and 02 are expected to be negative (why?), and a3 and 03 are expected to be positive (why?). As we know, the slope coefficients in the log-linear model are elasticity coefficients.
The regression results are as follows:
Log-linear model: ln Yt = ß1 + ß2 lnX2t + ß3 lnX3t + ut
Yt = 9734.2176 - 3782.1956X2t + 2815.2515X3t t = (3.3705) (-6.6069) (2.9712)
ln Yt = 9.2278 - 1.7607 lnX2t + 1.3398 lnX3t t= (16.2349) (-5.9044) (2.5407)
21This discussion is based on William H. Greene, ET. The Econometrics Toolkit Version 3, Econometric Software, Bellport, New York, 1992, pp. 245-246.
282 PART ONE: SINGLE-EQUATION REGRESSION MODELS
EXAMPLE 8.5 (Continued)
As these results show, both the linear and the log-linear models seem to fit the data reasonably well: The parameters have the expected signs and the t and R2 values are statistically significant.
To decide between these models on the basis of the MWD test, we first test the hypothesis that the true model is linear. Then, following Step IV of the test, we obtain the following regression:
Yt = 9727.5685 - 3783.0623X2t + 2817.7157X3t + 85.2319Z1t t = (3.2178) (-6.3337) (2.8366) (0.0207) (8.11.5)
Since the coefficient of Z1 is not statistically significant (the pvalue of the estimated tis 0.98), we do not reject the hypothesis that the true model is linear.
Suppose we switch gears and assume that the true model is log-linear. Following step VI of the MWD test, we obtain the following regression results:
\nYt = 9.1486 - 1.9699ln Xt + 1.5891 ln X2t - 0.0013Z2t t = (17.0825) (-6.4189) (3.0728) (-1.6612) (8.11.6)
The coefficient of Z2 is statistically significant at about the 12 percent level (p value is 0.1225). Therefore, we can reject the hypothesis that the true model is log-linear at this level of significance. Of course, if one sticks to the conventional 1 or 5 percent significance levels, then one cannot reject the hypothesis that the true model is log-linear. As this example shows, it is quite possible that in a given situation we cannot reject either of the specifications.
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