*Note: The nominal price was divided by the Consumer Price Index (CPI) for food and beverages, 1967 = 100.

Source: The data for Y are from Summary of National Coffee Drinking Study, Data Group, Elkins Park, Penn., 1981; and the data on nominal X (i.e., Xin current prices) are from Nielsen Food Index, A. C. Nielsen, New York, 1981.

I am indebted to Scott E. Sandberg for collecting the data.

*Note: The nominal price was divided by the Consumer Price Index (CPI) for food and beverages, 1967 = 100.

Source: The data for Y are from Summary of National Coffee Drinking Study, Data Group, Elkins Park, Penn., 1981; and the data on nominal X (i.e., Xin current prices) are from Nielsen Food Index, A. C. Nielsen, New York, 1981.

I am indebted to Scott E. Sandberg for collecting the data.

(Continued)

CHAPTER SEVEN: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF ESTIMATION 221

EXAMPLE 7.2 (Continued)

linear model?) The r2 value of about 0.74 means that about 74 percent of the variation in the log of coffee demand is explained by the variation in the log of coffee price.

Since the r2 value of the linear model of 0.6628 is smaller than the r2 value of 0.7448 of the log-linear model, you might be tempted to choose the latter model because of its high r2 value. But for reasons already noted, we cannot do so. But if you do want to compare the two r2 values, you may proceed as follows:

1. Obtain ln Ytfrom (7.8.9) for each observation; that is, obtain the estimated log value of each observation from this model. Take the antilog of these values and then compute r2 between these antilog values and actual Yt in the manner indicated by Eq. (3.5.14). This r2 value is comparable to the r2 value of the linear model (7.8.8).

2. Alternatively, assuming all Y values are positive, take logarithms of the Y values, ln Y. Obtain the estimated Y values, Yt, from the linear model (7.8.8), take the logarithms of these estimated Y values (i.e., ln Yt) and compute the r2 between (ln Yt) and (ln Yt) in the manner indicated in Eq. (3.5.14). This r2 value is comparable to the r2 value obtained from (7.8.9).

For our coffee example, we present the necessary raw data to compute the comparable r2's in Table 7.2. To compare the r2 value of the linear model (7.8.8) with that of (7.8.9), we first obtain log of (Yt) [given in column (6) of Table 7.2], then we obtain the log of actual Y values [given in column (5) of the table], and then compute r2 between these two sets of values using Eq. (3.5.14). The result is an r2 value of 0.7318, which is now comparable with the r2 value of the log-linear model of 0.7448. Now the difference between the two r2 values is very small.

On the other hand, if we want to compare the r2 value of the log-linear model with the linear model, we obtain ln Yt for each observation from (7.8.9) [given in column (3) of the table], obtain their antilog values [given in column (4) of the table], and finally compute r2 between these antilog values and the actual Y values, using formula (3.5.14). This will give an r2 value of 0.7187, which is slightly higher than that obtained from the linear model (7.8.8), namely, 0.6628.

Using either method, it seems that the log-linear model gives a slightly better fit.

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