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Obviously, these expressions are meaningless. Therefore, if we have data at the micro, or individual, level, we cannot estimate (15.6.1) by the standard

20This point is due to David Garson.

598 PART THREE: TOPICS IN ECONOMETRICS

OLS routine. In this situation we may have to resort to the maximum-likelihood (ML) method to estimate the parameters. Although the rudiments of this method were discussed in the appendix to Chapter 4, its application in the present context will be discussed in Appendix 15A, Section 15A.1, for the benefit of readers who would like to learn more about it.21 Software packages, such as Microfit, Eviews, Limdep, Shazam, PcGive, and Minitab, have built-in routines to estimate the logit model at the individual level. We will illustrate the use of the ML method later in the chapter.

### Grouped or Replicated Data

Now consider the data given in Table 15.4. This table gives data on several families grouped or replicated (repeat observations) according to income level and the number of families owning a house at each income level. Corresponding to each income level Xi, there are Ni families, ni among whom are home owners (ni < Ni). Therefore, if we compute n

that is, the relative frequency, we can use it as an estimate of the true Pi corresponding to each Xi. If Ni is fairly large, Pi will be a reasonably good estimate of Pi .22 Using the estimated Pi, we can obtain the estimated logit as

TABLE 15.4 HYPOTHETICAL DATAON X (INCOME), Ni (NUMBER OF FAMILIES AT INCOME X,), AND rn (NUMBER OF FAMILIES OWNING A HOUSE)

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