First, let us interpret this regression. The intercept of -0.9457 gives the "probability" that a family with zero income will own a house. Since this value is negative, and since probability cannot be negative, we treat this value as zero, which is sensible in the present instance.7 The slope value of 0.1021 means that for a unit change in income (here $1000), on the average the probability of owning a house increases by 0.1021 or about 10 percent. Of course, given a particular level of income, we can estimate the actual probability of owning a house from (15.2.10). Thus, for X = 12 ($12,000), the estimated probability of owning a house is
That is, the probability that a family with an income of $12,000 will own a house is about 28 percent. Table 15.2 shows the estimated probabilities, Y, forthe various income levels listed in the table. The most noticeable feature of this table is that six estimated values are negative and six values are in excess of 1, demonstrating clearly the point made earlier that, although E(Y | X) is positive and less than 1, their estimators, Y, need not be necessarily positive or less than 1. This is one reason that the LPM is not the recommended model when the dependent variable is dichotomous.
Even if the estimated Y were all positive and less than 1, the LPM still suffers from the problem of het-eroscedasticity, which can be seen readily from (15.2.8). As a consequence, we cannot trust the estimated standard errors reported in (15.12.10). (Why?) But we can use the weighted least-squares (WLS) procedure discussed earlier to obtain more efficient estimates of the standard errors. The necessary weights, wt, required for the application of WLS are also shown in Table 15.2. But note that since some Yi are negative and some are in excess of one, the wt corresponding to these values will be negative. Thus, we cannot use these observations in WLS (why?), thereby reducing the number of
7One can loosely interpret the highly negative value as near improbability of owning a house when income is zero.
CHAPTER FIFTEEN: QUALITATIVE RESPONSE REGRESSION MODELS 589
LPM: A NUMERICAL EXAMPLE (Continued)
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