As we have seen, the LPM is plagued by several problems, such as (1) non-normality of ui, (2) heteroscedasticity of ui, (3) possibility of Y lying outside the 0-1 range, and (4) the generally lower R2 values. But these problems are surmountable. For example, we can use WLS to resolve the heteroscedasticity problem or increase the sample size to minimize the non-normality problem. By resorting to restricted least-squares or mathematical programming techniques we can even make the estimated probabilities lie in the 0-1 interval.
But even then the fundamental problem with the LPM is that it is not logically a very attractive model because it assumes that Pi = E(Y = 1 | X) increases linearly with X, that is, the marginal or incremental effect of X remains constant throughout. Thus, in our home ownership example we found that as X increases by a unit ($1000), the probability of owning a house increases by the same constant amount of 0.10. This is so whether the income level is $8000, $10,000, $18,000, or $22,000. This seems patently unrealistic. In reality one would expect that Pi is nonlinearly related to Xi:
12D. Rubinfeld, "An Econometric Analysis of the Market for General Municipal Bonds,'' unpublished doctoral dissertation, Massachusetts Institute of Technology, 1972. The results given in this example are reproduced from Robert S. Pindyck and Daniel L. Rubinfeld, Econometric Models and Economic Forecasts, 2d ed., McGraw-Hill, New York, 1981, p. 279.
594 PART THREE: TOPICS IN ECONOMETRICS
At very low income a family will not own a house but at a sufficiently high level of income, say, X* it most likely will own a house. Any increase in income beyond X will have little effect on the probability of owning a house. Thus, at both ends of the income distribution, the probability of owning a house will be virtually unaffected by a small increase in X.
Therefore, what we need is a (probability) model that has these two features: (1) As Xi increases, Pi = E(Y = 1 | X) increases but never steps outside the 0-1 interval, and (2) the relationship between Pi and Xi is nonlinear, that is, "one which approaches zero at slower and slower rates as Xi gets small and approaches one at slower and slower rates as Xi gets very large.''13
Geometrically, the model we want would look something like Figure 15.2. Notice in this model that the probability lies between 0 and 1 and that it varies nonlinearly with X.
The reader will realize that the sigmoid, or S-shaped, curve in the figure very much resembles the cumulative distribution function (CDF) of a random variable.14 Therefore, one can easily use the CDF to model regressions where the response variable is dichotomous, taking 0-1 values. The practical question now is, which CDF? For although all CDFs are S shaped, for each random variable there is a unique CDF. For historical as well as practical reasons, the CDFs commonly chosen to represent the 0-1 response
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