An extension of the probit model is the tobit model originally developed by James Tobin, the Nobel laureate economist. To explain this model, we continue with our home ownership example. In the probit model our concern was with estimating the probability of owning a house as a function of some socioeconomic variables. In the tobit model our interest is in finding out the amount of money a person or family spends on a house in relation to socioeconomic variables. Now we face a dilemma here: If a consumer does not purchase a house, obviously we have no data on housing expenditure for such consumers; we have such data only on consumers who actually purchase a house.
Thus consumers are divided into two groups, one consisting of, say, w1 consumers about whom we have information on the regressors (say, income, mortgage interest rate, number of people in the family, etc.) as well as the regressand (amount of expenditure on housing) and another consisting of n2 consumers about whom we have information only on the regres-sors but not on the regressand. A sample in which information on the regressand is available only for some observations is known as a censored sample.35 Therefore, the tobit model is also known as a censored regression model. Some authors call such models limited dependent variable regression models because of the restriction put on the values taken by the regressand.
Statistically, we can express the tobit model as where RHS = right-hand side. Note: Additional X variables can be easily added to the model.
Can we estimate regression (15.11.1) using only n1 observations and not worry about the remaining n2 observations? The answer is no, for the OLS estimates of the parameters obtained from the subset of n1 observations will be biased as well as inconsistent; that is, they are biased even asymptotically.36 To see this, consider Figure 15.7. As the figure shows, if Yis not observed (because of censoring), all such observations (= n2), denoted by crosses, will
35A censored sample should be distinguished from a truncated sample in which information on the regressors is available only if the regressand is observed. We will not pursue this topic here, but the interested reader may consult William H. Greene, Econometric Analysis, Prentice Hall, 4th ed., Englewood Cliffs, N.J., Chap. 19. For an intuitive discussion, see Peter Kennedy, A Guide to Econometrics, The MIT Press, Cambridge, Mass., 4th ed., 1998, Chap. 16.
36The bias arises from the fact that if we consider only the n1 observations and omit the others, there is no guarantee that E(ui) will be necessarily zero. And without E(ui) = 0 we cannot guarantee that the OLS estimates will be unbiased. This bias can be readily seen from the discussion in App. 3A, Eqs. (4) and (5).
CHAPTER FIFTEEN: QUALITATIVE RESPONSE REGRESSION MODELS 617
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