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117

) = j[jit)dt

Using this notation we can write the likelihood function for the tobit model as

v,>0 CT \ CT / v,sO V O-

Maximizing this likelihood function with respect to p and ct, we get the maximum likelihood (ML) estimates of these parameters. We will not go through the algebraic details of the ML method here. Instead, we discuss the situations under which the tobit model is applicable and its relationship to other models with truncated variables.

Limitations of the Tobit Model

Consider the models of automobile expenditures in (8.20), of hours worked in (8.21), and of wages in (8.22). In each case there are zero observations on some individuals in the sample and thus the structure of the model looks very similar to that in (8.19). But is it really? Every time we have some zero observations in the sample, it is tempting to use the tobit model. However, it is important to understand what the model in (8.19) really says. What we have in model (8.19) is a situation where yj" can, in principle, take on negative values. However, we do not observe them because of censoring. Thus the zero values are due to nonobservability. This is not the case with automobile expenditures, hours worked, or wages. These variables cannot, in principle, assume negative values. The observed zero values are due not to censoring, but due to the decisions of individuals. In this case the appropriate procedure would be to model the decisions that produce the zero observations rather than use the tobit model mechanically.

Consider, for instance, the model of wages in (8.22). We can argue that each person has a reservation wage W, below which the person would not want to work. If W2 is the market wage for this person (i.e., the wage that employers are willing to pay) and > W, then we will observe the person as working and the observed wage W is equal to W2. On the other hand, if > W2, we observe the person as not working and the observed wage is zero.

If this is the story behind the observed zero wages (one can construct other similar stories), we can formulate the model as follows. Let the reservation wages and market wages W2, be given by

Wi, = Pi-i, + «1/

2, = / + «2/

For details see Maddala. Limited-Dependent, Chap. 6. A convenient computer program is the LIMDEP program by William Greene at New York University. This program can also be used to estimate the other models discussed later.



The observed W, is given by We can write this as

Wi, if Wi, 2 Wi, 0 otherwise

= I 2X2, + "2, if "2, - «I, - - , (

[ otherwise

Note the difference between this formulation and the one in equation (8.19). The criterion that W, = 0 is not given by M2, - 2X2, as in the simple tobit model but by Mj, - ,, < PiX,, - fiiXb- Hence estimation of a simple tobit model in this case produces inconsistent estimates of the parameters.

Estimation of the model given by (8.23) is somewhat complicated to be discussed here. However, the purpose of the example is to show that every time we have some zero observations, we should not use the tobit model. In fact, we can construct similar models for automobile expenditures and hours worked wherein the zero observations are a consequence of decisions by individuals. The simple censored regression model (or the tobit model) is applicable only in those cases where the latent variable can, in principle, take negative values and the observed zero values are a consequence of censoring and nonobservability.

The Truncated Regression Model

The term truncated regression model refers to a situation where samples are drawn from a truncated distribution. It is important to keep in mind the distinction between this model and the tobit model. In the censored regression model (tobit model) we have observations on the explanatory variable x, for all individuals. It is only the dependent variable y* that is missing for some individuals. In the truncated regression model, we have no data on either y* or x, for some individuals because no samples are drawn if yJ is below or above a certain level. The New Jersey negative-income-tax experiment was an example in which high-income families were not sampled at all. Families were selected at random but those with incomes higher than 1.5 times the 1967 poverty line were eliminated from the study. If we are estimating earnings functions from these data, we cannot use the OLS method. We have to take account of the fact that the sample is from a truncated distribution. Figure 8.5 illustrates this point. The observations above = F are omitted from the sample. If we use OLS, the estimated line gives a biased estimate of the true slope. A method of estimation often suggested is, again, the maximum likelihood method.* Suppose that the untruncated regression model is

y, = px, + u, ,- IN(0, u)

1 IS discussed in Maddala, Limited-Dependent, Sec. 6.11. "Maddala, Limited-Dependent, Chap. 6.



Trae line

-.-Estimated line

Figure 8.5. Truncated regression model.

Now, only observations with y* < L are included in the sample. The total area under the normal curve up to y, < L, that is, uja < (Z. - ,)/ is F[iL -Px,)/ct], where F(-) is the cumulative distribution function of the standard normal. The density function of the observed y, is the standard normal density except that its total area is F[{L - ,)/ ]. Since the total area under a probability density function should be equal to 1, we have to normalize by this factor. Thus the density function of the observations y, from the truncated normal is

1/y, - px,

(T \ CT

L - Px,

ify,<L otherwise

The log-likelihood is, therefore,

logL = -« log CT - 2 (y, - P,) - 2 log F

/ * V CT

Maximizing this with respect to p and ct, we get the ML estimates of p and . Again, we need not be concerned with the details of the ML estimation method. As an illustration of how different the OLS and ML estimates can be, we present the estimates obtained by Hausman and Wise" in Table 8.6. The dependent variable was log earnings. The results show how the OLS estimates are biased. In this particular case they are all biased toward zero.

"J. A. Hausman and D. A. Wise, "Social Experimentation, Truncated Distributions, and Efficient Estimation," Econometnca. Vol. 45, 1977, p. 319-339.



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