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Following James Tobin’s 1958 article “Estimation of Relationships for Limited Dependent Variables,” the tobit is a statistical model that is used to estimate the relationship between a limited dependent variable (y) and a vector of explanatory variables ( x), usually by the method of maximum likelihood. The tobit model is warranted when the variable y is censored (i.e., when it is observed for some values above or below a certain threshold, but not in the remainder of the data). The term “tobit” was derived from Tobin’s name and by adding the suffix “it,” as for the 1964 probit model by Arthur Goldberger.

For example, a tobit model can be used to estimate the relationship between the number of hours worked ( y ) and education, age, gender, race, number of children, and so on ( x). In this case, some individuals do not work at all (y = 0) while some work (y > 0), but education, age, gender, race, and the like are observed for all individuals, so that the data is censored. The advantage of using the tobit instead of the usual linear regression model is that it yields unbiased coefficient estimates for each of the variables in x. Note that the censoring need not occur at zero, nor does it need to occur below a specific value of y; the tobit also accommodates censoring above a specific value of y, and in a two-limit tobit, the dependent variable is censored both below and above a certain range of y.

Critiques Of The Tobit Model

The most important critique of the tobit model is that it does not allow for the set of variables used in explaining whether y is positive or zero (say, x1) to differ from the set of variables used in explaining the value of y conditional on y being strictly positive (say, x2). James Heckman introduced a model in his 1979 article “Sample Selection Bias as a Specification Error,” which allows such a specification as well as control for selection bias in applications. Heckman’s model (or “heckit”), however, requires the use of an instrumental variable that can be excluded on theoretical grounds from the estimated equation for the values of y that are strictly positive in order to explain selection into the noncensored sample. Additionally, the tobit model is inconsistent when the error term is not normally distributed (this critique also applies to the heckit model), or when the variance of the error term is not constant, (i.e., it has unequal variances).

In a thorough survey of the literature on the tobit, Takeshi Amemiya (1984) discussed the properties of the model and provided many examples and applications of the tobit. He also outlined various estimation techniques as well as possible departures from the usual statistical assumptions and discussed generalizations of—and extensions to—the basic tobit model. G. S. Maddala suggested in his 2001 book Introduction to Econometrics exercising caution when considering the use of the tobit model (i.e., researchers who wish to use the tobit model should make sure that the dependent variable is, indeed, censored).

Bibliography:

Amemiya, Takeshi. 1984. Tobit Models: A Survey. Journal of Econometrics 24 (1–2): 3-61.
Goldberger, Arthur. 1964. Econometric Theory. New York: J. Wiley.
Heckman, James. 1979. Sample Selection Bias as a Specification Error. Econometrica 47 (1): 153–161.
Maddala, G. S. 2001. Introduction to Econometrics. 3rd ed. New York: John Wiley.
Tobin, James. 1958. Estimation of Relationships for Limited Dependent Variables. Econometrica 26 (1): 24–36.