[1] The term was coined by Arthur Goldberger in reference to James Tobin,[2][a] who developed the model in 1958 to mitigate the problem of zero-inflated data for observations of household expenditure on durable goods.
[3][b] Because Tobin's method can be easily extended to handle truncated and other non-randomly selected samples,[c] some authors adopt a broader definition of the tobit model that includes these cases.
[4] Tobin's idea was to modify the likelihood function so that it reflects the unequal sampling probability for each observation depending on whether the latent dependent variable fell above or below the determined threshold.
[5] For a sample that, as in Tobin's original case, was censored from below at zero, the sampling probability for each non-limit observation is simply the height of the appropriate density function.
For any limit observation, it is the cumulative distribution, i.e. the integral below zero of the appropriate density function.
[7] For the truncated (tobit II) model, Orme showed that while the log-likelihood is not globally concave, it is concave at any stationary point under the above transformation.
, the resulting ordinary least squares regression estimator is inconsistent.
Takeshi Amemiya (1973) has proven that the maximum likelihood estimator suggested by Tobin for this model is consistent.
, as one would with a linear regression model; this is a common error.
[11] Variations of the tobit model can be produced by changing where and when censoring occurs.
Schnedler (2005) provides a general formula to obtain consistent likelihood estimators for these and other variations of the tobit model.
A common variation of the tobit model is censoring at a value
The rest of the models will be presented as being bounded from below at 0, though this can be generalized as done for Type I.
Type II tobit models introduce a second latent variable.
[13] In Type I tobit, the latent variable absorbs both the process of participation and the outcome of interest.
Type II tobit allows the process of participation (selection) and the outcome of interest to be independent, conditional on observable data.
[15] Type III introduces a second observed dependent variable.
is not normally distributed, one must use quantiles instead of moments to analyze the observable variable
Powell's CLAD estimator offers a possible way to achieve this.
[16] Tobit models have, for example, been applied to estimate factors that impact grant receipt, including financial transfers distributed to sub-national governments who may apply for these grants.
In these cases, grant recipients cannot receive negative amounts, and the data is thus left-censored.
For instance, Dahlberg and Johansson (2002) analyse a sample of 115 municipalities (42 of which received a grant).
[17] Dubois and Fattore (2011) use a tobit model to investigate the role of various factors in European Union fund receipt by applying Polish sub-national governments.
[18] The data may however be left-censored at a point higher than zero, with the risk of mis-specification.
Both studies apply Probit and other models to check for robustness.
Tobit models have also been applied in demand analysis to accommodate observations with zero expenditures on some goods.
In a related application of tobit models, a system of nonlinear tobit regressions models has been used to jointly estimate a brand demand system with homoscedastic, heteroscedastic and generalized heteroscedastic variants.