In statistics, a tobit model is any of a class of regression models in which the observed range of the dependent variable is censored in some way. The term was coined by Arthur Goldberger in reference to James Tobin, who developed the model in 1958 to mitigate the problem of zero-inflated data for observations of household expenditure on durable goods. Because Tobin's method can be easily extended to handle truncated and other non-randomly selected samples, some authors adopt a broader definition of the tobit model that includes these cases. 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. 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 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. The tobit likelihood function thus is a mixture of densities and cumulative distribution functions.
The log-likelihood is stated above is not globally concave, which complicates the maximum likelihood estimation. Olsen suggested the simple reparametrization and, resulting in a transformed log-likelihood, which is globally concave in terms of the transformed parameters. For the truncated model, Orme showed that while the log-likelihood is not globally concave, it is concave at any stationary point under the above transformation.
Consistency
If the relationship parameter is estimated by regressing the observed on, the resulting ordinary least squares regression estimator is inconsistent. It will yield a downwards-biased estimate of the slope coefficient and an upward-biased estimate of the intercept. Takeshi Amemiya has proven that the maximum likelihood estimator suggested by Tobin for this model is consistent.
Interpretation
The coefficient should not be interpreted as the effect of on, as one would with a linear regression model; this is a common error. Instead, it should be interpreted as the combination of the change in of those above the limit, weighted by the probability of being above the limit; and the change in the probability of being above the limit, weighted by the expected value of if above.
Variations of the tobit model
Variations of the tobit model can be produced by changing where and when censoring occurs. classifies these variations into five categories, where tobit type I stands for the first model described above. Schnedler provides a general formula to obtain consistent likelihood estimators for these and other variations of the tobit model.
The tobit model is a special case of a censored regression model, because the latent variable cannot always be observed while the independent variable is observable. A common variation of the tobit model is censoring at a value different from zero: Another example is censoring of values above. Yet another model results when is censored from above and below at the same time. 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. 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 and the outcome of interest to be independent, conditional on observable data. The Heckman selection model falls into the Type II tobit, which is sometimes called Heckit after James Heckman.
Type III introduces a second observed dependent variable. The Heckman model falls into this type.
Type IV
Type IV introduces a third observed dependent variable and a third latent variable.
Type V
Similar to Type II, in Type V only the sign of is observed.
Non-parametric version
If the underlying latent 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.
Dynamic unobserved effects tobit model
In a panel data tobit model, if the outcome partially depends on the previous outcome history this tobit model is called "dynamic". For instance, taking a person who finds a job with a high salary this year, it will be easier for her to find a job with a high salary next year because the fact that she has a high-wage job this year will be a very positive signal for the potential employers. The essence of this type of dynamic effect is the state dependence of the outcome. The "unobservable effects" here refers to the factor which partially determines the outcome of individual but cannot be observed in the data. For instance, the ability of a person is very important in job-hunting, but it is not observable for researchers. A typical dynamic unobserved effects tobit model can be represented as In this specific model, is the dynamic effect part and is the unobserved effect part whose distribution is determined by the initial outcome of individual i and some exogenous features of individual i. Based on this setup, the likelihood function conditional on can be given as For the initial values ,there are two different ways to treat them in the construction of the likelihood function: treating them as constant, or imposing a distribution on them and calculate out the unconditional likelihood function. But whichever way is chosen to treat the initial values in the likelihood function, we cannot get rid of the integration inside the likelihood function when estimating the model by maximum likelihood estimation. Expectation Maximum algorithm is usually a good solution for this computation issue. Based on the consistent point estimates from MLE, Average Partial Effect can be calculated correspondingly.
Applications
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 analyse a sample of 115 municipalities. Dubois and Fattore use a tobit model to investigate the role of various factors in European Union fund receipt by applying Polish sub-national governments. 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.
