Tobit model

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.^[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. The tobit likelihood function is thus a mixture of densities and cumulative distribution functions.^[6]

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