Tobit model

The Tobit model refers to 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. The Tobit model refers to 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 density, i.e. the integral below zero of the appropriate density function. The Tobit likelihood function thus is a mixture of densities and cumulative densities. Below are the likelihood and log likelihood functions for a type I Tobit. This is a Tobit that is censored from below at y L {displaystyle y_{L}} when the latent variable y j ∗ ≤ y L {displaystyle y_{j}^{*}leq y_{L}} . In writing out the likelihood function, we first define an indicator function I ( y j ) {displaystyle I(y_{j})} where: Next, let Φ {displaystyle Phi } be the standard normal cumulative distribution function and φ {displaystyle varphi } to be the standard normal probability density function. For a data set with N observations the likelihood function for a type I Tobit is