Tobit model

The Tobit model is a statistical model proposed by James Tobin (1958) to describe the relationship between a non-negative dependent variable ${\displaystyle y_{i}}$ and an independent variable (or vector) ${\displaystyle x_{i}}$. The term Tobit was derived from Tobin's name by truncating and adding -it by analogy with the probit model. The Tobit model shall not be confused with the truncated regression model, which is in general different and requires a different estimator.

The model supposes that there is a latent (i.e. unobservable) variable ${\displaystyle y_{i}^{*}}$. This variable linearly depends on ${\displaystyle x_{i}}$ via a parameter (vector) ${\displaystyle \beta }$ which determines the relationship between the independent variable (or vector) ${\displaystyle x_{i}}$ and the latent variable ${\displaystyle y_{i}^{*}}$ (just as in a linear model). In addition, there is a normally distributed error term ${\displaystyle u_{i}}$ to capture random influences on this relationship. The observable variable ${\displaystyle y_{i}}$ is defined to be equal to the latent variable whenever the latent variable is above zero and zero otherwise.

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