The Tobit Model is an econometric model proposed by James Tobin (1958) to describe the relationship between a dependent variable y_i which cannot take on values smaller than zero and an independent variable (or vector) x_i. James Tobin (March 5, 1918 – March 11, 2002) was a United States economist. ...

The model supposes that there is a latent unobservable variable . This variable linearly depends on x_i via a parameter (vector) β which determines the relationship between the independent variable (or vector) x_i and the latent variable (just as in a linear model). In addition, there is a normally distributed error term u_i to capture random influences on this relationship. The observable variable y_i is defined to be equal to the latent variable whenever the latent variable is above zero and zero else. In statistics the linear model can be expressed by saying where Y is an n×1 column vector of random variables, X is an n×p matrix of known (i. ...

where is a latent variable:

If the relationship parameter β is estimated by regressing the observed y_i on x_i, the resulting ordinary least squares estimator is inconsistent. Amemiya (1973) has proven that the likelihood estimator suggested by Tobin for this model is consistent. Least squares is a mathematical optimization technique that attempts to find a best fit to a set of data by attempting to minimize the sum of the squares of the ordinate differences (called residuals) between the fitted function and the data. ... Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. ...

The Tobit model is a special case of a censored regression model, because the latent variable cannot always be observed. Other examples are the Tobit type II-IV model. How to generally derive the likelihood estimator when a random variable (or vector) is censored is described by Schnedler (2005).

Bibliography

• Amemiya, Takeshi (1973). "Regression analysis when the dependent variable is truncated normal". Econometrica 41 (6), 997–1016.

• Tobin, James (1958). "Estimation for relationships with limited dependent variables". Econometrica 26 (2), 24–36.

• Schnedler, Wendelin (2005). "Likelihood estimation for censored random vectors". Econometric Reviews 24 (2),195–217.