Tobit model - Alchetron, The Free Social Encyclopedia

The Tobit model is a statistical model proposed by James Tobin (1958) to describe the relationship between a non-negative dependent variable y i and an independent variable (or vector) 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 different estimator.

The model supposes that there is a latent (i.e. unobservable) variable y i ∗ . 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 y i ∗ (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 otherwise.

y i = { y i ∗ if y i ∗ > 0 0 if y i ∗ ≤ 0

where y i ∗ is a latent variable:

y i ∗ = β x i + u i , u i ∼ N ( 0 , σ 2 )

Etymology

When asked why it was called the "Tobit" model, instead of Tobin, James Tobin explained that this term was introduced by Arthur Goldberger, either as a contraction of "Tobin probit", or as a reference to the novel The Caine Mutiny, a novel by Tobin's friend Herman Wouk, in which Tobin makes a cameo as "Mr Tobit". Tobin reports having actually asked Goldberger which it was, and the man refused to say.

Consistency

If the relationship parameter β is estimated by regressing the observed y i on x i , 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 (1973) has proven that the maximum likelihood estimator suggested by Tobin for this model is consistent.

Interpretation

The β coefficient should not be interpreted as the effect of x i on y i , as one would with a linear regression model; this is a common error. Instead, it should be interpreted as the combination of (1) the change in y i of those above the limit, weighted by the probability of being above the limit; and (2) the change in the probability of being above the limit, weighted by the expected value of y i if above.

Variations of the Tobit model

Variations of the Tobit model can be produced by changing where and when censoring occurs. Amemiya (1985, p. 384) classifies these variations into five categories (Tobit type I - Tobit type V), where Tobit type I stands for the first model described above. Schnedler (2005) provides a general formula to obtain consistent likelihood estimators for these and other variations of the Tobit model.

Type I

The Tobit model is a special case of a censored regression model, because the latent variable y i ∗ cannot always be observed while the independent variable x i is observable. A common variation of the Tobit model is censoring at a value y L different from zero:

y i = { y i ∗ if y i ∗ > y L y L if y i ∗ ≤ y L .

Another example is censoring of values above y U .

y i = { y i ∗ if y i ∗ < y U y U if y i ∗ ≥ y U .

Yet another model results when y i is censored from above and below at the same time.

y i = { y i ∗ if y L < y i ∗ < y U y L if y i ∗ ≤ y L y U if y i ∗ ≥ y U .

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

Type II Tobit models introduce a second latent variable.

y 2 i = { y 2 i ∗ if y 1 i ∗ > 0 0 if y 1 i ∗ ≤ 0.

Heckman (1987) falls into the Type II Tobit. In Type I Tobit, the latent variable absorb both the process of participation and 'outcome' of interest. Type II Tobit allows the process of participation/selection and the process of 'outcome' to be independent, conditional on x.

Type III

Type III introduces a second observed dependent variable.

y 1 i = { y 1 i ∗ if y 1 i ∗ > 0 0 if y 1 i ∗ ≤ 0. y 2 i = { y 2 i ∗ if y 1 i ∗ > 0 0 if y 1 i ∗ ≤ 0.

The Heckman model falls into this type.

Type IV

Type IV introduces a third observed dependent variable and a third latent variable.

y 1 i = { y 1 i ∗ if y 1 i ∗ > 0 0 if y 1 i ∗ ≤ 0. y 2 i = { y 2 i ∗ if y 1 i ∗ > 0 0 if y 1 i ∗ ≤ 0. y 3 i = { y 3 i ∗ if y 1 i ∗ > 0 0 if y 1 i ∗ ≤ 0.

Type V

Similar to Type II, in Type V only the sign of y 1 i ∗ is observed.

y 2 i = { y 2 i ∗ if y 1 i ∗ > 0 0 if y 1 i ∗ ≤ 0. y 3 i = { y 3 i ∗ if y 1 i ∗ > 0 0 if y 1 i ∗ ≤ 0.

The likelihood function

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 when the latent variable y j ∗ ≤ y L . In writing out the likelihood function, we first define an indicator function I ( y j ) where:

I ( y j ) = { 0 if y j ≤ y L 1 if y j > y L .

Next, let Φ be the standard normal cumulative distribution function and ϕ to be the standard normal probability density function. For a data set with N observations the likelihood function for a type I Tobit is

L ( β , σ ) = ∏ j = 1 N ( 1 σ ϕ ( y j − X j β σ ) ) I ( y j ) ( 1 − Φ ( X j β − y L σ ) ) 1 − I ( y j )

and the log likelihood is given by

log ⁡ L ( β , σ ) = ∑ j = 1 n I ( y j ) log ⁡ ( 1 σ ϕ ( y j − X j β σ ) ) + ( 1 − I ( y j ) ) log ⁡ ( 1 − Φ ( X j β − y L σ ) )

Note that this is different from the likelihood function of the truncated regression model.

Non-Parametric Version

If the underlying latent variable y i ∗ is not normally distributed, one must use quantiles instead of moments to analyze the observable variable y i . Powell's CLAD estimator offers a possible way to achieve this.

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 this left-censored. For instance, Dahlberg and Johansson (2002) analyse a sample of 115 municipalities (42 of which received a grant). Dubois and Fattore (2011) 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.

References

Tobit model Wikipedia

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