Tobit model

The Tobit model is a statistical model proposed by James Tobin (1958)^[1] 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.^[2]

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) $\beta$ 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.

${\displaystyle y_{i}={\begin{cases}y_{i}^{*}&{\textrm {if}}\;y_{i}^{*}>0\\0&{\textrm {if}}\;y_{i}^{*}\leq 0\end{cases}}}$

where $y_i^*$ is a latent variable:

$y_i^* = \beta x_i + u_i, u_i \sim N(0,\sigma^2) \,$

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.^[3]

Consistency

If the relationship parameter $\beta$ 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.^[4]

Interpretation

The $\beta$ 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.^[5]

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.^[6]

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:

${\displaystyle y_{i}={\begin{cases}y_{i}^{*}&{\textrm {if}}\;y_{i}^{*}>y_{L}\\y_{L}&{\textrm {if}}\;y_{i}^{*}\leq y_{L}.\end{cases}}}$

Another example is censoring of values above $y_U$.

${\displaystyle y_{i}={\begin{cases}y_{i}^{*}&{\textrm {if}}\;y_{i}^{*}<y_{U}\\y_{U}&{\textrm {if}}\;y_{i}^{*}\geq y_{U}.\end{cases}}}$

Yet another model results when $y_{i}$ is censored from above and below at the same time.

${\displaystyle y_{i}={\begin{cases}y_{i}^{*}&{\textrm {if}}\;y_{L}<y_{i}^{*}<y_{U}\\y_{L}&{\textrm {if}}\;y_{i}^{*}\leq y_{L}\\y_{U}&{\textrm {if}}\;y_{i}^{*}\geq y_{U}.\end{cases}}}$

The rest of the models will be presented as being bounded from below at 0, though this can be generalized as we have done for Type I.

Type II

Type II Tobit models introduce a second latent variable.

${\displaystyle y_{2i}={\begin{cases}y_{2i}^{*}&{\textrm {if}}\;y_{1i}^{*}>0\\0&{\textrm {if}}\;y_{1i}^{*}\leq 0.\end{cases}}}$

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.

${\displaystyle y_{1i}={\begin{cases}y_{1i}^{*}&{\textrm {if}}\;y_{1i}^{*}>0\\0&{\textrm {if}}\;y_{1i}^{*}\leq 0.\end{cases}}}$

${\displaystyle y_{2i}={\begin{cases}y_{2i}^{*}&{\textrm {if}}\;y_{1i}^{*}>0\\0&{\textrm {if}}\;y_{1i}^{*}\leq 0.\end{cases}}}$

The Heckman model falls into this type.

Type IV

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

${\displaystyle y_{1i}={\begin{cases}y_{1i}^{*}&{\textrm {if}}\;y_{1i}^{*}>0\\0&{\textrm {if}}\;y_{1i}^{*}\leq 0.\end{cases}}}$

${\displaystyle y_{2i}={\begin{cases}y_{2i}^{*}&{\textrm {if}}\;y_{1i}^{*}>0\\0&{\textrm {if}}\;y_{1i}^{*}\leq 0.\end{cases}}}$

${\displaystyle y_{3i}={\begin{cases}y_{3i}^{*}&{\textrm {if}}\;y_{1i}^{*}>0\\0&{\textrm {if}}\;y_{1i}^{*}\leq 0.\end{cases}}}$

Type V

Similar to Type II, in Type V we only observe the sign of $y_{1i}^*$.

${\displaystyle y_{2i}={\begin{cases}y_{2i}^{*}&{\textrm {if}}\;y_{1i}^{*}>0\\0&{\textrm {if}}\;y_{1i}^{*}\leq 0.\end{cases}}}$

${\displaystyle y_{3i}={\begin{cases}y_{3i}^{*}&{\textrm {if}}\;y_{1i}^{*}>0\\0&{\textrm {if}}\;y_{1i}^{*}\leq 0.\end{cases}}}$

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 $y_j^* \leq y_L$. In writing out the likelihood function, we first define an indicator function $I(y_j)$ where:

${\displaystyle I(y_{j})={\begin{cases}0&{\textrm {if}}\;y_{j}\leq y_{L}\\1&{\textrm {if}}\;y_{j}>y_{L}.\end{cases}}}$

Next, we mean $\Phi$ to be the standard normal cumulative distribution function and $\phi$ to be the standard normal probability density function. For a data set with N observations the likelihood function for a type I Tobit is

$\mathcal{L}(\beta, \sigma) = \prod _{j=1}^N \left(\frac{1}{\sigma}\phi \left(\frac{y_j-X_j\beta }{\sigma }\right)\right)^{I\left(y_j\right)} \left(1-\Phi \left(\frac{X_j\beta-y_L}{\sigma}\right)\right)^{1-I\left(y_j\right)}$

and the log likelihood is given by

$\log {\mathcal {L}}(\beta ,\sigma )=\sum _{{j=1}}^{n}I(y_{j})\log \left({\frac {1}{\sigma }}\phi \left({\frac {y_{j}-X_{j}\beta }{\sigma }}\right)\right)+(1-I(y_{j}))\log \left(1-\Phi \left({\frac {X_{j}\beta -y_{L}}{\sigma }}\right)\right)$

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 possibility to achieve this.^[7]

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)^[8] analyse a sample of 115 municipalities (42 of which received a grant). Dubois and Fattore (2011)^[9] 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.

See also

• Generalized Tobit
• Limited dependent variable
• Rectifier (neural networks)
• Truncated regression model
• Probit model, the name Tobit is a pun on both Tobin, their creator, and their similarities to probit models.

References

1. Tobin, James (1958). "Estimation of relationships for limited dependent variables". Econometrica. 26 (1): 24–36. doi:10.2307/1907382. JSTOR 1907382.
2. International Encyclopedia of the Social Sciences (2008)
3. The ET Interview: Professor James Tobin
4. Amemiya, Takeshi (1973). "Regression analysis when the dependent variable is truncated normal". Econometrica. 41 (6): 997–1016. doi:10.2307/1914031. JSTOR 1914031.
5. McDonald, John F.; Moffit, Robert A. (1980). "The Uses of Tobit Analysis". The Review of Economics and Statistics. The MIT Press. 62 (2): 318–321. doi:10.2307/1924766. JSTOR 1924766.
6. Schnedler, Wendelin (2005). "Likelihood estimation for censored random vectors". Econometric Reviews. 24 (2): 195–217. doi:10.1081/ETC-200067925.
7. Powell, James L (1 July 1984). "Least absolute deviations estimation for the censored regression model". Journal of Econometrics. 25 (3): 303–325. doi:10.1016/0304-4076(84)90004-6.
8. Dahlberg, Matz; Johansson, Eva (2002-03-01). "On the Vote-Purchasing Behavior of Incumbent Governments". American Political Science Review. null (01): 27–40. doi:10.1017/S0003055402004215. ISSN 1537-5943.
9. Dubois, Hans F. W.; Fattore, Giovanni (2011-07-01). "Public Fund Assignment through Project Evaluation". Regional & Federal Studies. 21 (3): 355–374. doi:10.1080/13597566.2011.578827. ISSN 1359-7566.

Further reading

• Amemiya, Takeshi (1984). "Tobit models: A survey". Journal of Econometrics. 24 (1–2): 3–61. doi:10.1016/0304-4076(84)90074-5.
• Amemiya, Takeshi (1985). "Tobit Models". Advanced Econometrics. Oxford: Basil Blackwell. pp. 360–411. ISBN 0-631-13345-3.
• Gouriéroux, Christian (2000). "The Tobit Model". Econometrics of Qualitative Dependent Variables. New York: Cambridge University Press. pp. 170–207. ISBN 0-521-58985-1.