Tobit model

If the relationship parameter ${\displaystyle \beta }$  is estimated by regressing the observed ${\displaystyle y_{i}}$  on ${\displaystyle x_{i}}$ , the resulting ordinary least squares regression estimator is inconsistent. It will yield a downwards-biased estimate of the slope coefficient and an upwards-biased estimate of the intercept. Takeshi Amemiya (1973) has proven that the maximum likelihood estimator suggested by Tobin for this model is consistent.

The ${\displaystyle \beta }$  coefficient should not be interpreted as the effect of ${\displaystyle x_{i}}$  on ${\displaystyle y_{i}}$ , 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 ${\displaystyle y_{i}}$  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 ${\displaystyle y_{i}}$  if above.^[2]

Variations of the Tobit model can be produced by changing where and when censoring occurs. Amemiya (1985) 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.

The Tobit model is a special case of a censored regression model, because the latent variable ${\displaystyle y_{i}^{*}}$  cannot always be observed while the independent variable ${\displaystyle x_{i}}$  is observable. A common variation of the Tobit model is censoring at a value ${\displaystyle y_{L}}$  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 ${\displaystyle y_{U}}$ .

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

Yet another model results when ${\displaystyle y_{i}}$  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 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 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 introduces a third observed dependent variable and a third latent variable.

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

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

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

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

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

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

Below are the likelihood and log likelihood functions for a type I Tobit. This is a Tobit that is censored from below at ${\displaystyle y_{L}}$  when the latent variable ${\displaystyle y_{j}^{*}\leq y_{L}}$ . In writing out the likelihood function, we first define an indicator function ${\displaystyle I(y_{j})}$  where:

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

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

${\displaystyle \prod _{j=1}^{N}\left({\frac {1}{\sigma }}\phi \left({\frac {Y_{j}-X_{j}\beta }{\sigma }}\right)\right)^{I\left(Y_{j}\right)}\left(1-\Phi \left({\frac {X_{j}\beta -y_{L}}{\sigma }}\right)\right)^{1-I\left(Y_{j}\right)}}$ 

• Generalized Tobit
• Limited dependent variable
• Truncated regression model

• Amemiya, Takeshi (1973), "Regression analysis when the dependent variable is truncated normal", Econometrica, 41 (6), The Econometric Society: 997–1016, doi:10.2307/1914031
• 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), Advanced Econometrics, Oxford: Basil Blackwell, ISBN 0-631-13345-3
• Schnedler, Wendelin (2005), "Likelihood estimation for censored random vectors", Econometric Reviews, 24 (2): 195–217, doi:10.1081/ETC-200067925
• Tobin, James (1958), "Estimation of relationships for limited dependent variables", Econometrica, 26 (1), The Econometric Society: 24–36, doi:10.2307/1907382

• Guided tour on Tobit models