Tobit model

The Tobit model is on James Tobin returning econometric model for analyzing limited dependent variables ( censored data ). Since the dependent variable only exists in a certain range of values , normal regression parameters are not the best possible estimators, so the estimator has to be corrected. This correction is implemented in the Tobit model.

model

The Tobit model describes the relationship between a non-negative dependent variable and an independent variable (or a vector) . The model assumes that there is a latent (i.e., unobservable) variable . This variable is linearly dependent on a parameter (or vector) that, like a linear regression, determines the relationship between the independent variable (or the vector) and the latent variable . ${\displaystyle y_{i}}$ ${\displaystyle x_{i}}$ ${\displaystyle y_{i}^{*}}$ ${\displaystyle x_{i}}$ ${\displaystyle \beta }$ ${\displaystyle x_{i}}$ ${\displaystyle y_{i}^{*}}$

In addition, there is a normally distributed error term that models the random influences on this relationship. ${\displaystyle u_{i}}$

By definition, the observable variable is equal to the latent variable if it is greater than zero; otherwise it is zero: ${\displaystyle y_{i}}$

${\displaystyle y_{i}={\begin{cases}y_{i}^{*}&\mathrm {f{\ddot {u}}r} \;y_{i}^{*}>0,\\0&\mathrm {f{\ddot {u}}r} \;y_{i}^{*}\leq 0,\end{cases}}}$

where a latent variable represents: ${\displaystyle y_{i}^{*}}$

${\displaystyle y_{i}^{*}=\beta x_{i}+u_{i},\quad u_{i}\sim {\mathcal {N}}(0,\sigma ^{2})}$

Parameter estimation

If the true parameter is estimated via a conventional regression of the observed variable at , the resulting least squares estimate is not consistent for . Amemiya (1973) has shown that the probability estimator proposed by Tobin for this model is consistent. ${\displaystyle \beta }$ ${\displaystyle y_{i}}$ ${\displaystyle x_{i}}$ ${\displaystyle \beta }$

generalization

The Tobit model is a special case of a truncated regression model because the latent variable cannot always be observed while the independent variable is observable. A common variant of the Tobit model is to restrict a variable to a non-zero value : ${\displaystyle y_{i}^{*}}$ ${\displaystyle x_{i}}$ ${\displaystyle y_{L}}$

${\displaystyle y_{i}={\begin{cases}y_{i}^{*}&\mathrm {f{\ddot {u}}r} \quad y_{i}^{*}>y_{L}\\0&\mathrm {f{\ddot {u}}r} \quad y_{i}^{*}\leq y_{L}.\end{cases}}}$

Another example concerns the restriction to values ​​over . ${\displaystyle y_{U}}$

${\displaystyle y_{i}={\begin{cases}y_{i}^{*}&\mathrm {f{\ddot {u}}r} \quad y_{i}^{*}<y_{U}\\0&\mathrm {f{\ddot {u}}r} \quad y_{i}^{*}\geq y_{U}.\end{cases}}}$

Another model results if there is a simultaneous restriction from above and below. ${\displaystyle y_{i}}$

${\displaystyle y_{i}={\begin{cases}y_{i}^{*}&\mathrm {f{\ddot {u}}r} \quad y_{L}<y_{i}^{*}<y_{U}\\0&\mathrm {f{\ddot {u}}r} \quad y_{i}^{*}\leq y_{L}\quad {\text{oder}}\quad y_{i}^{*}\geq y_{U}.\end{cases}}}$

Such generalizations are typically also referred to as Tobit models. Depending on where and when the restriction occurs, further variants of the Tobit model result. Takeshi Amemiya classifies these variants into five categories (Tobit regression type 1 – Tobit regression type 4), with Tobit regression type 1 representing the model described above. Schnedler provides a general formula to achieve consistent probability estimates for this and other variants of the Tobit model.

literature

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

• Tobin, James (1958). "Estimation of relationships for limited dependent variables". Econometrica 26 (1), 24-36.

Individual evidence

1. ↑ Takeshi Amemiya: Advanced Econometrics . Harvard University Press, Cambridge 1985, ISBN 0-674-00560-0 , pp. 360 ff .
2. ↑ Schnedler, Wendelin (2005). "Likelihood estimation for censored random vectors". Econometric Reviews 24 (2), 195-217.