Tobit model

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 :

: I(y) = \begin{cases} 0 & \text{if } y \leq y_L, \ 1 & \text{if } y y_L. \end{cases}

Next, let \Phi be the standard normal cumulative distribution function and \varphi 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}\varphi \left(\frac{y_j-X_j\beta }{\sigma} \right)\right)^{I(y_j)} \left(1-\Phi \left(\frac{X_j\beta-y_L}{\sigma}\right)\right)^{1-I(y_j)}

and the log likelihood is given by :\begin{align} \log \mathcal{L}(\beta, \sigma) &= \sum^n_{j = 1} I(y_j) \log \left( \frac{1}{\sigma} \varphi\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) \ &= \sum_{y_jy_L} \log \left( \frac{1}{\sigma} \varphi\left( \frac{y_j - X_j\beta}{\sigma} \right) \right) + \sum_{y_j=y_L} \log\left( \Phi\left( \frac{ y_L - X_j \beta}{\sigma} \right) \right) \end{align}

Reparametrization

The log-likelihood as stated above is not globally concave, which complicates the maximum likelihood estimation. Olsen suggested the simple reparametrization \beta = \delta/\gamma and \sigma^2 = \gamma^{-2}, resulting in a transformed log-likelihood,

:\log \mathcal{L}(\delta, \gamma) = \sum_{y_jy_L} \left{ \log \gamma + \log \left[ \varphi\left( \gamma y_j - X_j \delta \right) \right] \right} + \sum_{y_j=y_L} \log\left[ \Phi\left( \gamma y_L - X_j \delta \right) \right]

which is globally concave in terms of the transformed parameters.

For the truncated (tobit II) model, Orme showed that while the log-likelihood is not globally concave, it is concave at any stationary point under the above transformation.

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.

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;
2. the change in the probability of being above the limit, weighted by the expected value of y_i if above.{{Cite journal \frac{\partial \mathbb{E}[Y_i \mid X_i]}{\partial x_{ik}} = \frac{\partial \mathbb{E}[Y_i \mid Y_i 0, X_i]}{\partial x_{ik}}\cdot\mathbb{P}(Y_i 0 \mid X_i) + \frac{\partial \mathbb{P}(Y_i 0 \mid X_i)}{\partial x_{ik}} \cdot \mathbb{E}[Y_i \mid Y_i 0, X_i].

Variations of the tobit model

Variations of the tobit model can be produced by changing where and when censoring occurs. 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 = \begin{cases} y_i^* & \text{if } y_i^* y_L, \ y_L & \text{if } y_i^* \leq y_L. \end{cases}

Another example is censoring of values above y_U.

: y_i = \begin{cases} y_i^* & \text{if } y_i^* y_U & \text{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.

: y_i = \begin{cases} y_i^* & \text{if } y_L y_L & \text{if } y_i^* \leq y_L, \ y_U & \text{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 done for Type I.

Type II

Type II tobit models introduce a second latent variable.

: y_{2i} = \begin{cases} y_{2i}^* & \text{if } y_{1i}^* 0, \ 0 & \text{if } y_{1i}^* \leq 0. \end{cases}

In Type I tobit, the latent variable absorbs both the process of participation and the outcome of interest. Type II tobit allows the process of participation (selection) and the outcome of interest to be independent, conditional on observable data.

The Heckman selection model falls into the Type II tobit, which is sometimes called Heckit after James Heckman.

Type III

Type III introduces a second observed dependent variable. : y_{1i} = \begin{cases} y_{1i}^* & \text{if } y_{1i}^* 0, \ 0 & \text{if } y_{1i}^* \leq 0. \end{cases} : y_{2i} = \begin{cases} y_{2i}^* & \text{if } y_{1i}^* 0, \ 0 & \text{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. : y_{1i} = \begin{cases} y_{1i}^* & \text{if } y_{1i}^* 0, \ 0 & \text{if } y_{1i}^* \leq 0. \end{cases} : y_{2i} = \begin{cases} y_{2i}^* & \text{if } y_{1i}^* 0, \ 0 & \text{if } y_{1i}^* \leq 0. \end{cases} : y_{3i} = \begin{cases} y_{3i}^* & \text{if } y_{1i}^* \leq0, \ 0 & \text{if } y_{1i}^*
\end{cases}

Type V

Similar to Type II, in Type V only the sign of y_{1i}^* is observed. : y_{2i} = \begin{cases} y_{2i}^* & \text{if } y_{1i}^* 0, \ 0 & \text{if } y_{1i}^* \leq 0. \end{cases} : y_{3i} = \begin{cases} y_{3i}^* & \text{if } y_{1i}^* \leq 0, \ 0 & \text{if } y_{1i}^* 0. \end{cases}

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 thus 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. Tobit models have also been applied in demand analysis to accommodate observations with zero expenditures on some goods. In a related application of tobit models, a system of nonlinear tobit regressions models has been used to jointly estimate a brand demand system with homoscedastic, heteroscedastic and generalized heteroscedastic variants.

Notes

References

References

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