Tobit model

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

${\displaystyle \log {\mathcal {L}}(\delta ,\gamma )=\sum _{y_{j}>y_{L}}\left\{\log \gamma +\log \left[\varphi \left(\gamma y_{j}-X_{j}\delta \right)\right]\right\}+\sum _{y_{j}=y_{L}}\log \left[\Phi \left(\gamma y_{L}-X_{j}\delta \right)\right]}$

which is globally concave in terms of the transformed parameters.[7]

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.[8][9]

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 upward-biased estimate of the intercept. Takeshi Amemiya (1973) has proven that the maximum likelihood estimator suggested by Tobin for this model is consistent.[10]

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.[11]

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:

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

Another example is censoring of values above ${\displaystyle y_{U}}$.

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

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

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

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 tobit models introduce a second latent variable.[13]

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

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,[14] which is sometimes called Heckit after James Heckman.[15]

Type III introduces a second observed dependent variable.

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

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

The Heckman model falls into this type.

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

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

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

${\displaystyle y_{3i}={\begin{cases}y_{3i}^{*}&{\text{if }}y_{1i}^{*}\leq 0,\\0&{\text{if }}y_{1i}^{*}<0.\end{cases}}}$

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

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

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

If the underlying latent variable ${\displaystyle y_{i}^{*}}$ is not normally distributed, one must use quantiles instead of moments to analyze the observable variable ${\displaystyle y_{i}}$. Powell's CLAD estimator offers a possible way to achieve this.[16]