# Tobit model

In statistics, a tobit model is any of a class of regression models in which the observed range of the dependent variable is censored in some way.[1] The term was coined by Arthur Goldberger in reference to James Tobin,[2][lower-alpha 1] who developed the model in 1958 to mitigate the problem of zero-inflated data for observations of household expenditure on durable goods.[3][lower-alpha 2] Because Tobin's method can be easily extended to handle truncated and other non-randomly selected samples,[lower-alpha 3] some authors adopt a broader definition of the tobit model that includes these cases.[4]

Tobin's idea was to modify the likelihood function so that it reflects the unequal sampling probability for each observation depending on whether the latent dependent variable fell above or below the determined threshold.[5] For a sample that, as in Tobin's original case, was censored from below at zero, the sampling probability for each non-limit observation is simply height of the appropriate density function. For any limit observation, it is the cumulative density, i.e. the integral below zero of the appropriate density function. The tobit likelihood function thus is a mixture of densities and cumulative densities.[6]

## 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 ${\displaystyle y_{L}}$ when the latent variable ${\displaystyle y_{j}^{*}\leq y_{L}}$. In writing out the likelihood function, we first define an indicator function ${\displaystyle I}$:

${\displaystyle I(y)={\begin{cases}0&{\text{if }}y\leq y_{L},\\1&{\text{if }}y>y_{L}.\end{cases}}}$

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

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

and the log likelihood is given by

{\displaystyle {\begin{aligned}\log {\mathcal {L}}(\beta ,\sigma )&=\sum _{j=1}^{n}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_{j}>y_{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{aligned}}}

### Reparametrization

The log-likelihood is 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]

### Consistency

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]

### Interpretation

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]

## 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.[12]

### Type I

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}^{*}

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}

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.[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

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

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

${\displaystyle y_{1i}={\begin{cases}y_{1i}^{*}&{\text{if }}y_{1i}^{*}>0,\\0&{\text{if }}y_{1i}^{*}\leq 0.\end{cases}}}$
${\displaystyle y_{2i}={\begin{cases}y_{2i}^{*}&{\text{if }}y_{1i}^{*}>0,\\0&{\text{if }}y_{1i}^{*}\leq 0.\end{cases}}}$
${\displaystyle y_{3i}={\begin{cases}y_{3i}^{*}&{\text{if }}y_{1i}^{*}>0,\\0&{\text{if }}y_{1i}^{*}\leq 0.\end{cases}}}$

### Type V

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

${\displaystyle y_{2i}={\begin{cases}y_{2i}^{*}&{\text{if }}y_{1i}^{*}>0,\\0&{\text{if }}y_{1i}^{*}\leq 0.\end{cases}}}$
${\displaystyle y_{3i}={\begin{cases}y_{3i}^{*}&{\text{if }}y_{1i}^{*}>0,\\0&{\text{if }}y_{1i}^{*}\leq 0.\end{cases}}}$

### Non-parametric version

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]

### Dynamic unobserved effects tobit model

In a panel data tobit model,[17][18] if the outcome ${\displaystyle Y_{i,t}}$ partially depends on the previous outcome history ${\displaystyle Y_{i,0},\ldots ,Y_{t-1}}$ this tobit model is called "dynamic". For instance, taking a person who finds a job with a high salary this year, it will be easier for her to find a job with a high salary next year because the fact that she has a high-wage job this year will be a very positive signal for the potential employers. The essence of this type of dynamic effect is the state dependence of the outcome. The "unobservable effects" here refers to the factor which partially determines the outcome of individual but cannot be observed in the data. For instance, the ability of a person is very important in job-hunting, but it is not observable for researchers. A typical dynamic unobserved effects tobit model can be represented as

${\displaystyle Y_{i,t}=Y_{i,t}^{1}[Y_{i,t}>0];}$
${\displaystyle Y_{i,t}=z_{i,t}\delta +\rho y_{i,t-1}+c_{i}+u_{i,t};}$
${\displaystyle c_{i}\mid y_{i,0},\ldots ,y_{i,t-1}\sim F(y_{i,0}x_{i});}$
${\displaystyle u_{i,t}\mid z_{i,t},y_{i,0},\ldots ,y_{i,t-1}\sim N(0,1).}$

In this specific model, ${\displaystyle \rho y_{i,t-1}}$ is the dynamic effect part and ${\displaystyle c_{i}}$ is the unobserved effect part whose distribution is determined by the initial outcome of individual i and some exogenous features of individual i.

Based on this setup, the likelihood function conditional on ${\displaystyle \{y_{i,0}\}_{i-1}^{N}}$ can be given as

${\displaystyle \prod _{i=1}^{N}\int f_{\theta }(c_{i}\mid y_{i,0},x_{i})\left[\prod _{t=1}^{T}{\Bigl (}1[y_{i,t}=0][1-\Phi (z_{i,t}\delta +\rho y_{i,t-1}>0]{\frac {\varphi (z_{i,t}\delta +\rho y_{i,t-1}+c_{i})}{\Phi (z_{i,t}\delta +\rho y_{i,t-1}+c_{i})}}{\biggr )}\right]\,dc_{i}}$

For the initial values ${\displaystyle \{y_{i,0}\}_{i-1}^{N}}$ ,there are two different ways to treat them in the construction of the likelihood function: treating them as constant, or imposing a distribution on them and calculate out the unconditional likelihood function. But whichever way is chosen to treat the initial values in the likelihood function, we cannot get rid of the integration inside the likelihood function when estimating the model by maximum likelihood estimation (MLE). Expectation Maximum (EM) algorithm is usually a good solution for this computation issue.[19] Based on the consistent point estimates from MLE, Average Partial Effect (APE)[20] can be calculated correspondingly.[21]

## 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)[22] analyse a sample of 115 municipalities (42 of which received a grant). Dubois and Fattore (2011)[23] 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.[24]

## Notes

1. 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 portmanteau of "Tobin's 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. See Shiller, Robert J. (1999). "The ET Interview: Professor James Tobin". Econometric Theory. 15 (6): 867–900. doi:10.1017/S0266466699156056.
2. An almost identical model was independently suggested by Anders Hald in 1949, see Hald, A. (1949). "Maximum Likelihood Estimation of the Parameters of a Normal Distribution which is Truncated at a Known Point". Scandinavian Actuarial Journal. 49 (4): 119–134. doi:10.1080/03461238.1949.10419767.
3. A sample ${\displaystyle (y_{i},\mathbf {x} _{i})}$ is censored in ${\displaystyle y_{i}}$ when ${\displaystyle \mathbf {x} _{i}}$ is observed for all observations ${\displaystyle i=1,2,\ldots ,n}$, but the true value of ${\displaystyle y_{i}}$ is known only for a restricted range of observations. If the sample is truncated, both ${\displaystyle \mathbf {x} _{i}}$ and ${\displaystyle y_{i}}$ are only observed if ${\displaystyle y_{i}}$ falls in the restricted range. See Breen, Richard (1996). Regression Models : Censored, Samples Selected, or Truncated Data. Thousand Oaks: Sage. pp. 2–4. ISBN 0-8039-5710-6.

## References

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2. Goldberger, Arthur S. (1964). Econometric Theory. New York: J. Wiley. pp. 253–55.
3. Tobin, James (1958). "Estimation of Relationships for Limited Dependent Variables" (PDF). Econometrica. 26 (1): 24–36. doi:10.2307/1907382. JSTOR 1907382.
4. Amemiya, Takeshi (1984). "Tobit Models: A Survey". Journal of Econometrics. 24 (1–2): 3–61. doi:10.1016/0304-4076(84)90074-5.
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18. The model framework comes from Wooldridge, J. (2002). Econometric Analysis of Cross Section and Panel Data. Cambridge, Mass: MIT Press. p. 542. But the author revises the model more general here.
19. For more details, refer to: Cappé, O.; Moulines, E.; Ryden, T. (2005). "Part II: Parameter Inference". Inference in Hidden Markov Models. New York: Springer-Verlag.
20. Wooldridge, J. (2002). Econometric Analysis of Cross Section and Panel Data. Cambridge, Mass: MIT Press. p. 22.
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22. Dahlberg, Matz; Johansson, Eva (2002-03-01). "On the Vote-Purchasing Behavior of Incumbent Governments". American Political Science Review. null (1): 27–40. CiteSeerX 10.1.1.198.4112. doi:10.1017/S0003055402004215. ISSN 1537-5943.
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