tobit model

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 $y\_j^*\; \backslash leq\; y\_L$. In writing out the likelihood function, we first define an indicator function $I$:

: $I(y)\; =\; \backslash begin\{cases\}$

0 & \text{if } y \leq y_L, \\

1 & \text{if } y > y_L.

\end{cases}

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

: $\backslash mathcal\{L\}(\backslash beta,\; \backslash sigma)\; =\; \backslash prod\; \_\{j=1\}^N\; \backslash left(\backslash frac\{1\}\{\backslash sigma\}\backslash varphi\; \backslash left(\backslash frac\{y\_j-X\_j\backslash beta\; \}\{\backslash sigma\}\; \backslash right)\backslash right)^\{I(y\_j)\}\; \backslash left(1-\backslash Phi\; \backslash left(\backslash frac\{X\_j\backslash beta-y\_L\}\{\backslash sigma\}\backslash right)\backslash right)^\{1-I(y\_j)\}$

and the log likelihood is given by

:$\backslash begin\{align\}$

\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_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{align}

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

:$\backslash log\; \backslash mathcal\{L\}(\backslash delta,\; \backslash gamma)\; =\; \backslash sum\_\{y\_j>y\_L\}\; \backslash left\backslash \{\; \backslash log\; \backslash gamma\; +\; \backslash log\; \backslash left[\; \backslash varphi\backslash left(\; \backslash gamma\; y\_j\; -\; X\_j\; \backslash delta\; \backslash right)\; \backslash right]\; \backslash right\backslash \}\; +\; \backslash sum\_\{y\_j=y\_L\}\; \backslash log\backslash left[\; \backslash Phi\backslash left(\; \backslash gamma\; y\_L\; -\; X\_j\; \backslash delta\; \backslash right)\; \backslash right]$

which is globally concave in terms of the transformed parameters.{{cite journal |first=Randall J. |last=Olsen |title=Note on the Uniqueness of the Maximum Likelihood Estimator for the Tobit Model |journal=Econometrica |volume=46 |issue=5 |year=1978 |pages=1211–1215 |doi=10.2307/1911445 |jstor=1911445 }}

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.{{cite journal |first=Chris |last=Orme |title=On the Uniqueness of the Maximum Likelihood Estimator in Truncated Regression Models |journal=Econometric Reviews |volume=8 |year=1989 |issue=2 |pages=217–222 |doi=10.1080/07474938908800171 }}{{cite journal |first=Shigeru |last=Iwata |title=A Note on Multiple Roots of the Tobit Log Likelihood |journal=Journal of Econometrics |volume=56 |issue=3 |year=1993 |pages=441–445 |doi=10.1016/0304-4076(93)90129-S }}

If the relationship parameter $\backslash beta$ 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.{{Cite journal |last=Amemiya |first=Takeshi |year=1973 |title=Regression analysis when the dependent variable is truncated normal |journal=Econometrica |volume=41 |issue=6 |pages=997–1016 |jstor=1914031 |doi=10.2307/1914031 }}

The $\backslash beta$ 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; and

(2) the change in the probability of being above the limit, weighted by the expected value of $y\_i$ if above.{{Cite journal

|last1=McDonald |first1=John F.

|last2=Moffit |first2=Robert A.

|year=1980

|title=The Uses of Tobit Analysis

|journal=The Review of Economics and Statistics

|volume=62 |issue=2

|pages=318–321

|jstor=1924766

|doi= 10.2307/1924766

}}

Variations of the tobit model can be produced by changing where and when censoring occurs. {{harvtxt|Amemiya|1985|loc=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.{{Cite journal |last=Schnedler |first=Wendelin |year=2005 |title=Likelihood estimation for censored random vectors |journal=Econometric Reviews |volume=24 |issue=2 |pages=195–217 |url= http://www.uni-heidelberg.de/md/awi/forschung/dp417.pdf|doi=10.1081/ETC-200067925 |hdl=10419/127228 |s2cid=55747319 }}

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:

: $y\_i\; =\; \backslash begin\{cases\}$

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 $y\_U$.

: $y\_i\; =\; \backslash begin\{cases\}$

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.

: $y\_i\; =\; \backslash begin\{cases\}$

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 tobit models introduce a second latent variable.{{cite book |last=Amemiya |first=Takeshi |title=Advanced econometrics |publisher=Harvard University Press |location=Cambridge, Mass |year=1985 |isbn=0-674-00560-0 |oclc=11728277 |page=[https://archive.org/details/advancedeconomet00amem/page/384 384] |url-access=registration |chapter=Tobit Models |url=https://archive.org/details/advancedeconomet00amem}}

: $y\_\{2i\}\; =\; \backslash begin\{cases\}$

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,{{cite journal|last1=Heckman|first1=James J.|title=Sample Selection Bias as a Specification Error|journal=Econometrica|volume=47|issue=1|year=1979|pages=153–161|issn=0012-9682|doi=10.2307/1912352|jstor=1912352}} which is sometimes called Heckit after James Heckman.{{cite journal|last1=Sigelman|first1=Lee|last2=Zeng|first2=Langche|title=Analyzing Censored and Sample-Selected Data with Tobit and Heckit Models|journal=Political Analysis|volume=8|issue=2|year=1999|pages=167–182|issn=1047-1987|doi=10.1093/oxfordjournals.pan.a029811|jstor=25791605}}

Type III introduces a second observed dependent variable.

: $y\_\{1i\}\; =\; \backslash begin\{cases\}$

y_{1i}^* & \text{if } y_{1i}^* >0, \\

0 & \text{if } y_{1i}^* \leq 0.

\end{cases}

: $y\_\{2i\}\; =\; \backslash begin\{cases\}$

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.

: $y\_\{1i\}\; =\; \backslash begin\{cases\}$

y_{1i}^* & \text{if } y_{1i}^* >0, \\

0 & \text{if } y_{1i}^* \leq 0.

\end{cases}

: $y\_\{2i\}\; =\; \backslash begin\{cases\}$

y_{2i}^* & \text{if } y_{1i}^* >0, \\

0 & \text{if } y_{1i}^* \leq 0.

\end{cases}

: $y\_\{3i\}\; =\; \backslash begin\{cases\}$

y_{3i}^* & \text{if } y_{1i}^* \leq0, \\

0 & \text{if } y_{1i}^* <0.

\end{cases}

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

: $y\_\{2i\}\; =\; \backslash begin\{cases\}$

y_{2i}^* & \text{if } y_{1i}^* >0, \\

0 & \text{if } y_{1i}^* \leq 0.

\end{cases}

:$y\_\{3i\}\; =\; \backslash begin\{cases\}$

y_{3i}^* & \text{if } y_{1i}^* \leq 0, \\

0 & \text{if } y_{1i}^* > 0.

\end{cases}

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.{{cite journal|last1=Powell|first1=James L|title=Least absolute deviations estimation for the censored regression model|journal=Journal of Econometrics|date=1 July 1984|volume=25|issue=3|pages=303–325|doi=10.1016/0304-4076(84)90004-6 |citeseerx=10.1.1.461.4302}}

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).{{Cite journal|last1=Dahlberg|first1=Matz|last2=Johansson|first2=Eva|date=2002-03-01|title=On the Vote-Purchasing Behavior of Incumbent Governments|journal=American Political Science Review|volume=96|issue=1|pages=27–40|doi=10.1017/S0003055402004215|issn=1537-5943|citeseerx=10.1.1.198.4112|s2cid=12718473}} 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.{{Cite journal|last1=Dubois|first1=Hans F. W.|last2=Fattore|first2=Giovanni|date=2011-07-01|title=Public Fund Assignment through Project Evaluation|journal=Regional & Federal Studies|volume=21|issue=3|pages=355–374|doi=10.1080/13597566.2011.578827|s2cid=154659642|issn=1359-7566}} 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.{{Cite journal|last=Baltas|first=George|date=2001|title=Utility-consistent Brand Demand Systems with Endogenous Category Consumption: Principles and Marketing Applications|journal=Decision Sciences|language=en|volume=32|issue=3|pages=399–422|doi=10.1111/j.1540-5915.2001.tb00965.x|issn=0011-7315}}

• Truncated normal hurdle model
• Limited dependent variable
• Rectifier (neural networks)
• Truncated regression model
• {{section link|Dynamic unobserved effects model#Censored dependent variable}}
• Probit model, the name tobit is a pun on both Tobin, their creator, and their similarities to probit models.

{{Notelist}}

{{Reflist}}

• {{cite book|ref=none |last=Amemiya |first=Takeshi |chapter=Tobit Models |title=Advanced Econometrics |year=1985 |publisher=Basil Blackwell |location=Oxford |isbn=0-631-13345-3 |pages=360–411 |chapter-url=https://books.google.com/books?id=0bzGQE14CwEC&pg=PA360 }}
• {{cite book|ref=none |first=Richard |last=Breen |title=Regression Models : Censored, Samples Selected, or Truncated Data |location=Thousand Oaks |publisher=Sage |year=1996 |isbn=0-8039-5710-6 |chapter=The Tobit Model for Censored Data |pages=12–33 }}
• {{cite book|ref=none |first=Christian |last=Gouriéroux |author-link=Christian Gouriéroux |chapter=The Tobit Model |title=Econometrics of Qualitative Dependent Variables |location=New York |publisher=Cambridge University Press |year=2000 |isbn=0-521-58985-1 |pages=170–207 |chapter-url=https://books.google.com/books?id=dE2prs_U0QMC&pg=PA170 }}
• {{cite book|ref=none |last=King |first=Gary |author-link=Gary King (political scientist) |chapter=Models with Nonrandom Selection |title=Unifying Political Methodology : the Likehood Theory of Statistical Inference |publisher=Cambridge University Press |year=1989 |isbn=0-521-36697-6 |pages=208–230 |chapter-url=https://books.google.com/books?id=cligOwrd7XoC&pg=PA208 }}
• {{cite book|ref=none |first=G. S. |last=Maddala |author-link=G. S. Maddala |title=Limited-Dependent and Qualitative Variables in Econometrics |url=https://archive.org/details/limiteddependent00madd |url-access=limited |location=New York |publisher=Cambridge University Press |year=1983 |isbn=0-521-24143-X |chapter=Censored and Truncated Regression Models |pages=[https://archive.org/details/limiteddependent00madd/page/n82 149]–196 }}

{{Economics}}

Category:Regression models

Category:Single-equation methods (econometrics)