Optical cross section

2013-10-18T18:17:04Z

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The '''Tobit model''' is a [[statistical model]] proposed by [[James Tobin]] (1958) to describe the relationship between a non-negative dependent variable <math>y_i</math> and an independent variable (or [[Euclidean vector|vector]]) <math>x_i</math>. The term ''Tobit'' was derived from Tobin's name by truncating and adding ''-it'' by analogy with the [[probit model]].<ref>''[[International Encyclopedia of the Social Sciences]]'' (2008)</ref>

The model supposes that there is a [[latent variable|latent (i.e. unobservable) variable]] <math>y_i^*</math>. This variable linearly depends on <math>x_i</math> via a parameter (vector) <math>\beta</math> which determines the relationship between the independent variable (or vector) <math>x_i</math> and the [[latent variable]] <math>y_i^*</math> (just as in a [[linear model]]). In addition, there is a [[normal distribution|normally distributed]] [[errors and residuals in statistics|error term]] <math>u_i</math> to capture random influences on this relationship. The observable variable <math>y_i</math> is defined to be equal to the latent variable whenever the latent variable is above zero and zero otherwise.

: <math>y_i = \begin{cases}
y_i^* & \textrm{if} \; y_i^* >0 \\
0 & \textrm{if} \; y_i^* \leq 0
\end{cases}</math>

where <math>y_i^*</math> is a latent variable:

: <math> y_i^* = \beta x_i + u_i, u_i \sim N(0,\sigma^2) \, </math>

==Consistency==

If the relationship parameter <math>\beta</math> is estimated by regressing the observed <math> y_i </math> on <math> x_i </math>, the resulting ordinary [[least squares]] regression estimator is [[consistency (statistics)|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 <math>\beta</math> coefficient should not be interpreted as the effect of <math>x_i</math> on <math>y_i</math>, 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 <math>y_i</math> 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 <math>y_i</math> if above.<ref>{{Citation |last=McDonald |first=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 |url=http://www.jstor.org/stable/1924766 |issn= |doi= |publisher=The MIT Press}}</ref>

==Variations of the Tobit model==
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.

===Type I===
The Tobit model is a special case of a [[censored regression model]], because the latent variable <math>y_i^*</math> cannot always be observed while the independent variable <math> x_i </math> is observable. A common variation of the Tobit model is censoring at a value <math> y_L</math> different from zero:

: <math> y_i = \begin{cases}
y_i^* & \textrm{if} \; y_i^* >y_L \\
y_L & \textrm{if} \; y_i^* \leq y_L.
\end{cases}</math>

Another example is censoring of values above <math> y_U</math>.

: <math> y_i = \begin{cases}
y_i^* & \textrm{if} \; y_i^* <y_U \\
y_U & \textrm{if} \; y_i^* \geq y_U.
\end{cases}</math>

Yet another model results when <math> y_i </math> is censored from above and below at the same time.

: <math> y_i = \begin{cases}
y_i^* & \textrm{if} \; y_L<y_i^* <y_U \\
y_L & \textrm{if} \; y_i^* \leq y_L \\
y_U & \textrm{if} \; y_i^* \geq y_U.
\end{cases}</math>

The rest of the models will be presented as being bounded from below at 0, though this can be generalized as we have done for Type I.
===Type II===
Type II Tobit models introduce a second latent variable.

: <math> y_{2i} = \begin{cases}
y_{2i}^* & \textrm{if} \; y_{1i}^* >0 \\
0 & \textrm{if} \; y_{1i}^* \leq 0.
\end{cases}</math>

Heckman (1987) falls into the Type II Tobit. In Type I Tobit, the latent variable absorb both the process of participation and 'outcome' of interest. Type II Tobit allows the process of participation/selection and the process of 'outcome' to be independent, conditional on x.

===Type III===
Type III introduces a second observed dependent variable.
: <math> y_{1i} = \begin{cases}
y_{1i}^* & \textrm{if} \; y_{1i}^* >0 \\
0 & \textrm{if} \; y_{1i}^* \leq 0.
\end{cases}</math>
: <math> y_{2i} = \begin{cases}
y_{2i}^* & \textrm{if} \; y_{1i}^* >0 \\
0 & \textrm{if} \; y_{1i}^* \leq 0.
\end{cases}</math>
The [[Heckman correction|Heckman]] model falls into this type.
===Type IV===
Type IV introduces a third observed dependent variable and a third latent variable.
: <math> y_{1i} = \begin{cases}
y_{1i}^* & \textrm{if} \; y_{1i}^* >0 \\
0 & \textrm{if} \; y_{1i}^* \leq 0.
\end{cases}</math>
: <math> y_{2i} = \begin{cases}
y_{2i}^* & \textrm{if} \; y_{1i}^* >0 \\
0 & \textrm{if} \; y_{1i}^* \leq 0.
\end{cases}</math>
: <math> y_{3i} = \begin{cases}
y_{3i}^* & \textrm{if} \; y_{1i}^* >0 \\
0 & \textrm{if} \; y_{1i}^* \leq 0.
\end{cases}</math>
===Type V===
Similar to Type II, in Type V we only observe the sign of <math>y_{1i}^*</math>.
: <math> y_{2i} = \begin{cases}
y_{2i}^* & \textrm{if} \; y_{1i}^* >0 \\
0 & \textrm{if} \; y_{1i}^* \leq 0.
\end{cases}</math>
: <math> y_{3i} = \begin{cases}
y_{3i}^* & \textrm{if} \; y_{1i}^* >0 \\
0 & \textrm{if} \; y_{1i}^* \leq 0.
\end{cases}</math>

==The likelihood function==
Below are the [[likelihood function|likelihood]] and log likelihood functions for a type I Tobit. This is a Tobit that is censored from below at <math> y_L </math> when the latent variable <math> y_j^* \leq y_L </math>. In writing out the likelihood function, we first define an indicator function <math> I(y_j) </math> where:

: <math> I(y_j) = \begin{cases}
0 & \textrm{if} \; y_j = y_L \\
1 & \textrm{if} \; y_j \neq y_L.
\end{cases}</math>

Next, we mean <math> \Phi </math> to be the standard normal [[cumulative distribution function]] and <math> \phi </math> to be the standard normal [[probability density function]]. For a data set with ''N'' observations the likelihood function for a type I Tobit is

: <math> \prod _{j=1}^N \left(\frac{1}{\sigma}\phi \left(\frac{Y_j-X_j\beta }{\sigma
}\right)\right)^{I\left(y_j\right)} \left(1-\Phi
\left(\frac{X_j\beta-y_L}{\sigma}\right)\right)^{1-I\left(y_j\right)}</math>

==See also==
*[[Generalized Tobit]]
*[[Limited dependent variable]]
*[[Truncated regression model]]

==References==
{{Reflist}}

==Further reading==
*{{Cite journal |last=Amemiya |first=Takeshi |authorlink= |year=1973 |title=Regression analysis when the dependent variable is truncated normal |journal=[[Econometrica]] |volume=41 |issue=6 |pages=997–1016 |jstor=1914031 |issn= |doi=10.2307/1914031 }}
*{{Cite journal |last=Amemiya |first=Takeshi |authorlink= |year=1984 |title=Tobit models: A survey |journal=[[Journal of Econometrics]] |volume=24 |issue=1–2 |pages=3–61 |url= |doi=10.1016/0304-4076(84)90074-5 }}
*{{cite book |last=Amemiya |first=Takeshi |authorlink= |chapter= |title=Advanced Econometrics |year=1985 |publisher=Basil Blackwell |location=Oxford |isbn=0-631-13345-3 |pages= |url= }}
*{{Cite journal |last=Schnedler |first=Wendelin |authorlink= |year=2005 |title=Likelihood estimation for censored random vectors |journal=Econometric Reviews |volume=24 |issue=2 |pages=195–217 |url= |doi=10.1081/ETC-200067925 }}
*{{Cite journal |last=Tobin |first=James |authorlink= |year=1958 |title=Estimation of relationships for limited dependent variables |journal=Econometrica |volume=26 |issue=1 |pages=24–36 |jstor=1907382 |issn= |doi=10.2307/1907382 }}

==External links==
*[http://econ.la.psu.edu/~hbierens/EasyRegTours/TOBIT.HTM Guided tour on Tobit models]

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