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The Tobit Model

The Tobit model is used when the dependent variable is censored or limited to positive values for some observations. It defines a latent variable model where the observed dependent variable is equal to the latent variable if it is positive, and zero otherwise. The Tobit model relies on normality and homoskedasticity assumptions. Sample selection corrections can be used if the sample is selected non-randomly, by including the inverse Mills ratio as an additional regressor to account for sample selection bias. This requires the selection equation to depend on variables not included in the main regression equation.

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702 views13 pages

The Tobit Model

The Tobit model is used when the dependent variable is censored or limited to positive values for some observations. It defines a latent variable model where the observed dependent variable is equal to the latent variable if it is positive, and zero otherwise. The Tobit model relies on normality and homoskedasticity assumptions. Sample selection corrections can be used if the sample is selected non-randomly, by including the inverse Mills ratio as an additional regressor to account for sample selection bias. This requires the selection equation to depend on variables not included in the main regression equation.

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rishabh
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THE TOBIT MODEL

• Continuous over strictly positive values


but is zero for a nontrivial fraction of the
population.

• The Tobit model is most easily defined


as a latent variable model:

P(y*=y/x)
• partial effects of the xj on E(y*/x), where
y* is the latent variable: β j

BUT MORE INTERESTING

Substituting:
• This equation also shows why using
OLS only for observations where yi >0
will not always consistently estimate β ;
essentially, the inverse Mills ratio is an
omitted variable, and it is generally
correlated with the elements of x.

Partial Derivative:Conditional on being


Uncensored

• It can be shown that the adjustment


factor is strictly between zero and one

Partial Derivative:”Unconditional”
Substituting we get:

• The Tobit model, and in particular the


formulas for the expectations, rely
crucially on normality and
homoskedasticity in the underlying
latent variable model.

• One potentially important limitation of


the Tobit model, at least in certain
applications, is that the expected value
conditional on y >0 is closely linked to
the probability that y>0.
SAMPLE SELECTION CORRECTIONS

Define a selection indicator si for


each i by si = 1 if we observe all of
(yi,xi), and si =0 otherwise.

Consistency Requires:
CONSISTENT
• If s is a function only of the explanatory
variables, then sxj is just a function of x
exogenous sample selection

• If sample selection is entirely random in


the sense that si is independent of (xi,ui),
then E(sxju) =E(s)E(xju) =0, because
E(xju)=0

• If s depends on the explanatory variables


and additional random terms that are
independent of x and u, OLS is also
consistent and unbiased.

Incidental Truncation
• WAGE OFFERS
• the truncation of wage offer is
incidental because it depends on another
variable, namely, labor force
participation. (Observe all Xs for all
individuals)
• The selection equation, depends on
observed variables, zh and an
unobserved error, v.

• A standard assumption, which we will


make, is that z is exogenous

• HERE we will require that x be a strict


subset of z: any xj is also an element of
z, and we have some elements of z that
are not also in x.

• The error term v in the sample selection


equation is assumed to be independent
of z (and therefore x)
.

• We also assume that v has a standard


normal distribution
• Assume that (u,v) is independent of z.

Now, if u and v are jointly normal (with zero


mean), then

Therefore:
We do not observe v, but we can use this
equation to compute E(y/z,s) and then
specialize this to s =1.

Using selection equation and normality:

This equation shows that we get β using


only the selected sample, provided we
include the term as an additional
regressor.
• including all elements of x in z is not
very costly; excluding them can lead to
inconsistency if they are incorrectly
excluded.

• A second major implication is that we


have at least one element of z that is not
also in x. This means that we need a
variable that affects selection but does
not have a partial effect on y. This is not
absolutely necessary to apply the
procedure—in fact, we can mechanically
carry out the two steps when z = x—but
the results are usually less than
convincing unless we have an exclusion
restriction
• The reason for this is that while the
inverse Mills ratio is a nonlinear
function of z, it is often well-
approximated by a linear function. If z =
x, λ can be highly correlated with the
elements of xi. As we know, such multi-
collinearity can lead to very high
standard errors for the β j .

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