Econometric Analysis of Cross Section and Panel Data - Free

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Chapter 17
di¤erence: in censored regression, we observe the covariates x for all people, even
those for whom the response is not known. If we drop observations entirely when the
response is not observed, we obtain the truncated regression model. If in Example
16.1 we use the information in the top coded observations, we are in the censored
regression case. If we drop all top coded observations, we are in the truncated regression
case. (Given a choice, we should use a censored regression analysis, as it uses
all of the information in the sample.)
From our analysis of the censored regression model in Chapter 16, it is not surprising
that heteroskedasticity or nonnormality in truncated regression results in inconsistent
estimators of b. This outcome is unfortunate because, if not for the sample
selection problem, we could consistently estimate b under Eðy j xÞ ¼xb, without
specifying Varðy j xÞ or the conditional distribution. Distribution-free methods for
the truncated regression model have been suggested by Powell (1986) under the assumption
of a symmetric error distribution; see Powell (1994) for a recent survey.
Truncating a sample on the basis of y is related to choice-based sampling. Traditional
choice-based sampling applies when y is a discrete response taking on a finite
number of values, where sampling frequencies di¤er depending on the outcome of y.
[In the truncation case, the sampling frequency is one when y falls in the interval
ða 1 ; a 2 Þ and zero when y falls outside of the interval.] We do not cover choice-based
sampling here; see Manksi and McFadden (1981), Imbens (1992), and Cosslett
(1993). In Section 17.8 we cover some estimation methods for stratified sampling,
which can be applied to some choice-based samples.
17.4 A Probit Selection Equation
We now turn to sample selection corrections when selection is determined by a probit
model. This setup applies to problems di¤erent from those in Section 17.3, where the
problem was that a survey or program was designed to intentionally exclude part of
the population. We are now interested in selection problems that are due to incidental
truncation, attrition in the context of program evalution, and general nonresponse
that leads to missing data on the response variable or the explanatory variables.
17.4.1 Exogenous Explanatory Variables
The incidental truncation problem is motivated by Gronau’s (1974) model of the
wage o¤er and labor force participation.
Example 17.5 (Labor Force Participation and the Wage O¤er): Interest lies in estimating
Eðw o i j x i Þ, where w o i is the hourly wage o¤er for a randomly drawn individual