Research Method

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Research Method

• Lecture 15-2
• Censored regression

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Censored regression

• Sometimes, the dependent variable (y-variable) is
  right-censored (censored from above) or
  left-censored (censored from below).
• Example 1 In some household survey, when the
  wealth of a family exceeds 500,000, the data is
  recorded as 500,000 even if the actual wealth
  may be much higher than that amount. This is
  called the top coding. Top coding is done to
  protect the identity of the survey participants.
  In this case, wealth is right-censored.

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Top-coding example
Wealth
500,000
Educ of the head of the family
When wealth exceeds the threshold, say 500,000,
the data record it as 500,000
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• Example 2 Duration data. Suppose that a survey
  is conducted to measure the duration of
  unemployed workers to find a job. If the survey
  is conducted for 12 months, some survey
  participants may not have found a job. For those
  workers, the researcher only knows that the
  duration is greater than 12 months. Thus, the
  duration is right-censored.

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• When a variable is censored from above, you only
  know that the variable is at least equal to the
  threshold value.

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The censored regression model

• Here, I will explain the censored regression
  model for the case where the dependent variable
  is right censored.
• Let yi be the actual value of the dependent
  variable for the ith person.
• However, when yi exceeds certain threshold, ci,
  the data is recorded as ci. In such a case, the
  observation is said to be right-censored.

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• Let wi be the observed value of the dependent
  variable, which may be censored. Then the model
  is written as
• yiß0ß1xiui uN(0,s2)
• wiyi if yiltci
• ci if yici
• In the top-coding example, the threshold value is
  the same for all the people at 500,000. But, the
  threshold value can be different for different
  individual. Thus, we have i-subscript for the
  threshold values.

This can be also written as wimin(yi,ci)
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• It should be emphasized that, in the censored
  regression model, ci are known values. For
  example, in the top coding example, it is
  500,000 for all the observations.

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• When the person is not right censored, we have
  wiyi. Thus, we have uiwi-(ß0ß1xi). Thus, the
  likelihood contribution is the height of the
  density function, which is given by

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• If the person is right-censored, we only know
  that yici. Thus, the likelihood contribution of
  this person is given by

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• To summarize

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• Now, let Di be the dummy variable that takes the
  value 1 if the ith person is right-censored. Then
  the likelihood contribution for the ith person is
  conveniently written as

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• Note, that computation-wise, Tobit model is the
  same as the censored regression model where the
  actual dependent variable is left-censored, and
  ci0 for all observations.

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The partial effect

• In censored regression model, our interest is to
  estimate the effect of x-variable on y, not w.
  Since ß1 is the partial effect of x on y, you can
  use ß1 as the partial effect. No difficult
  computation is necessary. You can interpret the
  coefficients as if it were OLS.
• This is very different from the Tobit regression
  model. In Tobit model, our interest is to
  estimate the effect of x on w, not y. Thus, we
  had a very complicated partial effect formula in
  the case of Tobit.

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ExerciseDuration analysis of recidivism

• Recidivism is an act of a person repeating an
  undesirable behavior.
• The data RECID.dta contains the duration (in
  month) until an inmate who are released from the
  prison is imprisoned again.
• 1445 released inmates were followed for a certain
  period of time.

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• Among them, 893 of them had not been arrested
  again. Thus, the duration for those inmates are
  right-censored We only know that the duration
  until they would come back to prison is at least
  as long as the recorded duration.
• Now, we want to estimate the determinants of the
  duration of prisoner recidivism.
• Although modern duration analysis is mostly
  conducted using hazard function analysis or
  survival function analysis, censored regression
  is also a valid model for a duration analysis.

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• Using RECID.dta, estimate the censored regression
  model of the duration of recidivism. Explanatory
  variables are workprog, priors, tserved, felon,
  alcohol drugs, black, married, educ age. Use the
  log of duration as the dependent variable.

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Put the censoring indicator. This indicator must
be 1 if right censored, -1 if left censored, and
0 if uncensored.