SAMPLE SELECTION

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SAMPLE SELECTION

• Cheti Nicoletti
• ISER, University of Essex
• 2009

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Wage equation and labour participation for women

• Gourieroux C. (2000),  Econometrics of
  Qualitative Dependent Variables, Cambridge
  University Press, Cambridge
• Let y be the potential offered wage and let w be
  the reservation wage then the observed wage y is
  given by
• Let us consider the following very simple
  earnings profile equation

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Women in the labour force are not a random sample

• Womens labour force participation rates are
  highly dependent on age. Gourieroux (2000)
• Labour participation is in general lower for
  women aged
• 16-20 because some women are still studying
• 25-44 for work interruption linked to children
• 55-60 because some women prefer to retire early
• Presumably the earnings observed for women aged
• 16-20 are lower than if all women worked
• 25-44 are higher because women with higher
  earnings are less incline to work interruptions
• 55-60 are higher because women with higher
  earnings are less incline to retire early

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Sample selection model Labour participation
equation

• Probit model for labour participation

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Joint model for the log-earnings and the labour
participation equationsGeneralized TOBIT MODEL

• Possible candidates for x education dummies,
  age, work experience
• Possible candidates for z age, education, number
  of children, dummies for the presence of children
  lt5, for cohabiting, for widow, regional
  unemployment rate.

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Bivariate normal
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Truncated Normal

• Suggestions for the proof

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Sample selection problem

• E(yd1,x,z)x?E(?d1,x,z)
• E(?d1,x,z) E(?ugt-zd )
• E(yd1,x,z) X?

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Two-step estimation

• 1 STEP estimation of a probit model for the
  probability to be in the labour market,
• ? Pr(di1zi)di Pr(di0zi)1-di? ?(zi ?) di
  ?(-zi ?) 1-di
• 2 STEP estimation of the regression model with
  an additional variable (the inverse Mills
  ratio) using the subsample of individuals with
  di1 (and using some IV restrictions)

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Testing selectivity

• If the error terms ? and u are uncorrelated, then
  the selection problem is ignorable.
• H0 s?u 0
• Verifying H0 is equivalent to verify whether the
• coefficient of the additional variable in the
• equation is zero (using for ex. a Wald test)
• Notice that the errors are heteroskedastic so a
  proper estimation should be adopted to estimate
  the standard errors

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Generalized Tobit Maximum Likelihood Estimation
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heckman

• The heckman command is used to estimate
  Generalized Tobit or Tobit of the 2nd type using
  ML estimation (default option) or the two-step
  estimation (option twostep)
• heckman y x1 x2 xk, select(z1 z2 zs)
• heckman y x1 x2 xk, select(d z1 z2 zs)
• heckman y x1 x2 xk, select(z1 z2 zs) twostep

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Generalized Tobit Maximum Likelihood Estimation
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Joint model for log-income and response
probability

• Possible candidates for x education dummies,
  age, work experience
• d is the propensity to respond to the earnings
  question
• Z mode of interview, education, gender, age,
  etc.

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Item nonresponse for income equation or poverty
model in cross section sample surveys

• Potential explanatory variables
• Socio-demographic variables age, gender, level
  of education, number of adults, number of
  children.
• Situational economic circumstance labour status
  activity.
• Data collection characteristics mode of the
  interview, number of visits, duration of the
  interview. (These are plausible IV)

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Attrition in panel surveys has two possible
causes failed contact and refusal

• The potential variables explaining attrition
  (contact and cooperation) are lagged variables
  observed in the last wave.
• The equation of interest has to use lagged
  variables (otherwise we have missing explanatory
  variables too)
• Socio-demographic variables age, gender, level
  of education, number of adults, number of
  children.
• Social-integration talking often to neighbours,
  cohabitation, house ownership.
• Situational economic circumstance labour status
  activity, household equalised income.
• Data collection characteristics mode of the
  interview, number of visits, duration of the
  interview, same interviewer across wave, duration
  of the panel, length of the fieldwork. (These are
  plausible IV)

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Attrition due to lack of cooperation (BHPS
1994-96)
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Weighted estimation
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Weighted estimation
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• Conditioning and integrating out (marginalizing)
• with respect to z
• EZ (Ex(y-xß)dp-1x,z)
• EZ (Ex(y-xß)x,z,d1 Pr(d1x,z)p-1)
• EZ (Ex(y-xß)x,z)Ex(y-xß)x0

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How to use weights in Stata

• Most Stata commands can deal with weighted data.
  Stata allows four kinds of weights
• fweights, or frequency weights, are weights that
  indicate the number of duplicated observations.
• pweights, or sampling weights, are weights that
  denote the inverse of the probability that the
  observation is included due to the sampling
  design, nonresponse or sample selection.
• aweights, or analytic weights, are weights that
  are inversely proportional to the variance of an
  observation i.e., the variance of the j-th
  observation is assumed to be sigma2/w_j, where
  w_j are the weights.
• iweights, or importance weights, are weights that
  indicate the "importance" of the observation in
  some vague sense.

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Option pweights

• Usually sample surveys provide weights to take
  account of sampling design, nonresponse .
• Let p be individual weight
• Then we can run a regression with weighted
  observations
• regress y x1 x2 xk pweightp
• Let us assume to have a random sample affected by
  nonresponse, but weights to take account of unit
  nonresponse are not available
• A possible way to estimate your own weights is
  described in the following
• probit d z1 z2 zs
• predict prop
• gen invprop1/prop
• reg y x1 x2 xk pweightinvprop

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For complex survey design it is better to use

• svyset pweightp
• svy regress y x1 x2 xk
• svyset have options for cluster sampling designs
  or other complex design
• To declare survey design with stratum
• svyset pweightp, strata(stratid)

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Stata propensity score methods for evaluation of
treatment

• Abadie A., Drukker D., Herr J.L., Imbens G.W.
  (2001), Implementing Matching Estimators for
  Average Treatment Effects in Stata, The Stata
  Journal, 1, 1-18 http//ksghome.harvard.edu/.aaba
  die.academic.ksg/software.html
• Becker S.O., Ichino A. (2002), Estimation of
  average treatment effects based on propensity
  scores. The Stata Journal, 2, 358-377
  http//www.lrz-muenchen.de/sobecker/pscore.html
• Sianesi B. (2001), Implementing Propensity Score
  Matching Estimators with STATA, UK Stata Users
  Group, VII Meeting London, http//ideas.repec.org/
  c/boc/bocode/s432001.html

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Some references for regressions with sample
selection

• Buchinski, M. (2001) Quantile regression with
  sample selection Estimation women return to
  education in the U.S., Empirical Economics, 26,
  86-113.
• Ibrahim, J.G., Chen, M.-H., Lipsitz, S.R.,
  Herring, A.H. (2005) Missing-data methods for
  generalized linear models A comparative review,
  Journal of the American Statistical Association,
  100, 469, 332-346.
• Lipsitz, S.R., Fitzmaurice, G.M., Molenberghs,
  G., Zhao, L.P. (1997), Quantile regression
  methods for longitudinal data with drop-outs,
  Applied Statistics, 46, 463-476.
• Robins, J. M., Rotnitzky, A. (1995),
  Semiparametric Effciency in Multivariate
  Regression Models With Missing Data, Journal of
  the American Statistical Association, 90,
  122-129.
• Vella F. (1998), Estimating models with sample
  selection bias a survey', The Journal of Human
  Resources, vol. 3, 127-169.
• Wooldridge, J.M. (2007) Inverse probability
  weighted M-Estimation for General missing data
  problems, Journal of Econometrics, 141, 2,
  1281-1301.
• Wooldridge, J.M. (2007) Inverse probability
  weighted M-Estimation for General missing data
  problems, Journal of Econometrics, 141, 2,
  1281-1301.