More on Limited Dependent Variables

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More on Limited Dependent Variables

• Lecturer Zhigang Li

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Interpreting the effect of binary response models

• We are often more interested in the partial
  effect of Xi (the effect on the probability of
  success)
• The partial effect depends on X and can be easily
  calculated for different values of X.
• For example, when Xß0, the scaling factor is
  approximately .4 for probit and .25 for logit.
• Typically, we report the partial effect for the
  sample averages of X. This can be easily obtained
  using the dprobit command of Stata.

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Specification Issue 1 Neglected (omitted)
Heterogeneity

• Neglected (omitted) Heterogeneity
• Omitted variables are independent of the included
  explanatory variables.
• Consequence
• Parameters are inconsistent
• Partial effect for specific value of c would be
  incorrect
• Nevertheless, average partial effect (APE) for
  the whole distribution of c can be still be
  correctly estimated.

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Specification Issue 2 Continuous Endogenous
(Explanatory) Variables

• One possible solution is to estimate a linear
  probability model using 2SLS.
• Another solution is to estimate a probit model
  use the Rivers-Vuong two-step approach.

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Rivers-Vuong two-step approach

• y1z1d1a1y2u1
• y11y1gt0
• y2z1d21z2d22v2 zd2v2
• v2 and u1 may be correlated.
• Assume v2 and u1 follow a joint normal
  distribution, then u1?1v2e1. e1 is independent
  of v2
• Step 1 Estimate v2
• Step 2 Use probit model to regress
• y1z1d1a1y2 ?1v2e1

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Unobserved Effects Probit Models under Strict
Exogeneity

• P(yit1)F(xitßci)
• A fixed-effect approach that estimates ß and ci
  (like linear fixed-effect model) is infeasible.
• Solution 1 Estimate a random effect probit model
  (i.e. c and x are independent)
• Solution 2 (Chamberlains random effects probit
  model) Assume
• Solution 3 Estimate a fixed effects logit model

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On Heckman Selection Procedure with Endogenous
Explanatory variables

• y1z1d1a1y2u1
• y2 zd2v2
• y31zd3v3 gt0
• First estimate the selection equation to compute
  the inverse Mills ratio
• Estimate the first equation with the inverse
  Mills ratio and z as instrument for y2.
• At least two elements of z are not also in z1.

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Sample Selection in Linear Panel Data Models

• Two issues
• Endogenously unbalanced panel Units leave and
  enter the sample over time based on factors
  related to disturbance.
• Incidental truncation Units covered by the
  sample do not change but some variables are
  unobserved for some time periods based on factors
  related to disturbance.

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Fixed effects Estimation with Unbalanced Panels

• Sample selection is a problem for fixed effects
  estimation when the selection is related to the
  disturbances.
• yitxitßciuit
• sit1 if (xit, yit) is observed
• Key assumption for consistency
• Error term is mean independent of si for all t (
  E(uitxi,si,ci)0)
• Estimates from unbalanced panel and the balanced
  subsample are all consistent.

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Testing for Sample Selection Bias

• The Nijman and Verbeek approach
• Including the lagged or lead selection indicator
  (e.g. si,t-1 or si,t1) to the fixed effects
  model and test its significance.
• Panel Heckmans test
• yit1xit1ß1ci1uit1
• sit21xi?t2vit2gt0
• Estimate the selection equation using pooled
  probit and calculate the inverse Mills ratio.
• Include the inverse Mills ratio to the fixed
  effects model and test its significance.

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Correcting for sample selection bias Incidental
truncation with panel

• Chamberlains approach
• Key assumption
• ci1xip1ft1vit2
• With inverse Mills ratio, the following
  regression can be estimated consistently
• yit1xit1ß1xip1?t1?(xi?t2)eit
• Tobit selection equation

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Correcting for sample selection bias Attrition

• Focus on attrition in which units that leave the
  sample do not reenter.
• Approach Estimate the first-difference model
  with probit selection equation
• ?yit?xitß?uit
• sit1witdtvitgt0
• Key assumptions
• xit does not affect attrition once the elements
  in wit have been controlled for
• Strict exogeneity of xit (may be relaxed with IV)