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Limited Dependent Variable Models

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									      Binary Response Models
  Multivalued response models
            Truncated Models
             Sample Selection




Limited Dependent Variable Models

            Gabriel V. Montes-Rojas
            City University London




        Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Binary Response Models

   Assume that your dependent variable is an indicator/dummy
   variable that takes values 0 and 1. We adopt the convention that a
   value of 1 is called the “success” and 0 the “failure”
   Labour Force Participation: Consider a model when you want to
   estimate the effect of human capital on labour force participation,
   i.e. whether the individual actually works or not. Say you have the
   variable inlf that takes the value 1 if the individual is working and
   0 otherwise.
   Bankruptcy : Consider a model when you want to estimate effect
   of some firm charactaristics on the probability that a firm declares
   bankruptcy.


                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Linear Probability Model

   Let y = 0, 1 be the dependent variable. One option is to use a
   linear probability model of the form:

                               y = β0 + β1 X + u
   How do we interpret β1 ?

                   E[y|X] = β0 + β1 X = P [y = 1|X]
   Then β1 = ∂P [y=1|X] . In other words: β1 gives you the marginal
                 ∂X
   effect on the probability of obtaining a success (i.e. y=1).



                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Linear Probability Model



   There are some drawbacks of using a linear probability model:
    1. Predicted value: the model does not guarantee that
       0 ≤ y ≤ 1.
           ˆ
    2. Heteroskedasticity:
       V ar(y|X) = P [y = 1|X] ∗ (1 − P [y = 1|X])




                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Linear Probability Model




   Consider Example 7.12: A Linear Probability Model of Arrests
   http://fmwww.bc.edu/gstat/examples/wooldridge/wooldridge7.html

   Database: http://fmwww.bc.edu/ec-p/data/wooldridge/CRIME1.des




                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                      Binary Response Models
                  Multivalued response models
                            Truncated Models
                             Sample Selection



Logit and Probit Models
   An alternative specification uses the concept of a cumulative
   distribution function. Let u be a random variable, then its
   cumulative function is

                       P [u ≤ t] = F (t), 0 ≤ F (.) ≤ 1
   Then consider the following latent variable model:

                                y ∗ = β0 + β1 X + e
   But you don’t observe y ∗ , rather

                  y = 1[y ∗ > 0] = 1[e > −(β0 + β1 X)]
   Here 1[.] is an indicator function that takes the value of 1 if the
   argument in brackets is true, 0 otherwise.
                        Gabriel Montes-Rojas    Limited Dependent Variable Models
                  Binary Response Models
              Multivalued response models
                        Truncated Models
                         Sample Selection



Logit and Probit Models


      If we assume that e follows a normal distribution, i.e.
                  2
      e ∼ N (0, σe ), then have the probit model. In this case:
                             ∞
      F (z) = P [e ≤ z] = z φ(v)dv = Φ(z) where φ is the normal
      (or Gaussian) density function and Φ is the normal
      distribution (or cumulative) function.
      Then, P [y = 1|X] = P [e > −(β0 + β1 X)] =
      1 − F (−(β0 + β1 X)) = F (β0 + β1 X) = Φ(β0 + β1 X)
      Then, P [y = 0|X] = P [e ≤ −(β0 + β1 X)] =
      F (−(β0 + β1 X)) = 1 − F (β0 + β1 X) = 1 − Φ(β0 + β1 X)



                    Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



Logit and Probit Models


      If we assume that e follows a logistic distribution, then have
      the logit model. In this case:
                            exp(z)
      F (z) = P [e ≤ z] = 1+exp(z) = Λ(z), where Λ is the
      cumulative distribution function of a logit model.
      Then, P [y = 1|X] = P [e > −(β0 + β1 X)] =
      1 − F (−(β0 + β1 X)) = F (β0 + β1 X) = Λ(β0 + β1 X)
      Then, P [y = 0|X] = P [e ≤ −(β0 + β1 X)] =
      F (−(β0 + β1 X)) = 1 − F (β0 + β1 X) = 1 − Λ(β0 + β1 X)




                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



Logit and Probit Models

      How to interpret coefficients? Note that

            ∂P [y = 1|X]   ∂F (β0 + β1 X)
                         =                = f (β0 + β1 X)β1
                 ∂X             ∂X
      As a result β1 = ∂P [y=1|X] ... then you cannot interpret the
                            ∂X
      coefficients of a probit or logit model directly. For that you
      need f (.), i.e. the density function of your assumed e.
      You can though interpret the direction of the effect through
      the sign:

                                                 ∂P [y = 1|X]
                      sign(β1 ) = sign
                                                      ∂X


                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



Logit and Probit Models

      How to interpret coefficients? Note that

            ∂P [y = 1|X]   ∂F (β0 + β1 X)
                         =                = f (β0 + β1 X)β1
                 ∂X             ∂X
      As a result β1 = ∂P [y=1|X] ... then you cannot interpret the
                            ∂X
      coefficients of a probit or logit model directly. For that you
      need f (.), i.e. the density function of your assumed e.
      You can though interpret the direction of the effect through
      the sign:

                                                 ∂P [y = 1|X]
                      sign(β1 ) = sign
                                                      ∂X


                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



Logit and Probit Models




      For a probit model f (z) = φ(z) = (2π)−1/2 exp(−z 2 /2).
                                          exp(z)
      For a logit model f (z) =         (1+exp(z))2
      But what value of X we have to include in f (β0 + β1 X)? In
                  ¯
      general X = X.




                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



An introduction to Maximum Likelihood estimation
      The dependent variable data consists on {yi }n , 0s and 1s
                                                   i=1
      for each observation.
      If you observe a 1, say yi = 1, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
       ∗
      yi = β0 + β1 Xi + ei > 0, and since e was assumed to be
      probit/logit P [yi = 1|Xi ] = F (β0 + β1 Xi )
      If you observe a 0, say yi = 0, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
                             ∗
      ... this implies that yi = β0 + β1 Xi + ei ≤ 0, and since e was
      assumed to be probit/logit P [yi = 0|Xi ] = 1 − F (β0 + β1 Xi )

                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



An introduction to Maximum Likelihood estimation
      The dependent variable data consists on {yi }n , 0s and 1s
                                                   i=1
      for each observation.
      If you observe a 1, say yi = 1, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
       ∗
      yi = β0 + β1 Xi + ei > 0, and since e was assumed to be
      probit/logit P [yi = 1|Xi ] = F (β0 + β1 Xi )
      If you observe a 0, say yi = 0, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
                             ∗
      ... this implies that yi = β0 + β1 Xi + ei ≤ 0, and since e was
      assumed to be probit/logit P [yi = 0|Xi ] = 1 − F (β0 + β1 Xi )

                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



An introduction to Maximum Likelihood estimation
      The dependent variable data consists on {yi }n , 0s and 1s
                                                   i=1
      for each observation.
      If you observe a 1, say yi = 1, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
       ∗
      yi = β0 + β1 Xi + ei > 0, and since e was assumed to be
      probit/logit P [yi = 1|Xi ] = F (β0 + β1 Xi )
      If you observe a 0, say yi = 0, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
                             ∗
      ... this implies that yi = β0 + β1 Xi + ei ≤ 0, and since e was
      assumed to be probit/logit P [yi = 0|Xi ] = 1 − F (β0 + β1 Xi )

                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



An introduction to Maximum Likelihood estimation
      The dependent variable data consists on {yi }n , 0s and 1s
                                                   i=1
      for each observation.
      If you observe a 1, say yi = 1, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
       ∗
      yi = β0 + β1 Xi + ei > 0, and since e was assumed to be
      probit/logit P [yi = 1|Xi ] = F (β0 + β1 Xi )
      If you observe a 0, say yi = 0, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
                             ∗
      ... this implies that yi = β0 + β1 Xi + ei ≤ 0, and since e was
      assumed to be probit/logit P [yi = 0|Xi ] = 1 − F (β0 + β1 Xi )

                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



An introduction to Maximum Likelihood estimation
      The dependent variable data consists on {yi }n , 0s and 1s
                                                   i=1
      for each observation.
      If you observe a 1, say yi = 1, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
       ∗
      yi = β0 + β1 Xi + ei > 0, and since e was assumed to be
      probit/logit P [yi = 1|Xi ] = F (β0 + β1 Xi )
      If you observe a 0, say yi = 0, what is the associated
      probability that you would have got THIS PARTICULAR
      VALUE?
                             ∗
      ... this implies that yi = β0 + β1 Xi + ei ≤ 0, and since e was
      assumed to be probit/logit P [yi = 0|Xi ] = 1 − F (β0 + β1 Xi )

                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



An introduction to Maximum Likelihood estimation




      ... more generally:
      P [y|X] = [F (β0 + β1 X)]y [1 − F (β0 + β1 X)]1−y




                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                      Binary Response Models
                  Multivalued response models
                            Truncated Models
                             Sample Selection



An introduction to Maximum Likelihood estimation
   What about the whole sample altogether, i.e. {yi }n instead of a
                                                     i=1
   particular observation?
   REMEMBER THE STATISTICAL PROPERTY OF
   INDEPENDENCE. IF TWO EVENTS A AND B ARE
   INDEPENDENT, THEN P [A&B] = P [A] × P [B]

                                                      n
                        P [y1 , y2 , ..., yn |X] =         P [yi |Xi ]
                                                     i=1
                 n
             =         [F (β0 + β1 Xi )]yi [1 − F (β0 + β1 Xi )]1−yi
                 i=1

   This is the likelihood function.

                          Gabriel Montes-Rojas   Limited Dependent Variable Models
                      Binary Response Models
                  Multivalued response models
                            Truncated Models
                             Sample Selection



An introduction to Maximum Likelihood estimation
   What about the whole sample altogether, i.e. {yi }n instead of a
                                                     i=1
   particular observation?
   REMEMBER THE STATISTICAL PROPERTY OF
   INDEPENDENCE. IF TWO EVENTS A AND B ARE
   INDEPENDENT, THEN P [A&B] = P [A] × P [B]

                                                      n
                        P [y1 , y2 , ..., yn |X] =         P [yi |Xi ]
                                                     i=1
                 n
             =         [F (β0 + β1 Xi )]yi [1 − F (β0 + β1 Xi )]1−yi
                 i=1

   This is the likelihood function.

                          Gabriel Montes-Rojas   Limited Dependent Variable Models
                      Binary Response Models
                  Multivalued response models
                            Truncated Models
                             Sample Selection



An introduction to Maximum Likelihood estimation
   In general, it is easier to work with the log-likelihood function
   instead of the likelihood function.
                                                n
                                  L(β) =              i (β)
                                                i=1

   where

                                 i (β)   = log P [yi |Xi ]
      = yi × log F (β0 + β1 Xi ) + (1 − yi ) log[1 − F (β0 + β1 Xi )]
                                                      ˆ
   Then, the maximum likelihood estimator (MLE) is β that
                                                           ˆ
   maximises L(β). In other words, for every possible β, L(β) ≥ L(β)


                        Gabriel Montes-Rojas      Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Probit vs. Logit
   Density functions




                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Probit vs. Logit
   Cumulative distribution functions




                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                    Binary Response Models
                Multivalued response models
                          Truncated Models
                           Sample Selection



Logit and Probit Models


      probit y x1 x2 (probit model)
      logit y x1 x2 (logit model)
      Remember that the coefficients of these models cannot be
      interpreted except for the sign... If you want the marginal effect on
      the probability of success:
      dprobit y x1 x2 (probit model)
      logit y x1 x2 (logit model)
      mfx (this gives you the marginal effects)




                      Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Multinomial logit model
   What can be done if the dependent variable can take several values
   y = 0, 1, 2..., J, but the y values do not represent a particular
   ordering? This is a multinomial model.
   Example: y could be marital status
   y = 0 single
   y = 1 married
   y = 2 divorced
   y = 3 widow
   Example: discrete choice models. y could be place of holiday
   y = 0 Europe
   y = 1 Asia
   y = 2 America
   y = 3 Africa
   y = 4 Oceania
                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



Multinomial logit model


      Select a base group. By convention this corresponds to
      j = 0.
      Each outcome contains a different set of parameters
      βj , j = 1, 2, ..., J
      In a multinomial logit model each probability is of the form

                                                  exp(Xβj )
                     P [y = j|X] =                  J
                                             1+     h=1 exp(Xβh )




                     Gabriel Montes-Rojas     Limited Dependent Variable Models
                    Binary Response Models
                Multivalued response models
                          Truncated Models
                           Sample Selection



Multinomial logit model


      mlogit y x1 x2 x3 (multinomial logit model)
      Remember that the coefficients of these models cannot be
      interpreted except for the sign (similar to probit and logit models)
      mfx, predict(p outcome(1)) (computes the marginal effects for
      y = 1)
      mfx, predict(p outcome(2)) (computes the marginal effects for
      y = 2)




                      Gabriel Montes-Rojas    Limited Dependent Variable Models
                      Binary Response Models
                  Multivalued response models
                            Truncated Models
                             Sample Selection



Ordered probit model

   What can be done if the dependent variable can take several values
   y = 0, 1, 2..., J, and the values of y represent a particular ordering?
   Example: y could be monthly income range
   y = 0 no income
   y = 1 £1 to £500
   y = 2 £501 to £1000
   y = 3 £1001 to £2000
   y = 4 £2001 to £5000
   y = 5 greater than £5000
   Here it does not make much sense to run a OLS model with y as
   the dependent variable...


                        Gabriel Montes-Rojas    Limited Dependent Variable Models
                    Binary Response Models
                Multivalued response models
                          Truncated Models
                           Sample Selection



Ordered probit model


      oprobit y x1 x2 x3 (ordered probit model)
      Remember that the coefficients of these models cannot be
      interpreted except for the sign (similar to probit and logit models)
      mfx, predict(p outcome(1)) (computes the marginal effects for
      y = 1)
      mfx, predict(p outcome(2)) (computes the marginal effects for
      y = 2)




                      Gabriel Montes-Rojas    Limited Dependent Variable Models
                      Binary Response Models
                  Multivalued response models
                            Truncated Models
                             Sample Selection



Tobit Models


   Consider the following latent variable model:

                  y ∗ = β0 + β1 X + u, u|X ∼ N (0, σu )
                                                    2


   But you don’t observe y ∗ , rather

                                   y = max{0, y ∗ }
   Here the variable y ∗ is truncated at 0, i.e. it cannot take negative
   values.




                        Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Tobit Models



   Each observation log-likelihood for this model is

                  i (β, σ)   = 1[yi = 0] log[1 − Φ(xi β/σ)]
                 +1[yi > 0] log[(1/σ)φ ((yi − xi β)/σ)]
   Note that it has two components...




                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                       Binary Response Models
                   Multivalued response models
                             Truncated Models
                              Sample Selection



Tobit Models



   Then,

           E(y|X) = P [y > 0|X] ∗ E[y|y > 0, X] + P [y = 0|X] ∗ 0
                         = P [y > 0|X] ∗ E[y|y > 0, X]




                         Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Tobit Models

   Example: Hours worked. The number of hours you work cannot be
   negative, then h ≥ 0. However, if you consider the model

                                     h = βX + u
   certainly, there is the restriction that h cannot be negative. Then,

        E(h|X) = P [h > 0|X] ∗ E[h|h > 0, X] + P [h = 0|X] ∗ 0
                       = P [h > 0|X] ∗ E[h|h > 0, X]

   Example: Annual amount spent in electronic goods (i.e. TV, DVD
   players). Some years you may declare to have spent £0.


                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Tobit Models



   Here we need some mathematical statistics tools...
   If z ∼ N (0, 1), then E(z|z > c) = φ(c)/[1 − Φ(c)].
   Then,
   E(y|y > 0, X) = Xβ + E(u|u > −Xβ) = Xβ + σφ(Xβ)/Φ(Xβ).
   Here we have used φ(−c) = φ(c) and 1 − Φ(−c) = Φ(c).




                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Tobit Models


   Then,
                    E(y|y > 0, X) = Xβ + σλ(Xβ/σ)
   where λ is the inverse Mills ratio, the ratio of a standard normal
   pdf and cdf.
   Moreover,

      E(y|X) = Φ(Xβ)E(y|y > 0, X) = Φ(Xβ)[Xβ + σλ(Xβ/σ)]




                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                 Binary Response Models
             Multivalued response models
                       Truncated Models
                        Sample Selection



Marginal effects in Tobit models



                                       dλ
            ∂E(y|y > 0, X)/∂xj = βj + βj  (Xβ/σ)
                                       dc
             = βj {1 − λ(Xβ/σ) [Xβ + σλ(Xβ/σ)]}


                    ∂E(y|X)/∂xj = βj Φ(Xβ/σ)




                   Gabriel Montes-Rojas    Limited Dependent Variable Models
                   Binary Response Models
               Multivalued response models
                         Truncated Models
                          Sample Selection



Tobit Models




      tobit y x1 x2 (tobit estimation)
      mfx compute, predict(ystar(0,.))
                       ¯
      (∂E(y|y > 0, X = X)/∂xj )
                                             ¯
      mfx compute, predict(e(0,.)) (∂E(y|X = X)/∂xj )




                     Gabriel Montes-Rojas    Limited Dependent Variable Models
                    Binary Response Models
                Multivalued response models
                          Truncated Models
                           Sample Selection



Tobit Models




   Consider Example 17.2: Married Women’s Annual Labor Supply
   http://fmwww.bc.edu/gstat/examples/wooldridge/wooldridge17.html




                      Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Sample selection models
   Consider the following model.
   The outcome equation is

                          y ∗ = Xβ + u, E(u|X) = 0

   However, we only observe the dependent variable if something
   happens.
   The selection equation is

                                     Zγ + v > 0
   then

                            y = y ∗ × 1[Zγ + v > 0]

                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Sample selection models
   Consider the following model.
   The outcome equation is

                          y ∗ = Xβ + u, E(u|X) = 0

   However, we only observe the dependent variable if something
   happens.
   The selection equation is

                                     Zγ + v > 0
   then

                            y = y ∗ × 1[Zγ + v > 0]

                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                    Binary Response Models
                Multivalued response models
                          Truncated Models
                           Sample Selection



Sample selection models

   Under certain conditions OLS is biased. Assume that u and v are
   correlated, i.e. corr(u, v) = ρ. Then,


   E(y|y > 0, X) = E(y|y > 0, X, Zγ+v > 0) = Xβ+E(u|Zγ+v > 0)

                               = Xβ + ρσu λ(Zγ)
   Now,

             ∂E(y|y > 0, X)/∂xj = βj + ρσu ∂λ(Zγ)/∂xj




                      Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Sample selection models
   There are two ways of estimating these models:
   1. MLE
       heckman y x1 x2, select(c= z1 z2)

   2. Heckman’s two-step estimator (James Heckman won the Nobel
   Prize for this...)
       heckman y x1 x2, select(c= z1 z2) twostep

    1. Here the idea is that you estimate a probit model first, to get
                                          ˆ
       1[Zγ + e > 0], that is to estimate γ .
                                                  ˆ          γ
    2. Then you construct the inverse Mills ratio λ(Zγ) = λ(Zˆ ).
    3. Then you run a regression of

                                            ˆ
                                  y = Xβ + αλ(Zγ) + e
                       Gabriel Montes-Rojas    Limited Dependent Variable Models
                     Binary Response Models
                 Multivalued response models
                           Truncated Models
                            Sample Selection



Sample selection models
   There are two ways of estimating these models:
   1. MLE
       heckman y x1 x2, select(c= z1 z2)

   2. Heckman’s two-step estimator (James Heckman won the Nobel
   Prize for this...)
       heckman y x1 x2, select(c= z1 z2) twostep

    1. Here the idea is that you estimate a probit model first, to get
                                          ˆ
       1[Zγ + e > 0], that is to estimate γ .
                                                  ˆ          γ
    2. Then you construct the inverse Mills ratio λ(Zγ) = λ(Zˆ ).
    3. Then you run a regression of

                                            ˆ
                                  y = Xβ + αλ(Zγ) + e
                       Gabriel Montes-Rojas    Limited Dependent Variable Models

								
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