# More on limited Dependent Variable Models by hongkonguniv

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More on Limited Dependent Variables
Lecturer: Zhigang Li

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 Xβ=0, 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.

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.

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.

Rivers-Vuong two-step approach
y1*=z1δ1+α1y2+u1 y1=1[y1*>0] y2=z1δ21+z2δ22+v2= zδ2+v2  v2 and u1 may be correlated.

 Step 1: Estimate v2  Step 2: Use probit model to regress y1*=z1δ1+α1y2+ θ1v2+e1

 Assume v2 and u1 follow a joint normal distribution, then u1=θ1v2+e1. e1 is independent of v2

Unobserved Effects Probit Models under Strict Exogeneity
 P(yit=1)=Φ(xitβ+ci)  A fixed-effect approach that estimates β 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 (Chamberlain’s random effects probit model): Assume ci    xi  ai  Solution 3: Estimate a fixed effects logit model

On Heckman Selection Procedure with Endogenous Explanatory variables
y1=z1δ1+α1y2+u1 y2= zδ2+v2 y3=1[zδ3+v3 >0]  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.

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.

Fixed effects Estimation with Unbalanced Panels
 Sample selection is a problem for fixed effects estimation when the selection is related to the disturbances.
 yit=xitβ+ci+uit  sit=1 if (xit, yit) is observed

 Key assumption for consistency
 Error term is mean independent of si for all t ( E(uit|xi,si,ci)=0)  Estimates from unbalanced panel and the balanced subsample are all consistent.

Testing for Sample Selection Bias
 The Nijman and Verbeek approach
 Including the lagged or lead selection indicator (e.g. si,t-1 or si,t+1) to the fixed effects model and test its significance.

 Panel Heckman’s test
yit1=xit1β1+ci1+uit1 sit2=1[xiΨt2+vit2>0]  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.

Correcting for sample selection bias: Incidental truncation with panel
 Chamberlain’s approach
 Key assumption: ci1=xiπ1+φt1vit2  With inverse Mills ratio, the following regression can be estimated consistently: yit1=xit1β1+xiπ1+γt1λ(xiΨt2)+eit

 Tobit selection equation

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
Δyit=Δxitβ+Δuit sit=1[witδt+vit>0]

 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)

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