Sample Selection. In a UW Econ Ph.D. Model. April 25, Kevin Kasberg and Alex Stevens ECON 5360 Dr. Aadland

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1 Sample Selection In a UW Econ Ph.D. Model Kevin Kasberg and Alex Stevens ECON 5360 Dr. Aadland April 25, 2013

2 Sample Selection Bias Sample Selection Bias (Incidental Truncation): When estimates from a non-random sample are used to generalize about the wider population Requires the researcher to explain why certain observations are systematically not included in the sample through a selection equation. Ignoring the selection criteria results in similar issues as omitted variable bias (inconsistent estimates of β) Sample Selection Intro 2 / 10

3 Where Does it happen? Labor Supply Models: only individuals with observed wages are included in sample Macro-growth models: Lack countries that do not report relevant information Removing outliers in data when in reality they account for something critical Partitioning data non-randomly Publication Bias Every Contemporary Medical study Sample Selection Intro 3 / 10

4 How is it dealt with? Need to figure out what is causing participation in the sample. If participation can be determined by z i = w i γ + u i then it should be a part of our primary equation y i = x i β + ϵ i. By not including the participation equation in the primary equation, omitted variable bias would occur because corr(u i, ϵ i ) 0. With the sample rule that y i only has positive values when z i is greater than zero (for probit selection equation, but can have other forms). Sample Selection Intro 4 / 10 Greene, 874

5 Estimation E[y i y i is observed] =E[y i z i > 0] =E[y i u i > w i γ] =x i β + E[ϵ i u i > w i γ] =x i β + ρσ ϵλ i (α u ) =x i β + β λλ i (α u ) where λ(α u ) = ϕ(w i γ)/φ(w iγ), the inverse Mill's Ratio or hazard rate. The denominator is the probability (CDF) that observation i has data for the primary equation. The y i z i > 0 =E[y i z i > 0] + υ i =x i β + β λλ i (α u ) + υ i Sample Selection Estimation 5 / 10 Greene, 874

6 ML or Two-Step The associated log likelihood function lnl = z=1 ln [ exp ( ( 1/2) ϵ 2 i /σ 2 ϵ ) σ ϵ 2π Φ ( ) ρϵ i /σ ϵ + w i γ + [ ] ] 1 lnφ(w 1 ρ 2 i γ) z=0 can be estimated. Alternatively, there is a two step procedure explained in Heckman (1976), The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for Such Models. The procedure is similar to IV and as Kennedy put it, "practitioners are eager to find a practical alternative to ML". Greene, 874; Kennedy, 252 Sample Selection Estimation 6 / 10

7 Heckman Two Step 1. Estimate the Probit selection equation and obtain estimates of γ. 2. Create a new explanatory variable for each observation, λ i = ϕ(w i γ )/Φ(w i γ ) 3. Estimate β and βλ = ρσϵ. 4. With some algebra, obtain consistent estimates of ρ and σ ϵ 5. With more algebra, obtain asymptotic covariance matrix Greene, 876 Sample Selection Heckman 7 / 10

8 Extensions of the Heckman Important to test for selection bias (significance of β λ ) The Heckman Method has is often called the "Heckit" method to fit into the "it" estimators. The Heckman method is similar to hurdle models and the selection equation can be thought of as a Tobit model. There are many variations to the Heckman model: Endogenous Selection models, Qualitative dependent variables in the primary Sample Selection Heckman 8 / 10

9 Heckman vs. ML Heckman is inferior to ML because although it is consistent, it is inefficient. This is due to a measurement error in the expected value of the selection equation's error. Literature exploring the issues with Heckman (see Kennedy p.270) find that the MSE criterion of the Heckman method relative to no sample selection equation is not always better. Small sample sizes and identical regressors in the selection equation and the primary equation are not ideal. Some benefit of the Heckman method is getting explicit Inverse Mill's ratio (important in certain fields) and there are applications where ML has trouble getting the expected value of the error. Sample Selection Heckman 9 / 10 Kennedy, 270

10 UW Econ Ph.D. Degree Years Accepted to UW Econ. Ph.D. Program 1-Φ(w i γ) Fail Comps Φ(w i γ) Pass Comps OrderedProbit Years to degree Sample Selection Application 10 / 10