We compare the performance of LPM with probit and logit model for Heckman correction and propensity score matching respectively.

1. Selection Bias with
Linear Probability Models
(LPM)
Suneel Chatla
Galit Shmueli
Institute of Service Science,
National Tsing Hua University, Taiwan

2. Outline
Ø Introduction to self selection
Ø Popular methods for selection bias
correction
o Two step methods (2SLS)
o Matching methods (PSM)
Ø Incorporating LPM into 2SLS and PSM
Ø Simulation study
Ø Conclusions

3. Quasi-experiments
Like randomized
experimental designs
that test causal
hypotheses but
lack random
assignment
(=self selection)

4. Pros
• When random assignment is impractical
and/or unethical
• Easier to setup, greater external validity
• Minimize threats to ecological validity
Cons
• Estimates are subject to contamination
by confounding variables (Biased)
• Do not have total control over
extraneous variables
Why we need Quasi experiments?

5. Two Methods for Addressing Selection Bias

6. Two Methods for Addressing Selection Bias

7. Two step methods: Heckman vs Olsen
Stage 1:
Selection
model (T)
Adjustment
Stage 2:
Outcome
model (Y)
𝐸[𝑇|𝑋] = Φ(𝑋𝛾) 𝐼𝑀𝑅 =
𝜙(𝑋𝛾)
Φ(𝑋𝛾) 𝑌 = 𝑋𝜷 + 𝛿 𝐼𝑀𝑅 + 𝜀
Heckman
(1977)
𝐸[𝑇|𝑋] = 𝑋𝛾 𝜆 = 𝑋𝛾 − 1 𝑌 = 𝑋𝜷 + 𝛿 𝜆 + 𝜀
Olsen
(1980)
Probit
LPM

8. Heckman’s
• Bivariate normality
• Inconsistent second stage
standard errors
• Identification issues
• Expensive computation
• Convergence issues
Olsen’s
• Linear conditional expectation
• Inconsistent second stage
standard errors
• Identification issues
• Cheaper computation
• No convergence issues
In Short: For Continuous Outcome

9. Open Research Questions
1. Selection model with unequal sample sizes
(treat/control) - continuous outcome
2. Binary outcome model – coefficient consistency
3. Selection model with unequal sample sizes
(treat/control) + binary outcome model with
unequal sample sizes

10. Simulation Design
Selection model:
𝑆∗ = −0.5 + 0.5𝑥? − 0.5𝑥@ + 1.5𝑥A − 𝑥B + 𝜔
𝑇 = D
1 𝑖𝑓 𝑆∗
> 0
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Continuous Outcome model:
𝑌 = 0.5 − 1.5𝑥? + 0.5𝑥@ + 𝑥A + 𝜀
Binary Outcome:
𝑌O = D
1 𝑖𝑓 𝑌 > 0
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑁
0
0
,
0.5 −0.4
−0.4 0.5

11. Q1: Continuous outcome: treat/control sample size
ratio has no influence

12. Q2: Binary outcome - coefficients inconsistent
How about marginals?

13. Q3: Binary outcome - divergence of marginals with imbalance ratio
Outcome cut-off 50% Outcome cut-off 25% Outcome cut-off 5%
Selectioncut-off50%Selectioncut-off25%Selectioncut-off5%

14. Summary: Heckman Vs Olsen
Ø Continuous outcome: Heckman and Olsen
corrections are similar, even when unbalanced
Ø Binary outcome: marginal effects from Heckman
and Olsen corrections, diverge with imbalance
ØLPM in both stages provides consistent estimates
(OLS)
ØBut how about Probit?

15. Two Methods for Addressing Selection Bias

16. Matching Methods
Stage 1:
Selection
model (T)
Covariate
balance
Stage 2:
Outcome
model (Y)
𝑙𝑜𝑔𝑖𝑡(𝐸 𝑇 𝑋 ) = (𝑋𝛾)
| 𝑝 𝑇 = 1
− 𝑝 𝑇 = 0 |
< 𝜀
𝑌 = 𝑋𝜷 + 𝜀
Rosenbaum
and Rubin
(1985)
𝐸 𝑇 𝑋 = (𝑋𝛾)
| 𝑝 𝑇 = 1
− 𝑝 𝑇 = 0 |
< 𝜀
𝑌 = 𝑋𝜷 + 𝜀LPM

17. Propensity Score Matching (PSM)
ü Only accounts for observable/observed covariates
ü Requires large samples and substantial overlap
between treatment and control
ü What happens to ATE if we use LPM for matching?

18. Simulation Design
Selection model:
𝑇 = 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖( Z
Z[](^_[`))
Outcome model :
𝑌 = 𝑇 + 𝑋𝛽 + 𝜀
𝑁(0, {0.1,1,5})
𝑋 ~ 𝑁 0,1 and 𝛽 = 1

19. • Sample size
1000
•Standard
deviation0.1,1,5
• Bootstrap
50
• 𝑚𝑒𝑎𝑛 𝑌hi? −
𝑚𝑒𝑎𝑛 𝑌hij
ATE
Identical ATE from Logit and LPM matching

20. Summary & Future Research
ü LPM similar to logit in terms of estimated Average
Treatment Effect
ü Ongoing work: what about binary outcome
models?
ü Logit faces problems if insufficient overlap between
treat/control
ü Ongoing work: does LPM have overlap issues?

21. Thank you!