Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
for reasons of confidentiality, the dataset does not report any information for individuals with a large wage the dependent variable is truncated to the ...
Typology: Study notes
1 / 10
This page cannot be seen from the preview
Don't miss anything!
OLS and Heckman's modelTruncation Summary
Econometrics II
Ricardo Mora
Department of Economics Universidad Carlos III de Madrid Máster Universitario en Desarrollo y Crecimiento Económico
Ricardo Mora Heckman's Selection Model
Introduction OLS and Heckman's modelTruncation Summary
(^1) Introduction
(^2) Truncation
3 OLS and Heckman's model
Notes
Notes
OLS and Heckman's modelTruncation Summary
Investment in capital equipment q∗ i = xi β + εi
we observe qi =
q∗ i if q i∗ > 0 0 if q i∗ ≤ 0
rms only carry out investment decisions if their net discounted value is positive the censored dependent variable is a latent variable which is the result of our economic model this is the Tobit model
Ricardo Mora Heckman's Selection Model
Introduction OLS and Heckman's modelTruncation Summary
wi = xi β + εi we only observe (wi , xi ) if wi ≤ W
for reasons of con dentiality, the dataset does not report any information for individuals with a large wage the dependent variable is truncated to the right because of the data collection mechanism this is a truncated regression model
Notes
Notes
OLS and Heckman's modelTruncation Summary
de ne s = 1 (β 0 + β x + ε > 0 ) note that sy = sβ 0 + β sx + sε then E [(sx) (sε)] = E [sxε] (note that s^2 = s)
OLS is inconsistent because E [sxε] 6 = 0
Ricardo Mora Heckman's Selection Model
Introduction OLS and Heckman's modelTruncation Summary
The density of the sample is not a normal density because the population has been truncated We need the distribution of yi given xi AND given that yi > 0 Joint density for (yi , yi > 0 ) given xi :
σ
φ
( (^) εi σ
Pr (yi > 0 |xi ) = Φ
β xi σ
Li (β , σ ) =
( (^) σ^1 )φ
( (yi −β^ xi ) σ
)
Φ
( (^) β xi σ
)
Notes
Notes
OLS and Heckman's modelTruncation Summary
we observe wi if si = 1 output equation: w = β 0 + β x + ε participation equation: s = 1 (γ′z + v ) [ u v
σ (^) u^2 ρ ρ 1
we can generalize this model to include another output equation for those for whom s = 0
Ricardo Mora Heckman's Selection Model
Introduction OLS and Heckman's modelTruncation Summary
note that sw ∗ = sβ 0 + β sx + sε then E [sx ∗ sε |x, z ] = E [sε |x, z ] x because s^2 = s therefore, OLS will be biased if E [sε |x, z ] 6 = 0
OLS is inconsistent if ρ 6 = 0
Notes
Notes
OLS and Heckman's modelTruncation Summary
from the Tobit model, we know that E [w |x, z, s = 1 ] = xβ + ρλ (zγ)
where λ () is the inverse Mills ratio
λ is like a missing variable which is correlated with ε if ρ = 0, no problem with OLS
Ricardo Mora Heckman's Selection Model
Introduction OLS and Heckman's modelTruncation Summary
Heckman's two-step sample selection correction First Step: Using all observations, estimate a probit model of work on z and compute the inverse of Mills ratio, ˆλi = φˆi Φ^ ˆi Second Step: using the selected sample, ols wage on x and λˆ
β^ ˆ is consistent and asymptotically normal
Notes
Notes
OLS and Heckman's modelTruncation Summary
ML estimates of the participation equation are consistent ˆλ shifts the conditional expectations of those individuals more likely to work due to unobservable factors in the right direction assume that ρ > 0: a wage observation with a low index zγ (high λi ) is likely to work due to unobservable factors and also more likely to have higher wages in the sample due to unobservable factors: λi should be large a wage observation with a high index zγ (low λi ) is less likely to work due to unobservable factors and also less likely to have higher wages due to unobservable factors: λi should be small
Ricardo Mora Heckman's Selection Model
Introduction OLS and Heckman's modelTruncation Summary
OLS (Robust) Standard Errors in second step are invalid It is possible to test for sample selection: t test on ρˆ in second step If there are endogenous controls in wage equation, we replace OLS by 2SLS in second step The method works best if x ⊂ z (i.e. some variables appear only in participation equation)
Notes
Notes
OLS and Heckman's modelTruncation Summary
there is a variety of ways to account for sample selection Stata allows for estimation of Heckman's Selection Model both two-stage and ML estimation testing and prediction is computed as usual
Ricardo Mora Heckman's Selection Model
Notes
Notes
