Question

To test the effectiveness of a job training program on the subsequent wages of workers, suppose
we estimate the following model using individual level data:

log(wage_i) =b₀ + b₁train_i+ b₂ educ_i + b₃ exper_i + b₄ part-time_i + u_i (3)
where train=1 if a worker participated in the program and =0 otherwise, educ=the years of
education a worker has, exper=the number of years of work experience the worker has, and parttime=
1 if the worker held a part-time job(<30hrs/week) and zero otherwise. The error term, u,
contains unobserved worker ability (or motivation or enthusiasm).

a) If less able workers have a greater chance of being selected for (and participate in) the program,
and you use an OLS analysis, what can you say about the likely bias in the OLS estimator of b₁?
Explain/show how you arrived at this expectation of the bias. [3]

Answer 1

a) the likely bias in OLS estimate of b1 is negative. It means the effect of training is underestimated. let's see why.

The correct regression equation should have been:

log(wagei) =b0 + b1traini+ b2 educi + b3 experi + b4 part-timei + b5 abilityi + ui (3)

where ability i = c0 + c1 train i ( c1 is negtivehigher training means lower abilit of workers)

Putting thi value of ability i in the regression equaion we are actually estimating i.e

log(wagei) =b0 + b1traini+ b2 educi + b3 experi + b4 part-timei + ui (3)

we get

log(wagei) =b0 + b1traini+ b2 educi + b3 experi + b4 part-timei + b5(c0 + c1 train i ) + ui (3)

=> log(wagei) = (b0 + b5c0) + (b1 + b5c1)traini+ b2 educi + b3 experi + b4 part-timei + ui (3)

Now, since b5 is positive as higher ability should have positive impact on growth and c1 is negative (discussed above), b5c1 is negative. Thus, b1 + b5c1 < b1 and hence we have a downward bias.