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I am learning about Heckman selection model and confused with conditional expectation notation.

If the equation that determines sample selection is: enter image description here

The outcome equation is:

enter image description here

If error terms have bivariate normal distribution, then:

enter image description here

Can someone explain how it arrived from second to third line? Why are explanatory variables with coefficients written outside the brackets and epsilons are now written inside the conditional expectation brackets?

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By substitution and linearity of expectation

$$E[y_i \mid u_i \gt -w_i \gamma] $$ $$= E[x^\prime_i \beta + \epsilon_i \mid u_i \gt -w_i \gamma] $$ $$= E[x^\prime_i \beta \mid u_i \gt -w_i \gamma]+E[ \epsilon_i \mid u_i \gt -w_i \gamma]$$

But $x^\prime_i \beta$ is not random here, so $E[x^\prime_i \beta \mid u_i \gt -w_i \gamma] =x^\prime_i \beta$ and this make the expression equal to$$ x^\prime_i \beta+E[ \epsilon_i \mid u_i \gt -w_i \gamma]$$

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  • $\begingroup$ Do you have any links for beginners on this topic and rules when calculating with conditional expectation? $\endgroup$ – Quirik Jun 25 '16 at 12:26
  • $\begingroup$ Why is E(u | z,v) = E(u | v) if z is independent of u and v? $\endgroup$ – Quirik Jun 26 '16 at 13:26

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