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Heckman selection model assumes that

  1. error of both selection and main equation are correlated and distributed normally,
  2. explanatory variables in selection equation are independent of the error term,
  3. explanatory variables in main equation are independent of the error term.

Doubt: How can explanatory variables of both equations be independent of error terms, and at the same time, the errors terms be correlated?

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Error terms can be correlated because they represent a common shock. For example, in a model of wage determination, where there is selection into the labour market (work or not), a negative demand shock for labour can affect how much individuals desire to work, and their wages.

This scenario can exist even if regressors are exogenous (which is what you mean by explanatory variables being independent of error terms).

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