# Dealing with Heckman correction and sampling bias in an experiment without instruments

Imagine a randomised experiment where workers are offered a job for \$x, and they can choose to reject the offer, accept$x or propose some counteroffer x' > x. An equal number of workers are offered the job in the control and treatment groups, and it is hypothesised that workers will exhibit higher "labour-market discrimination" towards the employer in the treatment compared to the control.

To measure discrimination, the main analysis will be to test the number of offers that are not rejected (i.e. accepted or countered) by treatment. I would like to use the counteroffer data for a secondary analysis of discrimination, but I am worried that a comparison of the treatment and control means of the counteroffers x' runs into a selection issue. For example, some individuals who reject the offer may do so because they have an extremely high willingness-to-accept, which corresponds to a high counteroffer that is not observed.

My questions:

1. Can I directly compare the means of the counteroffers, or are my worries valid?
2. If there is a selection issue, a Heckman correction requires some instrument that affects whether an offer is rejected, but not the amount of the counteroffer. I don't have anything like this and no other control variables besides the treatment. Does that mean that I cannot run a two-stage Heckman correction?
3. Is there anything else I could do to test for discrimination using the counteroffer data, and what assumptions would I need?