I am trying to model contract outcomes, let's say for car sales and I am interested if a certain group of people is better in negotiating better deals (e.g. women). The dependent variable is price, which is the final effective price the sellers have in their records and the independent variables are the car characteristics, such as the mileage. Something like this equation:

$(1) \log(price_i) = \alpha + \beta Mileage_i + ... + \gamma D_{woman,i} + \epsilon_i$

where the $D_{woman,i}$ is a dummy indicating that buyer $i$ was a woman and $\gamma$ is my coefficient of interest.

My problem is that the data are incomplete and effective price includes "discounts", some in USD but others in other forms translated into monetary terms (e.g. the \$ equivalent of extended warranty). Essentially, the price paid for the buyer is missing.

I tested the model below for different discounts $x$ and found that certain car characteristics as well as $\gamma$ are significant, which makes intuitive sense (e.g. sellers give more warranty for younger cars, some sellers prefer to give non-monetary benefits). I would like to report these results as they are interesting.

$(2) Discount_x,i = \alpha + \beta Mileage_i + ... + \gamma D_{woman,i} + \epsilon_i $

However, it turns out that $\phi$ in the model below is also significant. Neglecting the causality discussion, it means that discounts and price are related. This is obvious, as higher discounts lead to lower effective prices.

$(3) \log(price_i) = \alpha + \phi * Discount_x,i + ...+ \epsilon_i $

Now my question is: I feel I should include the discounts in equation (1) otherwise I purposely omit variables. However, if I include the discounts in equation (1) and knowing about the results of equation (2), I have a multicollinearity problem.

I was considering to re-calculate transaction prices by excluding discounts, but it would not be 100% correct as I have to work with assumptions about the rate for non-monetary discounts. Also, the relationship in equation (3) might still hold for transaction prices. Unfortunately, I don't have any instrument for an IV analysis as the data are very special.

I would be happy to hear about any (paper) suggestions.



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