I have a regression that I have run on average ratings for some products (dependent variable) and their characteristics (Model 1). I have reason to believe there is a prejudice against a specific set of products (I will call these the unloved) and an alternative model that includes an indicator for these products provides some evidence for this with a large and statistically significant coefficient (Model 2).
My question is that, assuming that Model 2 is the correct one (that is, Model 1 is missing exactly one variable), can I take the mean of the predicted values of the unloved (observations for which I believe the prejudice exists) from Model 1 (the mean of the predicted values for the observations that would have got the indicator using the coefficients from the misspecified model), and compare them to the mean of the observed ratings for the unloved to quantify the prejudice.
— Extra details:
The difference between the two means falls within the 95% confidence interval of my estimate for the prejudice term in Model 2, but I could not come up with an analytical argument for whether or not the difference between the means should be the effect of the omitted variable (that is, did I just get lucky?)
I am interested in presenting this comparison instead of reporting the dummy in Model 2 due to the intended audience. I couldn’t convince myself this was acceptable but was interested in whether or not it could be done (and hopefully learn how to approach similar problems in the future)