If you have a data generation process y=f(x)+ϵ and mistakenly regress x on y will your estimate be consistent for f inverse? Will your estimator be less efficient then if you had chosen to regress y on x? I am interested in a general f and a general regression method.
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What is the nature of X here? Note that regression makes no assumptions about the distribution of X. So, eg, X could be binary, which would cause problems for regressing X on Y. It may interest you to read my answer here: What is the difference between linear regression on y with x and x with y? – gung - Reinstate Monica Feb 01 '16 at 03:14
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1Here X is a continuous variable distributed uniform over a range [a,b]. Thank you for the link to the other question. I had actual read that and thought it was very close to my question, but still slightly different. But it was that post that motivated me to post my question here. Thank you for your response. – A. Johnson Feb 01 '16 at 03:53