I have been trying to identify a good instrument for industry concentration that is uncorrelated with price, specifically related to the housing market.

So far, I have found various iterations of the following:

1. Using a variation of a Bartik instrument that predicts local industry concentration based on national shares.
2. Exploiting differences in local shares between two large merging national firms which then creates large shift in HHI at the local level.

If anyone has other examples of instruments they have seen in concentration-price studies, I would really appreciate hearing about them. Thanks!