Just curious - what's the problem with letting them run a credit check?
Every time someone checks your credit, you credit rating goes down slightly. The credit bureaus operate on the theory that if someone is checking your credit it is because you are trying to get a loan to buy something.
After many complaints, the credit bureaus have agreed not to penalize you "too much" for a large number of credit checks over a short period of time, something like two weeks, because that could indicate shopping for a large item, not the intent to buy many large items.
It still works out against the average guy, or at least me, since I take my time looking for large items and may spend months looking for a car, as opposed to the two weeks allowed. Many credit checks can result in a major downward movement in your score.
Another reason is that I am opposed to people running credit checks on me in principle. Until I get ready to actually sign on the dotted line, there is no reason at all for a dealer to check my credit. Especially if I am going to pay cash for an item.
When I finally decide to buy something I have no objection to one dealer doing one credit check just to decide whether he will take my check and let me drive off, or whether he needs to wait until the check clears.
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There is a very negative aspect about letting a car dealer check your credit. They know how much you can afford, and will try to sell you all sorts of things you don't really want.
I keep my credit rating high, and one of the ways I do that is by not buying fancy things I don't need. If you go into a dealership and negotiate price before you negotiate financing you almost always come out ahead. And, if they don't know that you can afford the luxury model instead of the plain jane one that fits your needs just fine, they won't be constantly trying sell you the fancy one.