I changed up the tags on this question so our immigration attorneys will respond as well.
What you are saying is "I want to do indirectly that which I am prohibited from doing directly." The answer is that you cannot. You cannot be an active principal in a US business while here on a student visa. You could own shares of a business as a passive investor, but this would mean passive in the literal sense NOT some fictitous sense. That would mean, for example, you cannot contract on behalf of the business, cannot pay for things, cannot conduct oridinary business activities, etc. So owning a single member-managed LLC would on its face be a problem.
I would first discuss this with immigration counsel so you are clear on what you can and cannot do. Then if you have a legal plan contact a business lawyer. Most of us here, including myself, offer a free phone consult.
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If you're looking to get into being an affiliate marketing publisher through a larger network, just have a family member set up a company back home. Larger networks like Clickbooth and A4D make payments to foreign pubs all the time. You should be fine.
Running a company = work = unauthorized employment.
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you may own anything you want to own. But you are suggesting using ownership as a cover for work. That is not allowed.
I am a pretty smart guy, but I do not know everything. My answers here come from the knowledge I carry around in my head--if you want me to look something up, you should come see me. Take any advice here as being general in nature. Please be careful with your life and make decisions that are smart.
That would entail engaging in employment in the US and would be a violation of the terms of your visa. You may also run afoul of tax laws since you must pay the Managers of the corporation a wage. A legal entity that does not employ anyone but produces profit would probably make you a target for an IRS audit.