Providing false ID info to webhosting service

I registered my Domain with whois-guard elsewhere and since namecheap provides a very cheap shared-hosting and also accepts Bitcoin, and since I'm not going to do anything illegal on my blog-to-be, how serious is it to provide them false ID for hosting registration? (I'm only overly concerned about privacy, nothing illegal.)

You should not provide them fake documents as this can result on a ban/suspension. You should provide your ID or if you do not like their terms you should look for another hosting company.