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I have changed Domain Registrars on a couple of domains that were ranking for not too competitive keywords with no change in rankings. I can't imagine it would affect any rankings IMO. Having a private registrations is nice to avoid spam and hundreds of call from people calling you as if they knew you trying to sell you their services.  As far as it affecting your rankings i doubt it does.

You can use this script, it will redirect all pages on the old domain to a new page in the new domain. this works is you are going to keep the same site structure and not change URLs.  If you do change URLs then i would suggest to 301 each old page to a new corresponding page on The new domain.

RewriteEngine on

RewriteCond %{HTTP_HOST} ^olddomain.com [nc]

RewriteRule (.*) http://www.olddomain.com/$1 [R=301,L]

Redirect 301 / http://www.newdomain.com/