As you will have already noticed the Sign in as a different user option has been deprecated from SharePoint 2013. Dirk Van den Berghe has written about it here.
So, I have found 3 options to get around this issue:
- Append the following string (from /_layouts) to the SharePoint URL in your browser http://mySPSite/_layouts/closeConnection.aspx?loginasanotheruser=true
- Launch your browser as a different user: http://www.little.org/blog/2012/07/17/launch-your-web-browser-as-another-user
- Add the menu item back in (not supported and may be overwritten in future upgrades): http://nickgrattan.wordpress.com/2012/07/23/sign-in-as-different-user-and-sharepoint-2013/