Instructions
Start a new game, then play through the opening until your rival has you follow him to Route 201 to teach you the basics about the game. After a certain point, Prof. Rowan will show up and speak to you about choosing your first Pokemon. His assistant will drop a briefcase, and Rowan will give you the opportunity to choose first.
Save the game at this point, so you don't have to go through the entire process over and over when you're trying to get a Shiny Starter. Open the briefcase and choose one of the three. If it's a Shiny Pokemon, a different sound will play when opening the Pokeball.
Restart the game if your chosen Pokemon isn't Shiny. This can take hundreds of tries, but is still possible because this attribute is randomly assigned every time a save is loaded. By saving at the moment right before you choose a starter, you can try multiple times much more quickly than normal.