Can I lose the first race?

Solution 1:

No it isn't possible to lose at all. The race will only end if you grab the statue or just stop playing the game.

And it would make sense too right? The game isn't about Groose (nor Cawlin nor Strich), it's about Link. The winner of the race will graduate and get all the goodies that comes with it, there is no second round. And you can't really progress through the game unless you become a knight so you may leave the grounds. If you're not, it's pretty much game over and a game over at this point in the game would be too soon.

I even started a new game just to test this out. As already mentioned, it will not end at all until you get the statue.

• Waiting at the start of the race (not diving off the platform to call your Loftwing) will go on endlessly.
• It is still possible to dive off the platform, not call your Loftwing and get rescued by a knight from the fall but nothing else happens.
• You cannot dive off of your Loftwing after you had mounted it.
• You cannot crash at the bottom of the clouds at all.
• You could fly off away from the others and will never be able to go beyond a certain point. You're all essentially leashed off in a small portion of the sky.
• Doing nothing in either phase of the race (when they start throwing eggs at you) lasts endlessly. I waited approximately 10 minutes in each stage.

There's no way to lose the race even if you wanted to.

Solution 2:

No, it's impossible to fail. The race will not end until you manage to catch the bird.