Sadly, estimates are a negotiation. Whoever provides a number first generally loses because of "The Wince".

"What's your estimate?" "I'm not sure." "Just ballpark it."

"Well, when would you want it by?"

This is the trap that new managers will fall into every time. If they give you an answer? Bingo. You give them "The Wince":

You suck air between your teeth, and with a frown say, "Oh, that's completely unrealistic. Where in the world did you get that number from??" Then you provide a number that's many multiples higher, or offer a reduced amount of work: "Oh, gosh. We'd only be able to get X feature done in that amount of time, and only if we got lucky and cut feature Y from the other project."

Regardless, whatever you do, don't be the first to provide a number. It takes a little bit to get the feel for it, like poker or buying a car.

Time is money, even in a big corporation. Treat it as such: It's a zero sum game.