In short I don't think a COA affects the price, it is mostly just to verify the legitimacy of the signature. It usually just helps the signed item to actually sell, because the buyer feels safer that it is "legitimate". I might be very wrong.
I don't know about specifics but in part it depends on WHO wrote the COA. Which company did the certification. As with any collectible remember it is only worth what someone who really wants to buy it is willing to pay. You hardly ever can get that top price, usually you are lucky to get 1/2 the top "value" price. There are two basic types of autographed baseball collectors, the kind that will buy almost anything signed from their favorite players certificate or not, and those that will only buy if there is a specific type of COA or they had the item autographed in person personally in front of them. The second kind NEED the proof the signature is legitimate. If selling on Ebay you almost always need that COA just for your item to sell, and in some cases it has to be from a specific certification company. I don't know all the companies so I can't help with which ones are the most popular, legitimate or reliable. Sometimes if you get something from Ebay the COA will be a home made thing from the person doing the selling so you might need to do some research into who authorized your COA.