From the Statalist FAQ (emphasis mine):
4.1 What is the correct way to pronounce ‘Stata’?
Stata is an invented word. Some pronounce it with a long a as in day (Stay-ta); some pronounce it with a short a as in flat (Sta-ta); and some pronounce it with a long a as in ah (Stah-ta). The correct English pronunciation must remain a mystery, except that personnel of StataCorp use the first of these. Some other languages have stricter rules on pronunciation that will determine this issue for speakers of those languages. (Mata rhymes with Stata, naturally.)
This of course means that there is a right answer, but that StataCorp doesn’t want to take sides because they might piss people off. People are amazingly passionate about how they say the rarely-spoken names of software and other technical terms. If you want to start a fistfight, ask a group of nerds how to pronounce the name of the document markup language “LaTeX” or the image format “GIF”.
EDIT: Nick Cox, who wrote the Statalist FAQ, provided the following corrective in the comments:
Not so; or not really so. I wrote that originally, tongue in cheek, without any prompting whatsoever from the company and I don’t think it appears on any documents that don’t have my name on them, and I am not a StataCorp employee. Naturally, it remains true that the company would not knowingly host on any of their websites statements that they thought inappropriate. But the main interpretation is not that the company don’t want to take sides — there is only one pronunciation used at StataCorp — but that they have a sense of humour about something that isn’t really very important, namely how users and others pronounce the name. Now the correct spelling of Stata: that really is a big deal.
So the correct answer really is that there is no right answer. I am leaving my original post up for posterity, and because of my irrational passion in favor of the “Stay-ta” pronunciation.