There are many dedicated Timestamping Services for this Use-Case, one that has been around since '95 and with a corrrespondingly solid reputation IMHO is http://www.itconsult.co.uk/stamper/stampinf.htm. But there are other services as well, just search for "Timestamping Service"... I'd steer clear of the Bitcoin-based ones, they tend to be... enthusiastic but not too proffessional.
Proving you made the statement at a given time is as simple as getting a trusted signed timestamp or inserting it in some blockchain-like ledger, but that's not even close to making yourself accountable for predictive accuracy.
Be sure to publish *all* your predictions so we don't get file-drawered (except on you as a person which we probably can't help).
A common technique is to publish a secure hash of your prediction rather than the text (in case you want to avoid it being self-fulfilled or anti-fulfilled or otherwise traded on) (crypto signed w/ your identity, too).
But if we don't see a stream of plaintext reveals and a means of identifying all such hashes you've published, we might suspect you of planting both positive and negative predictions.
Most people prefer to publish their prediction (+reasoning) clear-text because they want to persuade and they want credit for being smart before the verdict is in.
I guess the best way is just to post on a bunch of platforms and have those web pages back-up by various archiving services (notably the Wayback Machine).
Publishing a book is probably too much work for most used cases.
I've also heard one could publish text on the Bitcoin blockchain for example, but I'm not sure how well that works.