Hi Philip,
Julia doesn't have much in the way of time-series econometrics yet. However, I'll second the recommendation for RCall. I tried it out for the first time earlier this week wrapping R's arima functionaity and it has worked very well for me thus far. I'll post the code to github sometime next week (I can post a reply here with a link when I do so if you like), although there really isn't that much to it. Most of the work involves breaking down more complicated objects from either language into simpler objects (e.g. concrete types or arrays of concrete types) that can be passed between Julia and the embedded R without a problem).
While doing the above, I also had a look at wrapping some of Rob Hyndman's forecast library for R which (I think?) has the exact functionality you're after. It looks like it should be pretty straightforward.
As mentioned in other responses, just be aware of the difference between MIT license and GPL, as I think Rob Hyndman's forecast library is GNU version 3, which is a lot more restrictive than MIT.
Cheers,
Colin