This must come up—the forecasting of things that are stuck between 0 and 1.
In my series, I suspect an auto-regression component, and also a mean-reverting component, so I want something that I can interpret like an ARIMA—but I don't want it to shoot off to 1000% in the future.
Do you just use an ARIMA model as the parameter in a logistic regression to confine the result between 0 and 1?
Or I learned here that Beta regressions are more appropriate for (0,1) data. How would I apply this to a time series? Are there good R packages or Matlab functions that make fitting and forecasting this easy?
. A significant change point was detected at period 137 suggesting time-varying parameters. The remaining 668 observations suggest a pdq ARIMA Model (3,0,0) with a level.step shift supporting your preliminary conclusions about lag 3. 
. The Actual/Fit/Forecast graph is 
The Residual Plot 
and the acf of the residuals is 
. Since the acf of the residuals shows strong structure at periods 5 and 10 , 
you might further investigate seasonal structure at lag 5. I hope this helps.
Best Answer
I asked this a long time ago but SO just popped it back up. In the case I was looking at, I ended up forecasting numerator and denominator separately, which made more sense for the metric anyway.