# What's a time series model for forecasting a percentage bound by (0,1)?

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?

• I might begin by estimating a logit/probit type model by including the lags . However, I believe there are issues with correcting for autocorrelation in these types of models, so I would hesitate to draw any statistical inferences.
– John
May 29, 2012 at 21:16

## 2 Answers

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.

In my PhD Dissertation at Stanford in 1978 I constructed a family of first order autoregressive processes with uniform marginal distributions on $$[0,1]$$ For any integer $$r\geq 2$$ let $$X(t) = X(t-1)/r+e(t)$$ where $$e(t)$$ has the following discrete uniform distribution that is $$P(e(t) = k/r)=1/r$$ for $$k=0,1,..., r-1$$. It is interesting that even though $$e(t)$$ is discrete each $$X(t)$$ has a continuous uniform distribution on $$[0,1]$$ if you start out assuming $$X(0)$$ is uniform on $$[0,1]$$. Later Richard Davis and I extended this to negative correlation i.e. $$X(t) =-X(t-1)/r + e(t)$$. It is interesting as an example of a stationary autoregressive time series constrained to vary between $$0$$ and $$1$$ as you indicated you are interested in. It is a slightly pathological case because although the maximum of the sequences satisfies an extreme value limit similar to the limit for IID uniforms it has an extremal index less than $$1$$. In my thesis and Annals of Probability paper I showed that the extremal index was $$(r-1)/r$$. I didn't refer to it as the extremal index because that term was coined later by Leadbetter (most notably mentioned in his 1983 Springer text coauthored with Rootzen and Lindgren). I don't know if this model has much practical value. I think probably not since the noise distribution is so peculiar. But it does serve as a slightly pathological example.