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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?

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  • $\begingroup$ 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. $\endgroup$
    – John
    May 29, 2012 at 21:16

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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.

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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.

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