Modeling a non-stationary bounded series I'm trying to model a time series variable that represents a percentage, strictly bounded between 0 and 1, that is also non-stationary about the mean.
Is there a model form that is able to account for non-stationary behavior while producing forecasts within the [0,1] interval?
I came across the following work while researching the topic which I believe could be extended to answer this particular question (just a hunch based on a quick reading):

Park, Joon Y., and Peter CB Phillips. "Nonlinear regressions with integrated time series." Econometrica 69.1 (2001): 117-161. 
Park, Joon Y., and Peter CB Phillips. "Nonstationary binary choice." Econometrica 68.5 (2000): 1249-1280.
Hu, Ling, and Robert de Jong. Nonstationary Censored Regression. mimeo, Ohio State Uni, 2006.

Unfortunately, I'm unaware of any implementation (preferably in R) of the ML estimators discussed. 

Per Alecos' requests, below is a plot of the series that led to this question

and the de-trended series 

 A: Given the graph, the obvious first comments are:
1) No-intercept, or very close to zero.
2) A deterministic time trend. Start the $t$ variable at the value $0$, or $1$, and increase the index following the periodicity of the data.
3) A structural shift is detectable from 2010 onwards, that shifts the mean of the de-trended series, creating a non-zero constant term (without affecting the slope of the trend). You could include that as a dummy variable taking the value $0$ up to 2009, and the value $1$ afterwards (this is essentially a "partial" intercept).  
In all cases, de-trend the data, estimating by OLS a regression of the form
$$y_t = a + bt + \gamma d_t + u_t$$
where $d_t$ is the partial intercept I mentioned previously, obtain the series
$y^* = y_t - \hat b t -\hat \gamma d_t$
and graph it. And if you want, let us know.
ADDENDUM
the residual graph shows traces of varying variance, but not very strong. So probably you can treat the de-trended series as variance stationary. Then estimating an ARMA model 
$$y_t = a + bt + \gamma d_t + A(L)y_t+B(L)u_t$$
where $A(L)$ and $B(L)$ are lag-polynomials, forecasts will be bounded in (0,1), since the dominant term here is the deterministic trend, and given also that the process is still at ~$0.1$. Conceptually one could wonder what will happen if this upward trend continues indefinitely: will the process hit the ceiling and remain there, or will it start to descend? But this is very closely tided to the real-world phenomenon under study. For short term prediction, this is not a problem, since again, the process has still way to go in order to approach the ceiling.
A: If your observations are proportions $p_1, \ldots, p_T$ you can use Beta regression (GLM type) to model
$$ p_t \sim Beta(\mu_t, \phi_t),$$
where $\mu_t = g^{-1}(X_t \beta)$ is the conditional mean of the Beta distribution, and $g$ is the link function (typically, g = logit). The $\phi$ parameter is the precision and can be either set as a constant, or also modeled with covariates.  In your case a simple trend regression would already do the trick.
If you want to look into time series models, I can recommend this very recent AoAS paper on Beta regression, ARMA models, and Google Flu data. Very intuitive and nicely illustrated.
References


*

*Ferrari and Cibrari-Neto (2004): original paper on Beta regression (see here too)

*vignette of the betareg R package: this nicely works together with the strucchange package, so you can actually answer also questions about some of the other replies re structural shifts in the data around 2010.

*Guolo and Varin (2014) and their ARMA models for Beta regression (and here the officially published version)

