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


  • $\begingroup$ You will have to be more descriptive about what "very non-stationary" means. Does the series exhibit seasonality-cycles (non-stationary in the mean)? Does variability seems to change, and maybe cluster? (non-stationarity in the variance)? Both? Other? The best idea would be to graph the series and upload it here -visual inspection of time series is always important. $\endgroup$ – Alecos Papadopoulos Aug 30 '14 at 19:30
  • $\begingroup$ Sorry about that. I've included an image in the question. $\endgroup$ – MrT Aug 30 '14 at 19:43

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.

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.

  • $\begingroup$ Thanks Alecos. I've included the result you requested, however, I fail to see how this answers my question, as I'm interested in ensuring any forecast remains within the [0,1] interval. This is more likely due to my poor phrasing of the problem; I've rewritten my answer to highlight that requirement, and that I'm looking for a general solution to this kind of problem. $\endgroup$ – MrT Aug 30 '14 at 22:46
  • $\begingroup$ The conventional unit root test is not applicable in case of bounded series : see the paper Testing for unit roots in bounded time series $\endgroup$ – Metrics Aug 30 '14 at 23:24
  • $\begingroup$ @Metrics And why should we need a unit-root test here? $\endgroup$ – Alecos Papadopoulos Aug 30 '14 at 23:25
  • $\begingroup$ @ Alecos: to check whether the bounded series is stationary or not. $\endgroup$ – Metrics Aug 30 '14 at 23:31
  • 1
    $\begingroup$ @Metrics The phenomenon of the existence of a unit-root is just one of the ways that non-stationarity can be induced. In the specific series, a deterministic trend is clearly present. In theory, they could co-exist, so a unit-root test could be applied to the detrended series, but I don't see any real gain in that, since the deterministic trend is the leader here. $\endgroup$ – Alecos Papadopoulos Aug 30 '14 at 23:37

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.



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