# (multiple) fractional outcomes & autoregression

Let me start with a broad description of the problem and I will then describe my approach (that might be totally inappropiate). The big goal is to predict the distribution of population of a given age (say 15-24) across "educational states" e.g. undergrad in humanities, undergrad social sciences, high-school student, vocational school pupil in IT etc. (I define ~ 20 such states). I have a 20 years long time series of a small, but representative sample from the population.

My approach is to:

• calculate the ratios of states in the sample (that neccessarily sum up to one),
• somehow examine how those ratios evolve over time (problem: time-series is short)
• predict the ratios for the next few years
• multiply the ratios by the expected total population size

Apart from obvious problem with the sample siize and time-series length, how would I approach a question of trends in fractional outcomes? Since I have little intuition on what exogenous variables would explain the changes, I wanted to start with an autoregressive model. But does it make sense here? And how to include the restriction that the ratios must be postive and sum up to 1 in every prediction?

(edit) I was suggested to use Bayesian VAR - how sensible it is?
(further edit) After a long search and consideration, my current approach is as follows:
- transform ratios into log-odds (with logit function)
- use Bayesian VAR with Banbura prior to forecast (as it looks robust and consistent with a stylized fact that education choices do not change rapidly)
- transform back with inverse-logit to ratios.

My questions: does it make sense? What assumptions about the data I implictly employ? What do I need to check (some statistical tests?) to make sure this approach is valid?

• Can you clarify your objective? Distribution for a given age across... 20 years? How will you put all this together? Commented Feb 6, 2019 at 12:32
• The objective is to model (and predict) changes in the students' degree choices, e.g. to check whether graduate studies in humanities are becoming more or less popular over time (given the alternatives). Update: I currently sketched a version in which I transform ratios to log-odds, then perform a Bayesian VAR with very simple (Banbura) prior and re-transfrom the prediction into ratios again. However, the relevant question is how sensible/robust/statistically correct it is. Commented Feb 6, 2019 at 13:11
• If you are worried that since your DV is a fraction (and so bounded) this will cause problems with a regular linear model, then you can use beta regression, which is used when your DV is bounded in $(0,1)$. An easy transformation of your data can transform from $[0,1]$ to $(0,1)$. Commented Feb 19, 2019 at 9:34
• If your DV is bounded and you have a lot of values on the boundaries then a logit transformation will not save you, the assumptions of the model are not satisfied and you will get wrong values for SE, CI, p-values,... Commented Feb 21, 2019 at 9:05
• I don't think it exists. The closest thing I found is this stats.stackexchange.com/questions/232697/… Commented Feb 21, 2019 at 9:30