I am trying to model and forecast an industrial process, in which the agent has to choose the percentage to attribute to four products, which I will call y1, y2, y3 and y4.
They add up to 100% in the data, and must add to 100% in the forecast.
I have the four time series y1 to y4, and two explicative series for each one, so the system goes (in R code):
y1 ~ x1a + x1b y2 ~ x2a + x2b with sum (y1...y4) = 1
...and so on
Each xn* is uncorrelated with the others.
Regressing each y on his xs, I obtain meaningful relationships, but of course there is a simultaneity problem, as the agent's choice for each y depends on the other y, as increasing one goes at the expense of the other, since all yn must sum up to 100%.
I am using a Simultaneous Equations Model, which is fine for descriptive purposes.
The problem is, how can I forecast the four ys, based on my model, so that it satisfies the following constraint?
sum(y1...y4) = 1
Any help appreciated