What you might do is to develop a theoretical (guessed !) prior probability distribution (frequency distribution) for possible values of Y(t) and create/simulate via Monte Carlo a family of possible values of size N . Then use your equation to predict the next value for each of the simulated(possible) values and then generate a histogram of those forecasts. Confidence limits for these forecasts can the based upon the actual distribution of outcomes.


We have recently incorporated that very useful feature into AUTOBOX ( a time series analysis package that I have helped to develop) and are using it to compute probability distributions ala http://probabilitymanagement.org/ for both univariate and causal models where exogenous variables are used. See here https://www.youtube.com/watch?v=-eBkpr3P27M