I have two time series
S = s, s, s,..., s[t-1], s[t] R = r, r, r,..., r[t-1], r[t]
r[t] = p[t,1]*s + p[t,2]*s + p[t,3]*s ... + p[t,t-1]*s[t-1] r[t-1] = p[t-1,1]*s + p[t-1,2]*s + p[t-1,3]*s ... + p[t-1,t-2]*s[t-2] etc...
Where p[i,j] are unknown percentages.
My goal is to forecast the second time series R.
I could perform a straightforward forecast of R based on its own historical values:
r[t+1] = f(r r r ... r[t])
using ARIMA, Exponential smoothing, etc...but it seems that I would be loosing valuable information by discarding the values from S. I could go the other way and try to forecast R solely based on S:
r[t+1] = f(s s s ... s[t])
using some sort of regression or pattern recognition algorithm, but that seems like a bad idea, since it would dismiss any inherent patterns in R that are not dependent on S.
What is the best approach to forecast R? How can one 'blend' a straightforward time series forecast of R with information gleaned from S ? Assuming we have a good forecast of S, how can we include them in the forecast for R?