# How to forecast a time series which is dependent on different time series?

I have two time series

S = s[1], s[2], s[3],..., s[t-1], s[t]
R = r[1], r[2], r[3],..., r[t-1], r[t]


Such that:

r[t] = p[t,1]*s[1] + p[t,2]*s[2] + p[t,3]*s[3] ... + p[t,t-1]*s[t-1]
r[t-1] = p[t-1,1]*s[1] + p[t-1,2]*s[2] + p[t-1,3]*s[3] ... + 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[1] r[2] r[3] ... 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[1] s[2] s[3] ... 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?

• You can't fit an ARIMA model on R with S as a variable? – Jon Dec 12 '16 at 19:32
• You could try an ensemble model or build up your own regression model that's autoregressive based on both R&S. I think you will want to use interaction effects as well. – Chris Dec 12 '16 at 19:34
• I thought ARIM had to be "auto"-regressive, meaning that it works only with past observations from the same series? – Skander H. Dec 12 '16 at 19:34
• Not necessarily. Time series is not my strength, but here is a reference: people.duke.edu/~rnau/arimreg.htm – Jon Dec 12 '16 at 19:47
• when u have found your answer accept the one you like to close the question. – IrishStat Dec 19 '16 at 17:51