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

Question

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?

Answer 1

I believe what you are referring to is called a Transfer Function also called XARMAX and sometimes referred to as Dynamic Regression. See @analyst remarks here ARIMAX model's exogenous components? . Finally see How to include control variables in an Intervention analysis with ARIMA? for some of my over-arching reflections. Most responders are only aware of ARIMA models but the class of Box-Jenkins models is much more general than that.

@Jon

Astonishingly some practitioners teach " Hence, lagged values of other macroeconomic time series may have little to add to a forecasting model which has already fully exploited the history of the original time series. " . This view of leaning on the past first and then trying to add causal structure fails to realize the past of a series is a proxy for unknown causal variables as the past NEVER causes the future thus when you start with an ARIMA model you may have already proxied (naively accounted for) omitted causals . When you use Y(t-1) which may be dependent on X(t-1) to lean on.. you are essentially/implicitely accounting for the omitted variable BUT you inadvertently lean on history rather than using explicit causal dependency thus double jeopardy. ARIMA based practitioners lose sight of the fact that one needs to capture whatever structure is important from suggested causals and then as a secondary resort use ARIMA or Dummy Variables ( trends,level shifts,seasonal pulses,pulses) to render the remaining error process Gaussian. Anytime you use ARIMA , lagged values of Y or empirically identified Intervention Variables ( pulses/time trends,level shifts,seasonal pulses) it is a tacit statement of ignorance as the past is never the cause for the future. Restated in the spirit of Markov, if one has complete knowledge of all current and future system causal variables the past of the series being predicted is irrelevant.