Say I want to forecast a time series x. I believe that the values of time series a, b, and, c could be used to predict future values of x but I don't know the magnitude or reliability of their effect nor do I know what time lag(s) ought to be used.
As an example, say I want to test if the average monthly temperature in Dubai has an effect on the price of oil sometime in the future. From exploratory analysis, there appears to be some sort of relationship but I can't tell if it's a strong effect or if it's consistent over time. It's also possible that high temperatures could have an impact in the Summer months but not in the Winter. What would be the appropriate methodology to use here?