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I am not really strong with time series but I have a project I am working on..

I have a problem where I am trying to model a time series of the difference in pressure before and after oil has passed through an oil filter. I am trying to see if this oil filter can go longer without being changed. I also have data on the temperature of this oil. The temperature of the oil is a strong indicator of Difference in Pressure, 'PD'. I was thinking that I would like to remove the effect of the Temperature by modeling the residuals from a model with Temperature as a predictor and PD as the response. Would this work? Or is there a better approach?

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Yes . Not only can their be a dynamic response but anomalies may be present in the data suggesting the need for Intervention Detection schemes. ID can suggest Pulses/Level Shifts/Seasonal Pulses/Time Trends reflecting omitted deterministic structure.

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Your approach sounds reasonable. With time series there's often the lag issue. Maybe the pressure change responds to temperature change with the delay of some structure in time.

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