Is there a way or method to use older values (lagged) of independent variables with alternative lags to explain current value of dependent variable? For time series specific
Exploring relation between time series provides an example of Transfer Function Modelling which leads to equations that can be then expressed as PDL or ADL as suggested by @mlofton. A flow diagram for model building is here http://www.autobox.com/pdfs/A.pdf
It is also called an ARIMAX Model or ARMAX Model transfer function-noise modelling in R.
For a larger set of references on this site see https://stats.stackexchange.com/search?q=user%3A3382+transfer+function