The two variables are monthly stock prices for a period of 20 years.
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hard to answer without additional information on what kind of "relationship" you are interested in ; however, you could try a simple Engle-Granger test for co-integration of the two series ... as a starter !
http://en.wikipedia.org/wiki/Cointegration
and a gentle introduction here :
http://www.uta.edu/faculty/crowder/papers/drunk%20and%20dog.pdf
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If the series are cointegrated I could use the ECM, otherwise the Autoregressive Distributed Lags Model?
even if you venture cointegration in a later analysis you would need independent autoregressive models
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Engle-Granger test is also known as Engle-Granger two step error correction model (ECM) !
So, yes, you could try that as it is the "vanilla" approach to the extraction of a "relationship" between time series ; of course, cointegration is a very tight kind of "relationship" and the test may fail for a zillion of reasons (you can't expect two time series to behave as gently as the drunken lady and her dog !)
more details here if you are interested (but maybe you know this already and my suggestion is too simplistic!) :
http://springschool.politics.ox.ac.uk/archive/2008/OxfordECM.pdf
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