A situation I frequently encounter is that I have multiple time series of observations which I'd like to analyze using a linear regression model. The time series contain many very similar observations and some rather unusual phases, which are not to be considered as outliers but rather as providing particularly interesting information on the behavior of the underlying system. I'm primarily interested in the actual regression coefficients and less in a good prediction, although I'm not sure this is relevant to the question.
Which would be an appropriate method to give less weight to the frequently occurring, similar measurements, and more to the rather unusual measurements? I know I could do some kind of weighted linear regression for example, but how would I choose the weights to achieve something like this?