I have a bunch of time series (~2000) with monthly data stretching over one hundred years. My interest is not to do inference on them but simply to classify them as descending or not. There are two important features of my data: (1) a lot of data points are missing (~50%), and (2) there is a very strong seasonality to the data (~10x larger than the trend).
My first thought was to simply do a OLS regression on each one and if the second coefficient is negative I would classify the series as descending. But then I started thinking: Should I somehow remove the seasonality before the OLS or account for it in the OLS via dummy variables for months, or can I just trust in OLS to do a decent job anyway?