High-level time-series question: How does one study a series' trend? I want to understand a series' trend, not the deviations from the trend. I would like to do analysis on the trend, such as run a multivariate regression, but every time-series source I read online says that we must decompose (or difference) the data and run the analysis on the stationary data. But that's not what I want to do. I want to inspect the trend itself.
For example, if I want to study manufacturing jobs in the United States, I would see that it increased until around 1980, then started to decline. If I wanted to run an analysis on this, a source would tell me to detrend the data, etc. But what if I want to understand and explain the forces underlying trend? I want to look for the reasons for the increase and then the eventual reversal in the trend and subsequent decline. How would I run an analysis on that? Pretty much, I would like to study the serial correlation, not remove it.
I don't expect an in-depth lecture on time-series. If you could provide some explanation and links to sources, that would be great.
 A: Please see https://stats.stackexchange.com/search?q=user%3A3382+trends for a number of my posts on this subject. Generally speaking a trend may be due to some unspecified causal variable and can be treated as either a stochastic or deterministic model. The stochastic model for a trend is simply a differenced model with a steady state constant (drift) . A deterministic model for trend could contain predictor series containing the counting numbers eg. 1,2,3,4,... to deal with observed data like 11,12,13,14,15,17,19,21,23,25 ....
You ask "But what if I want to understand and explain the forces underlying trend? I want to look for the reasons for the increase and then the eventual reversal in the trend and subsequent decline." First one needs to identify the time points where the significant trend changes occurred and then attempt to investigate/understand/blame possible omitted causal factors "explaining" the observed breakpoints in trend. 
Differencing should not be done willy-nilly as the trend may be deterministic in form.
Deviations from trend are important to be be able to identify trends e.g. pulses and memory effects need to be considered in the identification of trends.
A: Maybe regime switching models are what you are looking for. 
As for the trends in a time series, you can put breakpoints on the regime switching points (e.g. reversals) and calculate the trend coefficients in a straightforward manner. Then, you could look into the relation between your factors and the sign and magnitude of the trends.
Be cautious in using non-detrended time-series directly though. The problem with that is that it tends to show spurious relationships.
