# 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.

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.

• Could you explicitly show how this responds to the question? In its current form it seems to brush aside the OP's stated concerns about the information lost in differencing and their objective of estimating rates of change rather than the values themselves.
– whuber
Commented May 16, 2018 at 11:21
• Thanks for your response. I checked the link and found some interesting hints, but the questions that alluded to similar issues seemed to deal with univariate models. Commented May 16, 2018 at 17:58
• Also, to clarify, it is the deterministic trend that I want to explain, not write off by using integers as a predictor. I'm also assuming that the trend is not stochastic. You state: "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." This touches on my question, as does orcmor's answer. Could you elaborate on this? Commented May 16, 2018 at 18:03
• And finally, more generally, if there were no change to the deterministic trend, would there still be a way to investigate what variable (real world mechanism) is behind the trend and to measure it, or them, using a multivariate analysis? Commented May 16, 2018 at 18:03
• no to your second comment . A deterministic trend (no counting numbers) is simply a permanent steady state constant . If you had a series 1,2,3,4,5,9,10,11,12.., the steady state constant would be a "1" with a pulse at period 6 of "3" representing a temporary change in trend. This "3" would be found via Intervention Detection schemes. If you wish to continue this discussion please feel free to contact me . Commented May 16, 2018 at 19:31

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.

• Thanks. Based on a cursory glance, I am not sure if this answers my question, but it is certainly a step in the right direction. I AM trying to explain changes in the underlying deterministic trend and this appears to touch on that. I'll need some more time to study the material before selecting this answer. I gave it an upvote but I don't yet have the points on CV for it to show up. Commented May 16, 2018 at 17:45