correlation - What is the most stable, non-trivial dependence structure in finance?

The highest rated answer to the question on What concepts are the most dangerous ones in quantitative finance work? is this one:

Correlation

Correlations are notoriously unstable in financial time series [...]

My question
My question is a little bit broader than just about linear dependence, it is:
What is the most stable, non-trivial dependence structure in financial data?

With non-trivial I mean that I don't want answers that are about direct connections, e.g. between derivative and underlying.

The dependence structure can be either cross-sectional or through time with univariate time series, it can also be non-linear.

The context of my question is that I am preparing the documentation for a new machine learning R package I wrote and I am looking for a good showcase in the financial sphere. Now this is not a trivial feat given that correlations are notoriously... see above ;-)

Answer

It is hard to find a stable non-trivial dependence structure in financial data. Usually when such is found it is hard to rationalize.

One of my favorite (although I am sure there are others) is the so called "Presidential Puzzle". This is an old finding by Santa-Clara and Valkanov (2003) They find that "

Excess return in the stock market is higher under Democratic than Republican presidencies: 9 percent for the value‐weighted and 16 percent for the equal‐weighted portfolio.

At the time the finding was very robust and did not seem to be explained by anything else. What is more impressive is that 12 years later the result still holds true. We now have a very good out-of-sample period. This is confirmed by Pastor and Veronesi (2017) recent work. More interesting, they racionalize the finding by building a continuous-time general equilibrium model and conclude that:

When risk aversion is high, agents are more likely to elect the party promising more fiscal redistribution. The model predicts higher average stock market returns under Democratic than Republican presidencies, explaining the well-known “presidential puzzle.”