# How important is the chronological ordering of historical returns?

The returns of asset $$A$$ in chronological order are

 0.03
0.01
-0.04
0.02
0.05
-0.10
0.02


The expected return, or sample mean, is $$-0.00143$$ while its sample volatility is $$\sigma = 0.051455$$

Next, I try to compute the same moments for a shuffled version of the time series which we'll call $$B$$

 0.05
-0.10
0.02
0.01
-0.04
0.02
0.03


and get the exact same mean and volatility, $$-0.00143$$ and $$0.051455$$, but with the added bonus that the shuffling has likely eliminated the autocorrelation found in the original dataset.

Given that many economic models characterize time series data based only on summary statistics, which are mean and variance, can it be said that the entire density function of a return series (the return distribution) is blind or insensitive to the ordering of time observations, just like how the moment estimators are blind?

Now taken from the perspective of the users of the data as an input: Does this imperviousness to data re-sorting mean that the exact date or when a sudden up- or down-tick occurred (regardless recently or 2 years ago) is statistically meaningless to a model that takes such historical data as an input?

Isn't this naive of historical-based estimators and models to not take into account the recency of events? What are the effects of this insensitivity to models models that assume stationarity? If nothing, then what other financial concepts/assumptions are impacted?

• Econometric models go further than mean/variance though. The series is viewed as a string, so the ordering is important, and that’s the very reason for the existence of time series techniques. – Magic is in the chain Aug 29 '20 at 17:09
• indifference to ordering might be called "permutation invariance", the property of being permutation invariant: changing the order of a model or estimator's inputs does not yield different outputs – develarist Sep 8 '20 at 5:05

This does not pretend to be a complete answer to the question posed (but that question is not itself, to my mind, completely posed ;-)

I'm just struck by how it resonates with the whole topic of "path dependency risk". This topic naturally captures a multitude of investment "sins".

The most obvious of which is the sequencing of returns if one saves and invests a constant amount. Imagine (simplistically) that I invest 1 in each period. So my 1 at end-0 becomes 1.5 at end-1 after +50%, becomes 2.5 with my savings then, becomes 1.25 at end-2 after -50%, or 2.25 with savings invested then. Reverse the sequencing and this becomes 1->0.5->1.5->2.25->3.25 instead of 2.25 beforehand. I'm cruising the Caribbean rather than Eastern Europe when I retire ;-)

From a simple pensions and asset allocation perspective, it is always better to take the losses earlier and cash out later gains than enjoy early wins and crash out later, given the same underlying return distribution (with different sequencing -> terminal outcomes that have path dependency risk).

Shuffling the sequencing of the same returns would indeed change the statistical properties of the sample, with respect to autocorrelation. It would also likely create in-sample differences to means and variances observed at any point of time, from which investors might wish to draw inferences (before the out-of-sample that conforms to the whole-sample occurs). Even if one did not allow one's forecasts of future mean returns to be influenced by the past, it could still be very true that one allowed one's perception of "risk" to be influenced by past volatility!

But even if one was ruthlessly disciplined and "looked through this" as well, then removing the autocorrelation (and the heteroskedasticity in the term structure of returns more broadly) would imply that one should have more confidence that any in-sample sub-sample was a fairer guess of the population dynamics than if they were the flawed autocorrelated/heteroskedastic kind... So one would still behave differently to the shuffled data than one would to the unshuffled!

If asset returns were TRULY random, we'd invest differently than if they were quasi-random and almost-random. So yes the sequencing matters!

[All above intended more as a stimulus for debate, than as an exhaustive answer/solution. Please do treat as such]

very best, DEM

• being absorbed in the impact shuffling would have on probability-based models that take the shuffled/unshuffled historical data as an input, i completely overlooked the behavioral aspects. these are just as important, thanks. based on answers so far, re-sorting is still meaningless to the model, but pretty meaningful to the human – develarist Aug 29 '20 at 21:17
• in a nutshell, bingo ;-) But even if the probabilities are unchanged, the absence of autoC and heteroS might lead an "objective" probability-based model to draw different inferences compared to the same with the same population dynamics, but with autoC and heteroS. Devil obviously in the detail of the model you're looking at here. I've assumed that these diminish inference confidence; which diminishes confidence in the investment thesis; which diminishes position sizes; which affects model performance. Which might, of course, be wrong here. – demully Aug 29 '20 at 21:43
• could the indifference to sample ordering be a factor as to why asset means are difficult to estimate? instead of equally-weighting observations regardless of order like how the sample estimator of the mean does, shouldn't a weighting scheme consistent with the decay in autocorrelation structure reduce estimation error while at the same time preserve the ordering of samples? quant.stackexchange.com/questions/57514/… – develarist Sep 1 '20 at 5:03
• Yes, exactly so. This is precisely the basis for including momentum as a factor in factor models, since it is stat significant (even if far from decisive). – demully Sep 1 '20 at 20:55