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?