The 4314 SPY daily returns 93-present were checked for mean and stdev:



4314 random daily returns were generated with normal distribution having the same mean and stdev as the actual series.  Both series were
ranked, then compared means of the top 5% of the real and simulated
returns (N=215):

Two-sample T for day ret vs SIM

                  N     Mean    StDev  SE Mean

day ret
 215  0.0290   0.0146  0.00099  T=2.49

SIM       215   0.0263   0.0050  0.00034

>> the real top 5% ("day ret") was indeed heavier than the
simulated top-tail.  Here is the comparison on the bottom 5% tail:

Two-sample T for day ret- vs SIM-

             N         Mean    StDev  SE Mean

day ret-
 215   -0.0293   0.0125  0.00086  T=-4.4

   215   -0.0252   0.0044  0.00030

The real bottom tail was even heavier than the top tail, compared to its normal counterpart. 
And here are the entire two series compared, showing that global means
and stdevs were the same:

Two-sample T for real day vs sim day

                 N    Mean   StDev  SE Mean

real day
  4316  0.0004  0.0125  0.00019  T=0.2

sim day  
4314  0.0003  0.0125  0.00019


Vixenophiles note that the volatility of the real tails was higher than the simulated/normal tails:

Test for Equal Variances: day ret, SIM

95% Bonferroni confidence intervals for standard deviations

             N        Lower      StDev      Upper

day ret
 215  0.0131569  0.0145864  0.0163495
   SIM  215  0.0045136  0.0050040  0.0056089

F-Test (normal distribution)
Test statistic = 8.50, p-value = 0.000

Levene's Test (any continuous distribution)
Test statistic = 24.26, p-value = 0.000

Begging the question whether tail obesity is related in some way to higher tail volatility.


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