Tail Weigh-In Update, from Kim Zussman
The 4314 SPY daily returns 93-present were checked for mean and stdev:
mean 0.0003742
stdev 0.0124951
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
SIM- 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
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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.