How Return Distributions Change Across Time Frames
Summary
The document discusses whether equity returns can be Gaussian at some sampling intervals even when they are not at others. It describes evidence and explanations for aggregational Gaussianity: the reported pattern is that short-interval returns tend to have fat tails, while distributions become more normal as returns are aggregated over longer periods. One cited account models changing returns as locally Gaussian processes whose variance shifts gradually, with different superstatistical distributions proposed for daily and minute-scale data.
A second explanation comes from the Central Limit Theorem: sums of independent returns with finite, nonzero variance tend toward a normal distribution. If returns instead have sufficiently heavy tails and infinite variance, a generalized limit theorem can lead to an alpha-stable distribution. These conclusions depend on assumptions such as independence and the tail behavior of returns. The discussion also cautions that available sample sizes may make normality tests unable to distinguish a stable distribution from a normal one, so apparent normality should be interpreted carefully.
Key ideas
- Short-interval equity returns are described as leptokurtic and fat-tailed.
- Return distributions may look increasingly normal as the sampling interval grows.
- Superstatistics explains changing distributions through locally Gaussian returns with a slowly varying variance.
- Independent returns with finite, nonzero variance aggregate toward a normal distribution under the Central Limit Theorem.
- Sufficiently heavy-tailed returns may aggregate toward an alpha-stable distribution instead.
- Limited sample sizes can make normality tests weak at distinguishing these possibilities.
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Full text
# Can Gaussianity of returns depend on the time frame?
# Can Gaussianity of returns depend on the time frame?
I would be interested in knowing if the fact that returns are Gaussian is disproved on all time frames, or if, for example, the 5 minute intra-day time frame could exhibits Gaussian returns assuming there are no micro-structure issues (low volume).
More generally is there any relationship between the time frame of equity returns and their generating distribution? Any known research in this direction?
## Answer by SiXUlm (score 3, accepted)
https://quant.stackexchange.com/a/18379
My main reference will be "Dan Xu, Christian Beck - Transition from lognormal to chi-square superstatistics for financial time series"
Non-equilibrium statistical mechanics (more specifically, superstatistics) gives some ideas of explaining the relation between time frame and its distribution: "...to regard the time series as a superposition of local Gaussian process weighted with a process of a slowly changing variance parameter"
In their article, the authors found out empirically that: "Chi-square superstatistics appears to best suitable for daily price change (assuming independent variation of volatility parameter in each interval), whereas on much smaller time scales of minutes, lognormal superstatistics seems preferrable"
There are couples of related articles on this topic:
- M. Ausloos and K. Ivanova - Dynamical model and nonextensive statistical mechanics of a market index on large time windows
- Katz, Y.A.; Tian, L. - Superstatistical fluctuations in time series of leverage returns
- S.M.D. Queirós and C. Tsallis - On the connection between financial processes with stochastic volatility and nonextensive statistical mechanics
- C. Becka, E.G.D. Cohen - Superstatistics
I'm not expert in this field, but hope the idea may help.
## Answer by Alexander Didenko (score 4)
https://quant.stackexchange.com/a/18375
Surely, there is; search for aggregational gaussianity in Google Scholar or ScienceDirect.
In fact, 5 minutes returns are leptokurtic and fat-tailed; then as you increase timeframe, returns become more and more normal. Yearly data is almost normal, if you have enough points.
## Answer by user1483 (score 1)
https://quant.stackexchange.com/a/18417
If high frequency returns are iid and the mean and variance are finite and vthe variance is greater than zero then the Central Limit theorem holds Then, regardless of the distribution of the high returns, when aggregated over time the aggregated returns will tend in distribution to a Normal distribution. The Lindeberg-Lévy-Feller version of the Central Limit Theorem gives a generalization of this result to independent random variables. If the higher frequency returns have fat tailed distributions with density such that \begin{equation} f(x)\sim \begin{cases} B_{-}|x|^{-(1+a)}\quad \text{as} \quad x \rightarrow - \infty \\ B_{+}|x|^{-(1+a)}\quad \text{as} \quad x \rightarrow \ \infty, \end{cases} \end{equation} where $0<a<2$ and $B_{-}$ and B_{+} are appropriate constants then the aggregated distributions will tend to $\alpha$-stable by the Generalized Central limit Theorem. One should note that with the sample sizes available no normality test has power against the alternative of an $\alpha$-stable distribution.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.