Stylized Facts Of Daily Return Series

In the class of Stat 956, we came across this interesting series of facts, termed as "stylized facts" that can be defined informally as a conclusion based on some empirical evidence that qualifies it as a fact but its non-intutive nature makes it stylized. The paper, referred by Dr. Steele in class was "Stulized Facts of Daily Return Series and Hidden Markov Model" from the Journal of Applied Econometrics.

A long return series was the subject of observation. Here, today's return refers to the logarthmic difference between today's and yesterday's price. Given such a series, its temporal and distributional properties were observed and empirically substantiated. Temporal properties refer to the properties shown by the return values in the series whereas distributional properties are with reference to the distribution of return in genral. Looking at one such property of each category helps appreciate the difference more.

It was also suggested that returns can be characterized by double exponential distribution. The present paper shows that these properties though seem to be very interesting are not special to return series. Infact a mixture of normal variables with zero mean can generate a series that exhibits such properties. Still, these are interesting facts to look at:



Temporal Properties:

- Returns are not aotucorrelated ( except at possibly lag one)



Distributional Properties:

- Mod(Rt) and sign(Rt) are independent.
- Mod(Rt) has the same mean and standard deviation
- The marginal distribution of mod(Rt) is exponential.



Leverage Effect:

- It is based on the persistance of volatility. Volatility is more persistant on a large fall of price than on a large increase in price.



More to add to this post:

- What is double exponential distribution.
- Temporal properties.
- Details behind some of these.