THE OBSERVATIONS
CHAPTER 3 · THE EMPIRICAL STUDY
The same estimators, on today’s markets
Chapter 1 measured one French equity over eleven years and found persistence that changes with the power measured. This chapter runs those estimators again, on ten years of daily observations from five markets that did not exist in that sample, or did not trade in that form.
MEASURED DATALoading measured series
WHAT THIS CHAPTER DOES
The chapter’s estimator, on a different series
Table 1.4 of Chapter 1 reports fractional-dependence estimates d̂(q) for power transformations of volatility and of trading volume. Those are reported constants, and they are a different quantity from the structure-function exponents ζ(q). Here the same estimator is applied to a measured series, so the two curves can be read on one axis.
The observations come first below — the series in level, its returns, and volatility under three of the definitions in use — and the estimates afterwards, so that each curve can be read against the sample it came from.
This is not a reproduction of the original study. The instrument, the period and the sampling frequency all differ. Agreement in shape would be a separate finding; disagreement is expected and is not evidence against Chapter 1.
THE OBSERVATIONS
Returns
THE OBSERVATIONS
Volatility
PERSISTENCE ACROSS POWERS
Measured d̂(q) against the reported table
Measured±2 standard errorsAlcatel volatility, reportedAlcatel volume, reported
MULTISCALING
Scaling exponent ζ(q) over the extended range
Measured ζ(q)Gaussian reference q/2
Chapter 1’s simulator caps its structure function at k = 64 observations. This view lifts that ceiling, which is what a longer sample buys; a wider scale range does not by itself establish multifractality.
Autocorrelation of |r|q
q = 0.5q = 1q = 2
Three power transformations at once, so the relative persistence across q is visible directly rather than one setting at a time.
PROVENANCE
Where these numbers came from
Method, corrections and what this is not
How a series reaches this page
Observations are loaded offline, turned into log returns in percentage points, and reduced to the curves this page draws. The browser fetches those curves and nothing else: it never queries a database, holds no credentials and estimates nothing at run time. The offline scripts under scripts/ are not part of the deployment.
Which corrections apply, and why that depends on the series
Overnight-gap removal and deseasonalisation are properties of an intraday sample, not of the pipeline. A daily series has no bar spanning a market close — consecutive trading days are the chapter’s own convention — and no time-of-day profile to estimate. Applying the intraday filter to a daily sample would discard every observation following a weekend: about a fifth of it, and systematically the Mondays. Each series therefore declares which corrections apply, and the preprocessing control offers only those.
Bandwidth
The regression uses the m lowest Fourier frequencies, with m defaulting to the square root of the sample size. On an intraday sample that keeps the diurnal harmonic, which sits at a Fourier index equal to the number of sessions, outside the band; wider choices such as n0.6 pull it inside. The reported standard error of the GPH slope is π/√(24m) and does not depend on the data.
What these series are not
The ten-year daily series carry roughly 2,600 observations each, which puts them close to Chapter 1’s own 2,633 Alcatel returns and gives a comparable estimator bandwidth. The estimation setup is therefore similar even though the instruments and the period are not. The US ten-year Treasury series is a yield in per cent: it passes through the same transformation as the price series, so its values are log-changes of a yield rather than returns in the chapter’s sense.
The reported Alcatel curves drawn alongside are transcribed constants at daily frequency on a different instrument over a different decade. They are a reference, not a target.