Semiparametric estimation and inference on the fractal index of Gaussian and conditionally Gaussian time series data

Abstract

This paper studies the properties of a particular estimator of the fractal index of a time series with a view to applications in financial econometrics and mathematical finance. We show how measurement noise (e.g., microstructure noise) in the observations will bias the estimator, potentially resulting in the econometrician erroneously finding evidence of fractal characteristics in a time series. We propose a new estimator which is robust to such noise and construct a formal hypothesis test for the presence of noise in the observations. A number of simulation exercises are carried out, providing guidance for implementation of the theory. Finally, the methods are illustrated on two empirical data sets; one of turbulent velocity flows and one of financial prices.

Keywords:

• Estimation
• fractal index
• fractional Brownian motion
• inference
• roughness
• stochastic volatility

JEL CLASSIFICATION:

• C12
• C22
• C51
• G12

MSC 2010 CLASSIFICATION:

• 60G10
• 60G15
• 60G17
• 60G22
• 62M07
• 62M09
• 65C05

Acknowledgments

The author wishes to thank Professor Asger Lunde and Dr. Mikko S. Pakkanen for insightful discussions relating to fractal processes.

Funding

The research has been supported by CREATES (DNRF78), funded by the Danish National Research Foundation.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 https://sites.google.com/site/mbennedsen/research/Code_EstimateAlpha.zip.

2 Following Bennedsen et al. (2019) this assumption is replaced by the following in the case α = 0: (A2’) $\frac{d^2}{d{x}^2}{\gamma }_2(x;X)=f(x)L_2(x),$ where L₂ is as in (A2), and the function f is such that $|f(x)|\le C{x}^{-\beta }$ for some constants C > 0 and $\beta >1/2.$

3 Recall that a process X is H-self-similar, or self-similar with self-similarity index H, if ${(X_{at})}_{t\ge 0}$ has the same distribution as ${(|a{|}^H{X}_t)}_{t\ge 0}$ for all constants a.

4 Again following Bennedsen et al. (2019), in the case α = 0 an alternative assumption is adopted: (BSSb’) $g\prime (x)=L_{g\prime }(x),$ where $L_{g\prime }$ is as in (BSSb).

5 Recall from the discussion in Section 2.1 that, except in the special case that X is self-similar, the smoothness properties of X will not be connected to the memory properties of X. In other words, the restriction that $\alpha \notin [1/4,1/2)$ does not restrict the memory properties of X. In particular, it is still possible for the correlation function of X to adhere to (2.3)(2.3) $$\rho (h)\sim h^{-\beta }{L}_{\infty }(h),\hspace{1em}h\to \infty ,$$(2.3) for all $\beta >0.$

6 Note that we also include a value of α in the interval $[1/4,1/2),$ namely $\alpha =0.40.$ In this case, we calculate the relevant variograms using second-order increments, cf. Section 2.4.1.

7 The fBm can be simulated efficiently using circulant embedding methods, see, e.g., Asmussen and Glynn (2007) Section XI.3.