# On Tail Dependence and Multifractality

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Multifractality, Its Estimators, and Its Measures

#### 2.2. Quantile Autoregression with Gaussian Copula

## 3. Simulations Setting

## 4. Results

#### 4.1. Baseline Settings

#### 4.2. Comparison to the Flat Correlation Structure

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**

**Multifractal range $\Delta \alpha $.**Multifractal ranges are shown for processes with Gaussian copulas with (

**top**) constant correlation coefficients, (

**middle**) symmetric tail dependence, and (

**bottom**) and asymmetric tail dependence for MF-DFA (

**left**), MF-DMA (

**center**), and GHE (

**right**). The solid lines represent the mean values of the 1000 simulations for the given settings and the dashed lines represent the 95% confidence bands as the respective quantiles of the simulated distribution. The lighter the color, the longer the generated time series (black for $T=500$ and light gray for $T=$ 10,000.

**Figure 2.**

**Multifractal range comparison.**The ratios between the singularity spectrum widths for the (a)symmetric quantile-dependent auto-correlation processes and their respective linear autoregressive counterparts are shown. The solid lines represent the ratios, and the lighter the color, the longer the generated time series.

**Figure 3.**

**Empirical example of the quantile auto-correlation structure.**Estimated quantile-dependent auto-correlation coefficients are reported for Bitcoin, EURUSD exchange rate, gold, and S&P 500 for the period between 1 Jan 2015 and 31 July 2020. Datasets have been obtained from finance.yahoo.com.

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**MDPI and ACS Style**

Avdulaj, K.; Kristoufek, L. On Tail Dependence and Multifractality. *Mathematics* **2020**, *8*, 1767.
https://doi.org/10.3390/math8101767

**AMA Style**

Avdulaj K, Kristoufek L. On Tail Dependence and Multifractality. *Mathematics*. 2020; 8(10):1767.
https://doi.org/10.3390/math8101767

**Chicago/Turabian Style**

Avdulaj, Krenar, and Ladislav Kristoufek. 2020. "On Tail Dependence and Multifractality" *Mathematics* 8, no. 10: 1767.
https://doi.org/10.3390/math8101767