# Long-Range Dependence in Financial Markets: A Moving Average Cluster Entropy Approach

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## Abstract

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## 1. Introduction

**Heterogeneity**. Volatility series have been analysed by using the cluster entropy approach over a constant temporal horizon (six years of tick-by-tick data sampled every minute). An information measure of heterogeneity, the Market Heterogeneity Index$I(T,n)$, where T and n are respectively the volatility and moving average windows, has been developed by integrating the cluster entropy curves of the volatility series over the cluster length $\tau $. It has been also shown that the Market Heterogeneity Index can be used to yield the weights of an efficient portfolio as a complement to Markowitz and Sharpe traditional approaches for markets not consistent with Gaussian conditions [22].**Dynamics**. Prices series have been investigated by using the cluster entropy approach over several temporal horizons (ranging from one to twelve months of tick-by-tick data with sampling interval between 1 up to 20 seconds depending on the specific market). The study has revealed a systematic dependence of the cluster entropy over time horizons in the investigated markets. The Market Dynamic Index$I(M,n)$, where M is the temporal horizon and n is the moving average window, defined as the integral of the cluster entropy over $\tau $, demonstrates its ability to quantify the dynamics of assets’ prices over consecutive time periods in a single figure [23].

## 2. Methods and Data

#### 2.1. Cluster Entropy Method

#### 2.2. Financial Data

#### 2.3. Artificial Data

#### 2.3.1. Geometric Brownian Motion

#### 2.3.2. Autoregressive Conditional Heteroskedasticity Models

#### 2.3.3. Fractional Brownian Motion

#### 2.3.4. Autoregressive Fractionally Integrated Moving Average

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Cluster entropy results for Geometric Brownian Motion (GBM) and Generalized Autoregressive Conditional Heteroskedastic (GARCH) series are reported respectively in the first and in the second row. GBM series are generated with following parameters: $\mu =1\times {10}^{-7}$ and $\sigma =5\times {10}^{-4}$. GARCH series are generated with the following parameters: $\alpha =0.475$, $\beta =0.1$ and $\omega =0.1$. Left and middle panels show cluster entropy curves for time horizons $M=1$ and $M=12$. Right panels show Market Dynamic Index $I(M,n)$ for different horizons M and moving average windows n. $I(M,n)$ is independent on M.

**Figure 2.**Cluster entropy results for Fractional Brownian Motion (FBM) series with $H=0.3$, $H=0.5$, $H=0.8$. First row shows results for time horizon $M=1$ (approximately equivalent to the first month (January 2018) of raw data for NASDAQ, S&P500, DIJA). The second row shows results for time horizon $M=12$ (approximately equivalent to twelve months of data in NASDAQ, S&P500, DIJA, i.e, the whole 2018 year).

**Figure 3.**Market Dynamic Index $I(M,n)$ for Fractional Brownian Motion series with Hurst exponent ranging from $H=0.2$ to $H=0.9$ respectively from $\left(a\right)$ to $\left(l\right)$.

**Figure 4.**Cluster entropy results for horizon $M=1$ for ARFIMA series with different combinations of the differencing parameter d, autoregressive parameter $\varphi $ and moving average parameter $\theta $. The differencing parameter takes values $d=0.05$, $d=0.15$, $d=0.25$ with a different combinations of autoregressive and moving average parameter. The full set of analysed values of d, $\varphi $ and $\theta $ is reported in Table 2.

**Figure 5.**Cluster entropy results for horizon $M=12$ on ARFIMA series with different combinations of the differencing parameter d, autoregressive parameter $\varphi $ and moving average parameter $\theta $. The differencing parameter takes values $d=0.05$, $d=0.15$ and $d=0.25$ with a different combination of autoregressive and moving average parameters. The full set of analysed values of d, $\varphi $ and $\theta $ is reported in Table 2.

**Figure 6.**Market Dynamic Index $I(M,n)$ for ARFIMA series with different combinations of the differencing parameter d, autoregressive parameter $\varphi $, and moving average parameter $\theta $. The differencing parameter takes values $d=0.05$, $d=0.15$, $d=0.25$, with a different combination of autoregressive and moving average parameters. The full set of analysed values of d, $\varphi $ and $\theta $ is reported in Table 2.

**Figure 7.**Cluster entropy results for horizon $M=1$ on ARFIMA series with different combinations of the differencing parameter d, autoregressive parameter ${\varphi}_{1}$, ${\varphi}_{2}$, and ${\varphi}_{3}$ and moving average parameter ${\theta}_{1}$, ${\theta}_{2}$ and ${\theta}_{3}$. The differencing parameter takes values $d=0.05$, $d=0.15$, $d=0.25$, with a different combination of autoregressive and moving average parameters. The full set of analysed values of d, $\varphi $ and $\theta $ is reported in Table 3.

**Figure 8.**Cluster entropy results for horizon $M=12$ on ARFIMA series with different combinations of the differencing parameter d, autoregressive parameter ${\varphi}_{1}$, ${\varphi}_{2}$, and ${\varphi}_{3}$ and moving average parameter ${\theta}_{1}$, ${\theta}_{2}$ and ${\theta}_{3}$. The differencing parameter takes values $d=0.05$, $d=0.15$ and $d=0.25$, with a different combination of autoregressive and moving average parameters. The full set of analysed values of d, $\varphi $ and $\theta $ is reported in Table 3.

**Figure 9.**Market Dynamic Index $I(M,n)$ for ARFIMA series with different combinations of the differencing parameter d, autoregressive parameter ${\varphi}_{1}$, ${\varphi}_{2}$, and ${\varphi}_{3}$ and moving average parameter ${\theta}_{1}$, ${\theta}_{2}$ and ${\theta}_{3}$. The differencing parameter takes values $d=0.05$, $d=0.15$ and $d=0.25$, with a different combination of autoregressive and moving average parameters as reported in Table 3.

**Table 1.**Example of lengths and time horizon M for NASDAQ data (2018). The 2nd column corresponds to the number of transactions over the horizon M. These lengths are used as a reference to generate artificial series to allow a direct comparison between results obtained on real and artificial data. The 3rd column corresponds to the sampled lengths used in the calculation of the cluster entropy. The 4th and 5th columns correspond respectively to the raw and rounded time intervals obtained dividing ${N}_{M}$ by the series of shortest length ${N}_{M}^{*}$.

M | ${\mathit{N}}_{\mathit{M}}$ | ${\mathit{N}}_{\mathit{M}}^{*}$ | ${\mathit{t}}_{\mathit{S}}$ | ${\mathit{t}}_{\mathit{S}}^{*}$ |
---|---|---|---|---|

1 | 586,866 | 586,866 | 1.0000 | 1 |

2 | 1,117,840 | 586,866 | 1.9048 | 1 |

3 | 1,704,706 | 586,866 | 2.9048 | 2 |

4 | 2,291,572 | 586,866 | 3.9048 | 3 |

5 | 2,906,384 | 586,866 | 4.9524 | 4 |

6 | 3,493,250 | 586,866 | 5.9524 | 5 |

7 | 4,069,315 | 586,866 | 6.9340 | 6 |

8 | 4,712,062 | 586,866 | 8.0292 | 8 |

9 | 5,243,029 | 586,866 | 8.9339 | 8 |

10 | 5,885,781 | 586,866 | 10.0292 | 10 |

11 | 6,461,845 | 586,866 | 11.0108 | 11 |

12 | 6,982,017 | 586,866 | 11.8971 | 11 |

**Table 2.**Full set of parameter range for the ARFIMA (1,d,1) processes simulated in this work. Specifically, D is the fractal dimension, H is the Hurst exponent and d is the differencing parameter (1st, 2nd and 3rd columns) which are related by Equation (20), $\varphi $ is the autoregressive parameter (4th column), and $\theta $ is the moving average parameter (5th column). Label refers to each parameter set (6th column). Specifically, cluster entropy results for the parameter sets: [a1], [b1], [e1], [f1], [i1], [l1] are plotted in Figure 4 (M = 1) and Figure 5 (M = 12), while the Market Dynamic Index is plotted in Figure 6.

D | H | d | $\mathit{\varphi}$ | $\mathit{\theta}$ | $\mathit{Label}$ |
---|---|---|---|---|---|

1.45 | 0.55 | 0.05 | 0.20 | 0.90 | a1 |

0.90 | 0.20 | b1 | |||

1.40 | 0.60 | 0.10 | 0.20 | 0.90 | c1 |

0.90 | 0.20 | d1 | |||

1.35 | 0.65 | 0.15 | 0.20 | 0.90 | e1 |

0.90 | 0.20 | f1 | |||

1.30 | 0.70 | 0.20 | 0.20 | 0.90 | g1 |

0.90 | 0.20 | h1 | |||

1.25 | 0.75 | 0.25 | 0.20 | 0.90 | i1 |

0.30 | 0.40 | j1 | |||

0.85 | k1 | ||||

0.90 | 0.20 | l1 | |||

0.40 | m1 | ||||

0.85 | n1 | ||||

1.20 | 0.80 | 0.30 | 0.20 | 0.90 | o1 |

0.90 | 0.20 | p1 | |||

1.02 | 0.98 | 0.48 | 0.30 | 0.40 | q1 |

0.85 | r1 | ||||

0.90 | 0.40 | s1 | |||

0.85 | t1 |

**Table 3.**Full set of parameter range for ARFIMA (3,d,2) and ARFIMA(1,d,3) processes simulated in this work. Specifically H is the Hurst exponent and d is the differencing parameter which are related by Equation (20) (1st and 2nd columns); ${\varphi}_{1}$, ${\varphi}_{2}$ and ${\varphi}_{3}$ are the autoregressive parameters (3rd, 4th and 5th columns); ${\theta}_{1}$, ${\theta}_{2}$ and ${\theta}_{3}$ are the moving average parameters (6th, 7th and 8th columns). Label refers to each parameter set (10th column). Specifically, cluster entropy results for the parameter sets: [a2], [b2], [e2], [f2], [i2], [j2] are plotted in Figure 7 (M = 1) and Figure 8 (M = 12), while the Market Dynamic Index is plotted in Figure 9.

D | H | d | ${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | ${\mathit{\varphi}}_{3}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | Label |
---|---|---|---|---|---|---|---|---|---|

1.45 | 0.55 | 0.05 | 0.20 | - | - | 0.90 | 0.90 | 0.90 | a2 |

0.90 | 0.90 | 0.90 | 0.20 | 0.20 | - | b2 | |||

1.40 | 0.60 | 0.10 | 0.20 | - | - | 0.90 | 0.90 | 0.90 | c2 |

0.90 | 0.90 | 0.90 | 0.20 | 0.20 | - | d2 | |||

1.35 | 0.65 | 0.15 | 0.20 | - | - | 0.90 | 0.90 | 0.90 | e2 |

0.90 | 0.90 | 0.90 | 0.20 | 0.20 | - | f2 | |||

1.30 | 0.70 | 0.20 | 0.20 | - | - | 0.90 | 0.90 | 0.90 | g2 |

0.90 | 0.90 | 0.90 | 0.20 | 0.20 | - | h2 | |||

1.25 | 0.75 | 0.25 | 0.20 | - | - | 0.90 | 0.90 | 0.90 | i2 |

0.90 | 0.90 | 0.90 | 0.20 | 0.20 | - | j2 | |||

1.20 | 0.80 | 0.30 | 0.20 | - | - | 0.90 | 0.90 | 0.90 | k2 |

0.40 | 0.16 | - | 0.90 | 0.81 | 0.73 | l2 | |||

0.90 | 0.90 | 0.90 | 0.20 | 0.20 | - | m2 | |||

1.15 | 0.85 | 0.35 | 0.20 | - | - | 0.90 | 0.90 | 0.90 | n2 |

1.02 | 0.98 | 0.48 | 0.40 | 0.16 | - | 0.90 | 0.81 | 0.73 | o2 |

**Table 4.**Probability p to reject the null hypothesis that the cluster entropy values for the ARFIMA processes at varying horizons M, have same mean and variance of the Fractional Brownian Motion with $H=0.5$. The probability p has been estimated by standard T-paired test. First column reports the temporal horizon M. The other columns refers to parameter sets [b1], [f1], [l1], [a2], [e2], [i2], [n2], [o2] of Table 2 and Table 3.

M | [b1] | [f1] | [l1] | [a2] | [e2] | [i2] | [n2] | [o2] |
---|---|---|---|---|---|---|---|---|

1 | 0.9597 | 0.7938 | 0.6013 | 0.8519 | 0.6779 | 0.4956 | 0.3542 | 0.2314 |

2 | 0.9863 | 0.8429 | 0.6985 | 0.9293 | 0.7883 | 0.6566 | 0.5414 | 0.4304 |

3 | 0.9820 | 0.8789 | 0.7743 | 0.938 | 0.8346 | 0.7362 | 0.6468 | 0.5576 |

4 | 0.9848 | 0.8922 | 0.8031 | 0.956 | 0.8689 | 0.7827 | 0.7147 | 0.6380 |

5 | 0.9878 | 0.9062 | 0.8325 | 0.9608 | 0.8809 | 0.8102 | 0.7528 | 0.6911 |

6 | 0.9940 | 0.9197 | 0.8517 | 0.9724 | 0.9043 | 0.8417 | 0.7840 | 0.7322 |

7 | 0.9785 | 0.9186 | 0.8633 | 0.9617 | 0.9038 | 0.8521 | 0.8036 | 0.7614 |

8 | 0.9930 | 0.9321 | 0.8775 | 0.9762 | 0.9229 | 0.8710 | 0.8333 | 0.7931 |

9 | 0.9867 | 0.9370 | 0.8890 | 0.9737 | 0.9273 | 0.8809 | 0.8438 | 0.8100 |

10 | 0.9813 | 0.9333 | 0.8952 | 0.9710 | 0.9261 | 0.8880 | 0.8533 | 0.8195 |

11 | 0.9816 | 0.9436 | 0.9011 | 0.9749 | 0.9326 | 0.8965 | 0.8643 | 0.8342 |

12 | 0.9853 | 0.9451 | 0.9072 | 0.9741 | 0.9353 | 0.9019 | 0.8764 | 0.8508 |

**Table 5.**Probability p to reject the null hypothesis that the cluster entropy values for the NASDAQ, DJIA and S&P500 at varying horizons M have same mean and variance of the Fractional Brownian Motion with $H=0.5$. First column reports the temporal horizon M. The probability p has been estimated by standard T-paired test [23].

M | NASDAQ | S&P500 | DJIA |
---|---|---|---|

1 | 0.5154 | 0.7399 | 0.8892 |

2 | 0.6026 | 0.8335 | 0.9257 |

3 | 0.6470 | 0.8588 | 0.9332 |

4 | 0.6631 | 0.8814 | 0.9283 |

5 | 0.6823 | 0.9018 | 0.9417 |

6 | 0.7124 | 0.9246 | 0.9534 |

7 | 0.7162 | 0.9224 | 0.9461 |

8 | 0.7288 | 0.9309 | 0.9618 |

9 | 0.7370 | 0.9479 | 0.9645 |

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## Share and Cite

**MDPI and ACS Style**

Murialdo, P.; Ponta, L.; Carbone, A.
Long-Range Dependence in Financial Markets: A Moving Average Cluster Entropy Approach. *Entropy* **2020**, *22*, 634.
https://doi.org/10.3390/e22060634

**AMA Style**

Murialdo P, Ponta L, Carbone A.
Long-Range Dependence in Financial Markets: A Moving Average Cluster Entropy Approach. *Entropy*. 2020; 22(6):634.
https://doi.org/10.3390/e22060634

**Chicago/Turabian Style**

Murialdo, Pietro, Linda Ponta, and Anna Carbone.
2020. "Long-Range Dependence in Financial Markets: A Moving Average Cluster Entropy Approach" *Entropy* 22, no. 6: 634.
https://doi.org/10.3390/e22060634