# Laplacian Split-BREAK Process with Application in Dynamic Analysis of the World Oil and Gas Market

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definition and Structure of the LSB Process

**Theorem**

**1.**

^{2}-continuity.

- (a)
- If the fluctuations of innovations (${\epsilon}_{t}$) were emphasized in the previous moment in time, it follows ${\theta}_{t-1}=0$. Thus, equality (9) becomes ${X}_{t}={\epsilon}_{t}$.
- (b)
- Fluctuations ${\epsilon}_{t-1}$ whose square do not exceed the critical value $c$ imply ${\theta}_{t-1}=1$. Thus, the value of ${X}_{t}$ is given as a linear, integrated MA(1) process ${X}_{t}={\epsilon}_{t}-{\epsilon}_{t-1}.$.

**Theorem**

**2.**

## 3. Distributional Features of the LSB Process

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

**Theorem**

**4.**

**Proof.**

**Remark**

**2.**

**Theorem**

**5.**

- (i)
- When$1\le \alpha \le 3/2$, both series${\stackrel{-}{M}}_{t;\alpha}$and${\stackrel{-}{Y}}_{t;\alpha}$have an asymptotically normal distribution, i.e., the following relations, when$t\to +\infty $, are valid:$${\stackrel{-}{M}}_{t;\alpha}~\mathcal{N}\left(\mu {t}^{1-\alpha},\frac{2{a}_{c}{\lambda}^{2}{t}^{3-2\alpha}}{3}\right),{\stackrel{-}{Y}}_{t;\alpha}~\mathcal{N}\left(\mu {t}^{1-\alpha},\frac{2{a}_{c}{\lambda}^{2}{t}^{3-2\alpha}}{3}\right).$$
- (ii)
- When$\alpha >3/2$, both series${\stackrel{-}{M}}_{t;\alpha}$and${\stackrel{-}{Y}}_{t;\alpha}$are asymptotically vanishing, i.e.,$${\stackrel{-}{M}}_{t;\alpha}\stackrel{d}{\u27f6}{I}_{0},{\stackrel{-}{Y}}_{t;\alpha}\stackrel{d}{\u27f6}{I}_{0},t\to +\mathrm{\infty}.$$

**Proof.**

**Remark**

**3.**

## 4. Procedures for Parameter Estimation

#### 4.1. Moments-Based Estimators

**Theorem**

**6.**

**Proof.**

**Remark**

**4.**

#### 4.2. Gauss–Newton and Maximum Likelihood Estimators

**Theorem**

**7.**

**Proof.**

**Remark**

**5.**

#### 4.3. Estimators of the Mean Value

## 5. Numerical Simulations of the LSB Estimators

## 6. Application in Dynamic Analysis of the World Oil and Gas Market

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AD test | Anderson–Darling test (statistics) |

AN | Asymptotic Normality |

AR process | Autoregressive process |

CDF | Cumulative Distribution Function |

CF | Characteristic Function |

CLD | Contaminated Laplacian Distribution |

DF | Degrees of Freedom |

GSB process | Gaussian Split-BREAK process |

IID | Independent Identical Distributed |

LSB process | Laplacian Split-BREAK process |

MA process | Moving Average process |

ML method | Maximum Likelihood method |

MSEE | Mean Square Estimated Error |

NASDAQ | National Association of Securities Dealers Automated Quotations |

Probability Density Function | |

RV | Random Variable |

Split-MA process | Splitting Moving Average process |

STOPBREAK | Stochastic Permanent Breaking |

W test | Cramér–von Mises test (statistics) |

## References

- Hardouvelis, G.A.; Kim, D. Margin Requirements, Price Fluctuations, and Market Participation in Metal Futures. J. Money Credit Bank.
**1995**, 27, 659–671. [Google Scholar] [CrossRef] - Fatás, A. Endogenous Growth and Stochastic Trends. J. Monet. Econ.
**2000**, 45, 107–128. [Google Scholar] [CrossRef][Green Version] - Chang, Y.; Schorfheide, F. Labor-Supply Shifts and Economic Fluctuations. J. Monet. Econ.
**2003**, 50, 1751–1768. [Google Scholar] [CrossRef] - Artis, M.J.; Hoffmann, M. Financial Globalization, International Business Cycles and Consumption Risk Sharing. Scand. J. Econ.
**2008**, 110, 447–471. [Google Scholar] [CrossRef][Green Version] - Lamey, L.; Deleersnyder, B.; Steenkamp, J.B.E.; Dekimpe, M.G. The Effect of Business-Cycle Fluctuations on Private-Label Share: What Has Marketing Conduct Got to Do with It? J. Mark.
**2012**, 76, 1–19. [Google Scholar] [CrossRef] - Blanchard, O.J.; L’Huillier, J.P.; Lorenzoni, G. News, Noise, and Fluctuations: An Empirical Exploration. Am. Econ. Rev.
**2013**, 103, 3045–3070. [Google Scholar] [CrossRef][Green Version] - Rodríguez, G.; Villanueva Vega, P.; Castillo Bardalez, P. Driving Economic Fluctuations in Peru: The Role of the Terms of Trade. Emp. Econ.
**2018**, 55, 1089–1119. [Google Scholar] [CrossRef] - Rebei, N.; Sbia, R. Transitory and Permanent Shocks in the Global Market for Crude Oil. J. Appl. Econom.
**2021**, 36, 1047–1064. [Google Scholar] [CrossRef] - Xu, Z.; Wang, H.; Zhang, H.; Zhao, K.; Gao, H.; Zhu, Q. Non-Stationary Turbulent Wind Field Simulation of Long-Span Bridges Using the Updated Non-Negative Matrix Factorization-Based Spectral Representation Method. Appl. Sci.
**2019**, 9, 5506. [Google Scholar] [CrossRef][Green Version] - Granero-Belinchón, C.; Roux, S.G.; Garnier, N.B. Information Theory for Non-Stationary Processes with Stationary Increments. Entropy
**2019**, 21, 1223. [Google Scholar] [CrossRef][Green Version] - Zhao, D.; Gelman, L.; Chu, F.; Ball, A. Novel Method for Vibration Sensor-Based Instantaneous Defect Frequency Estimation for Rolling Bearings under Non-Stationary Conditions. Sensors
**2020**, 20, 5201. [Google Scholar] [CrossRef] [PubMed] - Qu, C.; Li, J.; Yan, L.; Yan, P.; Cheng, F.; Lu, D. Non-Stationary Flood Frequency Analysis Using Cubic B-Spline-Based GAMLSS Model. Water
**2020**, 12, 1867. [Google Scholar] [CrossRef] - Aguejdad, R. The Influence of the Calibration Interval on Simulating Non-Stationary Urban Growth Dynamic Using CA-Markov Model. Remote Sens.
**2021**, 13, 468. [Google Scholar] [CrossRef] - Narr, C.F.; Chernyavskiy, P.; Collins, S.M. Partitioning Macroscale and Microscale Ecological Processes Using Covariate-Driven Non-Stationary Spatial Models. Ecological Applications
**2022**, 32, e02485. [Google Scholar] [CrossRef] [PubMed] - Engle, R.F.; Smith, A.D. Stochastic Permanent Breaks. Rev. Econ. Stat.
**1999**, 81, 553–574. [Google Scholar] [CrossRef][Green Version] - Diebold, F.X.; Inoue, A. Long Memory and Regime Switching. J. Econom.
**2001**, 105, 131–159. [Google Scholar] [CrossRef][Green Version] - Gonzalo, J.; Martínez, O. Large Shocks vs. Small Shocks. (Or does size matter? May be so.). J. Econom.
**2006**, 135, 311–347. [Google Scholar] [CrossRef][Green Version] - Kapetanios, G.; Tzavalis, E. Modeling Structural Breaks in Economic Relationships Using Large Shocks. J. Econ. Dynam. Control
**2010**, 34, 417–436. [Google Scholar] [CrossRef] - Dendramis, Y.; Kapetanios, G.; Tzavalis, E. Level Shifts in Stock Returns Driven by Large Shocks. J. Empir. Financ.
**2014**, 29, 41–51. [Google Scholar] [CrossRef] - Dendramis, Y.; Kapetanios, G.; Tzavalis, E. Shifts in Volatility Driven by Large Stock Market Shocks. J. Econ. Dynam. Control
**2015**, 55, 130–147. [Google Scholar] [CrossRef][Green Version] - Stojanović, V.; Popović, B.Č.; Popović, P. The Split-BREAK Model. Braz. J. Probab. Stat.
**2011**, 25, 44–63. [Google Scholar] [CrossRef] - Stojanović, V.; Popović, B.Č.; Popović, P. Stochastic Analysis of GSB Process. Publ. Inst. Math.
**2014**, 95, 149–159. [Google Scholar] [CrossRef] - Stojanović, V.; Popović, B.Č.; Popović, P. Model of General Split-BREAK Process. REVSTAT– Stat. J.
**2015**, 13, 145–168. [Google Scholar] - Stojanović, V.; Milovanović, G.V.; Jelić, G. Distributional Properties and Parameters Estimation of GSB Process: An Approach Based on Characteristic Functions. ALEA –Lat. Am. J. Probab. Math. Stat.
**2016**, 13, 835–861. [Google Scholar] [CrossRef] - Jovanović, M.; Stojanović, V.; Kuk, K.; Popović, B.; Čisar, P. Asymptotic Properties and Application of GSB Process: A Case Study of the COVID-19 Dynamics in Serbia. Mathematics
**2022**, 10, 3849. [Google Scholar] [CrossRef] - Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions; Wiley: Hoboken, NJ, USA, 1994; ISBN 0-471-58495-9. [Google Scholar]
- Williams, D. Probability with Martingales; Cambridge University Press: Cambridge, UK, 1991; Section 18.1. [Google Scholar]
- Stojanović, V.; Popović, B.Č.; Milovanović, G.V. The Split-SV model. Comput. Statist. Data Anal.
**2016**, 100, 560–581. [Google Scholar] [CrossRef] - Stojanović, V.; Kevkić, T.; Jelić, G. Application of the Homotopy Analysis Method in Approximation of Convolutions Stochastic Distributions. Univ. Politehnica Bucharest Sci. Bull.
**2017**, 79, 103–112. [Google Scholar] - Varadhan, S.R.S. Probability Theory, in: Courant Lecture Notes; American Mathematical Society: Providence, RI, USA, 2001; p. 131. [Google Scholar]
- Fuller, W.A. Introduction to Statistical Time Series; John Wiley & Sons: New York, NY, USA, 1996. [Google Scholar]
- Hoeffding, W.; Robbins, H. The central limit theorem for dependent random variables. Duke Math J.
**1948**, 15, 773–780. [Google Scholar] - Serfling, R.J. Approximation Theorems of Mathematical Statistics, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2002. [Google Scholar]
- Lawrence, A.J.; Lewis, P.A.W. Reversed Residuals in Autoregressive Time Series Analysis. J. Time Ser. Anal.
**1992**, 13, 253–266. [Google Scholar] [CrossRef] - Norton, R.M. The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator. Am. Stat.
**1984**, 38, 135–136. [Google Scholar] [CrossRef] - Sharma, D. On Estimating the Variance of a Generalized Laplace Distribution. Metrika
**1984**, 31, 85–88. [Google Scholar] [CrossRef] - Billingsley, P. The Lindeberg–Levy Theorem for Martingales. Proc. Am. Math. Soc.
**1961**, 12, 788–792. [Google Scholar] - Tests for Normality. R Package Version 1.0-2. 2013. Available online: http://CRAN.R-project.org/package=nortest (accessed on 31 March 2023).
- Nasdaq. Available online: https://nasdaq.com (accessed on 3 April 2023).
- Bessembinder, H.; Seguin, P.J. Futures-Trading Activity and Stock Price Volatility. J. Finan.
**1992**, 47, 2015–2034. [Google Scholar] [CrossRef] - So, M.K.; Chen, C.W.; Chiang, T.C.; Lin, D.S. Modelling Financial Time Series with Threshold Nonlinearity in Returns and Trading Volume. Appl. Stoch. Model. Bus. Ind.
**2007**, 23, 319–338. [Google Scholar] [CrossRef] - Ruckdeschel, P.; Kohl, M.; Stabla, T.; Camphausen, F. S4 Classes for Distributions. R News
**2006**, 6, 2–6. Available online: https://CRAN.R-project.org/doc/Rnews (accessed on 31 March 2023).

**Figure 1.**Histograms of the empirical distributions of log-returns fitted with Gaussian and Laplace distributions.

**Figure 2.**Dynamics of the non-stationary and stationary series of the LSB process (parameter values are: $\mu =0$ and $c=\lambda =1$).

**Figure 3.**Plots of CDFs (

**a**) and PDFs (

**b**) of increments $\left({X}_{t}\right)$, regarded as a mixed distribution of Laplacian distributed innovations $\left({\epsilon}_{t}\right)$, and two chi-square-based distributions (parameter values are: $\mu =0,$ $\lambda =1$, ${b}_{c}=0.5$).

**Figure 4.**Convergences of modulus of the characteristic functions ${\phi}_{m}\left(u/\sqrt{t},t\right)$ and ${\phi}_{y}\left(u/\sqrt{t},t\right)$, when $t=1,2,\dots ,500$ (parameter values are the same as in Figure 3).

**Figure 5.**(

**a**) Plots of the asymptotic variances of the estimators ${\stackrel{~}{b}}_{c}$ (dashed line) and ${\widehat{b}}_{c}$ (solid line). (

**b**) Plot of the ratio ${\widehat{V}}_{2}/{\stackrel{~}{V}}_{2}$ of the asymptotic variances of the estimators ${\widehat{\lambda}}^{2}$ and ${\stackrel{~}{\lambda}}^{2}$.

**Figure 6.**Histograms of empirical distributions of the estimated parameters (true parameter values are the same as in Table 1).

**Figure 7.**Dynamics of prices and trading volumes of energy products on the world market in the period from 2018 to 2023: (

**a**) crude oil; (

**b**) natural gas.

**Figure 8.**Dynamic diagrams of empirical and modelled data: (

**a**) log-volumes (solid lines) and martingale means (dashed lines); (

**b**) increments (solid lines) and innovation series (dashed lines). The diagrams above present the dynamics of Series A, and below are the dynamics of Series B.

**Figure 9.**Empirical distributions of actual data (given by histograms), along with the corresponding fitted PDFs (given by lines): (

**a**) Series A; (

**b**) Series B.

**Table 1.**Summary statistics of the estimated parameter values of the LSB process, along with the realized statistics of their normality tests (true parameter values are: $\mu =0$ and $c=\lambda =1$).

Parameter Estimators | Statistics | Estimated Values | AD (p-Value) | W (p-Value) | |
---|---|---|---|---|---|

Critical Value | $\stackrel{~}{c}$ | Min. | 0.5769 | 0.9908 * (0.0129) | 0.1791 ** (0.0099) |

Mean | 1.0257 | ||||

(MSEE) | (3.63 × 10 ^{−2}) | ||||

Max. | 2.0964 | ||||

$\widehat{c}$ | Min. | 0.7142 | 0.3202 (0.5320) | 0.0441 (0.6059) | |

Mean | 0.9944 | ||||

(MSEE) | (7.19 × 10 ^{−3}) | ||||

Max. | 1.2685 | ||||

Scale Parameter | $\stackrel{~}{\lambda}$ | Min. | 0.7829 | 0.5413 (0.1641) | 0.0702 (0.2773) |

Mean | 1.0026 | ||||

(MSEE) | (5.36 × 10 ^{−3}) | ||||

Max. | 1.2394 | ||||

$\widehat{\lambda}$ | Min. | 0.8592 | 0.4244 (0.3173) | 0.0588 (0.3920) | |

Mean | 1.0028 | ||||

(MSEE) | (2.52 × 10 ^{−3}) | ||||

Max. | 1.1702 | ||||

Mean Value | $\stackrel{~}{\mu}$ | Min. | −56.420 | 0.1843 (0.9088) | 0.0282 (0.8701) |

Mean | 0.3491 | ||||

(MSEE) | (252.01) | ||||

Max. | 55.208 | ||||

$\widehat{\mu}$ | Min. | −38.595 | 0.2508 (0.7417) | 0.0289 (0.8621) | |

Mean | 0.1647 | ||||

(MSEE) | (96.62) | ||||

Max. | 33.0875 |

**Table 2.**Basic statistical and market indicators of crude oil and natural gas in the last five years.

Statistics | Crude Oil | Natural Gas | ||||
---|---|---|---|---|---|---|

Price | Volumes | Log-Volumes | Price | Volumes | Log-Volumes | |

Mean | 65.42 | 4.22 × 10^{5} | 16.832 | 3.591 | 1.22 × 10^{5} | 12.672 |

Median | 64.62 | 3.88 × 10^{5} | 17.102 | 2.855 | 1.17 × 10^{5} | 12.789 |

Mode | 67.04 | N/A | N/A | 2.662 | 1.66 × 10^{5} | 12.717 |

Sample Variance | 397.1 | 5.41 × 10^{10} | 0.8172 | 3.274 | 3.56 × 10^{9} | 0.8194 |

Stand. Deviation | 19.94 | 2.33 × 10^{5} | 0.9040 | 1.809 | 5.97 × 10^{4} | 0.9052 |

Minimum | 9.060 | 1.23 × 10^{4} | 11.617 | 1.482 | 1.20 × 10^{3} | 8.281 |

Maximum | 123.7 | 1.69 × 10^{6} | 18.273 | 9.680 | 4.35 × 10^{5} | 14.56 |

Statistics | Series A | Series B | ||||
---|---|---|---|---|---|---|

$\left({\mathit{X}}_{\mathit{t}}^{\left(1\right)}\right)$ | $\left({\mathit{m}}_{\mathit{t}}^{\left(1\right)}\right)$ | $\left({\mathit{\epsilon}}_{\mathit{t}}^{\left(1\right)}\right)$ | $\left({\mathit{X}}_{\mathit{t}}^{\left(2\right)}\right)$ | $\left({\mathit{m}}_{\mathit{t}}^{\left(2\right)}\right)$ | $\left({\mathit{\epsilon}}_{\mathit{t}}^{\left(2\right)}\right)$ | |

Mean | 3.52 × 10^{−5} | 16.898 | −0.0196 | 1.17 × 10^{−5} | 12.688 | −0.0713 |

Median | −0.0577 | 17.127 | −0.0537 | −0.0660 | 12.830 | −0.0702 |

Mode | N/A | 16.588 | N/A | N/A | 12.709 | N/A |

Sample Variance | 0.9688 | 0.7227 | 0.7156 | 0.8492 | 0.8975 | 0.6484 |

Stand. Deviation | 0.8178 | 0.8501 | 0.8460 | 0.9215 | 0.9473 | 0.8053 |

Minimum | −3.5600 | 11.617 | −3.5560 | −4.0362 | 8.2808 | −3.8895 |

Maximum | 4.9542 | 18.185 | 4.9542 | 4.6061 | 14.560 | 4.2371 |

Range | 9.5142 | 6.5680 | 8.5142 | 8.6423 | 6.2794 | 8.1266 |

Skewness | 2.3966 | −2.1680 | 1.0276 | 1.5001 | −1.9954 | 0.6335 |

Kurtosis | 5.1593 | 5.3253 | 6.1547 | 7.7933 | 5.2876 | 6.0679 |

Parameter Estimates | Series A | Series B | |
---|---|---|---|

Mean Value | $\stackrel{~}{\mu}$ | 16.832 | 12.672 |

$\widehat{\mu}$ | 17.004 | 12.587 | |

Sample Correlation | ${\widehat{\rho}}_{X}\left(1\right)$ | −0.1911 | −0.3948 |

Threshold Parameter | ${\stackrel{~}{b}}_{c}$ | 0.2362 | 0.6523 |

${\widehat{b}}_{c}$ | 0.3766 | 0.5546 | |

Critical Value | $\stackrel{~}{c}$ | 0.0196 | 0.2868 |

$\widehat{c}$ | 0.0459 | 0.1487 | |

Scale Parameter | $\stackrel{~}{\lambda}$ | 0.5201 | 0.5069 |

$\widehat{\lambda}$ | 0.4520 | 0.5113 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stojanović, V.S.; Bakouch, H.S.; Ljajko, E.; Božović, I.
Laplacian Split-BREAK Process with Application in Dynamic Analysis of the World Oil and Gas Market. *Axioms* **2023**, *12*, 622.
https://doi.org/10.3390/axioms12070622

**AMA Style**

Stojanović VS, Bakouch HS, Ljajko E, Božović I.
Laplacian Split-BREAK Process with Application in Dynamic Analysis of the World Oil and Gas Market. *Axioms*. 2023; 12(7):622.
https://doi.org/10.3390/axioms12070622

**Chicago/Turabian Style**

Stojanović, Vladica S., Hassan S. Bakouch, Eugen Ljajko, and Ivan Božović.
2023. "Laplacian Split-BREAK Process with Application in Dynamic Analysis of the World Oil and Gas Market" *Axioms* 12, no. 7: 622.
https://doi.org/10.3390/axioms12070622