# Modeling and forecasting time series of precious metals: a new approach to multifractal data

## Abstract

We introduce a novel approach to multifractal data in order to achieve transcended modeling and forecasting performances by extracting time series out of local Hurst exponent calculations at a specified scale. First, the long range and co-movement dependencies of the time series are scrutinized on time-frequency space using multiple wavelet coherence analysis. Then, the multifractal behaviors of the series are verified by multifractal de-trended fluctuation analysis and its local Hurst exponents are calculated. Additionally, root mean squares of residuals at the specified scale are procured from an intermediate step during local Hurst exponent calculations. These internally calculated series have been used to estimate the process with vector autoregressive fractionally integrated moving average (VARFIMA) model and forecasted accordingly. In our study, the daily prices of gold, silver and platinum are used for assessment. The results have shown that all metals do behave in phase movement on long term periods and possess multifractal features. Furthermore, the intermediate time series obtained during local Hurst exponent calculations still appertain the co-movement as well as multifractal characteristics of the raw data and may be successfully re-scaled, modeled and forecasted by using VARFIMA model. Conclusively, VARFIMA model have notably surpassed its univariate counterpart (ARFIMA) in all efficacious trials while re-emphasizing the importance of co-movement procurement in modeling. Our study’s novelty lies in using a multifractal de-trended fluctuation analysis, along with multiple wavelet coherence analysis, for forecasting purposes to an extent not seen before. The results will be of particular significance to finance researchers and practitioners.

## Keywords

Continuous wavelet transform Multiple wavelet coherence Multifractal de-trended fluctuation analysis Vector autoregressive fractionally integrated moving average Forecast## Abbreviations

- ARFIMA
AutoRegressive Fractionally Integrated Moving Average

- COI
Cone of Influence

- CWT
Continuous Wavelet Transform

- DWT
Discrete Wavelet Transform

- MENA
Middle East and North African

- MF-DFA
Multifractal De-Trended Fluctuation Analysis

- MWC
Multiple Wavelet Coherence

- VARFIMA
Vector AutoRegressvie Fractionally Integrated Moving Average

## Introduction

Multifractal structure analysis has become more and more popular in financial studies. It is referred as one of the strong and dynamic techniques due to its ability to detect multifractal behavior in non-stationary time series. Especially, local Hurst exponents help point out the discontinuities in the financial time series. Hence, any asymmetric or inconsistent behavior in the time series, such as the failure of any economic system, can be captured. These irregularities are the main reason for the fat tail observations. Local Hurst exponents demonstrate that these irregular behaviors may be organized to be used in various models/methods.

Mandelbrot and Ness has laid the foundations of multifractal analysis by introducing fractional Brownian motions, fractional noises and its applications (Mandelbrot & Van Ness, 1968). Later, multifractal de-trended fluctuation analysis (MF-DFA) has been proposed as an alternative method in analyzing financial time series by (Kantelhardt et al., 2002). Heretofore, there have been many researchers using this method in their analysis. Zhang et al. have investigated asymmetric multiscale multifractal analysis of wind speed signals (Zhang et al., 2017). Multifractal and wavelet analysis of epileptic seizures have been studied by (Dick & Mochovikova, 2011). An in-depth analysis of stocks of the company GE was performed by Thomson and Wilson by contrasting the results with those obtained using multifractal de-trended fluctuation analysis and using conventional time series models (Thompson & Wilson, 2014). Benbachir and Alaoui employed the MF-DFA method in order to explore the multifractal properties of the Moroccan Dirham compared with the US Dollars (Benbachir & Alaoui, 2011). Zhu and Zhang studies the multifractal property of Chinese stock market in the CSI 800 index based on MF-DFA approach (Zhu & Zhang, 2017) and concluded that the shape and width of multifractal spectrum are dependent on the weighing order and Hurst exponents can account for the market crash. (Sensoy, 2013) studies the time-varying efficiency of 15 Middle East and North African (MENA) stock markets by generalized Hurst exponent analysis of daily data with a rolling window technique and concludes that all MENA stock markets exhibits different long range dependence. In other study, Sensoy and Tabak proposes an alternative index to model time-varying inefficiency in stock markets using generalized Hurst exponents calculations (Sensoy & Tabak, 2015). (Tiwari et al., 2017) in 2016 challenges efficient hypothesis using the generalized Hurst exponent and MF-DFA methods. In our study, we will show that obtaining coherent time series lead to more accurate forecasting results because not only the long-run effects but also the short and long term dynamics can be taken into considerations simultaneously.

The wavelet analysis is also one of the methods in scrutinizing the time series. Unlike Fourier transform, as Burrus et al. indicated, wavelet analysis does not need stationarity (Burrus et al., 1998) and is able to look deep into the frequency information of the series at different scales without losing time information (Reboredo & Rivera-Castro, 2014) which helps eliminate the weaknesses in Fourier transform (Gülerce & Ünal, 2016). Gencay’s study (Gencay, 2002) has become a pioneer work for many researchers using wavelet tools to analyze financial time series in many studies such as (Aguiar-Conraria & Soares, 2012; Aguiar-Conraria & Soares, 2013), (Barunik et al. 2013), and (McCarthy & Orlov, 2012). Multi-scale analysis is accepted as one of the main applications of wavelet methods in finance and economics (Haven, 2012).

There have been many investigations looking into the relation of precious metals using these types of tools. Kucher and McCoskey indicated that the long-run relationships between precious metal prices are strongly influenced by economic conditions using a vector error correction model (Kucher & McCoskey, 2017). A flexible modification of the dynamic conditional correlation model that accounts for asymmetry and long memory in variance is applied on precious metals by (Klein, 2017). He et al. uses wavelet analysis and autoregressive moving average model with higher accuracy to forecast prices of precious metals (He et al., 2017). Das et al. suggest intense and positive co-movement in Asian gold spot markets after the Asian financial crises of 1997 at all frequencies (Das et al., 2017). To the best of our knowledge, multifractal de-trended fluctuation analysis along with multiple wavelet coherence analysis of precious metals has not been studied to this extent heretofore.

In this paper, we will use a software package for wavelet coherence analysis provided by (Torrence & Compo, 1998) and (Grinsted et al., 2004) and a Matlab tool for multifractal de-trended fluctuation analysis developed by (Ihlen, 2012). The former will be used to analyze inter relations, co-movement dependencies, frequent and consistent signals in multiple financial time series of precious metals (gold, silver and platinum). The latter will be used to confirm their multifractal behavior, calculate local Hurst exponents and obtain the inter-calculated fractal function time series at specific scale determined.

Multiple wavelet coherence will be used to determine the specific time period and scale that possesses common long range dependence out of the time series. With the help of Matlab tool, the multifractal behavior of these time series will be validated and a new series out of local Hurst exponent calculations will be obtained. These new series will be modeled using multivariate methods and forecasted accordingly. The first multivariate model has been introduced by Quenouille in 1957 (Akaike, 1974) and later improved by Akaike (Dunsmuir & Hannan, 1976), Dunsmuir and Hannan (Hannan, 1981) and Hannan (Oral & Unal, 2017a) in order. Multivariate models are a dynamic system of equations that examine the impacts of fluctuations (shocks) or correlations (interactions) between financial variables (Oral & Unal, 2017a). Multivariate modeling is because of the fact that more information out of multiple highly correlated data can be used and low mean-squared errors compared to univariate models may be obtained (Oral & Unal, 2017b). Moreover, multifractal nature of our data urges us to employ vector autoregressive fractionally integrated moving average method.

Gold is treated as hedging instrument against inflation and exchange rates (Hammoudeh et al., 2010) and, in many papers, reported as an indicator of inflation (Ranson & Wainright, 2005). Likewise, Mahdavi and Zhou points that commodity prices respond to new information faster than any consumer price (Mahdavi & Zhou, 1997). Therefore, results obtained from a successful forecasting of gold with smaller error bands may help and support both finance researchers as well as many different players in financial world such as monetary policymakers, hedge fund managers, portfolio managers, centrals banks and investors while making investment decisions. As far as we know there is no general method about forecasting of data possessing multifractal nature and we believe our paper will serve well during these type of decision-making processes.

The rest of the paper is organized as follows: Section 2 covers the data and methodology used in this paper. The main equation of continuous wavelet transform and multiple wavelet coherence, basics of multifractal de-trended fluctuations analysis and vector autoregressive fractionally integrated moving average model will be included as a summary review. In section 3, multiple wavelet coherence will be utilized to detect the highly correlated time periods and frequencies. After the multifractal characteristics of the series are verified, a new series of fractional function will be obtained out of local Hurst exponent calculations at the specified scale. Section 4 will compare and discuss the forecasting results of both multivariate and univariate models. Section 5 will be ended with the discussion of the results. In Appendix 1 will display two forecasting results of each metal couple along with the plot of the data set extracted from the local Hurst exponents’ calculations at the specified period.

## Data & Methodology

### Data

Daily Data of Precious Metals from July 2011 to November 2016

Mean | Median | St. Dev. | Kurtosis | Skewness | |
---|---|---|---|---|---|

Gold | 4736.03 | 4478.74 | 736.49 | 1.8535 | 0.4855 |

Silver | 7805.07 | 6845.28 | 2478.29 | 2.4046 | 0.7307 |

Platinum | 4574.81 | 4843.27 | 872.87 | 2.0483 | −0.2758 |

Correlation of gold, silver and platinum

Gold | Silver | |
---|---|---|

Silver | 0.718556 | |

Platinum | 0.54741 | 0.8564173 |

In this paper, we will look deep into co-movement of metal prices in time and frequency space by using multiple wavelet coherence. Once highly correlated time interval and frequency is determined, the multifractal behavior of the real series will be validated. A new time series of fluctuation function at the specified scale will be obtained out of its local Hurst exponents calculations. Finally, we will compare and discuss the performance of modeling and forecasting using these series with the help of univariate and multivariate models.

### Methodology

#### Continuous wavelet transform (CWT) and multiple wavelet coherence

Where the transforming function, also known as the mother wavelet, is ^{"}*ψ*^{"}, the translation parameter is ^{"}*τ*^{"} and the scale parameter is “s”. The translation parameter is the time information in the transform domain and the scale parameter is the frequency of the corresponding information.

Where C denotes complex coherence, *ρ*_{ij} is the correlation factor and R^{2}ij is squared multiple wavelet coherencies.

#### Multifractal De-trended fluctuation analysis (MF-DFA)

According to Eq. (3), when H is greater than 0.5, the time series data possesses persistent behavior with long range dependence. This persistency indicates practicably the same sign for the next non-overlapping time segment in line. If H is less than 0.5, the time series data will exhibit anti-persistent behavior with short range dependence. This, however, anti-persistent behavior would mean that an increase (decrease) in the process is most likely to be followed by decrease (increase) in the next time segment.

Firstly, a non-overlapping segment of equal length scale s is formed. Then the profile y(i) is divided to the number N_{s} = int(N/s) rounding to the nearest integer, int(). The same procedure is repeated starting from the end of the profile in order to include a short part of the end of the profile, doubling the number of segments, 2N_{s.}

_{s}:

_{s}+ 1, …, 2N

_{s}:

^{th}order fitting polynomial in the segment order, v. For

*n*> 3, any order of polynomials can be used, linear, quadratic, cubic or higher. Next, q

^{th}order fluctuation functions will be obtained by averaging the variance over all segments. This calculation is shown in Eq. (7) as following, for q ≠ 0:

*F*

_{q}

*(s)*is the main goal of MF-DFA while taking into consideration the time scale s and various values of q. The Eq. 5 thru 8 should be run in loop for various values of time scales s. Hence, the multi-scaling behavior of the fluctuation functions

*F*

_{q}

*(s)*may be analyzed. In order to do that, the slope of log-log plots of

*F*

_{q}

*(s)*with respect to s for different values of q (such as − 3, − 2, − 1, 0, 1, 2, 3) should be estimated. The fluctuation function

*F*

_{q}

*(s)*will demonstrate the following power-law scaling behavior as shown in Eq. (9) depending on the analysis of long-range power-law correlation as fractal proprieties,

For different values of q, *h(q)* is regressed on the time series *F*_{q}*(s).* The time series is called monofractal, when a constant value for *h(q)* for all values of q is true. Conversely, the time series is said to be multifractal, when *h(q)* is a steadily decreasing function of q. The Hurst exponents of *h(q)* represent the scaling properties of small (large) fluctuations when the values of q are negative (positive). Ihlen pointed that the size of local Hurst exponents in the periods of the multifractal time series with local fluctuation of different magnitudes determines the structure of the local fluctuations (Ihlen, 2012). Small (large) local Hurst exponents mean large (small) noise like (random walk like) structure of local fluctuations. This is identical with the generalized Hurst exponents for negative and positive values of q, respectively. Regardless, the structural changes within time series are caught instantly with the local Hurst exponents and it is the major advantage compared to generalized Hurst exponent.

#### Vector autoregressive fractionally integrated moving average model (VARFIMA)

*y(t*) is the state output,

*e(t)*is the white noise input, and

*L*is the shift operator.

*a*

_{i}, and

*b*

_{j}are real coefficient matrices of n by n dimensions,

*d*is real integrating parameter between − 0.5 and + 0.5. The parameter d is also called memory parameter because it regulates the long memory property of the series.

*t(L*

^{− 1}

*)*where

*t(L)*function can be defined as (11)

*I*

_{n}is the identity matrix with dimensions n by n. Equation (11) can now be written in summary notation as

*L*is the lag operator and A(L) = (

*A*

_{0}−

*A*

_{1}

*L*− … −

*A*

_{p}

*L*

^{p}). (

*A*

_{0}−

*L*)

^{d}and B(L) =

*B*

_{0}+

*B*

_{1}

*L*+ … +

*B*

_{q}

*L*

^{q}. Equation (12) can be written as follows:

This means a two-step of calculation for *ξ(t)* first and then for *y(t).*

*A*(

*L*)| is the scalar valued determinant of A(

*L*) and

*A*

^{∗}(

*L*) is the adjoint matrix. The process becomes

*L*and co-integration fraction,

*d*.

It is important to underline that forecasting is an overwhelming task. Whilst a statistically sensible model may be obtained, it may still endure severe problems in forecasting. Hence in this paper, the validity of the model is determined by judging the performance of the forecasting results. A stochastic process vector ARFIMA (p, d, q) is already involved in our approach in modeling phase. Once we obtain statistically admissible model parameters (p, d, q), the standard best linear predictor of the program is used. The program automatically determines the best option by running three different methods, “Covariance” (exact covariance), “AR” (approximate with a large0order AR process) or “Kalman” (Kalman filter) method.

## Wavelet coherence and local Hurst exponents

Hurst Exponent of Real and Fluctuation function at scale 256

Gold | Silver | Platinum | |
---|---|---|---|

General Hurst Exponents | 1.5043 | 1.4714 | 1.4927 |

General Hurst Exponents of Fluctuation Function Time Series at Scale 256 | 1.3255 | 1.3237 | 1.4515 |

% Change | 11.89% | 10.04% | 2.76% |

When we calculated the local Hurst exponents at scale 256, the fluctuation function series obtained at the specified scale are still demonstrating multifractal behavior as strong as the raw data. There is only 10% difference for gold and silver and 2.7% for platinum. The main advantage of working with local Hurst exponents with respect to generalized Hurst exponent is the ability of local Hurst exponent to identify the time instant of structural changes within the time series (Ihlen, 2012).

Minimum and Maximum Values of Local Hurst exponents at scale 256

Gold | Silver | Platinum | |
---|---|---|---|

Min Hloc | 1.0342 | 0.8648 | 0.8657 |

Max Hloc | 1.8254 | 2.0615 | 2.133 |

Since we know that both the real data and the fluctuation function data exhibits multifractal behavior, autoregressive fractionally integrated moving average (ARFIMA) model with the real data and vector autoregressive fractionally integrated moving average (VARFIMA) model with the fluctuation function time series will be used to estimate a process and forecast for the next 30 days. Finally, we will compare the results accordingly.

## Forecasting results and discussion

Multivariate models lead to more accurate results compared to scalar counter models (Tsay, 2013) because not only the historical data of each series but also other fractionally co-integrated variables in between the series are taken into consideration in VARFIMA models. The study of Das et al. in 2018 concludes that diversification benefits in pacific developed markets are limited due to higher degrees of integration (Das et al., 2018). We believe otherwise that it is important to set up highly correlated data set in order to establish efficiently estimated processes with better forecasting results. The raw data and fluctuation function time series data will be picked for the same interval for forecasting purposes. Raw data using ARFIMA process and fluctuation function data using 2D VARFIMA process will be forecasted and the results will be compared accordingly.

The reader must note that when 2D VARFIMA model are run, the forecasting results are provided with upper and lower limits of two dimensional vectors. In our results, we have eliminated the highest value of the upper boundaries and the lowest value of the lower boundaries displaying the narrowest band possible.

Dates and couples used to forecast precious metals

# | From | To | Correlation | ||
---|---|---|---|---|---|

Gold | Platinum | GP1 | 25.02.2011 | 11.01.2012 | 0.7905 |

Gold | Platinum | GP2 | 19.12.2011 | 18.04.2012 | 0.6588 |

Gold | Silver | GS1 | 25.02.2011 | 06.01.2012 | 0.8538 |

Gold | Silver | GS2 | 06.03.2010 | 18.05.2012 | 0.8056 |

Silver | Platinum | SP1 | 19.04.2010 | 21.07.2012 | 0.7189 |

Silver | Platinum | SP2 | 26.02.2011 | 11.01.2012 | 0.8669 |

This has been the case for all metals and time intervals chosen. ARFIMA (univariate model) have been used with the raw data itself and VARFIMA (multivariate model) have been used with the fluctuation function time series provided out of local Hurst exponent calculations.

The difference between 30th day forecast and real value

The Difference Between 30th Day Forecast & Real Value | ||||||||
---|---|---|---|---|---|---|---|---|

ARFIMA | VARFIMA | % | ARFIMA | VARFIMA | % | |||

GP1 | Gold | 2511 | 82 | 3.2 | Platinum | 2463 | 104 | 4.2 |

GP2 | Gold | 1495 | 13 | 0.9 | Platinum | 2018 | 17 | 0.8 |

GS1 | Gold | 2279 | 273 | 11.9 | Silver | 4836 | 29 | 0.6 |

GS2 | Gold | 495 | 72 | 14.5 | Silver | 537 | 168 | 31.3 |

SP1 | Silver | 814 | 585 | 71.8 | Platinum | 1105 | 106 | 9.9 |

SP2 | Silver | 4254 | 561 | 13.2 | Platinum | 2467 | 63 | 2.5 |

Additionally, the present results may assist members of the financial world at varying degrees in their field. Vast number of the research papers report that the price of gold is an indicator of inflation (Ranson & Wainright, 2005). This is because of commodity prices can respond to new information faster than any consumer prices (Mahdavi & Zhou, 1997). Furthermore, Gold is treated as hedging instrument against inflation and exchange rates (Hammoudeh et al., 2010). Therefore, accurate forecasting (i.e. forecasts with smaller error bands) of the gold price will serve to monetary policymakers, hedge fund managers, international portfolio managers and central banks to make accurate investment decisions in the financial market. However, forecasting of gold prices is a formidable task. This is because of the multifractal nature of the gold prices. To the best of our knowledge there is no general approach for forecasting of multifractal data.

We have overcome this difficulty by determining the time scale in which gold, silver and platinum exhibit coherence for all times. This happens to be 256-day period. Furthermore, we fit VARFIMA model for this scale which leads to forecasting results with extremely narrow error bands. Although Silver and Platinum serve as coherent companions of Gold. We also have perfect forecasting performances since we consider them as a component of VARFIMA model. This will allow to make healthier assessment of inflation expectations. These empirical findings may particularly serve as decision aids for policy makers, hedge fund managers, international portfolio managers and gold exporters.

## Conclusion

Multiple wavelet coherence and multifractal de-trended fluctuation analysis (two important tools in the analysis of financial time series) have been employed to explore the inter dependency and multifractality of precious metals, gold, silver and platinum. Firstly, it is presented that multiple wavelet coherence provides higher resolution to visualize in-phase movement of different time series in time and frequency space compared to any other traditional correlation function analysis (Yilmaz & Unal, 2016). The direct observation allowed us to conclude that all three metals are highly correlated at higher periods throughout the time, especially around 256-day period.

We have observed that long range dependence is weaker over certain periods of time and frequency, which might be due to asymmetric error-correction mechanisms between precious metal prices during shocks or stresses. This could mean that it would take time at different rates for all metal prices to find equilibrium. This is parallel with the findings of (Kucher & McCoskey, 2017) whom stated that the long run relationships between precious metals are strongly influenced by economic conditions.

Compared to univariate counterparts (ARFIMA), it is depicted that the consistency and performance of forecasting with multifractal time series is remarkably increased with multivariate models (VARFIMA) (Durr et al., 1997). This was also true in spite of the size of the data set chosen (Dueker & Startz, 1998) where we have used set of data from 300 daily prices up to 1100 daily prices.

The generalized Hurst exponents of all precious metals have revealed that daily prices of precious metals do possess long range dependence with persistent structure and demonstrate multifractal behavior in general. Furthermore, local Hurst exponents of precious metals at 256-day period turned out to be multifractal as well. The local Hurst exponents are at the highest part of the scale during their latest timeframes. It indicates a strengthening trend in terms of long-range dependence with persistent structure (Ihlen, 2012), which means that the periods do not experience large variations and the prices are evolving in slower pace.

A new inter-calculated fluctuation function time series is generated during the calculations of local Hurst exponents. These fluctuation function time series are used to model and forecast the data for the next 30 days. Even though both of the methods (ARFIMA and VARFIMA) are adopted to integrate time series into the model fractionally and all series demonstrate multifractal behavior, ARFIMA model have almost never produced forecasting results as successful as VARFIMA model which have outperformed its univariate match conspicuously. This has proven the fact that obtaining data couples with higher correlation helps accomplish better forecasting results.

Jiang and Zhou developed multifractal detrended moving average cross correlation analysis (Jiang & Zhou, 2011) based on detrended fluctuation analysis. For future research, it may be utilized to compare with the current methodology.

## Notes

### Acknowledgements

We would like to thank our anonymous referees for their constructive comments and valuable contributions.

### Funding

Not Applicable.

### Availability of data and materials

The datasets generated and/or analyzed during the current study are available from Yahoo Finances! or from corresponding author on reasonable request.

### Authors` contributions

EO analyzed the data with multiple wavelet coherence and multifractal de-trended fluctuation analyses and generated forecasting results using vector autoregressive fractionally integrated moving average model. GU was the supervisor in construing the results, conclusion as well as in writing the manuscript. All authors read and approved the final manuscript. The content of the manuscript has not been published or submitted for publication elsewhere.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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