Quantifying information transfer among precious metals: A novel transfer entropy-based approach

Abstract

This study looks at the asymmetric information flow between bivariate pairs of daily returns for gold, silver, platinum, and palladium from January 1, 2000, to December 31, 2024. In order to obtain robust estimates while accounting for nonlinear, non-parametric, and asymmetric correlations in bivariate returns series, we employ Shannon and Rényi entropy transfer techniques rather than the widely utilized Granger causality approach. The findings reveal that there was a mixed information flow in the interactions between the precious metals, with gold leading the way in information transmission. Higher integration was found at the higher scale (lower frequency) via the wavelet multiple correlation whiles the wavelet multiple cross correlation reveal insignificant spillover effects since localization occurred at the point of symmetry. Investors and policymakers should understand the importance of time-sensitive information flow for risk assessment and portfolio management, as well as the dynamic interrelationships across market.

Keywords:

Precious Metals

Information Flow

Entropy Transfer

1. Introduction

For the overall macroeconomic performance and living standards of nations that rely heavily on commodities, the transmission effects in precious metals and commodities markets in general have significant ramifications (see Deaton, 1999; Cashin et al., 2002). When choosing and creating diversified investment portfolios, precious metals have taken on a significant and prominent role. Financial analysts and macroeconomists have concentrated on the dynamics and co-movement of prices across several commodities or market places. Nevertheless, there are few empirical investigations on the directionality and content of information transfer between precious metals. Despite the fact that the empirical literature on precious metals has grown over time, little research has been conducted in the metals market regarding how the content of information flow on individual metals affects the other.

A fascinating area of current empirical research in economics and finance is the study of information transfer among time series data. This cross-market information flow demonstrates how interdependent markets are. Market agents are very interested in quantifying such information transfer in order to identify information from dominant marketplaces.

After the March 2000 stock market crash, the outstanding value of commodity contracts which include agricultural commodities, metals, oils, and other resource commodities was more than US$7.6 trillion in June 2007 compared to US$9.2 trillion in equity-related contracts. A substantial US$0.5 trillion in outstandings is made up of gold and other important precious metals, such as silver, platinum, and palladium (BIS, 2008). Given the economic significance of the precious metals market, it is a surprise there is paucity of studies assessing the information flow among precious metals.

Studying the information flow between precious metals is crucial for several reasons that encompass economic, scientific, and environmental perspectives. Precious metals like Gold and silver have long been viewed as safe-haven assets during times of economic uncertainty (Tweneboah, 2019). Studying the information flow among these metals can help investors and economist gauge market sentiments and predict price movements (Baur & Lucey, 2010). Ciner et al., (2013) posit that understanding the correlation and interaction between different precious metals can provide insights into market dynamics, and changes in price of one metal often influence others, due to investment flows and industrial demand. Wang and Lee (2021) assert that understanding how information flows among these markets helps in predicting price reactions to global events, such as political instability or changes in trade policies.

The information flow between precious metals is essential for understanding market behaviour, investment strategies, and the broader implications of geopolitical and environmental factors. The literature supports the importance of this field across various dimensions, indicating that such research is not only timely but also critical for informed decision-making in finance, policy, and environmental management.

Different aspects of studies in stocks and precious metals have been investigated by different economists. Nevertheless, despite its crucial significance in economic progress, information transmission among precious metals is rarely studied (Yue et al., 2022). Various techniques have been used to determine the degree of correlation between precious metals and stock markets. These include non-parametric techniques such as generalized autoregressive conditional heteroskedasticity (GARCH) and vector autoregression (VAR) models. In this study, the information transmission between precious metals is measured using entropy based approach. Information flow is measured and quantified using the entropy-based technique, which uses a log-likelihood ratio based on the probability density function. The inability of VAR and GARCH models to quantify the direction of causality of information flow is one of its drawbacks. Decision-making cannot be adequately supported by merely detecting information flows rather than quantifying such information (Osei & Adam, 2020). The direction of spillovers inside the information network can be captured by the entropy-based approach (Huynh et al., 2022). The entropy approach's primary benefits are its capacity to capture non-linear dynamics and its ease of implementation and interpretation due to non-parametricity (Osei & Adam, 2020). It aids in determining the direction of the net flow when there is a bidirectional substantial information flow. We think this approach is a good and promising substitute for the conventional methods.

The main objective of this research is to contribute to the current understanding of the relationships between precious metal returns. We first look into the information flow from gold to silver, platinum, and palladium. Second, we evaluate the flow of information from silver to palladium, platinum, and gold. Third, we look at how information moves from platinum to palladium, silver, and gold. Finally, we examine how information moves from palladium to platinum, silver, and gold.

Additionally, our study makes a number of other contributions. First, there is lack of studies on the information transfer among precious metals, despite the fact that there have been several studies on precious metal returns. As far as we are aware, this is the first study to present actual empirical findings on the information flow across precious metal returns; nonetheless, it builds on Tweneboah's (2019) argument that precious metals show time-varying co-movement across frequencies and over time. Second, by concentrating on precious metals, we investigate the characteristics of the price relationship and arbitrage between gold and other precious metals, which have also been covered in earlier research. For instance, practitioners frequently contend that gold acts as a buffer against inflation since it functions similarly to surrogate money. Our empirical study will provide valuable evidence regarding the substitutability of gold and other precious metals, as indicated by their historical use as coinage, or whether they occupy distinct markets with distinct uses and functions, as suggested in recent finance literature (e.g. Ciner, 2000; Erb & Harvey, 2006).

By using a transfer entropy-based technique that is able to capture the direction and quantum of information flow among variables, in this case, precious metals, and this study closes an empirical gap in the body of knowledge. Given that it gives information regarding the direction and quantum of the information, this empirical approach is unquestionably appropriate. Given how important this information is to investors, speculators, and policymakers, it is highly helpful to extract the information content flow among these important metals. The transfer entropy-based approach used in this study closes an empirical gap in the literature by capturing the direction and quantum of information flow among variables. This is unquestionably an appropriate empirical procedure since it offers information regarding the direction and quantum of the information. For the analysis of economic time-series, entropy-based approaches are therefore the most suitable for capturing the quantum effect of a variable, which is important for market participants and investors when it comes to diversification and investment choices. As far as we are aware, this method is the first to be used to measure the dynamics of information flow between precious metals, and it would enable us to answer the following questions. First, can traders who invest in other precious metals learn more from the fluctuations in gold returns? Second, what propels the information flow—gold and other precious metals? Answers to these issues would give traders, investors, and other participants in the precious metals market more precise information on the kinds of positions they should take when deciding whether to buy, sell, or hold, as well as how to respond to dynamics of information flow among the precious metals. The findings reveal that there was a mixed information flow in the interactions between the precious metals, with gold leading the way in information transmission. Higher integration was found at the higher scale (lower frequency) via the wavelet multiple correlation whiles the wavelet multiple cross correlation reveal insignificant spillover effects since localization occurred at the point of symmetry. The theoretical and empirical literature is covered in Section 2. The methodology is described in Section 3. Section 4 presents a description of the data and statistical properties. The results and analysis of the information content flow between precious metals is captured in Section 5, and the last section is the conclusion and policy recommendations.

2. Literature review

Precious metals, such as gold, silver, platinum, and palladium, play pivotal roles in the global economy as stores of value, investment assets, and industrial materials. Understanding the information flow between these metals is essential for investors and policymakers, as it influences trading strategies and risk management practices. This review explores the theoretical frameworks and empirical research that analyze the interrelationships and information transmission among precious metals.

2.1 Theoretical Literature

2.1.1 Efficient Market Hypothesis (EMH)

The Efficient Market Hypothesis (EMH) asserts that asset prices reflect all available information at any given time (Fama, 1970). In the context of precious metals, this implies that price adjustments in one metal should immediately incorporate information that may affect others. For instance, the rapid transmission of price signals can be observed in the behavior of gold as a leading indicator for silver and platinum prices (Barberis et al., 1998).

2.1.2 Price Discovery

The process of price discovery in financial markets relates to how information is reflected in asset prices over time. In precious metals markets, gold is often viewed as the principal driver of price movements, with other metals following its lead. Empirical evidence indicates that gold price fluctuations significantly affect silver and platinum prices (Baffes & Koo, 2006), suggesting a unified price discovery process among these closely related assets.

2.1.3 Correlation and Causation

Understanding the correlation between the prices of different precious metals is crucial in evaluating their interdependencies. Research has demonstrated a strong positive correlation between gold and silver prices, indicating that changes in one often signal shifts in the other (Wang et al., 2018). However, correlation does not establish causation, necessitating further analyses, such as Granger causality tests, to elucidate the directional influence of price movements (Gao et al., 2020).

2.1.4 Psychological Factors and Investor Behavior

Investor behavior and market psychology also play crucial roles in the dynamics of precious metals trading. Theoretical frameworks from behavioral finance illustrate how perceptions, biases, and sentiments can drive price movements (Kahneman & Tversky, 1979). For example, during periods of economic uncertainty, investors may flock to gold as a safe haven, leading to price increases that subsequently influence silver and platinum prices through perceived risk aversion (Hwang & Kim, 2014).

2.1.5 Graph Theory

Recent advancements in network analysis facilitate a deeper understanding of the relationships among assets. Network approaches utilizing graph theory enable researchers to visualize and quantify the information flow between precious metals. By constructing correlation networks, it becomes possible to identify central metals in price transmission processes and highlight interdependencies within the market (Kok & Akiyama, 2022).

2.1.6 Algorithmic and Machine Learning Approaches

The application of machine learning techniques in financial markets has opened new avenues for evaluating complex relationships among assets. Recent studies employ these methods to identify patterns in price movements and to predict information transmission between precious metals (Abraham & Zhang, 2021). These innovative approaches demonstrate the growing significance of technological advancements in enhancing our understanding of market dynamics.

The theoretical literature on information flow between precious metals underscores a multifaceted relationship influenced by market dynamics, macroeconomic factors, and investor behavior. While gold often serves as the leading indicator, significant correlations exist among all precious metals, highlighting the importance of recognizing interconnectedness in trading strategies.

2.2 Empirical Literature

According to a research by Hillier et al. (2006), precious metals are now widely regarded as relevant assets for risk management, portfolio diversification, and hedging. Hillier et al., (2006) further posits that their properties have attracted much attention in modern literature. The literature on precious metals has been categorised into different strands. Some studies on precious metals have focused on the hedging properties (see for instance, McCown & Shaw, 2016; Bredina et al., 2017; ; Pierdzioch et al., 2016; among others). These studies came to the conclusion that precious metals like gold and silver have safe-haven qualities and can be used as a hedge against currency rate volatility and periods of ongoing inflation.

Another strand of literature on precious metals concentrates on the interdependence nature of these metals and their spillover effects. Sari et al. (2010) evaluated the co-movements and information transmission among Gold, Silver, Platinum, and Palladium using data covering the period 4 January 1999–19 October 2007 and utilizing closing spot prices of these precious metals, the price of oil, and the US dollar/euro exchange rate. An increase in the price of gold causes parallel changes in the prices of other precious metals, according to Sari et al. (2010). The study employed a number of econometric approaches, including the unit root, cointegration, the impluse response function, and variance decomposition. According to Sari et al. (2010), as gold is a leader in the transmission of information, a model that effectively explains gold prices could also contribute to models used in predicting the prices of other precious metals.

Some studies have also examined the volatility of price returns of precious metals such as gold, silver, platinum, and palladium. In order to investigate the correlation and volatility returns of these precious metals (gold, silver, platinum, and palladium), Hammoudeh et al. (2011) used daily closing returns data for the period 4 January 1995–12 November 2009 using the RiskMetrics and GARCH models. The primary goal of the study conducted by Hammoudeh et al. (2011) was to investigate risk management implications for hedging purposes. The study concluded that GARCH-t model should be used to calculate VaR in precious metals returns.

According to Batten et al. (2010), palladium, platinum, silver, and gold have a weak integration. Batten et al. (2010) employed the conditional standard deviations and incorporated a VAR framework and block exogeneity causality test into the estimation analysis to assess the integration of these precious metals. Batten et al. (2010) utilized data from 1986 to 2006 and apply these precious metals (gold, silver, platinum, and palladium) to investigate the relationships between these metals.

Oral and Unal (2017) evaluate the degree of interdependence among the daily prices of platinum, silver, and gold over the 2011–2015 timeframe. Bivariate and multiple wavelet coherence were employed in the investigation. According to Oral and Unal (2017), the multivariate results showed stronger integration than the bivariate situation, where the connection appeared to be negligible. The study highlights the bivariate model's drawbacks. Once more, the analysis shows that the precise correlation coefficient was not captured by the wavelet coherence approach.

Bhatia et al. (2018) examine the causal relationship between spot prices of gold, silver, platinum, and palladium through the mean and variance using the vector autoregressive (VAR) model. Using daily and weekly spot price data from 2000 to 2016, Bhatia et al. (2018) employed a model that enables the examination of causation between the metals during recessions, booms, and normal market situations. Strong causality was reported for the middle quantiles whiles bidirectional causality in mean and variance among the prices of precious metals were revealed.

Das et al. (2018) investigated the spot gold prices of China, Indonesia, India, Thailand, Vietnam, and South Korea using the bivariate and multivariate wavelet methods. The study made use of data from 1986 to 2016. According to the findings, there was a greater degree of price integration following the 1997 Asian financial crisis. According to the wavelet multiple cross correlation data, Thailand leads the market in spot gold prices across all wavelet scales with the exception of one that is dominated by India. Das et al. (2018)'s study's primary drawback is its focus on the Asian market.

Huynh (2020) investigated the association between precious metals prices and uncertainty. The study used the Economic Policy Uncertainty (EPU) and the Chicago Board Options Exchange Volatility Index (CBOEVIX) to represent uncertainty and used gold, silver, palladium, and platinum to represent precious metals. Huynh (2020) employed two cutting-edge methodologies, the multilayer perceptron neural network nonlinear Granger causality and transfer entropy for the estimation analysis and find gold to be the most popular safe-haven asset for hedging purposes. The study reveals precious metals had the propensity to influence EPU and VIX, although they are resistant (unresistant) to EPU (VIX) shocks.

2.3 Research gap

The reviews above make it evident that there are gaps in the body of knowledge. The results are far from definitive, despite the fact that numerous studies have looked into the causality and dynamic relationship among precious metals. Additionally, majority of the approaches used in these studies prominently ignore the nonlinearities in the linear relationship while capturing it. The majority of approaches were GARCH-based models, variance decomposition, unit roots, impulse response functions, causality tests, vector autoregressive models, and cointegration. The quantum of information transmission between the precious metals is not captured by these methods. The transfer entropy developed by Shannon (1949) and Renyi (2007) is not used in any of the reviewed studies above to investigate the information flow between the precious metals. These gaps in the existing literature serve as the impetus for this investigation. Answers that are relevant to the precious metals pricing system must be provided. The information transfer among these precious metals and whether there is a well-known metal that leads the pricing dynamics have not been sufficiently addressed by either economic theory or empirical research. While some economists, like Tweneboah (2019), have suggested that silver has more industrial uses than gold and hence frequently outperforms gold, the majority of economists, like Sari et al. (2010), have firmly maintained that gold is the most valuable metal among silver, platinum, and palladium. In its current form based on extant literature, the study acknowledges that there are intriguing concerns that remain unanswered. Does the price of gold determine the price of other precious metals, or does the price of other precious metals like silver determines the price of Gold? To the best of our knowledge, the study is the first to apply both the Shannon and the Renyi transfer entropies to analyze the information flow among the most valuable metals (gold, silver, platinum, and palladium), filling a vacuum in the body of existing literature.

3. Data and research design

3.1. Data and variables

In this study, we collected data from the Bloomberg Financial database. The sample we used spans the time frame of January 31, 2000, to November 29, 2024. When choosing the top four precious metals (gold, silver, platinum, and palladium) for the information flow study, we adhered to Tweneboah and Alagidede's (2018) methodology. Tweneboah and Alagidede (2018) contend that all of the major financial databanks lack information on other valuable metals, including iridium, rhodium, ruthenium, osmium, and rhodium. For the same reason—a paucity of data—we were unable to find an antidote in this investigation. The daily metal prices were transformed into returns by taking the first difference of the logarithms as as r =

$\:{lnp}_{t}$

$\:{lnp}_{t-1,}\:where\:{p}_{t}\:and\:{p}_{t-1}$

are are spot prices (in US dollars) of the metals at day t and t-1 respectively.

3.2. Data description

Table 1 displays the descriptive statistics. We present the variables' mean, standard deviation, skewness, Kurtosis, Jarque-Bera, and observations. Gold has the highest mean return among all the metals, which are all positive. Following gold, platinum had the second-highest mean, followed by palladium. Silver has the least mean. According to the standard deviation, which measures volatility, silver has the least volatility and gold is the second most volatile metal after palladium. The positive skewness of all the returns indicates that investors are more likely to receive sizable positive daily returns than negative ones. The distribution displays a peaked distribution, as indicated by the kurtosis. Once more, the returns are leptokurtic, with gold suffering the least amount and palladium the most, suggesting a wider or flatter shape with fatter tails which by implication to investors indicate a greater chance of extreme positive or negative events. The Jarque–Bera (JB) statistics for all the returns fails to conform to the Guassian distribution.

Table 1
Summary statistics
Metal	Mean	St dev	Skewness	Kurtosis	JB	Observation
Gold	1133.943	586.330	0.110	2.207	8.425*	299
Silver	16.714	8.642	0.450	2.959	10.135*	299
Platinum	1065.525	357.264	0.573	2.871	16.592*	299
Palladium	831.395	634.000	1.373	4.169	111.063*	299
Note: * indicates rejection at 5 percent level

3.3. Transfer entropy technique

Transfer Entropy techniques for analyzing data has become prominent in economics and finance studies since its introduction by Schreiber in 2000. Several authors have argued that transfer entropy provide more robust estimates than the conventional Granger Causality especially when the returns exhibit tails distributions (Wang & Wang, 2021). A non-parametric, transfer entropy offers asymmetric information transmission across returns and—more significantly—does not assume subject-specific constraints related to the underlying stochastic processes (Nyakurukwa & Seetharam, 2022). Techniques based on the Shannon and Renyi entropy have gained popularity for measuring the information flow between financial time series. To measure the information flow among the top precious metals, we use both the Shannon and R´enyi entropies in this work (see Tweneboah & Alagidede, 2018). Black points (blue for Shannon and red for Renyi) on the bars represent the effective transfer entropy. The bars' endpoints display the 95% confidence boundaries. This provides information on whether the precious metals have a statistically significant information flow. A rejection of the null hypothes signals that the boundaries are either positive or negative. Information flow among the metals becomes insignificant when an overlap at the origin is detected.

3.3.1. Shannon transfer entropy (STE)

According to Boateng et al. (2022), the Shannon transfer entropy method of quantifying the flow of information among time series is a measure of the uncertainty upon which the transfer entropy is encapsulated in information theory. Following Hartley (1928), the average of each symbol can be shown as:

H =

$\:\sum\:_{j=1}^{n}{p}_{j}$

$\:{log}_{2}$

$\:\left(\frac{1}{{p}_{j}}\right)$

bits (1)

In Eq. (1), n denotes the number of different symbols in relation to the probabilities Pj. Assuming the log denotes the logarithm of a number to base 2, Shannon (1948) posits that for a discrete random variable J with a probability distribution p (j), where j is presumed to stand for all possible outcomes, j can take the average number of bits which is needed to encode the independent draws from the distribution of j. It can be written as:

$\:{H}_{J}\:=\:-\sum\:_{j=1}^{n}p\left(j\right){log}_{2}p\left(j\right)$

In measuring and quantifying the information flow between the time series data, it is critical to merge the concepts of Shannon entropy and Kullback-Liebler distance (Kullback & Leibler, 1951). Schreiber (2000) also assumes that the underlying process evolves through a Markov process. If I and J stands for two discrete random variables with a marginal probability distribution p (i) and p (j) respectively, in addition to a joint distribution p (i, j), whose dynamic structure is similar or equivalent to stationarity, Markov process of order k and j, then the Markov property suggest that the probability of observing I and t + 1 in state i conditional on the k previous observation is: P

$\:\left({i}_{t+1}│{i}_{t}\dots\:\dots\:\dots\:{i}_{t-k+1}\right)\:$

= P

$\:\left({i}_{t+1}│{i}_{t}\dots\:\dots\:\dots\:{i}_{t-k}\right)$

The average number of bits which is needed to encode the observation in t + 1 if the previous K values are identified is represented as:

$\:{h}_{j}$

(k) = -

$\:{\sum\:}_{1}p\left({i}_{t+1}\:{i}_{t}^{\left(k\right)}\right)$

logP

$\:\left(({i}_{t+1}│{i}_{t}^{\left(k\right)}\right)$

(3)

Where

$\:{i}_{t}^{\left(k\right)}$

$\:\:\left({i}_{t}\dots\:\dots\:\dots\:{i}_{t-k+1}\right)$

is subject to the corresponding direction for process J.

From a bi-variate perspective, and in comformity with Kullback-Leibler distance, the information flow from process (J) to process (I) is derived by dicovering the departure from the generalised property

$\:\left({i}_{t+1}│{i}_{t}^{\left(k\right)}\right)$

$\:\left({\:i}_{t+1}│{i}_{t\:\:\:\:\:\:\:,}^{\left(k\right)}\:\:{j}_{t}^{\left(i\right)}\right)$

. The Shannon transfer entropy is therefore written as:

$\:{T}_{J\to\:1}$

(k, l) =

$\:{\sum\:}_{i,\:j}p\:\left({i}_{t+1}\:\:,\:\:{i}_{t}^{\left(k\right)}\:{j}_{t}^{\left(i\right)}\right)$

. Log

$\:\left(\frac{\text{p}\left({\:i}_{t+1}│{i}_{t\:\:\:\:\:\:\:,}^{\left(k\right)}\:\:{j}_{t}^{\left(i\right)}\right)\:}{P\:\left({i}_{t+1}│{i}_{t}^{\left(k\right)}\right)\:}\right)$

(4)

Where

$\:{T}_{J\to\:1}$

calculates the flow of information from J to I, in the same scenario,

$\:{T}_{1\to\:J}$

measures the information flow from I to J. The net information flow from I to J is derived by differencing

$\:\:\left(\text{t}\text{a}\text{k}\text{i}\text{n}\text{g}\:\text{t}\text{h}\text{e}\:\text{d}\text{i}\text{f}\text{f}\text{e}\text{r}\text{e}\text{n}\text{c}\text{e}\:\text{b}\text{e}\text{t}\text{w}\text{e}\text{e}\text{n}\:{T}_{J\to\:1}\text{a}\text{n}\text{d}{T}_{1\to\:J}\right)$

3.3.2. R´enyi Transfer Entropy (RTE)

In this study, we employ the Renyi transfer entropy to serve as a robustness check on the results of the Shannon transfer entropy since the Shannon information theory is not wide in scope and appears idealised in its information transmission appearing only when the buffer memory or a transmitting channel is infinite (Jizba et al., 2022). This limitation has led to improvement of the Shannon transfer entropy by information theorist, notable among them is the Renyi entropy. The advantage of the Renyi transfer entropy is its ability to extend more weights to tail events concerning its contribution to the whole system of the information flow (Nyakurukwa, 2021). The Renyi transfer entropy comes with a weighting parameter p > 0, which is derived from the R´enyi-based entropy for the individual probabilities p (j). It is specified as:

$\:{H}_{j}^{q}\:=\:\frac{1}{1-q}\:log\left(\sum\:_{j}{p}^{q}\:\:\left(j\right)\right)$

With ད>0. For ད→ 1, Renyi transfer entropy and Shannon converge. For 0 <ད< 1, events with low profitability of occurrence are given more weights, while for ད>1, the weights is tilted towards outcomes j with a probability of occurrence. RTE is a dynamic mechanism which measures the uncertainties by observing distinct distribution areas based on the parameter ད. using the escort distribution

$\:{\varnothing\:}_{q}\left(j\right)$

$\:\frac{{p}_{n\:\:\:\:\:\:\:\:\left(j\right)}}{\sum\:_{j}{p}^{q}\:\:\:\:\left(j\right)}$

with a normalisation of the distribution with ད>0. Jizba et al. (2012) proceed to derive RTE as:

$\:{\:RT}_{J\to\:I}$

(k, L) =

$\:\frac{1}{1-q}$

log

$\:\left(\frac{{\sum\:i\varnothing\:}_{q\:\:}\left({i}_{t}^{\left(k\right)}\right)\:{{p}^{q}\:\left(\left({\:i}_{t+1}|{i}_{t\:\:\:\:\:\:\:}^{\left(k\right)}\right)\right)}^{\:\:\:\:\:\:\:\:\:\:\:\:\:}}{{\sum\:}_{ij}{\:\:\:\:\varnothing\:}_{q\:\left({i}_{t}^{k\:\:,{j}_{t}^{l}\:\:}\right)}\:{p}^{q\:}\:\:\left({i}_{t+1│{\:\:i}_{t}^{\left(k\right)}\:{\:\:j}_{t}^{l}}\right)\:}\right)$

(6)

Where

$\:{\:RT}_{J\to\:I}$

estimates the information flow from process J to process I. It is assumed that the calculation of RTE has the tendency to produce negative values. With that scenario, the history of J, if identified, shows the extent of uncertainty and not just the history of I. The results from the TE are efficient especially when the sample size is large, though it can produce biased results when the sample size is too small (Marschiniski & Kantz, 2002). This is corrected with a shuffled version specified below:

$\:{ETE}_{j\to\:I}$

(K, I) =

$\:{T}_{j\to\:I}$

(K, l)

$\:-{T}_{jshuffled\to\:I}$

(K, I)

$\:{T}_{jshuffled\to\:I}$

(k, I) is TE using shuffled version of the time series J.

3.3.3. Wavelet multiple correlation & Wavelet multiple cross correlation

The wavelet multiple correlation and wavelet multiple cross correlation commence with the maximal overlap discrete wavelet transform (MODWT) which is defined as follows:

Let

${X_t}={x_{1t,}}{x_{2t,}},...,{x_{nt}}$

be a multivariate random process and

${W_{jt}}={w_{1jt,}}{w_{2jt,}},...,{w_{njt}}$

represent the corresponding scale

${\lambda _j}$

coefficients of the wavelet, which is derived by employing the MODWT. The wavelet multiple correlations (𝑊𝑀𝐶) 𝜑𝑋 (𝜆𝑗) is defined as a single set of multiscale correlations from Eq. (7) subsequently.

𝜑𝑋 (𝜆𝑗) =

$\:\sqrt{1-\frac{1}{maxdiag{P}_{{J}^{-1}}}}$

(7)

Where

${P_j}$

is the

$(n \times n)$

correlation matrix of

${W_{jt}}={w_{1jt,}}{w_{2jt,}},...,{w_{njt}}$

and

$\operatorname{maxdiag} (.)$

elects the maximum element in the diagonal argument. The coefficient of determination in regression theory is the squared correlation between the observed

$\:{Z}_{t}\:$

and the fitted

${\hat {z}_i}$

values. Thus, the wavelet multiple correlations can also be expressed as Eq. (8), where

${w_{ij}}$

is selected to maximize 𝜑𝑋 (𝜆𝑗) and

${\hat {w}_{ijt}}$

are the fitted values in the regression of

${w_{ij}}$

on the rest of the wavelet coefficients at scale

${\lambda _j};$

$\:\:$

(8)

The wavelet multiple cross-correlations are computed by allowing a lag

$\:\tau\:$

between the observed and fitted values of the variables at each scale, and can be specified as:

$\psi X({\lambda _j})=Corr({w_{ijt}},{\hat {w}_{ijt+\tau }})$

$=\frac{{Cov({w_{ijt}},{{\hat {w}}_{ijt+\tau }})}}{{\sqrt {Var({w_{ijt}})Var({{\hat {w}}_{ijt+\tau }}} )}}$

4. Empirical results and discussions

4.1. Transfer entropy analysis

The Shannon and Renyi transfer entropy results are shown in this section, with the 95% confidence bound shown by the ends of the red and blue bars. The entire blue or red bar should be in the positive or negative zone under the Shannon and Renyi transfer entropies in order to reject the null hypothesis that there is no information movement between the precious metals. When there is an overlap, the information flow is negligible. A negative information flow indicates interdependence, contagion, or the transfer of shocks, whereas a positive information flow indicates the transfer of low risks, according to the Shannon and Renyi transfer entropies. The information flow becomes insignificant when there are no significant exchanges showing no potential for positive or negative transfers. Tweneboah and Alagidede (2018) argued that precious metals are integrated if there is a contagion or interdependence. We can infer integration from a strong positive or negative information flow. When there is weak as well as an insignificant information flow, investors can explore avenues for diversification and hedging and insignificant transfers signify potential avenues for diversification and hedging.

4.1.1 Results of ShannonTransfer entropy

The quantification of information flow between gold and other precious metals (palladium, platinum, and silver) is shown in each panel of Fig. 1. Information flowing from gold to another precious metal is represented by the lines on the left of each panel, while information flowing from other precious metals to gold is represented by the lines on the right. The findings show that there are both positive and negative information exchanges on both sides of the market when gold moves toward other precious metals. With the exception of silver, the majority of information exchanges were negligible; however, the results show that there is no bidirectional information flow between gold and silver. The result of the information flow from Gold to silver is not surprising since several studies have established such findings using the conventional methodologies (see for instance, Tweneboah & Alagidede, 2018; Oral & Unal, 2017). We find no significant information flow from Silver and Platinum to Gold except Palladium which recorded positive information flow to Gold signifying low risk. The low integration of information flow among the precious metals is supported by Batten et al. (2010) whose studies concluded that there is a weak integration between gold, silver, platinum, and palladium. In summary, except for Gold and Silver, we found no significant relationship between gold, platinum, and palladium using the Shannon entropy transfer method. We infer from this findings that, the weak and insignificant transfers signify potential avenues for diversification and hedging for investors.

On information flow from silver to other precious metals, we notice a slight positive information flow from silver to gold which signifies silver emits low risks to the gold market. We however found no significant information flow between silver, platinum, and palladium. Concerning the information flow towards silver, we found platinum and palladium to have a positive information flow towards silver indicating low risks hence investors can find safe haven in platinum, and Palladium when prices of silver, and Gold becomes unstable. Meanwhile, information flows from Gold to Silver was insignificant when combined with other precious metals. We found the relationship between Gold and Silver to be more prononce as suggested by Wang et al., (2018) whose study indicated a strong positive correlation between gold and silver prices, indicating that changes in one often signal shifts in the other.

When the analysis is stretched to information flow from platinum to other precious metals, we found insignificant information flow between platinum, silver, and gold, however the study reveal a significant negative information flow between platinum and palladium indicating high risk between these two precious metals. This indicates that platinum and palladium have higher contagion and interdependence than silver and gold. We note that on the information flow from other precious metals to platinum, we found insignificant information flow from palladium, silver to platinum; however, we noticed a significant negative information flow from gold to platinum. This butrress the assertion by Erb and Harvey (2006), when they argued that gold is the leader in spillover in the precious metals terrane.

We observed bi-directional insignificant information flow between Palladiun and other precious metals except information flow from Palladium to Silver which depicted a positive significant information flow indicating low risk. This indicates that silver and palladium can serve as hedging purposes. Figure 1 below display the output of the information flow among the precious metals using the Shannon Transfer Entropy method.

Panel A

Fig. 1

Shannon Transfer Entropy Outputs displaying the information flows between bivariate pairs of precious metals

4.1.2 Results of Renyi Transfer entropy

The Shannon Transfer entropy usually portrays distinct findings as compared to the Renyi transfer entropy. This is because the Shannon transfer entropy sometimes fails to capture fat details and tail dependence. We however observed there is no significant difference between the results of the Shannon transfer and the Renyi transfer entropy results. The Shannon Transfer entropy results are our base for this study, while the Renyi Transfer entropy results are presented for comparison. We present the results of the Renyi transfer entropy for comparison (displayed in Fig. 2).

Panel B

Fig. 2

Renyi Transfer Entropy Outputs displaying the information flows between bivariate pairs of precious metals

In Table 2, we summarize the information flow between the precious metals, our study reveal gold to be the leader of the information flow among the precious metals. Generally, we discovered mixed findings on the flow of information among the precious metals. This could be due to several interralated factors such as market structure, behavioural dynamics, and methodological variations in research approaches. According to McDermott (2010), gold which is often perceived as a safe-haven asset tends to follow different information flows compared to industrial metals like silver or platinum which are influenced more by economic activity. Corbet et al, (2018) suggest that during times of economic uncertainty, investor behaviour toward gold may be more pronounced as a hedge, while silver could be seen as a riskier asset. This could lead to conflicting conclusions about the interconnectedness of the precious metals. Again studies that rely on different timeframes, statistical models, or data sources may yield constrasting results. For example some researchers may utilize cointegration analyses whiles others may apply vector autogression models, leading to disparate conclusions regarding the nature and extent of the information flow (Johansen, 1991; Granger, 1969).

Table 2
Summary of Information Transfers between Precious metals
GOLD	Shannon Transfer Entropy		Renyi Transfer Entropy
	Flow towards other precious metals	Flow towards Gold	Flow towards other precious metals		Flow towards Gold
Silver	High risk	Insignificant	High risk		Insignificant
Platinum	Insignificant	Low risk	High risk		Insignificant
Palladium	Insignificant	Insignificant	Insignificant		Insignificant
SILVER	Shannon Transfer Entropy		Renyi Transfer Entropy
	Flow toward other precious metals	Flow towrds Silver	Flow towards other precious metals	Flow towards Silver
Platinum	Insignificant	Low risk	Insignificant		Insignificant
Palladium	Insignificant	Low risk	High risk		Insignificant
Gold	Insignificant	Insignificant	Insignificant		High risk
PLATINUM	Shannon Transfer Entropy		Renyi Transfer Entropy
	Flow toward other precious metals	Flow towrds Platinum	Flow towards other precious metals		Flow towards Platinum
Palladium	High risk	Insignificant	High risk		Insignificant
Silver	Insignificant	Insignificant	Insignificant		Insignificant
Gold	Insignificant	High risk	Insignificant		High risk
PALLADIUM	Shannon Transfer Entropy		Renyi Transfer Entropy
	Flow toward other precious metals	Flow towards Palladium	Flow toward other precious metals		Flow towards Palldium
Gold	Insignificant	Insignificant	Insignificant		Insignificant
Platinum	Insignificant	Insignificant	Insignificant		High risk
Silver	Low risk	Insignificant	Insignificant		High risk
Source: Authors’ Construct 2025

4.2. Wavelet multiple analysis

Wavelet multiple correlation and wavelet multiple cross correlation results are shown in Figs. 3 and 4, respectively. The multiple correlations between the precious metals are displayed in Fig. 3. The multiple cross-correlations for the same period are shown in Fig. 4. Multiple correlations were shown to have accelerated in this investigation from higher frequencies (lower scale) to lower frequencies (higher scale). In particular, the greatest frequency correlation increased from 0.80 in the short term to 0.83 in the long term, after reaching 0.79 in the medium period. In comparison to the medium and long term, the short-term correlation's weakness suggests that risk is low. This suggests that short-term precious metals hedging and diversification are feasible. This is similar to the findings of Tweneboah and Alagidede (2018), who confirmed a short-term diversification opportunity among the precious metals.

Fig. 3

Wavelet Multiple Correlations between the precious metals

Figures 4 show the wavelet multiple cross-correlations between the precious metals at different frequencies and across leads and lags up to 20 days. The time lag at which the strongest or exact wavelet correlation coefficients are localised is displayed by the dashed lines (Fig. 4). We observed that cross-correlations are stronger at lower frequencies (higher scaler) which confirm the results of the wavelet multiple correlations. In anaysing the localisation Vis a Vis their time lags, we find less spillover effects among the precious metals since localisation occur at the point of symmetry (zero time lag). This evidence that correlations increase from lower scales to higher scales indicate no discrepancy in the integration of precious metal price returns across the wavelet spectrum thus promising more diversification and hedging in the short term. Our results corroborate the findings of Tweneboah and Alagidede (2018).

Fig. 4

Wavelet Multiple Cross Correlations between the precious metals

5. Conclusions and recommendations

5.1 Conclusions

The paper investigated the information flow among the leading precious metals (Gold, Silver, Platinum, and Palladium) using the Shannon transfer entropy (1948) and the Renyi transfer entropy (2007). Our base model is the Shannon transfer entropy whiles the Renyi serves as a robustness check. To factor in the time horizons of investors, we employed the wavelet multiple correlations and the wavelet multiple cross correlations by Fernando Macho (2012) to assess the appropriate time for diversification and hedging. Previous studies have used the conventional granger causality test, cointegration, VAR, and the GARCH models to examine the correlation of precious metals (see for instance, Bhatia et al., 2018; Oral & Unal, 2017; Pierdzioch et al., 2016; Bredina et al., 2017) However, the limitations of these methods is its failure to quantify the strength of information flow among these precious metals. This is where our study significantly contributes to the literature by filling this lacuna by using the transfer entropy to assess the tramsmission of information among these metals.

In this study, we utilized daily metal prices which were gleaned from the Bloomberg financial database for the period 2000–2024 to ascertain the level of information flow among these precious metals. The general observation was that information interactions among the precious metals displayed mixed information flow with Gold serving as the leader in transmitting information. The study reveals insignificant information flow among platinum, and Palladium. This indicates diversification avenues for investors when the prices of Gold and Silver become very volatile.

Theoretically, we contribute to the literature by emphasizing the theories (Efficient Market Hypothesis, Price Discovery, Correlation and Causation, Graph Theory and among others) highlighted in the study to explain and understand the information flow between the precious metals as well as assessing the possible reasons for the inconsistent findings when the conventional methods are employed for the relationship in question. Again, these contributions will help policymakers to formulate better regulatory policies to protect the precious metal market.

In this study, we observed significant findings from the entropy methods. First, with the Renyi method, we find Gold to have a significant negative information flow with silver indicating high risk. This gives little room for diversification though the Shannon entropy results depicted an overlap signifing insignificant information flow among these two precious metals. The results from the Renyi transfer entropy coincide with the finds of McDermott (2010), Tweneboah and Alagidede (2018).

Concerning the findings from the wavelet multiple correlation, we posit high chance of diversification at the short-run. Whereas in the long term which is termed as the lower frequencies, integration is high this suggests less opportunity for diversification. The wavelet multiple cross correlation revealed higher integration at the higher scale (lower frequency). This corroborates the findings of the wavelet multiple correlation. However, in analysing the localisation vis a vis their time lags, we find less spillover effects among the precious metals since localisation occur at the point of symmetry (zero time lags). The results from the WMC and WMCC indicate largely that the extent of integration of these precious metals varied across timescales and suggest that the integration begins to accelerate at lower frequencies (higher scales).

5.2 Recommendations

Based on the results of the transfer entropy, we recommend policy makers to create a regulatory body to monitor trading activities in the precious markets to mitigate manipulation, improve transparency and information flow, and enhance market integrity. Since the precious metals industry operates at the intersection of economic viability, environmental protection, and social responsibility, we recommend policy makers to promote sustainable practices, enhance market transparency, protect consumers, and encourage responsible mining and investment behaviours as this can increase the flow of information among industry players. We again recommend that if industry players adopt a holistic and integrated approach, the precious metal industry can thrive by contributing positively to both the economy and the environment.

Based on the WMC and WMCC findings, we recommend investors to explore portfolio diversification in the short run since integration is weak whiles in the long run, the avenue for hedging and diversification disappears.

6. Limitations and future studies

The limitation of this study is that we rely on only gold, silver, platinum, and palladium for the analysis since there is lack of data on the other precious metals such as rhodium, ruthenium, osmium, rhenium, and iridium on all the prominent financial databanks.

Future studies could establish robust datasets that incorporate various dimensions of information flow, including news sentiments analysis, social media indicators, and regional economic reports. Again, future research should explore the interconnectedness of precious metals with other asset classes (e.g., equities, commodities, and crytocurrencies) and how information flows between these markets influence precious metal prices. Lastly, research should focus on how information flows across different global markets and time zones, assessing how information disseminated in one region can influence prices and trading behaviours in another.

Author Contribution

Author contributions statementConception and design: R. E (Corresponding author)Analysis and interpretation of the data: R. E (Corresponding author) The drafting of the paper: R. E (Corresponding author)Revising it critically for intellectual content: R. E (Corresponding author) Final approval of the version to be published: R E

Data Availability

Data is sourced from the Bloomberg financial database

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