Marwa Talbi
email: marwetalbi@gmail.com

Rihab Bedoui

Christian De Peretti

Lotfi Belkacem

Is the role of precious metals as precious as they are? A vine copula and BiVaR approaches

Keywords: Precious metals, G-7 stock markets, hedge, safe haven, diversification, vine copula, BiVaR. JEL classification: C02, C58, G1

This paper revisits the international evidence on hedge, safe haven, and diversification properties of precious metals-namely gold, silver, and platinum-for the G-7 stock markets.

Therefore, this study proposes a multivariate vine copula-based GARCH model to assess the hedge and safe haven properties of precious metals and a Bivariate Value at Risk-based copula (BiVaR) measure to analyse the diversification potential of precious metals. Our empirical results suggest that; (1) gold is the strongest hedge and safe haven asset in almost all the G-7 stock markets, (2) silver and platinum results show that they may act as weak hedge assets, (3) silver bears the potential of a strong safe haven role only for Germany's and Italy's stock markets; however, platinum provides a weak safe haven role for most developed stock markets, (4) precious metals appear as interesting assets for diversifying a portfolio for G-7 stock markets investors. Overall, our findings provide noteworthy practical implication for investors.

1

Introduction

In the aftermath of the Global Financial Crisis, the interest in holding precious metals has been increased, triggering a rise in their prices. This interest is due to the role of precious metals, practically gold, as important stores of value and in diversifying risk [START_REF] Adrangi | Economic activity, inflation, and hedging: the case of gold and silver investments[END_REF][START_REF] Lucey | Seasonality, risk and return in daily COMEX gold and silver data 1982-2002[END_REF]. The literature on the hedging and safe haven potentials of precious metals, especially gold, is among the faster growing fields of the financial literature since the turmoil period of the last decade. Therefore, the aim of this study is to move forward the academic debate on the precious metals-stock markets nexus by examining the hedge, safe haven and diversification properties of precious metals, namely gold, silver, and platinum for the G-7 countries1 using more flexible copula-based models, named vine copula, that allows a finer analysis. We also deepen the analysis of our results using the BiVaR method.

The existing literature provides evidence that gold may serve as a hedge asset against stock market in normal periods and as a safe haven during turmoil periods (e.g., Baur and Lucey, 2010;Baur and McDermott, 2010;[START_REF] Ciner | Hedges and safe havens: An examination of stocks, bonds, gold, oil and exchange rates[END_REF][START_REF] Reboredo | Is gold a safe haven or a hedge for the US dollar? Implications for risk management[END_REF]Beckmann et al., 2015, among others). Recently, attention has been shifted from gold in favour of other precious metals, which have often similar properties to gold. Precious metals may serve as hedge assets against various market risks such as exchange rates (e.g., [START_REF] Ciner | Hedges and safe havens: An examination of stocks, bonds, gold, oil and exchange rates[END_REF]Reboredo, 2013b;[START_REF] Bedoui | Diamonds versus precious metals: What gleams most against USD exchange rates?[END_REF][START_REF] Nguyen | Hedging and safehaven characteristics of Gold against currencies: An investigation based on multivariate dynamic copula theory[END_REF], inflation (e.g., [START_REF] Hoang | Is Gold a Hedge Against Inflation? New Evidence from a Nonlinear ARDL Approach[END_REF][START_REF] Salisu | Assessing the inflation hedging of gold and palladium in OECD countries[END_REF],), oil prices (e.g., [START_REF] Rehman | Precious metal returns and oil shocks: A time varying connectedness approach[END_REF] and stock market indices (e.g., [START_REF] Hood | Is gold the best hedge and a safe haven under changing stock market volatility[END_REF][START_REF] Mensi | Are Sharia stocks, gold and US Treasury hedges and/or safe havens for the oil-based GCC markets?[END_REF][START_REF] Ali | Revisiting the valuable roles of commodities for international stock markets[END_REF]. Some other studies have documented the stochastic properties of precious metals, their dynamic interlinkages, and their volatility spillover (e.g., [START_REF] Lucey | What precious metals act as safe havens, and when? Some US evidence[END_REF][START_REF] Balcilar | The volatility effect on precious metals price returns in a stochastic volatility in mean model with time-varying parameters[END_REF][START_REF] Talbi | Dynamics and causality in distribution between spot and future precious metals: A copula approach[END_REF]. These studies are of key importance regarding information about hedging strategies for investors.

Prior studies on the hedge and safe-haven properties of precious metals for stock markets highlights a heterogeneous role of precious metals against equity movement. A number of papers including Baur and Lucey (2010), Baur and McDermott (2010), [START_REF] Hood | Is gold the best hedge and a safe haven under changing stock market volatility[END_REF], [START_REF] Ciner | Hedges and safe havens: An examination of stocks, bonds, gold, oil and exchange rates[END_REF], [START_REF] Bredin | Does gold glitter in the long-run? Gold as a hedge and safe haven across time and investment horizon[END_REF] and [START_REF] Shahzad | Dependence of stock markets with gold and bonds under bullish and bearish market states[END_REF] has particularly focused on gold which is traditionally accepted as hedge and safe haven asset. While recent studies have expanded the set of potential hedges and safe havens to other precious metals (e.g., [START_REF] Lucey | What precious metals act as safe havens, and when? Some US evidence[END_REF][START_REF] Low | Diamonds vs. precious metals: What shines brightest in your investment portfolio[END_REF][START_REF] Li | Reassessing the role of precious metals as safe havens-What colour is your haven and why[END_REF][START_REF] Ali | Revisiting the valuable roles of commodities for international stock markets[END_REF]. On the whole, although the literature has documented the role of precious metals, the results are quite mixed for different markets. This is somewhat to be expected, given the use of different market variables, different countries, different time periods and different methods. Therefore, this incites us to further explore this topic.

To examine the interactions between precious metals and stock markets, various econometric methods have been used, which can be divided into generalized autoregressive conditional heteroskedasticity (GARCH)-type models (e.g., [START_REF] Hood | Is gold the best hedge and a safe haven under changing stock market volatility[END_REF][START_REF] Low | Diamonds vs. precious metals: What shines brightest in your investment portfolio[END_REF]Li and[START_REF] Li | Reassessing the role of precious metals as safe havens-What colour is your haven and why[END_REF][START_REF] Wu | Economic evaluation of asymmetric and price range information in gold and general financial markets[END_REF][START_REF] Wu | Economic evaluation of asymmetric and price range information in gold and general financial markets[END_REF], vector autoregression (VAR)-type models (e.g., [START_REF] Wan | Interactions between oil and financial markets-do conditions of financial stress matter?[END_REF], wavelet models (e.g., [START_REF] Bredin | Does gold glitter in the long-run? Gold as a hedge and safe haven across time and investment horizon[END_REF], bivariate copulas ( e.g., [START_REF] Nguyen | Gold price and stock markets nexus under mixed-copulas[END_REF]) and quantile regression methods ( e.g., [START_REF] Shahzad | Dependence of stock markets with gold and bonds under bullish and bearish market states[END_REF][START_REF] Ali | Revisiting the valuable roles of commodities for international stock markets[END_REF].

In particular, the contribution of this present research consists of three main aspects. Firstly, we extend the analysis of precious metals hedging and safe haven properties by modeling the multivariate dependence using vine copula-based GARCH model to study whether precious metals are strong or weak safe haven and/or hedge for the stock markets. The use of copula in higher dimensions is challenging, where standard multivariate copulas, such as the multivariate Gaussian or Student-t copulas, suffer from inflexibility in modelling the dependence structure among larger numbers of variables and exhibit a problem of parameters restriction so they do not allow for different dependency structures between pairs of variables.

Hence, the use of vine copulas overcomes the restrictive characteristics of the bivariate copulas by providing flexible and conditional dependence structure between the variables [START_REF] Brechmann | Risk management with high-dimensional vine copulas: an analysis of the Euro Stoxx 50[END_REF][START_REF] Ji | Portfolio diversification strategy via tail-dependence clustering and arma-garch vine copula approach[END_REF]. Therefore, the vine copula approach has been widely used in the context of time series models, risk management and so on (see the review about the financial applications of vine copulas provided by [START_REF] Aas | Pair-Copula Constructions for Financial Applications: A Review[END_REF]). Nevertheless, in this paper we use the information of dependence on average and in times of extreme market conditions provided by the vine copula to assess, respectively, for the hedge and safe haven properties of precious metals. Secondly, we propose a new definition of "strong safe haven" property. In fact, the copula can test only for weak safe haven since it provides a non-zero probability of extreme price movements to test for uncorrelated series. So, to overcome this limitation, we propose to use simulated data from the best-fitting copula model to compute the tail correlation to rigorously test for strong safe haven property of precious metals. Finally, a BiVaR novel method proposed by [START_REF] Bedoui | On the study of conditional dependence structure between oil, gold and USD exchange rates[END_REF], will be employed to check for the diversification potential of precious metals for G-7 investors. As far as we know, we are the first to apply the BiVaR based copula method. The importance of this measure remains in combining the copula and the VaR techniques. In fact, the use of copulas allows to construct a level graph of two-dimensional Value at Risk and examine for a risk level, the marginal rate of substitution (TMS) between the Value at Risk of precious metal and the stock index. So, this method illustrates the level graph of a two-dimensional Value at Risk in a graphical way which enable us to have a clearer vision on the dependence structure between variable and their positioning with regard to the independence, comonotonicity and anti-monotonicity cases which enable as to analyse the diversification property.

The remainder of this paper is structured as follows: Section 2 presents the state of the art. Section 3 develops the data and the methodology. Section 4 reports and discusses the empirical results of our analysis. Finally, Section 5 concludes.

Literature review

In the recent literature, a stream of research emerges focusing on the role of gold and other precious metals as hedges, safe haven and diversifier assets for stock markets.

Notwithstanding the vast existing literature on precious metal markets, the gold market has received the most extensive study.

As defined in Baur and Lucey (2010) who provide the first operational definition of hedge and safe haven, an asset is considered to be a strong (weak) hedge instrument when it is negatively correlated (uncorrelated) on average with another asset. For safe haven property, an asset is considered to be a strong (weak) safe haven instrument when it is negatively correlated (uncorrelated) during the extreme market conditions. However, these hypotheses are based on linear models which generally look at linear correlation and cannot capture rare events on the tails of distributions.

In their paper, they examine the static and time-varying relations between United States (US), United Kingdom (UK), and German stock return and gold return using daily data from 1995 to 2005 to evaluate gold as a hedge and a safe haven. They find that gold serves as an effective hedging tool for stocks and has a role as a safe haven in extreme stock market conditions. As an extension of Baur and Lucey (2010)'s work, Baur and McDermott (2010) study the relationship between gold and stock markets in developed and developing countries using multiple-frequency data from 1979 until 2009. They apply rolling window regression to analyse the time-varying relationship between gold return and the world portfolio index. They find that gold serves as a hedge and safe haven only in European and US markets but not in the BRICs markets (Australia, Canada and Japan). Adopting the same methodology, [START_REF] Hood | Is gold the best hedge and a safe haven under changing stock market volatility[END_REF] study the role of precious metals (gold, silver, and platinum) relative to the Volatility Index (VIX) as a hedge and safe haven against the US stock market; and they find that, unlike gold, platinum and silver serve neither as a hedge nor as a safe haven for the US stock market. Similarly, [START_REF] Arouri | World gold prices and stock returns in China: insights for hedging and diversification strategies[END_REF] use a bivariate VAR-GARCH model to study return and volatility spillovers between world gold prices and the Chinese stock market over the period from 22 March 2004, through 31 March 2011. They find significant return and volatility cross effects between the variables, and gold may serve as a safe haven for the Chinese stock market. [START_REF] Shen | The dependence structure analysis among gold price, stock price index of gold mining companies, and Shanghai composite index[END_REF] investigate the dependence structure among gold price, stock price index of gold mining companies, and the Shanghai Composite Index in China using bivariate copula based GARCH models and find that the gold return price has positive correlation with stock market returns, which differs from the findings of other research (gold price typically has negative correlation with stock market returns). [START_REF] Kumar | Return and volatility transmission between gold and stock sectors: Application of portfolio management and hedging effectiveness[END_REF] investigates the mean and volatility transmission between gold and Indian industrial sectors. Using a generalised VAR-ADCC-BVGARCH model, he finds unidirectional and significant return spillover from gold to stock sectors and claims that gold can be considered a valuable asset class that can improve the risk-adjusted performance of a well-diversified portfolio of stocks and acts as a hedge against different markets.

Applying the Dynamic Conditional Correlation (DCC) Multivariate GARCH Model, [START_REF] Lucey | What precious metals act as safe havens, and when? Some US evidence[END_REF] study the safe haven properties of four precious metals (gold, silver, platinum, and palladium) using daily data from 1989 to 2013. They find evidence that in some periods of time, silver and platinum can act as safe haven when gold does not, and the effect can sometimes be stronger. [START_REF] Nguyen | Gold price and stock markets nexus under mixed-copulas[END_REF] investigate the role of gold as a safe haven in seven countries: UK, US, Indonesia, Japan, Malaysia, Philippines, Singapore, and Thailand. Using bivariate copulas, they find that gold may be a safe haven asset during a market crash for the cases of Malaysia, Singapore, Thailand, the UK and US markets but not for the Indonesian, Japanese, and the Philippine markets.

Expanding the work of [START_REF] Lucey | What precious metals act as safe havens, and when? Some US evidence[END_REF], [START_REF] Li | Reassessing the role of precious metals as safe havens-What colour is your haven and why[END_REF] examine the safe haven properties of precious metals versus equity market movements across a wide variety of countries using daily data from January 1994 to July 2016. Applying the standard approach as outlined in Baur and Lucey (2010) and Baur and McDermott (2010), they find that each metal may play a safe haven role against the stock market during tail events. [START_REF] Klein | Dynamic correlation of precious metals and flight-to-quality in developed markets[END_REF] examines the connection of developed markets and precious metal prices using daily data from Jan 2000 to Dec 2016. Applying DCC-GARCH model, he finds that gold and silver act as safe haven assets while platinum serves as a temporal surrogate safe haven in extreme market conditions. [START_REF] He | Is Gold a sometime Safe Haven or an Always Hedge for Equity Investor ? A Markov Swithching CAPM Approach for US and UK stock indices[END_REF] re-examines whether gold is a safe haven asset for UK and US investors.

Applying a Markov-switching CAPM (Capital asset pricing model) approach, they find that gold is consistently a hedge but that no distinct safe haven state exists between gold and UK or US stock markets. [START_REF] Junttila | Commodity marketbased hedging against stock market risk in times offinancial crisis: The case of crude oil and gold[END_REF] study the hedging property of gold and oil against stock market risk in times of financial crisis. They find that stock and gold markets become negatively correlated during times of financial crises. Thus, the gold market provides a better hedge than the oil market against stock market risks.

Recently, [START_REF] Ali | Revisiting the valuable roles of commodities for international stock markets[END_REF] re-examine the safe haven, hedge, and diversification potentials of 21 commodities (including precious metals) for 49 international stock markets. Using the cross-quantilogram approach (the quantile dependence across the whole range of quantiles), they find that precious metals in general and gold in particular provide strong safe havens for developed and frontier stock markets.

Table 1 summarises leading works, covering the period 2010 to 2020, that dealt with the hedge, safe haven, and diversification properties of precious metals. In the existing literature, linear models, GARCH-type models, and quantile regressions are frequently used methods.

Our contribution to the literature is two-fold. First, we apply a vine copula to analyse the hedge and safe haven properties. Then, we propose a novel method, the BiVaR measure, to examine the diversification potential of precious metals.

Data and methodology

Data description

This study considers daily prices data for precious metals and stock indices.

Vine copula model

Copulas have found many successful applications in a various empirical works to model joint distributions of random variables. 4 In his theorem, [START_REF] Sklar | Fonctions de Répartition à n Dimensions et Leurs Marges[END_REF] states that any n-dimensional multivariate distribution can be decomposed into n marginal distributions and a unique copula. More formally:

, , … . . , = , , … , (1)
where is a joint distribution of , , … . . , with marginal distributions = for = 1, 2,… , , and :[0,1] → [0,1] is a copula function. Suppose that and are differentiable. Then, the joint density function is defined as:

, , … . . , = , , … , (2)
where = is the (unconditional) density of and ! is the density of the copula.

Hence, the copula function separates the joint distribution into two contributions: the marginal distributions of each variable and the copula that combines these marginal distributions into a joint distribution.

Given the rich variety of bivariate copulas, they are limited to only one or two parameters to describe the dependence structure among variables. Hence, even though it is simple to generate multivariate Elliptical or Archimedean copulas, they cannot adequately capture the dependence in the multivariate scale.5 Therefore, we may go beyond these standard multivariate copulas by using vine copula approach, which is a more flexible alternative measure to capture the dependence structure among assets. [START_REF] Joe | Multivariate models and dependence concepts[END_REF]Bedford andCooke, 2001, 2002;[START_REF] Kurowicka | Uncertainty Analysis with High Dimensional Dependence Modelling[END_REF][START_REF] Aas | Pair-Copula Constructions of Multiple Dependence[END_REF] Technically, a vine copula consists of building a multivariate joint distribution from a cascade of unconditional and conditional bivariate copulas. It is well known that any multivariate density function can be represented as a product of unconditional and conditional densities:

, … , = . | . # | , … | , … , (3)
Bedford and Cooke ( 2002) introduced two types of vine copulas: canonical vine copulas (Cvine) and drawable vine copulas (D-vine).

In the C-Vine copula one variable plays a pivotal role. The general -dimensional C-vine copula can be written as:

10 ! , … , = ! , $%| ,…, & | , … , , $% | , … , ' % (4)
For instance, a 4-dimensional C-vine density decomposition and its hierarchical tree structure are represented as follows:

, , # , ( = .

. (5)

Similarly, D-vine copulas are also constructed by choosing a specific order for the variables in which variables are connected in a symmetric way. The general -dimensional D-vine copula can be written as:

! , … , = ∏ ∏ ! %,%$ |%$ ,…,%$ & % | %$ , … , %$ , %$ | %$ , … , %$ ' % . ( 6
)
We illustrate an example of 4-dimensional D-vine density decomposition and its hierarchical tree structure as follows:

, , # , ( = .

.

For the selection of the appropriate vine tree structure, pair-copula families and their parameter values, we follow the sequential procedure proposed by [START_REF] Dissmann | Selecting andestimating regular vine copulae and application to financial returns[END_REF], which is summarised in Table 2.

Table 2. Sequential method to select a vine copula model Algorithm. Sequential method to select a vine copula model 1-Calculate the empirical Kendall's τ for all possible variable pairs. 2-Select the tree that maximises the sum of absolute values of Kendall's τ.

3-Select a copula for each pair and fit the corresponding parameters. 4-Transform the observations using the copula and parameters from Step 3. To obtain the transformed values. 5-Use transformed observations to calculate empirical Kendall's τ's for all possible pairs. 6-Proceed with Step 2. Repeat until the vine copula is fully specified.

The estimation of vine copulas is a two-step separation procedure which is called the inference functions for the margin method (IFM). It implies that the joint log-likelihood is simply the sum of univariate log-likelihoods, and the copula log-likelihood is given as follows:

+ = , + + + &! , … , '. (8)
To full fill the aim of our study, we investigated the conditional dependence structure. As a first step, we filtered the returns using AR(1)-GARCH (1,1) processes. 6 Then, we applied the empirical cumulative distribution function (ECDF) for the standardized residuals. As a final step, we estimated the parameters of the vine copula, mainly the C-vine and D-vine copulas, using the sequential maximum likelihood estimation procedure described in Table 2 in order to examine the multivariate dependence and analysis of the hedge and safe haven properties of precious metals against stock markets.

Testing for hedge and safe haven properties

The use of copulas is crucial since it gives us information about both the average dependence and the dependence in times of extreme market movements (independent from how the marginal distributions are modelled). On one hand, the average dependence is given by correlation measures (Kendall's tau or Spearman's rho) which are obtained from the dependence parameter of the copula. On the other hand, the dependence in terms of extreme market movements is obtained from the copula tail dependence coefficients.

On the basis of copula dependence information, we can formulate the conditions under which a precious metal is characterised as a strong (weak) hedge and/or strong (weak) safe haven for the stock indexes under study as in Table 3.

Table 3. Hypothesis testing Conditions Conclusion

. /0,1 < 3 Precious metal is a strong hedge . /0,1 = 3 or strongly near to 0 Precious metal is a weak hedge

4 5 = 3
Precious metal is a weak safe haven 4 5 = 3 and . /0, 161 < 1 7 < 3 Precious metal is a strong safe haven Notes: . /0,1 is the average dependence measure (given by the copula parameter or the Kendall tau) between precious metals (PM) and stock market indexes (I) and 8 9 is the lower tail dependence coefficient for the joint distribution of precious metals and stock market indexes. We define . /0, 161 < 1 7 as the correlation between precious metals (PM) and stock market indexes (I) at the lower tail of stock returns distribution, where 1 7 denotes the q th percentile of I.

Based on the values of Kendall's tau, the precious metal serves as a strong hedge when the value of Kendall's tau is negative or equal to zero. Meanwhile, if Kendall's tau is positive and near to zero, the precious metal is considered as a weak hedge. Then, based on the lower tail dependence coefficient 4 5 , the precious metal is a weak safe haven asset when it shows zero lower tail dependence ( 4 5 = 0) with the stock market index. This means that in this case investors will not experience a loss in their precious metal holdings when the stock market crashes. And, if 4 5 > 0 , this implies that there is a positive probability of concurrent losses in precious metal and stock market indexes during periods of turmoil. However, positive lower tail dependence does not strictly signify a null probability of positive precious metal returns during the time of significant equity downturns since 4 5 is only a conditional probability measure rather than a tail correlation. The reason is that it is more reasonable to compute the value of correlation between stock market (I) and precious metal (PM) returns at the lower tail of the stock returns distribution. Hence, the precious metal is a strong safe haven asset for the stock market if λ @ = 0 and A B, C|C < C < 0 at the same time.

BiVaR based copula model

Based on the dependence structure computed using copulas, the bivariate Value at Risk (BiVaR) risk measure is implemented to examine the diversification benefits of precious metals. Copulas allow us to draw the level curves of the two-dimensional Value at Risk and examine, for a given threshold level (D=5%), the marginal rate of substitution (TMS) between the VaR of the two univariate risks. Hence, for a given marginal distribution of precious metals and stock returns, it is possible to draw the contours corresponding to the minimum copula (anti-monotonicity case), maximum copula (comonotonicity case), and the independence copula.

Let E F and E G be the precious metal and stock market index return series with univariate distribution functions, F and G , respectively. Hence, ∀ the threshold R[0,1], we have:

V E W , E X ; max W E W + X E X -1,0 = D], anti-monotonicity case (10) V E W , E X ; W E W . X E X = D], independence case (9) ^ E W , E X ; min W E W , X E X = D_ , comonotonicity case (11)
The level curves from the empirical copula are given by: ^ E W , E X ; W E W , X E X = D_ .

(12)

The level curves are used to determine the TMS between the two univariate VaR. The higher the level of the empirical curves (approaching the anti-monotonicity case), the more that dependence between the precious metals and stock market indices returns is negative. Hence, there is the presence of a compensation effect. However, the closer the curves are to their lower limit, corresponding to the case of comonotonicity (positive dependence) where the returns tend to move in the same direction, the more the correlation between losses is therefore extremely high. Regarding the curves of multiplication, they correspond to the diversification case (see [START_REF] Cherubini | Value-at-risk Trade-off and Capital Allocation with Copulas[END_REF][START_REF] Bedoui | Copulas and bivariate risk measures: An application to hedge funds[END_REF].

Empirical results and discussions

This section discusses the empirical results of this study. The preliminary analyses results are firstly presented. Then, we discuss our main results about the hedge, safe haven and diversification potentials of precious metals for the G-7 stock markets.

Preliminary analyses results

The descriptive statistics for the daily returns of precious metals and equity indexes are reported in Table 4. Results show that all the means are close to zero, and the standard deviations are small, which means that all series are around the mean. Likewise, among the precious metals, silver has the highest standard deviation, while gold has the lowest, which implies that silver is the most volatile. Moreover, asymmetry and fat tails in the returns were evident for all series. Thus, the results for the skewness and the excess of kurtosis reinforce the rejection of normality. The Jarque-Bera (JB) test rejects the normality hypothesis, while the Ljung-Box (LB) statistic confirmed the absence of serial correlation, except for the Nikkei 225, gold, and platinum. Likewise, the ARCH effect test indicates the presence of ARCH effects in all series.

Main results

In the first step, we filter the returns using GARCH model, aiming to obtain the residuals since copula model request independent and identically distributed (i.i.d) uniform data (the results of AR-GARCH model are presented in Appendix A, Table A 2). Parameter estimates for marginal models of precious metals and stock market returns). Then, we filter the standard innovation by t-distribution. Thereafter, we apply the vine copula to the transformed standardised residuals of each asset returns, mainly the C-vine and D-vine copulas. 7

Table 5 presents the best vine copula model that fits our data using the Log Likelihood (LL) values, AIC, and BIC criteria. The results show that the D-vine structure for Canada, Japan, UK, and US markets is more appropriate than the C-vine, whereas the latter is more suitable for France, Germany, and Italy than the former.

Table 5. Vine copula model selection

7 Before estimating copula models, we calculate the empirical Kendall's a for all possible pairs of variables to select the tree that maximises the sum of absolute values of Kendall's a. (Appendix A, Table A 3) The results shows that platinum has the strongest dependency in terms of the empirical value of pairwise Kendall's tau; hence, we consider platinum as the first root node. Thereby, the order of variables is as follows. For France, Canada, and US, the order of variables should be the following: Platinum (order 1), Gold (order 2), Silver (order 3), and the Stock Index (order 4). And for the rest of countries, the order is: Platinum (order 1), Silver (order 2), Gold (order 3), and the Stock Index (order 4). Tables 6-7 represent results of the parameter estimation of the selected C-vine and D-vine copulas for each of the G-7 stock markets.

For France, Germany, and Italy, the first tree of Table 6 shows symmetric upper and lower tail dependence between all pairs, indicating similar dependence during upward and downward periods and an average dependence near to zero, particularly, between platinum and the stock market indexes. This result indicates that platinum may serve as a weak hedge for French, German, and Italian investors. Taking the second tree, results show that gold may serve as both hedge and safe haven on the stock market of France, and silver may have the role of a hedge and a safe haven asset in the stock markets of Germany and Italy. Finally, the third tree shows zero upper tail for France, Germany, and Italy, a weak dependence on average for France, and negative dependence on average for Germany and Italy. This result reveals that silver is a weak hedge and safe haven asset for French investors, and gold serves as a strong hedge and a safe haven asset for German and Italian investors. In the first tree of Table 7, we observe symmetric upper and lower tail dependence given by the t-copula. We also observe a dependence on average strongly near to zero between silver and stock market indexes for Canada and US and between gold and stock market indexes for UK and Japan, indicating that silver (gold) is only a weak hedge asset for Canadian and American (UK and Japanese) investors. In the second tree, results indicate that gold may serve as a strong hedge and a safe haven asset (b c = 0 against the stock markets of Canada and US. However, silver is only a weak hedge for Japan and UK. Finally, the third tree shows weak dependence on average between platinum and stock market indexes, confirming the role of platinum as a weak hedge for the Canada, US, UK, and Japan stock markets. Regarding lower tail dependence, results indicate zero tail dependence for Canada and US, arguing that platinum is a safe haven for Canadian and American investors.

Table 7. Results of estimated parameters for D-vine copulas

Notes: P= platinum, G=gold, S=silver and I= stock index for each country. /,k denotes copula between platinum and gold. k,l|/ denotes copula between gold and silver given platinum. m is the Kendall's tau of the specified copulas, n is the copula parameter and o is the degree of freedom.

Canada

Japan

In the next step, we estimate the tail correlation between precious metals and stock market index returns to find out if precious metals are strong safe haven assets or just weak safe haven assets for the stock markets under study. For that end, we performed a Monte Carlo simulation with N = 10 5 simulations from the joint distributions characterised by the best fitting copula functions for each country where precious metals are shown to be weak safe haven assets. Then, we compute the correlation between precious metals and stock market index returns at the 1% tail of stock market returns. Notes: P= platinum, G=gold, S=silver and I= stock index for each country. g,h denotes copula between platinum and gold. g,i|h denotes copula between platinum and silver given gold. a is the Kendall's tau of the specified copulas, e is the copula parameter and f is the degree of freedom.

Table 9 summarises our empirical findings regarding hedge and safe haven properties of precious metals.

Table 9. Hedge and safe haven analysis results summary

In Fig 2 we present the 95% level curves of the BiVaR between precious metals and the S&P500 index. As we can see, the 95% level curve of the empirical copula is closer to the level curve of multiplication or independence, which means that the losses for precious metals and the stock market are uncorrelated.

As we expected before, the low correlations between the precious metals and equity market indices support the diversification properties of metals. Thus, in order to guarantee benefit from diversification, it is preferred to put precious metal and equity indexes in the same portfolio. This result is similar for the other stock markets. 8

8 The others level curves of the BiVaR are contained in Appendix A, Error! Reference source not found.

Discussion

By leading this study, additional insights are made regarding the literature debate on the interaction between precious metals and stock markets with a special concern on the valuable roles of precious metals; namely gold, silver and platinum as hedge, safe haven and diversification assets. Indeed, this study adopts the combination of copulas and VaR techniques. On one hand, we apply vine copulas to assess for the hedge and safe haven properties of precious metals for the G-7 stock markets and propose a new definition of "strong safe haven" asset by computing the tail correlation using simulated data from the bestfitting copula model. Our estimation results show that gold serves as a strong hedge and safe haven for European countries as well as the US and Canada, but only weak hedge for UK and Japan. These findings are in line with previous studies Baur and[START_REF] Baur | Is gold a safe haven? International evidence[END_REF][START_REF] Bredin | Does gold glitter in the long-run? Gold as a hedge and safe haven across time and investment horizon[END_REF] Gold is considered as safe haven asset for several reasons. To start with, gold served historically as a currency and still remains a monetary asset. Second, it is the most liquid precious metal and easiest to trade. Also, our results confirm that gold does not comove with stocks during extreme market conditions. Last but not least, it is an international asset, and its value is independent of the decision of a particular State.

Regarding silver, our results reveal that it may acts as strong hedge and safe haven asset in German and Italian stock markets, which means that it may be suitable and affordable alternative safe haven since it is cheaper than gold. However, silver is not considered as a safe haven asset for the rest of G-7 stock markets since it is more thinly traded, making it more volatile and illiquid. For the case of platinum, it is a weak hedge for all G-7 stock markets and only a weak safe haven for American, Canadian, and Japanese stock markets. As we know, an

Fig 2. BiVaR level curves between S&P 500 and Precious metals

asset is considered to be a safe haven when its market is extremely liquid, and platinum is relatively illiquid. This can explain the fact that platinum cannot be a safe haven for most of the G-7 stock markets. However, it is a cheap alternative to gold and may be profitable for investors if the price of gold continues to rise. On the other hand, we apply a novel methodthe BiVaR based copula method-to analyse the diversification potential of precious metals.

Our findings confirm that precious metals provide a useful means of diversification for G-7 investors.

Overall, our results suggest that there is some degree of heterogeneity regarding the role of precious metals between the G7 countries, which is due to the fact that each country has its own financial risk exposure.

To sum up, even though gold acts as a better hedge and safe haven for the G-7 stock markets, investors can find a valuable investment benefit in silver and platinum with different degrees (weak or strong).

Conclusions

In this study, we analyse the hedge, safe haven, and diversification potential of precious metals-namely gold, silver, and platinum-for the G-7 stock markets. Indeed, the vine copula method is used to test the hedge and safe haven hypotheses and the BiVaR is applied to assess the diversification benefits of precious metals. Our empirical results show that precious metal hedge and safe haven behaviors vary by country.

First, gold provides the strongest safe haven property for all G-7 countries, which is consistent with previous literature. This result means that gold may be used to offset losses in equity markets during turmoil periods. Also, silver bears the potential of a strong safe haven role for German and Italian stock markets. However, platinum provides a weak safe haven role for most developed markets. Furthermore, in line with existing studies, our study suggests that gold has a strong hedging property in developed stock markets. For silver and platinum, results show that they may act as weak hedge assets. Finally, the results of the BiVaR analysis argue that all precious metals exhibit diversification benefits for G-7 stock markets investors.

Our findings provide a noteworthy practical implication for investors in the G-7 countries in building their investment strategies. We suggest that investors may hedge their equity investments in normal times by investing in these precious metals and ensure their portfolios from losses during periods of turbulence by investing in gold.

Further research might analyze the out-of-sample forecasting of expected returns in precious metals and how precious metals investment should fit into a diversified portfolio.

Survival version of copula

p, q = p + q -s + s -p, s -q

Dependence measures

The Kendall's tau can be written as a function of the copula as follows:n m = -˜˜ p, q ™ p, q Notes: 8 ˆ and 8 9 denote the lower and upper tail dependence, respectively. For the Gaussian copula r s p et r s q are the standard normal quantile functions and r is the bivariate standard normal cumulative distribution function with correlation parameter .. For the tcopula • s p and • s q are the quantile functions of the univariate Student-t distribution and T is the bivariate Student-t cumulative distribution function with o the degree-of-freedom and . the correlation parameter. For the SJC copula, ‰ = s zv¢ • m ˆ and Š = s …£¤m 9 . m ˆ and m 9 denote the upper and lower tails of the SJC and the Joe-Clayton copulas.

Table A 2. Parameter estimates for marginal models of precious metals and stock market returns

The AR (1)-GARCH (1,1) model can be written as: = ¥ + ∅ + § , § = ¨ ©ª , ¨ ~¬-®¯° ---f , ª = ± + D § + ² ª , where ± >0, D , ² Ÿ 0, and are the logarithmic returns at time t, ¥ is a constant term of the return equation, § represents the return residuals, and ¨ are the standardised residuals which have a student-t distribution with f degrees of freedom.

CAC40

  Fig 1 displays the dynamics between precious metals and the S&P 500 index prices during the sampling period. 3 It reflects that precious metals and the US stock market are poorly correlated. The same is true for the other markets shown in Fig. A 1 in the Appendix. For almost all countries, a falling stock market results in a rise in precious metals demand.

Fig 1 .

 1 Fig 1. Precious metals and S&P500 dynamics

  The upper (right) and lower (left) tail dependence measures are, respectively,

Fig.A 1

 1 Fig.A 1 Precious metals and stock market indexes dynamics

Table 1 . Literature review summary Authors Precious metal Stock markets data Examined Property Model Findings Baur and Lucey (2010)

 1

		Gold	Germany,	Hedge and	Asymmetric	Gold is a hedge and a safe
			US and UK	safe haven	GARCH	haven
	Baur and	Gold	Developed	Hedge and	Quantile	Gold is a hedge and a safe
	McDermott (2010)		and emerging	Safe haven	regression	haven in Europe and the
			markets		approach	US and strong safe haven
						in developed countries
	Ibrahim and	Gold	Malaysia	Diversifier,	EGARCH	Gold is a diversifier
	Baharom (2011)			Hedge and	model
				safe haven
	Ciner et al. (2013)	Gold	US and UK	Safe haven	DCC model	Gold is a safe haven
	Hood and Malik	Gold, silver	US	Hedge and	GARCH model	Gold serves as hedge and
	(2013)	and		safe haven		weak safe haven
		platinum
	Flavin et al. (2014)	Gold	US	Safe haven	Regime-	Gold is a safe haven
					switching
					Model
	Bredin et al. (2015) Gold	Germany,	Hedge and	Wavelet	Gold is a hedge and a safe
			US and UK	safe haven	analysis	haven
	Arouri et al. (2015) Gold	China	Diversifier,	VAR-GARCH	Gold is a hedge and a safe
				hedge and		haven
				safe haven
	Lucey and Li	Precious	US	Safe haven	DCC model	Silver,	platinum	and
	(2015)	metals				palladium serve as safe
						haven assets
	Chkili (2016)	Gold	BRICS	Hedge and	Asymmetric	Gold is a hedge and a safe
				safe haven	DCC model	haven
	Mensi et al. (2016)	Gold	Gulf	Hedge and	Quantile	Gold is a strong hedge and
				safe haven	regression and	safe haven at various
					wavelet	investment horizons.
					decomposition
	Low et al. (2016)	Gold,	Brazil,	Hedge and	GJR-GARCH	Gold, silver, platinum and
		sikver,	Australia,	safe haven	model	palladium serve as hedges
		platinum	China,			and safe haven assets
		and	Germany,
		palladium	France, UK
			and US

Table 1 .

 1 (Continued)

	Authors	Precious	Stock		Examined	Model	Findings
		metals	markets data	Property
		data
	Nguyen et al.	Gold	Japan,		Hedge and safe	Copula	Gold serve as a safe
	(2016)		Singapore,		haven			haven asset for US,
			Malaysia,					UK, Singapore and
			Thailand,					Thailand
			Philippines,
			US and UK
	Li and Lucey	Gold,	Developed and	Safe haven	Asymmetric	Precious metals serve
	(2017)	sikver,	emerging				GARCH	as safe haven assets for
		platinum	markets					all countries under
		and						study clustered during
		palladium						some periods.
	Shahzad et al.	Gold	Developed and	Diversifier,	Quantile-on-	Gold is a strong hedge
	(2017)		Eurozone		hedge and safe	quantile method	and a diversifier asset
			peripheral		haven
	Bekiros et al.	Gold	BRICS		Diversifier,	Multi-scale	Gold is a diversifier
	(2017)				hedge and safe	wavelet copula-	asset
					haven		GARCH
	Wu and Chiu	Gold	US		Diversifier and	Asymmetric	Gold is a safe haven
	(2017)				safe haven		GARCH model
	Wen and Cheng	Gold	BRICS,		Safe haven	Copula	Gold is a safe haven for
	(2018)		Chechnya,					emerging markets
			Malaysia, and
			Thailand
	Raza et al. (2019)	Gold	Developed,		Hedge	and	DCC model	Gold serves as a hedge
			emerging		diversifier			and diversifier asset
			markets
			Europe, Asia
			Pacific	and
			Islamic stock
			markets
	Ali et al. (2020)	Gold,	49		Safe	haven,	Cross-	Precious metals and
		sikver,	international	hedge,	and	quantilogram	gold	in	particular
		platinum	stock markets	diversification	approach	provide strong safe
		and						havens for developed
		palladium						and	frontier	stock
								markets.

Table 4 . Descriptive statistics for log-returns

 4 Dev denotes the standard deviation, JB denotes the Jarque-Bera statistic for normality testing. Q (20) denotes the Ljung-Box statistic for autocorrelation testing, LM stat denotes Engle's LM test statistic for heteroskedasticity testing. (*) indicates rejection of the null hypothesis at the 5% level.

		CAC40	DAX	FTSE	FTSE	NIKKEI	S&P\	S&P500	Gold	Silver	Platinum
				100	MIB	225	TSX
	Mean(10 -2 )	0.0102	0.0255	0.0054	-0.0016	0. 0211	-0.1195	0.0232	0.0370	0.0270	0.0178
	Std. Dev	0.0155	0.0156	0.0134	0.0167	0.0140	0.0983	0.0115	0.0110	0.0200	0.0148
	Max	0.1214	0.1236	0.1221	0.1238	0.1164	0.0992	0.1095	0.0684	0.1828	0.0843
	Min	-0.117	-0.0960	-0.1150	-0.1542	-0.1211	-6.6025	-0.0947 -0.0960	-0.3535	-0.1728
	Skewness	0.0926	0.0573	-0.0203	-0.1592	-0.4742	-0.5417	-0.2356 -0.3902	-1.7524	-0.9114
	Kurtosis	8.8223	8.2855	10.7229	9.1053	9.6310	11.2394 11.3305	7.5531	31.2669	12.1964
	JB stat	4757,6* 3918,7* 8362,7* 5240,3* 6291,1*	9682,9* 9761,1*	2992*	113751*	12323,7*
	Q(20)	63.41*	40.09*	81.20*	46.17*	22.31	68.80*	73.77*	26.53	84.10*	20.89
	LM stat	198.79* 171.45* 354.85* 114.19* 243.51*	345.60* 365.35*	40.81*	117.90*	155.82*

Table 6 . Results of estimated parameters for C-vine copulas

 6

	France

Table 8 . Correlation between stock and precious metals returns at 1% tail of stock returns Copula Tail correlation Conclusions

 8 As shown in Table8, the tail correlation is negative only for gold and silver. Hence, based on our definition of a strong safe haven asset, we conclude that gold may act as a strong safe haven instrument against extreme losses in France, Germany, Italy, Canada, and US stock markets. Silver is a strong safe haven asset in Germany and Italy. However, platinum is a weak safe haven asset against extreme losses in Canada and US stock markets.

	Tree Blocks 1 2 g,h|i g,i i,h h,j i,j|h 3 g,j|i,h	Family t-Student t-Student t-Student t-Student Survival Gumbel t-Student	a 0.39 0.4 0.02 0.25 0.06 0.09	Parameter 0.5805 e 8.1081 f 0.5842 10.601 0.0379 8.7998 0.3883 11.645 1.0617 0 0.1433 19.984	b c 0.1540 0.1074 0.0133 0.0350 0.0789 0.0007	b d 0.1540 0.1074 0.0133 0.0350 0 0.0007
	France	Gold	Clayton		-0.0005	Strong safe haven
	Germany	Gold	Rotated Gumbel	-0.0001	Strong safe haven
		Silver	Rotated Joe		-0.0003	Strong safe haven
	Italy	Gold	Rotated Gumbel	-0.0021	Strong safe haven
		Silver	Rotated Joe		-0.0014	Strong safe haven
		Platinum Student-t		0.0021	Weak safe haven
	Canada	Gold	Rotated Joe		-0.0010	Strong safe haven
		Platinum Survival Clayton	0.0002	Weak safe haven
	US	Gold	Gaussian		-0.0001	Strong safe haven
		Platinum Rotated Joe		0.0042	Weak safe haven

Table A 3. Empirical Kendall's tau matrices for G-7 counties

 A

			France
		CAC 40	Gold	Silver	Platinum
	CAC 40	1	-0.02249074	0.03375731	0.08429327
	Gold	-0.02249074	1	0.39800740	0.41186940
	Silver	0.03375731	0.39800740	1	0.39854344
	Platinum	0.08429327	0.41186940	0.39854344	1
	Sum	1,14054132	1,83236754	1,830308	1,89470611
			Germany
		DAX	Gold	Silver	Platinum
	DAX	1	-0.02068612	0.04223904	0.08117698
	Gold	-0.02068612	1	0.39800740	0.41186940
	Silver	0.04223904	0.39800740	1	0.39854344
	Platinum	0.08117698	0.41186940	0.39854344	1
	Sum	1,14410214	1,8305629	1,83878988	1,89158982
			UK
		FTSE100	Gold	Silver	Platinum
	FTSE100	1	0.01770941	0.06112196	0.1128933
	Gold	0.01770941	1	0.39800740	0.4118694
	Silver	0.06112196	0.39800740	1	0.3985434
	Platinum	0.11289327	0.41186940	0.39854344	1
	Sum	1,19172464	1,827586	1,8576728	1,923306
			Italy
		FTSE MIB	Gold	Silver	Platinum
	FTSE MIB Gold	1 -0.01972463	-0.01972463 1	0.03392975 0.39800740	0.08201692 0.41186940
	Silver	0.03392975	0.39800740	1	0.39854344
	Platinum	0.08201692	0.41186940	0.39854344	1
	Sum	1,135671	1,829601	1,83048059	1,89242976
			Japan
		NIKKEI225	Gold	Silver	Platinum
	NIKKEI225	1	0.02487169	0.05709541	0.09598458
	Gold	0.02487169	1	0.39800740	0.41186940
	Silver	0.05709541	0.39800740	1	0.39854344
	Platinum	0.09598458	0.41186940	0.39854344	1
	Sum	1,17795168	1,83474849	1,85364625	1,90639742
			Canada
		S&P/TSX	Gold	Silver	Platinum
	S&P/TSX Gold	1 -0.002292608	-0.002292608 1	0.005024797 0.398007396	0.008544629 0.411869405
	Silver	0.005024797	0.398007396	1	0.398543445
	Platinum	0.008544629	0.411869405	0.398543445	1
	Sum	1,015862034	1,812169409	1,801575638	1,818957479
			US
		S&P500	Gold	Silver	Platinum
	S&P500	1	-0.006665807	0.01759668	0.06968461
	Gold	-0.006665807	1	0.39800740	0.41186940
	Silver	0.017596684	0.398007396	1	0.39854344
	Platinum	0.069684607	0.411869405	0.39854344	1
	Sum	1,093947098	1,816542608	1,81414752	1,88009745

The G-7 countries are Canada, France, Germany, Italy, Japan, U.K and U.S

The fixing occurs twice a day, except for silver, which is fixed at noon each day. The price fixing done in the morning is called the AM Fix, while the afternoon fixing is called the PM Fix.

Plots of precious metals and other stock market indexes are reported in Appendix A, Error! Reference source not found..

We refer the reader to[START_REF] Joe | Multivariate models and dependence concepts[END_REF] andNelson (2006) for more details about copulas.

The main characteristics of copula functions used in this study are summarised in Appendix A, TableA 1.

We refer readers to see TableA2, in Appendix A

FigA.1(Continued)

Acknowledgments

This work was financially supported by the "PHC Utique" programme of the French Ministry of Foreign Affairs and Ministry of Higher Education and Research and the Tunisian Ministry of Higher Education and Scientific Research in the CMCU project number 18G0411.

Rotated version of copula

There are three rotated forms, with angles 90 degrees, 180 degrees, and 270 degrees, defined as follows:

7 show that the D coefficients, which measure the adjustment to past shocks, and the ² coefficients, which measure the volatility persistence of the process, are significant for all series, which indicates that the conditional volatility is persistent over time and is past-dependent. As usual, all series are described by significant GARCH effects. The Ljung-Box (Q statistic) and ARCH (LM statistic) statistics indicate that neither autocorrelation nor ARCH effects remained in the residuals of the marginal models. We also checked the adequacy of the Student-t distribution model, testing the null hypothesis that the standardised model residuals were uniform (0,1) by comparing the empirical and theoretical distribution functions using the Kolmogorov-Smirnov test. The p-values for those tests indicate that the null hypothesis could not be rejected at the 5% significance level for either of the marginal models. Hence, the marginal models are correctly specified, and the copula model can correctly capture dependence between precious metals and equity markets.