Modelling Time Varying Volatility Spillovers and Conditional Correlations Across Commodity Metal Futures

This paper examines how the most prevalent stochastic properties of key metal futures returns have been affected by the recent financial crisis using both mapped and unmapped data. Our results suggest that copper and gold futures returns exhibit time-varying persistence in their corresponding conditional volatilities over the crisis period; in particular, such persistence increases during periods of high volatility compared with low volatility. The estimation of a bivariate GARCH model further shows the existence of time-varying volatility spillovers between these returns during the different stages of such a crisis. Our results, which are broadly the same in relation to the use of mapped or unmapped data, suggest that the volatilities of copper and gold are inherently linked, although these metals have very different applications.


Introduction
The …nancial crisis of 2007-08 and the European sovereign debt crisis that occurred afterwards sent a wave of panic throughout …nancial and commodity markets around the globe. Given the macroeconomic slowdown and the widespread fear of an international systemic …nancial collapse, an interesting issue is whether the main stochastic properties of the underlying …nancial time series of these markets and their cross-shock and volatility spillovers have been a¤ected by the crisis. Karanasos et al. (2014) do indeed …nd a time-varying pattern in the persistence of the volatility of stock market returns, as well as their correlations, cross-shock and volatility spillovers during the period.
Surprisingly, the aforementioned impact in relation to the commodity futures markets has drawn less attention. To the best of our knowledge, the studies by Vivian and Wohar (2012) and Sensoy (2013) are the only ones to date to have examined the impact of the recent crisis on the volatility of commodity returns, even though they consider spot price data. Moreover, such studies have limitations in that they ignore the impact of the crisis on the cross-shock and volatility spillovers between the corresponding returns.
In this paper, we examine the impact of the recent …nancial crisis on two metals futures' volatility dynamics and their associated cross-linkages: copper and gold. These metal futures are considered due to their sheer daily volumes. Gold is the main precious metal and has mixed demand characteristics. Its demand is determined by …nancial factors as it is a reserve currency for the world, as well as being a traded commodity whose price is longed and shorted continually in huge volumes. Gold is also a¤ected by its pure consumer and market application in jewellery and electronics. Copper, on the other hand, is the main industrial metal, with huge applications in electronics, mainly in wiring. It is far more abundant in comparison to other metals, and hence it is a useful candidate metal to be considered for this analysis.
Consequently, the present paper makes several broad contributions to the existing literature.
First, we make use of several modern econometric approaches for univariate and multivariate time series modelling, amongst which we consider the possibility of breaks taking place in the volatility dynamics of these metal futures returns to capture the di¤erent stages of the recent …nancial crisis. More speci…cally, we use a battery of tests to identify the number and estimate the timing of breaks, both in the mean and volatility dynamics. Then, we use these breaks in the univariate context, by adopting an asymmetric generalised autoregressive conditional heteroscedasticity (AGARCH) model, to determine changes in the volatility persistence and in the multivariate one, by employing the recently developed unrestricted extended dynamic conditional correlation (UEDCC) AGARCH model of Karanasos et al. (2014), to analyse the volatility transmission and the correlation structure. It follows that the adopted univariate and multivariate frameworks are completely time-varying, and more strikingly, unlike the methods used in the existing literature the adopted bivariate model is su¢ ciently ‡exible and allows for volatility spillovers of either positive or negative sign.
Moreover, both chosen univariate AGARCH and bivariate (UEDCC) AGARCH models are further employed to examine respectively how the volatility persistence of the two considered returns is a¤ected by their corresponding positive (e.g., increases in these metal futures) and negative (e.g., declines in these metal futures) returns and whether there are any regime-dependent shock and volatility spillovers between such returns. The former analysis will show the extent to which positive returns versus negative ones impact on volatility persistence for the considered metals, while the latter will help to discern shock and volatility spillovers associated with the exact movements of each metal future (e.g., upward or downward) to the other, and vice versa.
All in all, knowledge of the time-varying volatility persistence and the spillovers mechanism adopted in this paper could prove to be very valuable to investors since they could give rise to time-varying trading strategies, thereby minimising the risk exposure and maximising the returns. interactions using two types of data: unmapped and mapped. The unmapped data is comprised of prices that have not been adjusted for di¤erences in prices due to rollover or 'basis'. 1 Taking into account the roll or basis alters the time series in such a way that econometric models'best …t may change as a result. The use of the front month contract prices (at the time of trading in real time) indicates the time series as it would appear to a trader at the particular point in time. However, the use of mapped data will allow us to observe the true interactions between the commodities. The di¤erences in the time series (mapped and unmapped) may be large or small and sometimes cancel each other out. Yet, they should be considered if a true 'live'trading time series is to be created.
Our results suggest that both copper and gold futures returns exhibit time-varying persistence in their corresponding conditional variances over the recent crisis, speci…cally such persistence is shown to increase during periods of high volatility compared with low volatility. The results of the bivariate UEDCC-AGARCH(1; 1) model, on the other hand, show the existence of a bidirectional mixed feedback between the volatilities of the two returns; that is, the conditional variance of copper returns a¤ects that of gold returns negatively whereas the reverse e¤ect is of the opposite sign. This mixed feedback between the volatilities of copper and gold is consistent with the fact that these two metals are so di¤erent in their values and uses. The results also suggest that the volatility transmission from gold returns to those of copper is time-varying; it shifts on the onset of the high uncertainty period induced by the European sovereign debt crisis along with the downgrade of the US government debt status and also over the low volatility period ensued afterwards based on optimism to resolving the debt crisis. Finally, the regimedependent volatility spillovers analysis suggests that declines in copper prices induce positive volatility spillovers to gold returns. These time-varying volatility spillovers between the two metals further con…rm the sensitivity of these metals and so are their associated cross-linkages to structural changes in volatility …ltered through the …nancial system.
Overall, our results are broadly the same in terms of whether mapped or unmapped data are employed and, moreover, they are robust when di¤erent model speci…cations are considered, i.e., using constant conditional correlation instead of dynamic conditional correlation in the bivariate GARCH model, and by including an exogenous control variable, i.e., the VIX volatility index or squared returns of the US dollar exchange rate against the euro, of the US'S&P 500 stock more details, see Samuelson, 1965). In this …rst analysis, therefore, the data have not been mapped to account for the rollover values. It has been discovered that taking into account the roll can signi…cantly change the time series since these roll values can be signi…cant in the commodities considered (Margaronis, 2015). market index or of oil prices.
The remainder of this paper is as follows. Section 2 reviews the relevant literature. Section 3 describes our employed data and methodology. Sections 4 and 5 present our empirical results and a discussion, respectively. The …nal Section contains the summary and our concluding remarks.

A Review of the Relevant Literature
Modelling the stochastic properties of …nancial and commodity returns as well as their crossshock and volatility spillovers has drawn much attention to the …elds of …nancial and energy economics, given their important practical implications for investors. For example, understanding the stochastic properties of returns may help investors in terms of forecasting market movements, while strong linkages between …nancial and/or commodity returns would imply limited portfolio diversi…cation opportunities for them.
Although there is a large body of literature that has examined the returns properties of international …nancial markets such as those of equity, foreign exchange, and bond, and their cross-shock and volatility spillovers (see, e.g., Aloui  Some studies have also considered the linkages across commodity prices and their returns and volatility. Ciner (2001) reports that gold and silver futures contracts traded in Japan are not cointegrated, using daily data over the period 1992 to 1998. Erb and Harvery (2006) further argue that commodity futures returns have been largely uncorrelated with one another, especially across the di¤erent sectors. However, using daily data of gold, platinum, and silver futures contracts traded in both the US and Japanese markets, Xu and Fung (2005)  As the existing literature suggests, unlike copper, empirical evidence in relation to gold has drawn much attention along with silver and some other metals and, more importantly, evidence related to exploring cross-linkages between copper and gold, speci…cally, is sparse compared to, for example, other metal pairs (e.g., gold and silver). Further, a few studies have analysed the impact of the recent crisis on the stochastic properties of metal returns; however, they consider spot price data and also disregard the time-varying cross-shock and volatility spillovers among such returns during the period. This paper aims to …ll in the existing gaps by analysing the impact of the recent crisis on the volatility dynamics and the associated cross-linkages of two metal futures, namely copper and gold, and by using alternative econometric speci…cations and data compared to the wide existing literature, speci…cally the bivariate (UEDCC) AGARCH model (which is su¢ ciently ‡exible and allows for volatility spillovers of either positive or negative sign) and two types of data: mapped and unmapped.

Data and Methodology
This Section overviews the data we have used and outlines the methodology we have employed to study the di¤erent properties of the stochastic processes associated with gold and copper futures returns over the 2007-8 crisis. First, we provide a brief description of our data and the breaks identi…cation method which we have adopted. Then, we describe the univariate and bivariate models which we have estimated.

Data Description and Breaks Detection Procedure
We use daily (mapped and unmapped) data on gold and copper futures prices which span the period January 3, 2007 to April 27, 2012. The unmapped data have been retrieved from Bloomberg.

Gold versus Copper
The precious metals are, and for many years have been, used as a reserve currency in times of …nancial turmoil where uncertainty lingers within economies (see, for example, O'Connor et al., 2015, for a recent survey on the …nancial economics of gold). When consumers are not con…dent in their currency they often buy gold or other precious metals. The reason for this is the precious metals'value and demand. The increased volatility, liquidity and use as a reserve currency mean that gold prices will react to the market with little to no lag time. Precious metals are not really consumed (and if they are it is usually a small percentage, which is often recycled e.g. jewellery, watches, and used as wiring in expensive earphones or sound systems) and neither do they tarnish or rust. They also have value and demand worldwide, making them a very good substitute for a currency. Their price is therefore very di¢ cult to be determined as they are traded very frequently by countless companies and individuals. The use of gold to hedge currencies has become increasingly popular lately, which adds yet another demand dynamic to its already complex demand characteristics. The induced demand that results from uncertainty in …nancial markets can cause behavioural changes in the price, hence impacting volatility.
In the case of copper and its heavy industrial use, the demand characteristics are very di¤erent. Rather than being exposed to many market participants who trade lower volumes each, the copper market tends to consist of fewer market participants who trade larger volumes each, e.g. mining companies, electronics companies, of which there are limited numbers. Financial instability can be a major factor in ‡uencing the price of copper. Decreased demand for copper as world demand falls (especially for consumer goods in which copper is a major raw material) is therefore expected but as the non-industrial utilisation of copper rises, its demand characteristics are also subject to major changes. Over the years, the copper price has been subject to a huge amount of speculative trading (although far less signi…cant than in the gold market) and this, combined with the uncertainty of …nancial markets, which typically causes the demand for copper to fall, can induce signi…cant levels of volatility in the copper price. With a lower number of market participants, despite the very large volumes, the net positions placed in the copper market will di¤er signi…cantly from those of gold due to the lower speculative nature and far less complex demand characteristics of the copper market. The recyclable nature of copper also makes it an interesting prospect to be analysed.

Mapping Procedure
Various procedures have been used to construct continuous futures series (see Ma et al., 1992). For example, Coakley et al. (2011) and Gutierrez (2013) roll contracts over to the next ones on the …rst business day of the contract month in analysing a wide range of futures. Martikainen and Puttonen (1996) roll the contract over to the next a week before the contract expires in analysing the Finnish stock index futures market. Hou and Li (2016) roll contracts over to the next ones ten working days before maturity in analysing both the S&P 500 and the CSI 300 stock index futures markets.
By contrast, the mapping procedure adopted in this paper is achieved by a specialist computer programme where the input for the programme is the entire set of monthly futures contract.
The programme then takes the last (expiry) price of each contract and lines it up by date to the price of the second month contracts. As the programme uses a counter for both the price series and date series, mapping occurs when the counters match on the day before expiry. The front and second month prices on that date are then lined up and their di¤erence gives the basis or rollover for that contract. Each roll value or basis value is stored and accumulated in order for a calculation of the cumulative roll or basis to be made (see, for details, Margaronis, 2015). Finally, we use continuously compounded returns (r t ) on these metal futures calculated as r t = (log p t log p t 1 ) 100; where p t is the metal futures price at time t.

Structural Breaks
Since the employed data span includes various economic and …nancial events causing behavioural changes due to con…dence alterations in economies as a result of the …nancial crisis, the considered returns series are likely to contain breaks associated with such events. Examples may include the collapse of Lehman Brothers, the collapse and buy-out of Bearn Sterns and AIG, increased unemployment, quantitative easing and many more.
Given this, to account for the possibility of breaks in the mean and/or volatility dynamics of these returns we use a set of parametric and non-parametric data-driven methods to identify the number and timing of the potential structural breaks. In particular, we employ the procedures in Bai and Perron (2003) and Lavielle and Moulines (2000), 2 and …nd that the stochastic behaviour of both returns yields four breaks during the sample period, roughly one every one and a half years on average (see Table 1). The predominant feature of the underlying segments is that it is mainly changes in variance that are found to be statistically signi…cant. Moreover, all four breakdates for the two series are very close to one another, which apparently signi…es economic events with a global impact. It follows that the detected breaks contrast to those of Vivian and Wohar (2012), who …nd limited evidence of common breaks for spot precious and industrial metals using the AIT (adjusted Inclan and Tiao, 1994) test statistics. Finally, the breaks on August 10, 2011 and November 03, 2011 for gold and copper returns respectively do not exactly coincide with speci…c events. However, given the signi…cance of the events prior to these dates, it is clear that at some point the economies of the world would begin recovering from the global …nancial crisis and also the uncertainty associated with the European sovereign debt crisis had eased based on optimism to resolving the debt crisis following these dates. Therefore, such dates may represent the beginning of some stability in markets, and hence the start of a relatively lower volatility regime.

Univariate Models
The conditional mean of the considered metal futures returns (r t ) is speci…ed as: where the innovation " t j F t 1 N (0; h t ) is conditionally normal with zero mean and variance the conditional variance is speci…ed as follows: with where S t 1 = 1 if " t 1 < 0; and 0 otherwise. The breaks for metal futures returns, l = 1; ::::; n (where n = 4), are given in Table 1, and D l are dummy variables de…ned as 0 in the period before each break, and 1 afterwards. Note that failure to reject H 0 : = 0 and l = 0; l = 1; ::; n (where n = 4), implies that the conditional variance follows a simple GARCH(1; 1) process.
Furthermore, the stability conditions require P 0 ; P 4 < 1 where P n = + + 2 + n X l=1 ( l + l + l =2), n = 0; : : : ; 4 (we use the convention P n l=1 ( ) = 0 for l < n). Clearly in the time invariant case only P 0 < 1 is required, which, when there are no asymmetries, is reduced to the well known condition: Alternatively, to examine how the persistence of the conditional variances is a¤ected by upward and downward shifts in these metal futures, we consider a simple GARCH(1; 1) model which allows the dynamics of the conditional variances to switch across positive and negative returns. This is given by: where D t 1 = 1 if r t 1 < 0, and 0 otherwise.

Bivariate Models
Having de…ned the univariate modelling, in this Section we use a bivariate model to simultaneously estimate the conditional means, variances, and covariances of returns. Let y t = (r 1;t r 2;t ) 0 represent the 2 1 vector of the two returns of metal futures. As before F t 1 = (y t 1 ; y t 2 ; : : :) is the …ltration generated by the information available up through time t 1. That is, we estimate the following bivariate AGARCH(1; 1) model Let h t = (h 1;t h 2;t ) 0 denote the 2 1 vector of F t 1 measurable conditional variances. The residual vector is de…ned as " t = (" 1;t " 2;t ) 0 = e t h^1 =2 t , where the symbols and^denote the Hadamard product and the elementwise exponentiation, respectively. The stochastic vector e t = (e 1;t e 2;t ) 0 is assumed to be i.i.d with zero mean, …nite second moments, and 2 2 correlation matrix R t = diagfQ t g 1=2 Q t diagfQ t g 1=2 with diagonal elements equal to one and o¤-diagonal elements being absolutely less than one. Q t is speci…ed as follows (see Engle, 2002): where Q is the unconditional covariance matrix of " t , and DCC and DCC are non-negative scalars ful…lling DCC + DCC < 1: A typical element of R t takes the form ij;t = q ij;t = p q ii;t q jj;t for i; j = 1; 2 and i 6 = j.
where Moreover, we also amend the UEDCC-AGARCH(1; 1) model by allowing shock and volatility spillovers to vary across positive and negative returns: where

Empirical Results
In this Section we outline our analysis, which is based on the breaks that we have identi…ed, to discuss …rst the …ndings from the univariate modelling and then from the bivariate one.

Univariate Modelling Results
The QML estimates of the AGARCH(1; 1) model for copper and gold returns using mapped and unmapped data are displayed in Table 2 (the insigni…cant parameters are excluded). We allow the 'numerator of the unconditional variance'(the !'s) as well as the ARCH and GARCH parameters to change across the identi…ed breaks, as in Eq. (2). The estimated models, at the 5% level, appear to be well-de…ned: there is no evidence of further linear or nonlinear dynamics to be captured. In a broad sense, the results seem not to be dissimilar with regard to the type of data used, mapped or unmapped. Margaronis (2015) …nd that small rolls or basis prove to yield similar time series for mapped and unmapped data sets. The di¤erences in the results may be due to the explanations expressed earlier in this paper whereby small compensations required over time to map data sets can accumulate to, and result in, large cumulative changes in the time series. The unmapped data are likely to include arti…cial 'price jumps'when contract roll over occurs, which are of course re ‡ected in the returns.
Another remark is that copper returns are shown to exhibit asymmetric responses regardless of using mapped or unmapped data; however, this is not the case for gold returns. This …nding is consistent with that of Hammoudeh and Yuan (2008) Table 2). Moreover, as is shown from Table 3, the time-variation of the ARCH and GARCH parameters is also observed by allowing the dynamics of a GARCH (1, 1) process to switch across positive and negative metal futures returns (see the estimated and parameters).

Bivariate Modelling Results
We also apply the bivariate UEDCC-AGARCH(1; 1) time-varying model to estimate the shock and volatility spillovers structure between copper and gold returns using mapped and unmapped data. The results, reported in Table 6  The results of the regime-dependent volatility spillovers between the two metal futures returns, reported in Table 7, on the other hand, suggest that declines in copper prices generate positive volatility spillovers to gold, using mapped and unmapped data (the estimated 21 parameter is positive and signi…cant at the 5% level). This result indicates that negative shocks to copper result in an increase in the volatility of gold. Moreover, the corresponding dynamic conditional correlations (not displayed) were not much di¤erent from those shown in Figure 2.
[Insert Figure 2 about here] Finally, it is noteworthy to indicate that we have further tested the robustness of our univariate and bivariate …ndings by including an exogenous control variable in the conditional variance equations of the considered metal returns such as the Chicago Board Options Exchange Volatil-ity index (VIX), or squared returns of (i) the US dollar exchange rate against the euro, (ii) the US' S&P 500 stock market index, or (iii) the West Texas Intermediate (WTI) crude oil spot prices. 6 The empirical univariate and bivariate results (available upon request) were found to remain broadly unchanged. Furthermore, copper returns volatility showed a signi…cant positive response to each of the considered exogenous control variables (where the impact was stronger in the mapped compared to the unmapped data), but this was not the case for gold returns volatility, which had no response to any of the considered control variables.

Discussion
From both the mapped and unmapped data results it is clear that there are bidirectional volatility spillovers between the two metals, where the conditional variance of copper returns a¤ects that of gold returns negatively whereas the e¤ect in the opposite direction is positive. This means that when the price of copper exhibits greater volatility the price of gold becomes more stable and its volatility falls. This is in line with the di¤erences in the demand characteristics between the two metals, explained previously.
During times of …nancial turmoil, where uncertainty lingers and individuals and organisations tie their capital up in gold as a reserve currency, the price of gold is suddenly in ‡uenced more by all the new demand. Rather than trading gold to make pro…t on its price changes, people are suddenly inclined to buy gold and keep it until there is con…dence and stability in the economies of the world. Also, the fact that gold is a precious metal and copper is a base means that the ‡uctuations in these metal prices will di¤er simply because of the di¤erences in uses and therefore demand and demand characteristics.
This can also be understood by considering the products based on each of the metals. Products based on copper are generally less dear and are replaced with new ones at a much greater rate, which is not the case for products containing gold or made of gold. Since copper prices depend signi…cantly on the state of the Australian mining sector, Chinese and South-East Asian demand and the demand of large world economies, the volatility exhibited can be due to uncertainties in these. 6 The data for the exogenous control variables were obtained from Datastream.
The positive spillovers from the conditional variance of gold returns to that of copper returns are consistent with the sheer volume and signi…cance of gold in the world economy. Induced volatility in gold prices will almost certainly in ‡uence a wide range of world economic factors.
With gold being a reserve currency, an increase in the volatility of gold implies an increased uncertainty in world economies. Copper, being the main industrial metal, is therefore hugely impacted by such uncertainty as industrial demand is based on economic and business con…dence worldwide, hence the connection can be made. Uncertainty in such factors does not usually occur when economies are booming. In the case of the gold price, however, the opposite e¤ect is seen due to its establishment as a reserve currency and its non-consumable nature. This could therefore explain the inverse relationship observed in the cross-volatility e¤ects. The

Summary and Conclusions
In this paper, we have analysed how the recent …nancial crisis a¤ected the principal time series properties of the underlying series of two metal futures, namely copper and gold. In particular, we have employed several univariate and multivariate models to examine how the volatility dynamics, including the volatility persistence and volatility spillovers structures of these two metal futures returns have changed due to the recent …nancial crisis, and based our analysis on non-parametrically identi…ed breaks.
Our …ndings suggest that the volatility persistence of these metal futures returns exhibit a substantial time-variation over the recent …nancial crisis; in particular, such persistence is shown to increase during periods of high volatility compared with low volatility. This time-variation appears consistent across both metal futures returns and irrespective of whether we allow for positive or negative changes in the corresponding asset.       Notes: Robust-standard errors are used in parentheses. l and l indicate the estimated parameters of the break dummies where the break l = 1; ::; 4 (see Table 1). Insigni…cant parameters are excluded. LB (5) and LB 2 (5) are Ljung and Box (1978) tests for serial correlations of …ve lags on the standardised and squared standardised residuals, respectively (p-values are reported in brackets). a ; b and c indicate statistical signi…cance at the 1%, 5%, and 10% levels, respectively.   Notes: Robust-standard errors are used in parentheses. The estimated model is speci…ed as h t = ! + ! D t 1 + " 2 t 1 + D t 1 " 2 t 1 + h t 1 + D t 1 h t 1 , where D t 1 = 1 if r t 1 < 0, and 0 otherwise.
LB (5) and LB 2 (5) are Ljung and Box (1978) tests for serial correlation of …ve lags on the standardised and squared standardised residuals, respectively (p-values are reported in brackets). a and b indicate statistical signi…cance at the 1% and 5% levels, respectively. Notes: State 0 covers the period preceding all breaks, while state 1 covers the period between breaks 1 and 2, state 2 covers the period between breaks 2 and 3, and so on (see Table 1 for the dates of the breaks). The persistence is given by: P n = + + 2 + P n l=1 ( l + l + l =2), n = 0; : : : ; 4. Notes: r + (r ) indicates the persistence of the conditional variance generated from positive (negative) returns. The persistence of the positive returns is calculated as + , while that of the negative returns is calculated as + + ( + 2 ): Table 6 Estimates of the bivariate UEDCC-AGARCH models allowing for shifts in shock and volatility spillovers between copper and gold returns Unmapped Mapped LB (5) and LB 2 (5) are Ljung and Box (1978) tests for serial correlation of …ve lags on the standardised and squared standardised residuals, respectively (p-values are reported in brackets). a , b and c indicate statistical signi…cance at the 1%, 5%, and 10% levels, respectively. Table 7 Estimates of the bivariate UEDCC-AGARCH models allowing spillovers between copper and gold to vary across positive and negative returns

Unmapped Mapped
Conditional Mean Equation  Notes: Robust-standard errors are used in parentheses. Subscripts of the estimated parameters are de…ned as 1= copper and 2=gold. Therefore, 12 ( 12 ) indicates shock (volatility) spillovers from gold to copper, whilst 21 ( 21 ) indicates shock (volatility) spillovers in the reverse direction. 21 reports the shift in volatility spillovers from copper to gold (induced by negative copper returns). Insigni…cant parameters are excluded. LB (5) and LB 2 (5) are Ljung and Box (1978) tests for serial correlation of …ve lags on the standardised and squared standardised residuals, respectively (p-values are reported in brackets). a , b and c indicate statistical signi…cance at the 1%, 5%, and 10% levels, respectively.