On the Comovements Among Gold and Oil: A Multivariate Time-Varying Asymmetric Approach

Previous studies have reported that there is a relationship among gold and oil prices. This research analyses how gold and oil prices variables interact focusing on different Global financial crisis (GFC) phases, we adopt a multivariate asymmetric dynamic conditional correlation GARCH framework, during the period spanning from January 1 st , 2000 until December 31 th , 2017. Our empirical results suggest correlations’ asymmetric responses among them. Moreover, the results indicate a correlations increase of gold and oil, during the crisis periods, suggesting different prices vulnerability.

A sufficient condition for the conditional variance ℎ to be positive almost surely for all t is that 0 > −1and the parameter combination 7, 4, ' satisfies the inequality constraints provided in Conrad and Haag (2006) and Conrad (2010). When 0 >0 , negative shocks have more impact on volatility than positive shocks.

Bivariate FIAPARCH model with dynamic conditional correlations
In what follow, we introduce the multivariate FIAPARCH process (M-FIAPARCH) taking into account the dynamic conditional correlation (DCC) hypothesis (see Dimitriou  In the first stage, we fit a univariate FIAPARCH(1,d,1) model in order to obtain the estimations of ℎ :: The daily data are assumed to be generated by a multivariate AR(1) process of the following form: The residual vector is given by: = K ⨀ℎ ∧ * N (7) ⨀: the Hadamard product; ⋀: the elementwise exponentiation. ℎ = >ℎ :, ? :@ ,…J is Σ measurable and the stochastic vector K = >K :, ? :@ ,…J is independent and identically distributedwith mean zero and positive definite covariance matrix P = >P :Q ? :,Q@ ,…J with P :Q = 1 for = V . Note that W /ℱ = 0 and where| | is the vector with elements stripped of negative values.
In the second stage, we estimate the conditional correlation using the transformed stock return residuals, which are estimated by their standard deviations from the first stage. The multivariate conditional variance is specified as follows: 9 = e f e (9) Where e = 4EFG(ℎ * ⁄ , … , ℎ JJ * ⁄ denotes the conditional variance derived from the univariate AR(1)-FIAPARCH(1,d,1) model and f = 1 − g − g * f + g H + g * f is the conditional correlation matrix 1 . Engle (2002) derives a different form of DCC model. The evolution of the correlation in DCC is given by: h = 1−∝ −' h j + kK + 'h (10) In addition, g and g * are the non-negative parameters satisfying g + g * < 1 and R= lP :Q m is a time-invariant symmetric B × B positive definite parameter matrix with P :Q = 1 and H is the B × B correlation matrix of n for o = − p, − p + 1, … , − 1. The E, V − ℎ element of the matrix H is given as follows: Where K : = : / ℎ :: is the transformed stock return residuals by their estimated standard deviations taken from the univariate AR(1)-FIAPARCH(1,d,1) model. The matrix H could be expressed as follows: To ensure the positivity of H and therefore of f a necessary condition is thatp ≤ B.Then f itself is a correlation matrix if f is also a correlation matrix. The correlation coefficient in a bivariate case is given as: P *, = 1 − g g * P * + g * P *, + g ∑ r z,stu v },stu x uyz

Data and preliminary analyses
The data comprises daily gold prices and oil (wti) prices. All the data are taken from DataStream. The study period spans from 01/01/2000 until 31/12/2017, leading to a sample size of 8274 observations. For each gold and oil prices, the continuously compounded return is calculated as Summary statistics of gold and oil prices are displayed in Table 1. From this table, gold and oil prices are volatile, as measured by the standard deviation of 2.0556% and 2.7614%,. Besides, we note that gold and oil prices have the highest level of kurtosis, indicating that extreme changes tend to occur more frequently for gold and oil markets.
As well, gold and oil prices exhibit high values of excess kurtosis. To accommodate the existence of "fat tails", we assume T-Student distributed innovations. Furthermore, the Jarque-Bera statistic rejects normality at the 1% level for all gold and oil prices.  .81*** Notes: The superscripts ***, ** and * denote the statistical significance at 1%, 5% and 10% levels, respectively.
In Table 2 which displays the results of Serial correlation and LM-ARCH Test, the Ljung-Box test for correlating series rejects the null hypothesis of autocorrelations at 1%, 5% and 10% levels, respectively. .2396*** Notes: The superscripts ***, ** and * denote the statistical significance at 1%, 5% and 10% levels, respectively. Engle and Ng (1993) propose a set of volatility asymmetry tests, known as the sign and size of bias tests. Engle and Ng tests should be used to decide if an asymmetrical model is necessary for a given series or if the symmetrical GARCH model can be judged adequate. In practice, Engle and Ng tests are generally applied to the residue of a GARCH adjustment to the return data. Defining " as variable indicators model as: The test of bias sign is based on the importance or not ∅ in the following regression: K̂ * = ∅ = + ∅ " + ‹ (15) Where ‹ is an independent and identically distributed error term. If positive and negative shocks on K̂ . The impact of conditional variance is different, so ∅ will be statistically significant.
It could also be the case that the greatness or the size of the shock will affect whether or not the volatility answer to shocks is symmetrical. In this case, a test of negative size bias would be made, based on a regression where " is used as a binary variable. Negative size bias is argued to be present if ∅ is statistically significant in the following regression: K̂ * = ∅ = + ∅ " ; + ‹ (16) Finally, we define " OE = 1 − " , so that " OE selects its comments with positive innovations. Engle and Ng (1993) propose a test for partiality cause of bias signs and size based on the following regression: K̂ * = ∅ = + ∅ " + ∅ * " ; + ∅ • " OE ; + ‹ (17) ∅ significance indicates the existence of signs bias, where positive and negative shocks have different effects on the future volatility, compared to the symmetrical response required by the standard formulation of GARCH. However, the meaning of ∅ * or of ∅ • suggests the existence of size bias, where not only the sign, but the magnitude of the shock is important. A common test statistic is formulated in standard mode by calculating Žf * regression, which will be asymptotically follow a • * distribution with 3 freedom degrees under the null assumption of no asymmetric effect.  21.7551*** (0.0000) Notes: The superscripts ***, ** and * denote the statistical significance at 1%, 5% and 10% levels, respectively. The results in Table 3 show that symmetric GARCH model residues for oil price do not suffer from sign biases and have negative size biases. But they display a positive size bias. These results also show that symmetric GARCH model residuals for gold price variable exhibit sign bias, negative size bias and positive size bias. The joint effect χ2 (3) at significant values of 1% for all these variables, which demonstrates a rejection of the null hypothesis of non-asymmetries. The overall results would therefore suggest a motivation for estimating an asymmetric volatility model for these variables.  Table 4. Based on these results, we reject the null hypothesis of no long memory for absolute and squared returns at 1% significance level. Subsequently, all volatilities proxies seem to be governed by a fractionally integrated process. Thus, FIAPARCH seem to be an appropriate specification to capture volatility clustering, long-range memory characteristics and asymmetry.   6 (2012) used key financial and economic events, he estimated excessive volatility to identify the crisis period, and he studied the transmission of the global financial crisis from the financial sector to the real economy.
In this study, we specify the duration of global financial crisis and their phases according to economic and statistical approaches. We follow a statistical approach based on a Markov-dynamic regression model (MS-DR), which takes into account the endogenous structural breaks and thus allows us defining the beginning and the end of each crises phase.   8.8146*** 0.0000 6.1827*** 0.0000 In table 5 the ARCH and GARCH parameters (Phi1 and Beta1) are statistically significant and non-negative for all the returns of the oil and gold which justifies the relevance of the specification FIAPARCH (1, d, 1). The tstudent degree of freedom parameter (df) is very significant for all returns. This result confirms our preliminary analysis and, subsequently, the choice of t-student as an appropriate distribution. In addition the term (γ) leverage estimates are statistically significant, indicating an asymmetric response of volatilities to positive and negative shocks. Estimates of the power term (δ) are very significant for prices.

Estimation results
Conrad, Karanasos and Zeng (2011) show that when the series is very likely to follow a non-normal error distribution, the superiority of a squared term (δ = 2) is lost and other power transformations can be more appropriate. In addition, all currencies display a significant fractional (4) parameter, which indicates a high degree of persistence behavior. This implies that the impact of negative shocks and their persistence on the conditional volatility of oil and gold returns. Table 6 reports the estimation results of the bivariate FIAPARCH (1, d, 1)-DCC model. The ARCH and GARCH parameters of the DCC (1,1) model capture, respectively, the effects of standardized lagged shocks and the lagged dynamic conditional correlations effects on current dynamic conditional correlation. They are statistically significant. Moreover, they are non-negative, justifying the appropriateness of the FIAPARCH model.  (20) 342.089 1.0000 9'1'EƒG 2 (20) 530.901 0.0000 Ep ˆ'4 (20) 341.051 0.0000 Ep ˆ'4 2 (20) 729.343 1.0000 As shown in Table 6, the estimated coefficients are significantly positive for the pair of GOLD /WTI. Besides, the t-student freedom degrees parameters are highly significant, supporting the choice of this distribution. The statistical significance of the DCC parameters reveals a considerable time-varying co-movement and thus a high persistence of the conditional correlation. This implies that the volatility displays a highly persistent manner.
The multivariate FIAPARCH-DCC model is so important to consider in our analysis since it has some key advantages. First, it captures the long range dependence property. Second, it allows obtaining all possible pairwise conditional correlation coefficients for GOLD /WT in the sample. Third, it is possible to investigate their behavior during periods of particular interest, such as the global financial crises period. Finally, it is crucial to check whether the selected GOLD /WTI display evidence of bivariate long memory ARCH effects and to test ability of the bivariate FIAPARCH specification to capture the volatility linkages between gold and oil. In our study, we refer to the most broadly used diagnostic tests, namely the Hosking's and Li   . Nevertheless, the different path of the estimated DCC displays fluctuations for GOLD /WTI during the global financial crises phases, suggesting that the assumption of constant correlation is not appropriate. The above findings motivate a more extensive analysis of DCC, in order to capture contagion dynamics during different phases of the two crises.

The DCC behavior during crisis periods
We next provide further results on the contagion effects during the crises. Using various dummy variables allows us to identify which of the sub-periods exhibit contagion effects of gold and oil price. We create dummies, which are equal to unity for the corresponding crisis phase and zero otherwise, to the following mean equation in order to describe the behavior of DCCs over time: where = is a constant term, P :Q, is the pairwise conditional correlation k =1… λ is the number of dummy variables corresponding to crises, which are identified based on an economic and a statistical approach. Furthermore, the conditional variance equation is assumed to follow an asymmetric GARCH(1,1) specification of Glosten, Jagannathan and Runkle (1993) including the dummy variables identified by the two approaches : ℎ :Q, = k = + k ℎ :Q, + ∑ › • -•@ 4-˜˜™ •, + oe š :Q, * + k * š :Q, *`( š :Q, < 0-(19) As the model implies, estimated dummy coefficients significance indicates structural changes in mean or/and variance shifts of the correlation coefficients due to external shocks during the crises. According to Dimitriou and Kenourgios (2013), a positive and statistically significant dummy coefficient in the mean equation indicates that the correlation during a specific phase of the crisis is significantly different from that in the previous phase, supporting the existence of spillover effects among gold and oil prices. Furthermore, a positive and statistically significant dummy coefficient in the variance equation indicates a higher volatility of the correlation coefficients. This suggests that the stability of the correlation is less reliable, causing some doubts on using the estimated correlation coefficient as a guide for portfolio decisions.  (20) 10.6071 0.8665 Table 7 shows the estimations of the mean and variance equations, setting a dummy variable for each crisis phase according to the economic approach. The constant terms = and the autoregressive term (H ) are both statistically significant for all DCCs, with the latter taking values close to unity, indicating a strong persistence in the conditional correlations among the examined prices. For the mean equation, dummy coefficient ' for the global financial crisis phase 1 is positive and significantly. This evidence suggests that the DCCs between gold and oil have amplified during phase 1, supporting the existence of a difference in prices vulnerability. At the global financial crisis phase 2, the dummy coefficient ' * is positive and not statistically significant for the GOLD /WTI prices, supporting a decrease in DCCs.
This suggests that the relationship between gold and oil prices actually decreased during this phase. We could define this finding as a "currency contagion effect". During the global financial crisis phase 3, positive and statistically significant dummy coefficient ' • exists for only the prices pairs, implying an increase of DCCs. Finally, the variance estimations have been reported in Table 7. The dummy coefficients › •, where k = 1, 2, 3, 4 for gold and oil are positive and statistically significant across several crisis phases. This finding means that the volatility of correlation coefficients is increased, implying that the correlations stability is less reliable for investment strategies implementation.

Conclusion
Whereas time fluctuating correlations of gold and oil prices have seen large research, reasonably little attention has given to correlations dynamics within a market. This research analyses how gold and oil prices variables interact with each other. In this paper, we evaluate the dynamic conditional correlation between the within gold and oil markets by means of the Dynamic Conditional Correlation (DCC-FIAPARCH) model. We used this model to examine and analyze contagion risk between them. Our empirical results point out that gold and oil prices exhibit asymmetry in the conditional variances. For that reason, the results point to the importance of applying a suitably flexible modeling framework to truthfully estimate the interaction between them.
The conditional correlation surrounded by pairs gold and oil displays higher dependency when it was driven by negative expansions to variations than it is by positive improvements. In addition, market correlations turn out to be more volatile throughout the global financial crisis. The time-varying correlation coefficients empirical analysis, during the main crisis periods, provides contagion approval evidence. Our empirical results seem to be essential to researchers and practitioners and mainly to active investors and portfolio managers who include gold and oil in their equities portfolios. Actually, the high correlation coefficients, during crises periods, involve that the international diversification advantage, by holding an involving diverse portfolio from the contagious markets, drop.
The findings lead to essential implications for investors' and policy makers' perception. They have a great consequence on international investors' financial choices on managing their risk disclosures to gold and oil and on winning advantages of potential diversification opportunities that may arise due to released dependence amongst the market. Markets linkages' growth correlation throughout crisis periods shows the different prices vulnerability and implies a portfolio diversification benefits decline, meanwhile holding a diversified portfolio with gold and oil will be less subject to systematic risk. As pointed out by Sephton and Mann (2018), it is very Journal of Energy Technologies and Policy www.iiste.org ISSN 2224-3232 (Paper) ISSN 2225-0573 (Online) Vol. 9, No.7, 2019 important to appreciate those variables simultaneity therefore portfolio managers, investors and policy makers can make better decisions. Additionally, correlations' behaviors considered as confirmation of non-cooperative monetary policies nearby the world and highlight the need for some form of policy organization among central banks. As a final point, dynamic linkages' different patterns between gold and oil prices might influence the intercontinental trade flows and the multinational corporations' accomplishments, as they generate ambiguity with concern to exports and imports.