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Structural Volatility Impulse Response Function and Asymptotic Inference

Structural Volatility Impulse Response Function and Asymptotic Inference Abstract This paper traces contemporaneous effects of independent shocks on predicted future volatilities through time. The shocks are identified through time-varying heteroscedasticity. I obtain the explicit functions of volatility impulse responses for a structural system of simultaneity. The asymptotic distributions of the derived functions are also studied in analytical forms for statistical inference. The potential of this new framework is empirically illustrated in a structural system of the U.S. commodity and stock markets. In recent years, financial regulations have been reorienting toward safeguarding financial stability.1 Much of the post-crisis reform agenda has been centered on macroprudential policies to enhance the resilience of financial institutions and markets to adverse shocks (Fischer, 2015; Tarullo, 2015). To undertake this challenge, it is crucially important to model underlying volatility transmission mechanisms among financial institutions, markets, and assets. In this regard, causality-in-volatility inference can be considered based on volatility impulse responses derived from a structural system of simultaneity, where financial instability can thus be monitored by forecasting future volatilities through time in response to shocks. The goal of this paper is to provide such a modeling framework. The current econometric development in the volatility impulse response analysis is limited to a small strand of recent literature including Gallant and Rossi (1993), Lin (1997), and Hafner and Herwartz (2006). While Lin (1997) derives volatility impulse responses, independent shocks are not identified in his work, so that a sound inference of causality-in-volatility cannot be pursued. However, the research by Hafner and Herwartz (2006) identifies independent shocks via Jordan decomposition that avoids typical orthogonality and ordering problems. The comparison between Hafner and Herwartz (2006) and Gallant and Rossi (1993) shows several drawbacks in the conditional moment profile of Gallant and Rossi (1993), including the difficult and somewhat arbitrary choice of baselines and shocks. However, the asymptotic properties are not provided in Hafner and Herwartz (2006) for statistical inference. Traditional identification methods for independent shocks, such as exclusion restrictions, sign restrictions, and long-run restrictions, are very difficult to justify mainly because financial assets of high frequencies are highly correlated and usually subject to common influences (Rigobon, 2003). To address this econometric issue, Rigobon (2003) identifies structural vector autoregressive (SVAR) models through heteroscedasticity, for example, by using variance–covariance matrices characterized from exogenous structural breaks to provide additional determining equations. His method also avoids a priori restrictions and typical orthogonality and ordering problems. More recently, Weber (2010a,b) further extends the approach of Rigobon (2003) to identify structural conditional correlation models through time-varying heteroscedasticity. Basically, Weber (2010a) parsimoniously specifies a distinct variance state for each observation (a quasi-continuum of distinct regimes) so that additional determining equations can be obtained from data to identify a structural system. However, the analysis of structural volatility impulse responses still remains untapped in the literature including Rigobon (2003) and Weber (2010a,b). To fill this literature gap, utilizing the structural conditional correlation models of Weber (2010a) identified through time-varying heteroscedasticity, I derive the explicit functions of the structural volatility impulse response functions (SVIRFs) to trace the time profile of predicted future volatilities in response to independent shocks. Specifically, this paper defines a structural volatility impulse response as the marginal effect of a one-unit innovation shock to a variable at time t on future volatility values of included variables, holding all other innovations at all times constant. Therefore, in order to perform a sound inference of causality-in-volatility, the simultaneous transmission effects among included variables must be modeled to obtain uncorrelated structural innovations. In addition to simultaneous transmissions among conditional means, information flows might cause volatility spillovers contemporaneously in financial data of high frequencies (Weber, 2013). This particular concern is also discussed in the next section for the derived SVIRFs. Hafner and Herwartz (2008) show that analytically computing the score function and the Hessian for asymptotic inference in multivariate GARCH and dynamic conditional correlation (DCC) models is clearly preferable to numerical methods. Therefore, I derive the analytical solutions for the information matrix and the Hessian of the proposed structural volatility model, which are further applied to construct confidence intervals for the estimated structural volatility impulse responses. An additional important feature of the proposed SVIRFs is to incorporate common comovements among included variables into the analysis of volatility impulse responses. Allowing for common comovements in structural innovations is motivated by the fact that financial assets and markets are subject to common influences. Recall that Weber (2010a) models instantaneous covariances in structural innovations identifiable within the parsimonious specifications of the structural conditional correlation models. In addition to simultaneous transmission effects, common driving forces among included variables can thus be modeled as common comovements.2 This proposed framework will enhance the understanding of economic and financial instabilities with a more systematic structure. The rest of this article is organized as follows. Section 1 defines and analytically derives the SVIRFs based on the specified structural dynamic conditional comovement (SDCC) model. The asymptotic distribution of the derived SVIRFs is also studied for asymptotic inference. Section 2 applies the proposed SVIRFs to the U.S. daily commodity and stock markets. Several novel findings and useful implications are discussed. Section 3 concludes this paper. 1 Structural Volatility Impulse Response Function Let yt be an N × 1 vector of endogenous random variables. A basic structural model for contemporaneous transmission effects among yt can be specified as   Ayt=ɛt, (1.1) where A, an N × N structural coefficient matrix with normalized ones on its main diagonal, represents simultaneous transmission impacts among endogenous variables. ɛt, an N × 1 vector of (idiosyncratically) uncorrelated structural innovations, is assumed to follow a multivariate normal distribution with zero mean and time-varying structural volatility, ɛt|Ft−1∼N(0,Ht), where Ft−1 is the past information up to time t – 1.3 Assume A is an invertible matrix and Ht is a measurable function of a suitably chosen σ-field. Then, the reduced-form covariance matrix can be obtained from Equation (1.1) as   Σt=A−1HtA′−1. (1.2) In this structural system of simultaneity, Equations (1.1) and (1.2), not only have the contemporaneous effects among conditional means been identified through A, but the simultaneous volatility spillovers can be captured in Equation (1.2) as well. To see this, Equation (1.2) can be rewritten as vech(Σt)=DN+(A−1⊗A−1) DNvech(Ht) in which the matrix, DN+(A−1⊗A−1)DN, mirrors the contemporaneous spillovers between the volatilities, similar to that in Weber (2013).4 However, compared with the shock to a latent process of stochastic volatilities as in Weber (2013), Equations (1.1) and (1.2) identify independent shocks from ɛt, which are transmitted to conditional volatilities through Equation (1.2). In this regard, economic/financial sources associated with these identified shocks are known. In addition to conditional variances, Equation (1.2) also captures the endogeneities of conditional covariances, whereas Weber (2013) estimates only simultaneous variance spillovers. In particular, I identify this structural system through time-varying heteroscedasticity using the SDCC model proposed by Weber (2010a). Basically, the time-varying heteroscedasticity provides additional determining equations obtained from data to identify a structural system, so that traditional exclusion restrictions and orthogonality issues can be avoided.5 See Weber (2010a) for identification details. Since financial assets and markets are generally subject to common influences and shocks, Weber (2010a) also estimates instantaneous comovement among yt. Intuitively, even though structural innovations are idiosyncratically uncorrelated (see the discussion in footnote 2 of this paper), they might react to common influences so as to cause a common comovement among them. In addition, the SDCC model allows the common comovement to be time-varying because, in times of economic turbulence, the estimation might easily understate the influence of third-party common factors as it is not constant. 1.1 Definition From Equation (1.2), the future volatility can be obtained as   Σt+k|t=A−1Ht+k|tA′−1, (1.3) where Ht+k|t is the k-period-ahead forecast of Ht conditional on Ft. Extending the impulse response definition in Hamilton (1994), I define in this paper the SVIRF as the marginal effect of a one-unit change in a variable’s structural innovation at time t on the future volatility values of included variables, holding all other structural innovations at all times constant. Hence, I formulate the SVIRF as the partial derivative of Σt+k|t w.r.t. ɛt,   Qk=∂vechΣt+k|t∂ɛt′, (1.4) which is an n ×  N derivative matrix with n = N(N + 1)/2. Let ut2=(ɛ1t2,…,ɛNt2)′ which contains the main diagonal elements of ɛtɛt′. Note that the structural volatility specified in Equation (1.7) is a linear function of ut2, implying that the responses of future volatilities to a one-unit shock from ɛt depend on the impact filtered through ut2. In this regard, ut2 can be used as a composite function for applying the chain rule of matrix derivatives to Equation (1.4) to obtain the following expression6:   Qk=∂vechΣt+k|t∂ut2′×∂ut2∂ɛt′=2∂vechΣt+k|t∂ut2′×diag(ɛ0), (1.5) where diag(ɛ0) is an N × N diagonal matrix which has ɛ0=(ɛ10,…,ɛN0)′ on its main diagonal as the chosen initial shocks for ɛt. If a one-unit shock is considered at time t, then diag(ɛ0) = IN becomes an identity matrix in Equation (1.5). Remark 1 The symmetric composite function, ut2, eventually leads to a symmetric SVIRF in Equation (1.5). In particular, this composite function should be applied with the assumption of E(ɛitɛjt2)=0, implying that ɛit is uncorrelated with the second moment of ɛjt for i,j = 1,…,N and i  ≠  j. This assumption generally holds due to the assumed (idiosyncratically) uncorrelated structural innovations and the joint normality assumption in Equation (1.19). In addition, the assumptions of E(ɛit2ɛjt2)=1 and E(ɛitɛjt3)=0 should also be considered as discussed in Hafner and Herwartz (2006),. Empirically, these assumptions have been tested for the SDCC model based on the approach applied in Liu and Luger (2015),and Frazier and Liu (2016)(see the Online Supplementary Material for detailed test results). Finally, the SVIRF can be obtained by applying Equations (1.3)–(1.5) to yield   Qk=2DN+(A−1⊗A−1)DNϒkdiag(ɛ0), (1.6) where ϒk=∂vech(Ht+k|t)/∂ut2′, an n × N matrix, is the impulse response function of the structural conditional volatility. 1.2 The Analytical Solution to the Structural Volatility Impulse Response Function Clearly, ϒk in Equation (1.6) requires the k-period-ahead forecast of Ht. Following Weber (2010a), I specify a multivariate GARCH(p, q) process for conditional variances as   ht2=C+∑i=1qBiut−i2+∑j=1pGjht−j2, (1.7) where ht2=(h1t2,...,hNt2), an N × 1 vector, contains the main diagonal elements of Ht with hl,t2=Et−1(ɛl,t2) for l  = 1, … , N. C is an N × 1 vector of parameters, B and G are N × N matrices of parameters. Note that the matrices of B and G allow for non-zero off-diagonal elements. In the sense of Conrad and Karanasos (2010) and Conrad and Weber (2013), among others, the off-diagonal elements of Bi for i = 1, … , q measure the ARCH spillovers, i.e. that is, the effect of the squared shock ɛd,t−i2 on hl,t2, for d, l = 1, …, N and d  ≠  l. Similarly, the off-diagonal elements of Gj for j = 1, … ,p represent the GARCH spillovers, that is, the effect of the conditional variance hd,t−j2 on hl,t2, for d, l = 1, … , N and d ≠  l.7Conrad and Weber (2013) further show that the GARCH spillovers can be interpreted as determinants of the persistence of volatility innovations that are transmitted between included variables. Note that Equation (1.7) nests the diagonal constant and DCC specifications of Bollerslev (1990), Engle (2002), and Tse and Tsui (2002) as special cases if B and G are assumed diagonal matrices (Conrad and Karanasos, 2010, p. 842).8 To obtain the k-period-ahead forecast of ht2, I define vt≡ut2−ht2 for Equation (1.7), so that ht2 has an autoregressive moving average (ARMA) presentation9  ht2=C+∑j=1s(Bj+Gj)ht−j2+∑j=1qBjvt−j, (1.8) where s = max(p,q). Provided that {Et(vt+i)=0}i=1∞, the k-period-ahead forecast of the structural variances, ht+k|t2, can then be derived as follows conditional on Ft [see also Lin (1997)]   ht+k|t2=C+∑j=1k*(Bj+Gj)ht+k−j|t2+∑j=ks(Bj+Gj)ht+k−j2+∑j=kqBjvt+k−j, (1.9) where k*=min(k−1,s), Bj = 0 if j > q and Gj = 0 if j > p. Owing to common influences in financial markets, Weber (2010a) further specifies an instantaneous common comovement among yt recovered by the dynamic conditional comovement assumption as   hijt=cijthithjt, for i ≠ j and i,j = 1, … ,N. cijt∈(−1,1) denotes the common comovement between the ith and jth variables due to common influences (see the discussion in footnote 2). Let Rt designate the time-varying common comovement matrix for yt, holding ones on its main diagonal and cijt as its off-diagonal elements. The structural volatility matrix is then expressed as   Ht=UtRtUt, (1.10) where Ut = diag(ht) is an N ×  N diagonal matrix with ht=(h1t,…,hNt)′ on its main diagonal, an N ×  1 vector of standard deviations of ɛt. Building on Tse and Tsui (2002), I define the time-varying conditional common comovement matrix Rt as10  Rt=(1−φ1−φ2)R¯+φ1Rt−1+φ2Ψt−1, (1.11) where R¯ is the unconditional comovement matrix as a sample comovement target, φ1 and φ2 are assumed to be non-negative with the constraint that φ1 +  φ2 ≤ 1, and Ψt−1 is an N × N symmetric matrix whose elements are functions of the lagged observations of yt. Hence, if Ψt−1 and R0 are well-defined correlation matrices (i.e., positive definite with unit diagonal elements), Rt will also be a well-defined correlation matrix. Following Tse and Tsui (2002), I denote Ψt={ψijt}, and consider the following specification for Ψt−1  ψij,t−1=∑l=1Mɛ˜i,t−lɛ˜j,t−l∑l=1Mɛ˜i,t−l2∑l=1Mɛ˜j,t−l2,  1≤i<j≤N, (1.12) where ɛ˜t=(ɛ˜1t,…,ɛ˜Nt)′=Ut−1ɛt. Thus, Ψt−1 is the sample correlation matrix of {ɛ˜t−1,…,ɛ˜t−M}. Furthermore, define Γt−1=(ɛ˜t−1,...,ɛ˜t−M)(ɛ˜t−1,...,ɛ˜t−M)′ as an N  × N matrix, and Λt−1 as a diagonal matrix where its ith diagonal element is (∑l=1Mɛ˜i,t−l2)1/2 for i = 1, … , N. Then, Equation (1.12) can be rewritten in a matrix form as   Ψt−1=Λt−1−1Γt−1Λt−1−1. (1.13) Note that Equations (1.12) and (1.13) ensure that the main diagonal elements of Ψt have values of 1. In practice, imposing M ≥ N is necessary for Ψt−1 to be positive definite. Following Tse and Tsui (2002), I set M = N in this paper. Similar to Equation (1.8), Rt specified in Equation (1.11) can also be rewritten in an ARMA presentation   Rt=(1−φ1−φ2)R¯+(φ1+φ2)Rt−1+φ2ϑt−1, (1.14) where ϑt=Ψt−Rt. Then, the k-period-ahead forecast, Rt + k|t, can be obtained from Equation (1.14) as follows:   Rt+k|t=(1−φ1−φ2)R¯+(φ1+φ2)Rt+k−1 (1.15) with Rt+1|t=(1−φ1−φ2)R¯+(φ1+φ2)Rt+φ2ϑt and {Et(ϑt+i)=0}i=1∞. Finally, from Equations (1.9) and (1.15), I have the k-period-ahead forecast, Ht+k|t, as   Ht+k|t=Ut+k|tRt+k|t Ut+k|t, (1.16) where Ut+k|t=diag(ht+k|t). And then, the following proposition gives the solution to ϒk. Proposition 1 The impulse response function of the structural conditional volatility based on the SDCC model can thus be derived as  ϒk=DN+[(Ut+k|tRt+k|t⊗IN+IN⊗Ut+k|tRt+k|t)(Ut+k|t⊗IN+IN⊗Ut+k|t)−1DNDN−ϖk+(Ut+k|t⊗Ut+k|t)Πk], (1.17)where ϖk=∂ht+k|t2/∂ut2′is an N  × N derivative matrix given by   ϖk=∑i=1k*(Bi+Gi)ϖk−i+Bk (1.18) with k*=min(k−1,s), Bi = 0 if i > q, Gi = 0 if i > p, and ϖ1 = B1. If k > q, then Bk = 0, and Πk=∂vecRt+k|t/∂ut2′ is an N2 × N derivative matrix given by Πk=(φ1+φ2)k−1φ2ΠΨt. ΠΨt=∂vecΨt/∂ut2′ is an N2 × N derivative matrix derived as ΠΨt=(Λt−1⊗Λt−1)ΠΓt−(Λt−1ΓtΛt−1⊗Λt−1+Λt−1⊗Λt−1ΓtΛt−1)ΠΛt.11 ΠΛt=∂vecΛt/∂ut2′=(Λt⊗IN+IN⊗Λt)−1 (Ut−1⊗Ut−1)DNDN− and ΠΓt=∂vecΓt/∂ut2′=(Ut−1⊗Ut−1)DNLn,N are N2 × N derivative matrices. Ln,N=∂vech(ɛtɛt′)/∂ut2′ is an n ×  N matrix with zero and one elements [see Lin (1997)]. Proof See the Online Supplementary Material for detailed proof. Remark 2 If considering a constant common comovement, such as Rt+k|t = R, then Πk = 0 in Equation (1.17). If the common comovement is not modeled in the system, so that Rt+k|t = IN and Πk = 0, then ϒk=DN−ϖk in Equation (1.17). The SDCC model can directly be estimated by maximizing the log-likelihood for a sample of T observations under the assumption of conditional normality   L(Θ)=∑t=1Tlog(lt(Θ))=−12∑t=1T(Nlog2π+log|Σt|+yt′Σt−1yt)=−12∑t=1T[Nlog2π+log|A−1HtA′−1|+yt′(A−1HtA′−1)−1yt]. (1.19) All the listed parameters are stacked in Θ=(A,C,B1,…,Bq,G1,…,Gp,φ1,φ2). 1.3 Asymptotic Inference Comte and Lieberman (2003) and Hafner and Preminger (2009) have proved the asymptotic theories for multivariate GARCH models, including the strong consistency of the quasi-maximum-likelihood estimator (QMLE) under mild regularity conditions, and its asymptotic normality when the initial state is either stationary or fixed. In this paper, the SDCC model is identified through heteroscedasticity and estimated with the assumption of conditional normality as in Equation (1.19). Based on Comte and Lieberman (2003) and Hafner and Preminger (2009), I further conduct analytical inference for the derived SVIRF using Delta method. To establish asymptotic normality for Qk, the following assumptions are made. A1. The SDCC model is identifiable. A2.  φ1,φ2 are positive with φ1+φ2≤1, and M ≥ N for the dynamic process of Ψt−1. A3. All roots of the characteristic polynomials, det(IN−∑i=1qBiλi−∑i=1pGiλi)=0, lie outside a unit circle. A4. Qk and ϒk are continuously differentiable functions of Θ. Assumption 2 ensures that Rt is well-behaved. If Assumption 3 holds, then the impulse responses of conditional volatilities to a shock die out eventually as the step goes to infinity. Under conditions listed by Hafner and Preminger (2009) and the assumptions of A1–A3, the QML estimator Θ^ that maximizes Equation (1.19) is assumed to be consistent and asymptotically normally distributed. Let vec(Θ)=(vec(A)′,C′,vec(B1)′,…,vec(Bq)′,vec(G1)′,…,vec(Gp)′,φ1,φ2)′ be the m × 1 vector of parameters in the SDCC model with m=[(1+q+p)N+1]N+2. The following theorem provides the asymptotic distribution of Qk. Theorem 1 Apply the Delta method with the asymptotic normality of the QMLE and the assumption of A4. The SVIRF, Qk, is thus asymptotically normally distributed as  T1/2(vec Q^k−vec Qk)∼N(0,ΩkJ−1VJ−1Ωk′), (1.20)where V=E[(∂loglt(Θ)/∂vecΘ)(∂loglt(Θ)/∂vec(Θ)′)] is the information matrix, J=−E [∂2log(lt(Θ))/∂vec Θ ∂vec(Θ)′] is the negative expectation of the Hessian, lt(Θ) is the likelihood function defined in Equation (1.19), and Ωk=∂vecQk/∂vec(Θ)′ is an nN × m Jacobian matrix with respect to vec(Θ). A consistent estimator of the covariance matrix for the SVIRF structural volatility impulse response function can be obtained in two steps. First, obtain the QMLE which is robust to the non-normal density as proposed by Bollerslev and Wooldridge (1992). Second, evaluate the Jacobian matrix at the QMLE values. Since typical numerical procedures that are used to compute the score and the Hessian are numerically unstable, Hafner and Herwartz (2008) provide analytical formulas for the score and Hessian for a variety of multivariate GARCH models including DCC models. Their Monte Carlo simulation study shows that employing analytical derivatives for inference is clearly preferable to numerical methods. In the context of this paper, this is also crucially important when one constructs confidence intervals for statistical inference. Therefore, I derive the analytical solutions for both V and J for the SDCC model (see the Online Supplementary Material for detailed derivation), and the following proposition further provides the analytical Jacobian matrix for Equation (1.20). Proposition 2 Based on Theorem 1, the Jacobian matrix for the SVIRF’s asymptotic distribution of the SDCC model can be derived as  Ωk=2{[diag(ɛ0)⊗DN+(A−1⊗A−1)DN]∂vecϒk/∂(vecΘ)′−(diag(ɛ0)ϒk′DN′⊗DN+)(IN⊗KNN⊗IN)(IN2⊗vecA−1+vecA−1⊗IN2)(A′−1⊗A−1)∂vecA/∂vec(Θ)′}, (1.21)where KNN is an N2 × N2 commutation matrix, such that KNNvec(X)=vec(X′), ∂vecA/∂vec(Θ)′ is an N2 × m matrix with zero and one elements,12 and ∂vecϒk/∂vec(Θ)′ is an nN ×  m matrix. Proof See the Online Supplementary Materialfor detailed proofs Remark 3 As a special case, the SDCC(1,1) model is a parsimonious and widely applied volatility model in the literature by choosing p = q = 1. This special case has ht+k|t2=C+(G+B)ht+k−12 with ht+1|t2=C+(G+B)ht2+Bvt and ϖk=(G+B)k−1B for Equation (1.17), and the analytical solution to ∂vecϒk/∂(vecΘ)′ in Equation (1.21) is provided in the Online Supplementary Material. Particularly, I apply this SDCC(1,1) model to the empirical study in the next section. 2 Empirical Application As an empirical illustration, this section estimates the proposed SVIRF for the U.S. commodity and stock markets. Over the last few decades, massive inflows into commodity investments rely on low correlation with traditional asset classes (i.e., stocks and bonds), positive comovement with inflation, and a tendency to backwardation in the futures curve, so as to allowing for portfolio diversification benefits (Gorton and Rouwenhorst, 2006; Chong and Miffre, 2010). The underlying rationale is that the price of commodities is driven by different fundamentals—such as weather conditions, supply constraints in the physical production, geopolitical events—which determine different price patterns and dynamics in respect to traditional assets (Baldi, Peri, and Vandone, 2016). However, commodity financialization has generated a gradual integration between commodity and financial markets, which, in turn, has risen levels of correlation, convergence of risk-adjusted returns, and volatility spillovers between markets, see for example, Tang and Xiong (2012), Silvennoinen and Thorp (2013), Adams and Gluck (2015), among others. Simon (2013) finds that conditional correlations between commodity and equity index returns moved higher over the sample period from 1991 to 2011, and sharply spiked during the financial panic that began in 2008. In addition, Baldi, Peri, and Vandone (2016) report that volatility spillovers increased significantly after the 2008 financial crisis, signaling a rising interconnection between financial and commodity markets. Using non-mean–variance spanning tests, Daskalaki and Skiadopoulos (2011) find that the increased financialization of commodities is impairing diversification benefits of commodities. Contributing to this recent challenge in asset allocation, I consider daily prices of the S&P 500 stock index and the S&P GSCI index (formerly known as the Goldman Sachs Commodity Index) taken from Yahoo.Finance and Financial Times, respectively.13 The S&P GSCI index is a benchmark for investment in commodity markets and as a measure of commodity performance over time.14Table 1 summarizes the logarithm returns of GSCI and S&P 500 market indexes for the sample period from December 17, 1992 to March 17, 2017 with 6087 daily observations. Both commodity and stock market returns are positive on average, while the stock market with an average return of 2.8 basis points and a standard deviation of 115.2 basis points outperforms the commodity market which has a lower average return of 1.4 basis points and a higher standard deviation of 136.5 basis points over the sample period. This sample performance is in line with Simon (2013) who assesses the conditional relationship between weekly commodity and equity indexes for the sample period from 1991 to 2011. Table 1 Data summary statistics Data statistics    GSCI      S&P 500  Mean (%)    0.014      0.028  Median (%)    0.030      0.050  Maximum (%)    10.42      10.96  Minimum (%)    −9.190      −9.470  Standard deviation (%)    1.365      1.152  Skewness    −0.126      −0.254  Excess kurtosis    3.506      8.738  ρ1    −0.030      −0.063  ρ2    0.003      −0.043  ρ3    0.012      0.005  Q2(26)    <0.001      <0.001  Cross-correlations  S&P500t−2  S&P500t−1  S&P500t  S&P500t+1  S&P500t+2  GSCIt  −2.6%  7.1%  20.9%  −3.9%  0.6%  Data statistics    GSCI      S&P 500  Mean (%)    0.014      0.028  Median (%)    0.030      0.050  Maximum (%)    10.42      10.96  Minimum (%)    −9.190      −9.470  Standard deviation (%)    1.365      1.152  Skewness    −0.126      −0.254  Excess kurtosis    3.506      8.738  ρ1    −0.030      −0.063  ρ2    0.003      −0.043  ρ3    0.012      0.005  Q2(26)    <0.001      <0.001  Cross-correlations  S&P500t−2  S&P500t−1  S&P500t  S&P500t+1  S&P500t+2  GSCIt  −2.6%  7.1%  20.9%  −3.9%  0.6%  Notes: The daily sample period is from December 17, 1992 to March 17, 2017 (6087 observations). p-values of the Ljung–Box test (Q2(26)) are reported for squared returns. Autocorrelation coefficients, ρ1, ρ2, and ρ3, are also reported. GSCI and S&P 500 are the commodity and stock returns computed from the S&P GSCI and S&P 500 price indexes, respectively. Table 1 Data summary statistics Data statistics    GSCI      S&P 500  Mean (%)    0.014      0.028  Median (%)    0.030      0.050  Maximum (%)    10.42      10.96  Minimum (%)    −9.190      −9.470  Standard deviation (%)    1.365      1.152  Skewness    −0.126      −0.254  Excess kurtosis    3.506      8.738  ρ1    −0.030      −0.063  ρ2    0.003      −0.043  ρ3    0.012      0.005  Q2(26)    <0.001      <0.001  Cross-correlations  S&P500t−2  S&P500t−1  S&P500t  S&P500t+1  S&P500t+2  GSCIt  −2.6%  7.1%  20.9%  −3.9%  0.6%  Data statistics    GSCI      S&P 500  Mean (%)    0.014      0.028  Median (%)    0.030      0.050  Maximum (%)    10.42      10.96  Minimum (%)    −9.190      −9.470  Standard deviation (%)    1.365      1.152  Skewness    −0.126      −0.254  Excess kurtosis    3.506      8.738  ρ1    −0.030      −0.063  ρ2    0.003      −0.043  ρ3    0.012      0.005  Q2(26)    <0.001      <0.001  Cross-correlations  S&P500t−2  S&P500t−1  S&P500t  S&P500t+1  S&P500t+2  GSCIt  −2.6%  7.1%  20.9%  −3.9%  0.6%  Notes: The daily sample period is from December 17, 1992 to March 17, 2017 (6087 observations). p-values of the Ljung–Box test (Q2(26)) are reported for squared returns. Autocorrelation coefficients, ρ1, ρ2, and ρ3, are also reported. GSCI and S&P 500 are the commodity and stock returns computed from the S&P GSCI and S&P 500 price indexes, respectively. The summary statistics in this table also show that the U.S. commodity and stock market returns are negatively skewed. The much higher excessive kurtosis from the stock market return implies fatter tails than those of the commodity return. The return autocorrelations are insignificant and quite small in magnitude, except that the first-order autocorrelation of S&P 500 returns is still small in magnitude but significant at 5% level. I thus filter the stock return for estimating Equation (1.1) by an AR(1) model. In addition, the p-values of the Ljung–Box test statistics (Q2(26)) demonstrate that the squared returns are highly persistent, suggesting that autoregressive specifications might be appropriate for modeling conditional volatilities of the commodity and stock market returns. The bottom panel shows sample cross-correlations of GSCI returns with S&P 500 returns. The unconditional contemporaneous correlation of GSCI and S&P 500 index returns is 20.9%, which is modest positive over the sample period considered. Nonetheless, the cross-serial correlations of the first and second lags and leads of the returns are either positive or negative and quite low. Furthermore, Figure 1 plots the time series of the asset prices and returns. The figure shows the spike of the GSCI price on July 3, 2008, and then the substantial drop during the financial panic, where the index reached the lowest price on February 18, 2009, when it lost roughly 70% of its peak value. The figure also shows that the S&P 500 price index peaked on October 9, 2007, before the commodity index, and then fell to its lowest price on March 9, 2009, during the financial crisis period. The comovement of the asset price drops can thus easily be observed from July 2008 to February 2009. The returns also exhibit the stylized fact of financial return clustering as widely documented in literature. Figure 1 View largeDownload slide Time series data plots. Daily prices of the S&P 500 stock index and the S&P GSCI index (formerly known as the Goldman Sachs Commodity Index) are taken from Yahoo.Finance and Financial Times, respectively. The sample period is from December 17, 1992, to March 17, 2017. The returns are computed in logarithm. Figure 1 View largeDownload slide Time series data plots. Daily prices of the S&P 500 stock index and the S&P GSCI index (formerly known as the Goldman Sachs Commodity Index) are taken from Yahoo.Finance and Financial Times, respectively. The sample period is from December 17, 1992, to March 17, 2017. The returns are computed in logarithm. 2.1 Model Estimation Results Table 2 reports the model estimation results for (GSCI, S&P 500)′. t-Statistics are computed using the derived analytical results of the information matrix and the Hessian (see the Online Supplementary Material), evaluated at the QMLE values. The results show that the off-diagonal elements of the structural coefficient matrix, A, are statistically significant, indicating the significant simultaneous transmission effects among these asset markets.15 In particular, the contemporaneous effects among the commodity and stock market returns are positive. In addition, the positive instantaneous effect of the stock market on the commodity market is much stronger than that of the commodity market on the stock market. Table 2 Model empirical estimation results Estimates and statistics  Parameter estimates  t-Statistics  A  (1.000−0.149−0.0591.000)  (∼−8.248−3.335∼)  C  (0.689,1.194)×10−6  (6.025, 7.784)  B  (0.0390.0260.0020.095)  (7.5144.0812.0978.519)  G  (0.957−0.0240.0030.888)  (31.13−6.1994.17748.07)  φ={φ1,φ2)  (0.984, 0.015)  (33.35, 5.992)  Joint significance test of all off-diagonal elements of A  H0: aij=0  <0.001  ∀i≠j and i,j = 1, … ,N    Joint significance test of all off-diagonal elements of B and G  H0: bij=gij=0  <0.001  ∀i ≠ j and i,j = 1, … ,N    Tests for volatility spillovers:      (1) GSCI does not spillover to S&P 500  H0:b21=g21=0  <0.001  (2) S&P 500 does not spillover to GSCI  H0:b12=g12=0  <0.001  Estimates and statistics  Parameter estimates  t-Statistics  A  (1.000−0.149−0.0591.000)  (∼−8.248−3.335∼)  C  (0.689,1.194)×10−6  (6.025, 7.784)  B  (0.0390.0260.0020.095)  (7.5144.0812.0978.519)  G  (0.957−0.0240.0030.888)  (31.13−6.1994.17748.07)  φ={φ1,φ2)  (0.984, 0.015)  (33.35, 5.992)  Joint significance test of all off-diagonal elements of A  H0: aij=0  <0.001  ∀i≠j and i,j = 1, … ,N    Joint significance test of all off-diagonal elements of B and G  H0: bij=gij=0  <0.001  ∀i ≠ j and i,j = 1, … ,N    Tests for volatility spillovers:      (1) GSCI does not spillover to S&P 500  H0:b21=g21=0  <0.001  (2) S&P 500 does not spillover to GSCI  H0:b12=g12=0  <0.001  Notes: In this table, Wald test statistics are used for joint significance tests, and the corresponding p-values are reported. t-Statistics are computed using the derived analytical results of the information matrix and the Hessian, evaluated at the QMLE values. Table 2 Model empirical estimation results Estimates and statistics  Parameter estimates  t-Statistics  A  (1.000−0.149−0.0591.000)  (∼−8.248−3.335∼)  C  (0.689,1.194)×10−6  (6.025, 7.784)  B  (0.0390.0260.0020.095)  (7.5144.0812.0978.519)  G  (0.957−0.0240.0030.888)  (31.13−6.1994.17748.07)  φ={φ1,φ2)  (0.984, 0.015)  (33.35, 5.992)  Joint significance test of all off-diagonal elements of A  H0: aij=0  <0.001  ∀i≠j and i,j = 1, … ,N    Joint significance test of all off-diagonal elements of B and G  H0: bij=gij=0  <0.001  ∀i ≠ j and i,j = 1, … ,N    Tests for volatility spillovers:      (1) GSCI does not spillover to S&P 500  H0:b21=g21=0  <0.001  (2) S&P 500 does not spillover to GSCI  H0:b12=g12=0  <0.001  Estimates and statistics  Parameter estimates  t-Statistics  A  (1.000−0.149−0.0591.000)  (∼−8.248−3.335∼)  C  (0.689,1.194)×10−6  (6.025, 7.784)  B  (0.0390.0260.0020.095)  (7.5144.0812.0978.519)  G  (0.957−0.0240.0030.888)  (31.13−6.1994.17748.07)  φ={φ1,φ2)  (0.984, 0.015)  (33.35, 5.992)  Joint significance test of all off-diagonal elements of A  H0: aij=0  <0.001  ∀i≠j and i,j = 1, … ,N    Joint significance test of all off-diagonal elements of B and G  H0: bij=gij=0  <0.001  ∀i ≠ j and i,j = 1, … ,N    Tests for volatility spillovers:      (1) GSCI does not spillover to S&P 500  H0:b21=g21=0  <0.001  (2) S&P 500 does not spillover to GSCI  H0:b12=g12=0  <0.001  Notes: In this table, Wald test statistics are used for joint significance tests, and the corresponding p-values are reported. t-Statistics are computed using the derived analytical results of the information matrix and the Hessian, evaluated at the QMLE values. The Wald tests in Table 2 support statistically significant ARCH and GARCH spillovers between the commodity and stock markets. The magnitudes of the volatility spillovers from the commodity market to the stock market are small, that is, b21 = 0.002 and g21 = 0.003, respectively, for the ARCH and GARCH spillovers. Note that the GARCH spillover from the stock market to the commodity market takes a negative value, g12=−0.024. Conrad and Weber (2013) show that a negative GARCH spillover is a necessary condition for ensuring that the effect of an own volatility innovation on conditional variance is more persistent than that of a foreign innovation. Furthermore, the common comovement between the commodity and stock markets caused by common influences, such as economic growth, monetary policy, and inflation, and so on, varies over time, as the estimates of φ1 and φ2 are highly significant. 2.2 Structural Volatility Impulse Responses This subsection discusses the structural volatility impulse responses computed for the U.S. commodity and stock markets using the SDCC model estimates. Specifically, six volatility impulse responses are obtained including the impulse responses of two conditional variances and one conditional covariance to a shock to each market. In particular, unity shocks are employed as the initial shocks (diag(ɛ0) = I), and the 95% confidence intervals are calculated for the structural volatility impulse responses using Theorem 1 and the analytical solutions derived in the Online Supplementary Material. Figure 2 plots the predicted future conditional variances which respond to a shock to GSCI. As expected, the commodity market positively responds to its own shock, implying an immediate increase in the commodity market instability. Observably, the GSCI variance response to its own shock is persistent and exhibits a slow-decaying speed.16 The figure also shows that a shock to GSCI quickly increases the instability of the U.S. stock market within a short period of time, and then the shock effect slowly decays to zero. Figure 2 View large Download slide Conditional variance impulse responses to a shock to GSCI. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 2 View large Download slide Conditional variance impulse responses to a shock to GSCI. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 3 plots the predicted future conditional variances which respond to a shock to S&P 500. As seen, a shock to S&P 500 immediately and significantly worsens the stabilities of both U.S. commodity and stock markets. Compared with the variance response of the S&P 500 to a shock to GSCI shown in Figure 2, the immediate variance response of GSCI to a shock to S&P 500 appears to be much stronger, that is, about four times higher in magnitude. In addition, the S&P 500 variance response to its own shock decays to zero much faster than the GSCI variance response to its own shock, mainly due to its persistent estimate, g^22=0.886, lower than g^11=0.957 for GSCI. Figure 3 View large Download slide Conditional variance impulse responses to a shock to S&P 500. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 3 View large Download slide Conditional variance impulse responses to a shock to S&P 500. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 4 plots the predicted future conditional covariance responding to the shocks to GSCI and S&P 500, respectively. The figure shows that the shock to either GSCI or S&P 500 would immediately increase the covariance between the U.S. commodity and stock markets. A shock to S&P 500 positively impacts on the covariance much stronger than a shock to GSCI. In addition, the positive effect of a shock to S&P 500 also lasts longer than that from a shock to GSCI. However, the shock effects of both GSCI and S&P 500 shortly reverse to be negative, and thus decrease the covariance level. The negative shock effects are very persistent and slowly decay to zero. Figure 4 View large Download slide Conditional covariance impulse responses to shocks to GSCI and S&P 500. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 4 View large Download slide Conditional covariance impulse responses to shocks to GSCI and S&P 500. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Eventually, the sign change of the covariance response in Figure 4 seems to suggest a nonlinear relationship between the U.S. commodity and stock markets, which is crucial for explaining the dynamics of their conditional correlation and common comovement. This finding is consistent with Delatte and Lopez (2013) who have identified nonlinear dependence structures between equity and commodity returns using a copula approach. Moreover, the sign of the covariance response changing from positive to negative might also indicate a correlation mean-reverse process, which adjusts short-run deviations toward a long-term correlation equilibrium. In this regard, theoretical equilibrium frameworks should be developed in future research for an in-depth analysis. In the next subsection, I perform a more detailed empirical investigation for the market comovement, in which a potential structural break has found partly explaining the sign change of the covariance response to shocks to GSCI and S&P 500. 2.3 A Structural Break in Correlation and Common Comovement Figure 5 plots the common comovement obtained from Rt (the red dashed line) along with the reduced-form counterpart (the black solid line) calculated from Σt. The figure shows that prior to the recent financial crisis that began in 2008, the reduced-form correlation fluctuates in a range between −20% and 30% centered close to zero. However, both reduced-form correlation and common comovement experience a sharp increase during the recent financial crisis of 2007–2009. Notably, these comovements between the asset markets continuously remain at the higher level even 8 years (2009–2017) after the crisis, supporting the recent findings in, that is, Silvennoinen and Thorp (2013), Simon (2013), among others. Figure 5 View largeDownload slide Conditional correlation (the black solid line) and common comovement (the red dashed line). Gray areas indicate the NBER-dated business cycles, including the dot.com bubble bust period from April 2, 2001 to November 30, 2001 and the recent financial crisis period from January 2, 2008 to June 30, 2009. Figure 5 View largeDownload slide Conditional correlation (the black solid line) and common comovement (the red dashed line). Gray areas indicate the NBER-dated business cycles, including the dot.com bubble bust period from April 2, 2001 to November 30, 2001 and the recent financial crisis period from January 2, 2008 to June 30, 2009. As seen in Figure 5, the turning point occurred on August 26, 2008 when the market correlation reached the lowest negative value of −16.3% during the financial crisis period of 2007–2009. The market correlation, nonetheless, quickly became positive on September 30, 2008 right after Lehman Brothers Bankruptcy on September 15, 2008 when the global panic was triggered. The correlation reached a much higher level at 54.7% on June 17, 2009. And, this relatively high correlation level remained until December 30, 2010 before it was adjusted to be lower around 30%. Importantly, this mean-reverse adjustment that can repeatedly be observed over the time period of 2009–2017 might be explained by the sign change of the covariance response discussed in Figure 4. The time-varying correlation between the asset markets is vital for asset allocation decisions. To this end, Table 3 reports the averaged comovements over the full and subsample periods, including (i) the prior-crisis period from February 1, 1995 to December 31, 2007, (ii) the during-crisis period from January 2, 2008 to June 30, 2009, and (iii) the after-crisis period from July 1, 2009 to March 17, 2017. Table 3 Correlations and comovements between the asset markets Sample periods  Reduced-form correlations (%)  Common comovements (%)  Full sample period  15.5  −3.4  Prior-crisis  2.9  −16.9  During-crisis  17.4  −1.4  After-crisis  39.7  22.4  Sample periods  Reduced-form correlations (%)  Common comovements (%)  Full sample period  15.5  −3.4  Prior-crisis  2.9  −16.9  During-crisis  17.4  −1.4  After-crisis  39.7  22.4  Notes: Entries in this table are the averaged comovements on the full and subsample periods. Full sample period is December 17. 1992–March 17, 2017. Prior-, during-, and after-crisis periods are December 17, 1995–December 31, 2007, January 2, 2008–June 30, 2009, and July 1, 2009–March 17, 2017, respectively. Table 3 Correlations and comovements between the asset markets Sample periods  Reduced-form correlations (%)  Common comovements (%)  Full sample period  15.5  −3.4  Prior-crisis  2.9  −16.9  During-crisis  17.4  −1.4  After-crisis  39.7  22.4  Sample periods  Reduced-form correlations (%)  Common comovements (%)  Full sample period  15.5  −3.4  Prior-crisis  2.9  −16.9  During-crisis  17.4  −1.4  After-crisis  39.7  22.4  Notes: Entries in this table are the averaged comovements on the full and subsample periods. Full sample period is December 17. 1992–March 17, 2017. Prior-, during-, and after-crisis periods are December 17, 1995–December 31, 2007, January 2, 2008–June 30, 2009, and July 1, 2009–March 17, 2017, respectively. Table 3 shows that, over the full sample period, the reduced-form correlation is positive with a modest magnitude (15.5%), while the common comovement appears to be negative with a relatively small magnitude (−3.4%). For the prior-crisis period, the correlation is positive but low and close to zero on average. However, during the crisis period, the average correlation increases to 17.4% from 2.9%, and further rises to 39.7% during the after-crisis period. Remarkably, the average common comovement has also changed to be positive (22.4% on average after the crisis period), in contrast to the negative comovements, −16.9% and −1.4%, respectively, from the prior- and during-crisis periods. 2.3.1 The common comovement between the U.S. commodity and stock markets In Figure 5, the common comovement is mostly negative during the prior-crisis period, indicating that the U.S. commodity and stock markets move in opposite directions due to the different reactions to common influence, such as changes in economic fundamentals. For instance, the inflationary shocks cause commodity prices to rise. Nonetheless, the equity–inflation relation is most likely non-linear. Low-but-positive inflation levels appear optimal for real earnings growth and equity returns, whereas deflation or higher inflation typically reduces real earnings growth potential and possibly requires higher equity risk premiums (Ilmanen, 2003). On the other hand, a positive common comovement might occur during the after-crisis period due to monetary policy easing in response to the economic great recession, which tends to boost the performance of both asset classes. Recall that the common comovement defined in this paper captures common influences in the asset markets. The reduced-form correlation is thus a joint of the common comovement and the idiosyncratic correlation through the contemporaneous coefficient matrix, A. In this regard, the difference between the reduced-form correlation and the common comovement should to some extent reflect the proportion of the reduced-form correlation contributed by idiosyncratic effects. In the context of this paper, idiosyncratic effects are generally attributed to market-specific changes, such as weather and geopolitical conditions, the impact of increased speculative traders in commodity markets who view commodities as “financial assets” rather than “real assets” (Buyuksahin and Robe, 2014; Haase, Zimmermann, and Zimmermann, 2016; Dimpfl, Flad, and Jung, 2017; Oglend and Kleppe, 2017), and so on. Figure 6 plots this difference computed from the results in Figure 5. The figure shows that prior to the recent financial crisis idiosyncratic changes positively contribute about 20–30% correlation between the U.S. commodity and stock markets to the reduced-form correlation, which is non-negligible. During the economic recession period, the idiosyncratic correlation is, nonetheless, reduced to about 15%. This sharp downward-trend is because both asset markets strongly react to the U.S. economy-wide adverse shocks to, that is, growth and unemployment. However, for the last 3 years, the idiosyncratic correlation has gradually been adjusted toward the prior-crisis level. This adjustment in the idiosyncratic correlation can still be explained by the sign change of the covariance response discussed in Figure 4, which implies a correlation mean-reverse process. Figure 6 View largeDownload slide The difference between the reduced-form correlation and the common comovement. Gray areas indicate the NBER-dated business cycles, including the dot.com bubble bust period from April 2, 2001 to November 30, 2001 and the recent financial crisis period from January 2, 2008 to June 30, 2009. Figure 6 View largeDownload slide The difference between the reduced-form correlation and the common comovement. Gray areas indicate the NBER-dated business cycles, including the dot.com bubble bust period from April 2, 2001 to November 30, 2001 and the recent financial crisis period from January 2, 2008 to June 30, 2009. 2.3.2 Structural volatility impulse responses before and after the structural break Both Figures 5 and 6 indicate a potential structural break in the relationship between the U.S. commodity and stock markets during the period of the recent financial crisis. Statistically, November 7, 2008 is detected in this paper as a structural break date.17 This statistical evidence raises the concern about the robustness of the volatility impulse responses estimated from the full sample period. Therefore, I reestimate the SDCC model for the subsample periods before and after the detected structural break date. Figure 7 plots the subsample conditional covariance impulse responses.18 The figure shows that the results from the after-break period are very similar to but less persistent than those obtained from the full sample period. However, for the before-break period, the impulse responses of the conditional covariance are very different from those in Figure 4. For instance, before the structural break, both shocks to GSCI and S&P 500 immediately reduce the level of the conditional covariance between the U.S. commodity and stock markets. The shock originated from the stock market has much stronger negative effect on the conditional covariance than the one from the commodity market, but this covariance reduction effect decays to zero much faster than those estimated from the full and after-break sample periods. In fact, this change in the covariance impulse responses of before- and after-break periods raises strategical cautions for asset allocation decisions. Figure 7 View largeDownload slide Subsample conditional covariance impulse responses to shocks to GSCI and S&P 500. The conditional covariance impulse responses in (a) and (b) are estimated from the periods before and after the break date on November 7, 2008, respectively. Figure 7 View largeDownload slide Subsample conditional covariance impulse responses to shocks to GSCI and S&P 500. The conditional covariance impulse responses in (a) and (b) are estimated from the periods before and after the break date on November 7, 2008, respectively. 3 Conclusions This paper derives an explicit function to describe over time the impulse responses of predicted future volatilities to independent shocks, which are identified through time-varying heteroscedasticity. The corresponding information matrix and the Hessian of the specified structural volatility model are analytically derived for statistical inference. The proposed SVIRF is further illustrated for the U.S. commodity and stock markets. Empirical results show that shocks to the asset markets positively increase their covariance, but after a short period of time the shock effects reverse to negatively reduce the level of the covariance. Eventually, this sign change in the conditional covariance response implies that: (i) the correlation between the U.S. commodity and stock markets is nonlinear due to different reactions to common influences, that is, changes in growth, inflation, monetary policy, and so on; and (ii) the correlation mean-reverse process exists and adjusts short-term deviations toward a long-run correlation equilibrium. Importantly, this mean-reverse process has been found very useful for explaining correlation dynamics over time, especially after the financial panic that began in 2008. Moreover, a significant structural break has been detected in the correlation and common comovement of the asset markets. Both the correlation and the common comovement have positively increased after the structural break date of November 7, 2008, which challenges the alleged diversification benefits of including commodities in portfolios. Supplementary Data Supplementary data are available at Journal of Financial Econometrics online. Footnotes 1 The April 2014 Global Financial Stability Report by International Monetary Fund assesses the challenging transitions that the global financial system is currently undergoing on the path to greater stability, but are far from complete and normal. Since the recent financial crisis of 2007–2009, financial stability has been considered as the regulatory goal to achieve by many international organizations, such as International Monetary Fund, Bank of International Settlement, Federal Reserve Bank, the Department of the U.S. Treasury (Office of Financial Stability), and central banks, etc. 2 Conceptually, this paper corrects the terminology of fundamental correlation used in Weber (2010a) as common comovement being more accurate. In general, two different types of relations may exist. One is the correlation which describes the direct influence of a change in one variable on other variables, whereas the other might occur when variables, which are uncorrelated, react to common influences so as to cause a comovement among them. Therefore, I refer to the latter relation as common comovement among uncorrelated variables due to common influences, while the former is referred to idiosyncratic correlation as the simultaneous transmission effects are the responses to idiosyncratic shocks. 3 Recently, Lanne, Meitz, and Saikkonen (2017) identify SVAR models with non-Gaussian errors. They show that the Gaussian case is an exception in that a SVAR model whose error vector consists of independent non-Gaussian components is, without any additional restrictions, identified and leads to essentially unique impulse responses. While their approach is quite general, it does not cover identification through heteroscedasticity proposed by Rigobon (2003) and extended by Weber (2010a). 4 DN+ is an N(N + 1)/2 × N2 Moore–Penrose inverse, which transforms vec(X) into vech(X) as DN+vec(X)=vech(X). DN is an N2 × N(N + 1)/2 duplication matrix, which transforms a symmetric matrix X from vech(X) into vec(X) as DNvech(X) = vec(X). ⊗ is the Kronecker product. vec as an operator that transforms a matrix X into a vector by stacking the columns of the matrix one underneath the other, and vech as an operator that eliminates all supradiagonal elements of a symmetric matrix X from vecX. 5 It is desirable to avoid imposing exclusion restrictions since they are difficult to be justified for financial assets, especially for highly correlated financial data (Rigobon, 2003). 6 See, for example, Magnus and Neudecker (2007, Ch. 5, Theorems 8 and 12), among others. 7 Conrad and Karanasos (2010) show that this model is identified in the sense of Jeantheau (1998, Definition 3.3). And, Corollary 4 of Conrad and Karanasos (2010) suggests that for N  =  2 at least one of the two GARCH spillover parameters can take a negative value, and this restriction is further relaxed for higher dimensions. 8 In the multivariate GARCH literature, BEKK is generally considered more flexible than DCC, while DCC is more parsimonious than BEKK. The framework proposed in this paper can readily be extended to BEKK. However, in this paper, I focus on deriving the SVIRF from the structural DCC model for the following reasons. First, in the terminology of Conrad and Karanasos (2010) and Conrad and Weber (2013), the unrestricted and extended DCC (UEDCC) specification of Equation (1.7) has generated a higher level of flexibility in modeling volatility transmissions than conventional diagonal DCC models. Second, SDCC allows for directly modeling common comovements among included variables, which reveals insightful implications in the empirical application of this paper, while BEKK is used indirectly to forecast conditional correlations (Caporin and McAleer, 2012, p. 737). Third, prior to deriving structural volatility impulse responses, the identification of BEKK in a structural system must be studied, which is not the focus of this paper. By contrast, the structural DCC model has already been studied and identified by Weber (2010a), which can directly be applied in this paper to derive structural volatility impulse responses. 9 vt is a martingale difference sequence, that is, E(vt) = 0 and cov(vt,vt−j)=0 for j ≥ 1. Nonetheless, {vt} in general is not an i.i.d. sequence. See, for example, Lin (1997) and Tsay (2010, §3.5, p. 132), among others. 10 This paper is in favor of the DCC specification of Tse and Tsui (2002), different from Weber (2010a) who uses Engle (2002). The following reasons are considered. In Engle (2002), Rt involves a transformation to ensure that its main diagonal elements have values of 1. This transformation makes the derivations of SVIRFs and their asymptotic distributions more complicated than necessary. In addition, the DCC model of Engle (2002) has an issue of inconsistency (Caporin and McAleer, 2012; Aielli, 2013). 11 Since the main diagonal elements in both Rt+k|t and Ψt take the value of 1, Πk=∂vecRt+k|t/∂ut2′ and ΠΨt=∂vecΨt/∂ut2′ have the value of 0 for the rows of N(i−1)+i for i = 1, … ,N. 12 Since the main diagonal elements in A is normalized to take the value of 1, ∂vecA/∂(vecΘ)′ has the value of zero for the rows of N(i−1)+i for i = 1, … ,N. 13 The S&P GSCI index data are available at https://markets.ft.com/data/indices/tearsheet/historical?s= GNX:IOM (Accessed 22 September 2017). 14 The S&P GSCI contains as many commodities as possible, with rules excluding certain commodities to maintain liquidity and investability in the underlying futures markets. The index currently comprises 24 commodities from all commodity sectors—energy products, industrial metals, agricultural products, livestock products, and precious metals. The S&P GSCI contains a much higher exposure to energy (the weight is 78.65%) than other commodity price indices such as the Dow Jones-UBS Commodity Index (the energy subgroup weight is 36.69%). The wide range of constituent commodities provides the S&P GSCI with a high level of diversification, across subsectors and within each subsector. 15 The Wald test rejects the null hypothesis of joint zero off-diagonal elements in A at all conventional confidence levels. 16 This high persistence feature of the volatility impulse responses might be caused by the high persistence of conditional variances as widely documented in finance literature for high frequency data when the values of b11  +  g11 and/or b22  +  g22 approach to one. In this paper, b^11+g^11=0.996 and b^22+g^22=0.983. 17 Particularly, I conduct the tests for structural breaks with unknown breakpoints based on Andrews (1993), Andrews and Ploberger (1994), and Bai and Perron (1998, 2003). The observations from Figure 5 and 6 suggest that the tests with one break during the recent financial crisis period can be considered. In other cases, multiple breaks can certainly be tested as well. An exogenous break date, such as August 26, 2008 or September 30, 2008 when the lowest or highest correlation is observed, is not automatically considered, as these dates do not necessarily imply the time when the structural break occurs in the correlation between the asset markets. In fact, any time point between these two dates is possibly the structural break date. To this end, I consider an endogenous break date tested with an unknown breakpoint. 18 Basically, the subsample impulse responses of the conditional variances to shocks to GSCI and S&P 500 are very similar to those obtained from the full sample period, except that they are more persistent for the before-break period but less persistent for the after-break period. 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Structural Volatility Impulse Response Function and Asymptotic Inference

Liu, Xiaochun
Journal of Financial Econometrics , Volume 16 (2) – Apr 1, 2018
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Oxford University Press
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© The Author, 2017. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com
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1479-8417
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Abstract

Abstract This paper traces contemporaneous effects of independent shocks on predicted future volatilities through time. The shocks are identified through time-varying heteroscedasticity. I obtain the explicit functions of volatility impulse responses for a structural system of simultaneity. The asymptotic distributions of the derived functions are also studied in analytical forms for statistical inference. The potential of this new framework is empirically illustrated in a structural system of the U.S. commodity and stock markets. In recent years, financial regulations have been reorienting toward safeguarding financial stability.1 Much of the post-crisis reform agenda has been centered on macroprudential policies to enhance the resilience of financial institutions and markets to adverse shocks (Fischer, 2015; Tarullo, 2015). To undertake this challenge, it is crucially important to model underlying volatility transmission mechanisms among financial institutions, markets, and assets. In this regard, causality-in-volatility inference can be considered based on volatility impulse responses derived from a structural system of simultaneity, where financial instability can thus be monitored by forecasting future volatilities through time in response to shocks. The goal of this paper is to provide such a modeling framework. The current econometric development in the volatility impulse response analysis is limited to a small strand of recent literature including Gallant and Rossi (1993), Lin (1997), and Hafner and Herwartz (2006). While Lin (1997) derives volatility impulse responses, independent shocks are not identified in his work, so that a sound inference of causality-in-volatility cannot be pursued. However, the research by Hafner and Herwartz (2006) identifies independent shocks via Jordan decomposition that avoids typical orthogonality and ordering problems. The comparison between Hafner and Herwartz (2006) and Gallant and Rossi (1993) shows several drawbacks in the conditional moment profile of Gallant and Rossi (1993), including the difficult and somewhat arbitrary choice of baselines and shocks. However, the asymptotic properties are not provided in Hafner and Herwartz (2006) for statistical inference. Traditional identification methods for independent shocks, such as exclusion restrictions, sign restrictions, and long-run restrictions, are very difficult to justify mainly because financial assets of high frequencies are highly correlated and usually subject to common influences (Rigobon, 2003). To address this econometric issue, Rigobon (2003) identifies structural vector autoregressive (SVAR) models through heteroscedasticity, for example, by using variance–covariance matrices characterized from exogenous structural breaks to provide additional determining equations. His method also avoids a priori restrictions and typical orthogonality and ordering problems. More recently, Weber (2010a,b) further extends the approach of Rigobon (2003) to identify structural conditional correlation models through time-varying heteroscedasticity. Basically, Weber (2010a) parsimoniously specifies a distinct variance state for each observation (a quasi-continuum of distinct regimes) so that additional determining equations can be obtained from data to identify a structural system. However, the analysis of structural volatility impulse responses still remains untapped in the literature including Rigobon (2003) and Weber (2010a,b). To fill this literature gap, utilizing the structural conditional correlation models of Weber (2010a) identified through time-varying heteroscedasticity, I derive the explicit functions of the structural volatility impulse response functions (SVIRFs) to trace the time profile of predicted future volatilities in response to independent shocks. Specifically, this paper defines a structural volatility impulse response as the marginal effect of a one-unit innovation shock to a variable at time t on future volatility values of included variables, holding all other innovations at all times constant. Therefore, in order to perform a sound inference of causality-in-volatility, the simultaneous transmission effects among included variables must be modeled to obtain uncorrelated structural innovations. In addition to simultaneous transmissions among conditional means, information flows might cause volatility spillovers contemporaneously in financial data of high frequencies (Weber, 2013). This particular concern is also discussed in the next section for the derived SVIRFs. Hafner and Herwartz (2008) show that analytically computing the score function and the Hessian for asymptotic inference in multivariate GARCH and dynamic conditional correlation (DCC) models is clearly preferable to numerical methods. Therefore, I derive the analytical solutions for the information matrix and the Hessian of the proposed structural volatility model, which are further applied to construct confidence intervals for the estimated structural volatility impulse responses. An additional important feature of the proposed SVIRFs is to incorporate common comovements among included variables into the analysis of volatility impulse responses. Allowing for common comovements in structural innovations is motivated by the fact that financial assets and markets are subject to common influences. Recall that Weber (2010a) models instantaneous covariances in structural innovations identifiable within the parsimonious specifications of the structural conditional correlation models. In addition to simultaneous transmission effects, common driving forces among included variables can thus be modeled as common comovements.2 This proposed framework will enhance the understanding of economic and financial instabilities with a more systematic structure. The rest of this article is organized as follows. Section 1 defines and analytically derives the SVIRFs based on the specified structural dynamic conditional comovement (SDCC) model. The asymptotic distribution of the derived SVIRFs is also studied for asymptotic inference. Section 2 applies the proposed SVIRFs to the U.S. daily commodity and stock markets. Several novel findings and useful implications are discussed. Section 3 concludes this paper. 1 Structural Volatility Impulse Response Function Let yt be an N × 1 vector of endogenous random variables. A basic structural model for contemporaneous transmission effects among yt can be specified as   Ayt=ɛt, (1.1) where A, an N × N structural coefficient matrix with normalized ones on its main diagonal, represents simultaneous transmission impacts among endogenous variables. ɛt, an N × 1 vector of (idiosyncratically) uncorrelated structural innovations, is assumed to follow a multivariate normal distribution with zero mean and time-varying structural volatility, ɛt|Ft−1∼N(0,Ht), where Ft−1 is the past information up to time t – 1.3 Assume A is an invertible matrix and Ht is a measurable function of a suitably chosen σ-field. Then, the reduced-form covariance matrix can be obtained from Equation (1.1) as   Σt=A−1HtA′−1. (1.2) In this structural system of simultaneity, Equations (1.1) and (1.2), not only have the contemporaneous effects among conditional means been identified through A, but the simultaneous volatility spillovers can be captured in Equation (1.2) as well. To see this, Equation (1.2) can be rewritten as vech(Σt)=DN+(A−1⊗A−1) DNvech(Ht) in which the matrix, DN+(A−1⊗A−1)DN, mirrors the contemporaneous spillovers between the volatilities, similar to that in Weber (2013).4 However, compared with the shock to a latent process of stochastic volatilities as in Weber (2013), Equations (1.1) and (1.2) identify independent shocks from ɛt, which are transmitted to conditional volatilities through Equation (1.2). In this regard, economic/financial sources associated with these identified shocks are known. In addition to conditional variances, Equation (1.2) also captures the endogeneities of conditional covariances, whereas Weber (2013) estimates only simultaneous variance spillovers. In particular, I identify this structural system through time-varying heteroscedasticity using the SDCC model proposed by Weber (2010a). Basically, the time-varying heteroscedasticity provides additional determining equations obtained from data to identify a structural system, so that traditional exclusion restrictions and orthogonality issues can be avoided.5 See Weber (2010a) for identification details. Since financial assets and markets are generally subject to common influences and shocks, Weber (2010a) also estimates instantaneous comovement among yt. Intuitively, even though structural innovations are idiosyncratically uncorrelated (see the discussion in footnote 2 of this paper), they might react to common influences so as to cause a common comovement among them. In addition, the SDCC model allows the common comovement to be time-varying because, in times of economic turbulence, the estimation might easily understate the influence of third-party common factors as it is not constant. 1.1 Definition From Equation (1.2), the future volatility can be obtained as   Σt+k|t=A−1Ht+k|tA′−1, (1.3) where Ht+k|t is the k-period-ahead forecast of Ht conditional on Ft. Extending the impulse response definition in Hamilton (1994), I define in this paper the SVIRF as the marginal effect of a one-unit change in a variable’s structural innovation at time t on the future volatility values of included variables, holding all other structural innovations at all times constant. Hence, I formulate the SVIRF as the partial derivative of Σt+k|t w.r.t. ɛt,   Qk=∂vechΣt+k|t∂ɛt′, (1.4) which is an n ×  N derivative matrix with n = N(N + 1)/2. Let ut2=(ɛ1t2,…,ɛNt2)′ which contains the main diagonal elements of ɛtɛt′. Note that the structural volatility specified in Equation (1.7) is a linear function of ut2, implying that the responses of future volatilities to a one-unit shock from ɛt depend on the impact filtered through ut2. In this regard, ut2 can be used as a composite function for applying the chain rule of matrix derivatives to Equation (1.4) to obtain the following expression6:   Qk=∂vechΣt+k|t∂ut2′×∂ut2∂ɛt′=2∂vechΣt+k|t∂ut2′×diag(ɛ0), (1.5) where diag(ɛ0) is an N × N diagonal matrix which has ɛ0=(ɛ10,…,ɛN0)′ on its main diagonal as the chosen initial shocks for ɛt. If a one-unit shock is considered at time t, then diag(ɛ0) = IN becomes an identity matrix in Equation (1.5). Remark 1 The symmetric composite function, ut2, eventually leads to a symmetric SVIRF in Equation (1.5). In particular, this composite function should be applied with the assumption of E(ɛitɛjt2)=0, implying that ɛit is uncorrelated with the second moment of ɛjt for i,j = 1,…,N and i  ≠  j. This assumption generally holds due to the assumed (idiosyncratically) uncorrelated structural innovations and the joint normality assumption in Equation (1.19). In addition, the assumptions of E(ɛit2ɛjt2)=1 and E(ɛitɛjt3)=0 should also be considered as discussed in Hafner and Herwartz (2006),. Empirically, these assumptions have been tested for the SDCC model based on the approach applied in Liu and Luger (2015),and Frazier and Liu (2016)(see the Online Supplementary Material for detailed test results). Finally, the SVIRF can be obtained by applying Equations (1.3)–(1.5) to yield   Qk=2DN+(A−1⊗A−1)DNϒkdiag(ɛ0), (1.6) where ϒk=∂vech(Ht+k|t)/∂ut2′, an n × N matrix, is the impulse response function of the structural conditional volatility. 1.2 The Analytical Solution to the Structural Volatility Impulse Response Function Clearly, ϒk in Equation (1.6) requires the k-period-ahead forecast of Ht. Following Weber (2010a), I specify a multivariate GARCH(p, q) process for conditional variances as   ht2=C+∑i=1qBiut−i2+∑j=1pGjht−j2, (1.7) where ht2=(h1t2,...,hNt2), an N × 1 vector, contains the main diagonal elements of Ht with hl,t2=Et−1(ɛl,t2) for l  = 1, … , N. C is an N × 1 vector of parameters, B and G are N × N matrices of parameters. Note that the matrices of B and G allow for non-zero off-diagonal elements. In the sense of Conrad and Karanasos (2010) and Conrad and Weber (2013), among others, the off-diagonal elements of Bi for i = 1, … , q measure the ARCH spillovers, i.e. that is, the effect of the squared shock ɛd,t−i2 on hl,t2, for d, l = 1, …, N and d  ≠  l. Similarly, the off-diagonal elements of Gj for j = 1, … ,p represent the GARCH spillovers, that is, the effect of the conditional variance hd,t−j2 on hl,t2, for d, l = 1, … , N and d ≠  l.7Conrad and Weber (2013) further show that the GARCH spillovers can be interpreted as determinants of the persistence of volatility innovations that are transmitted between included variables. Note that Equation (1.7) nests the diagonal constant and DCC specifications of Bollerslev (1990), Engle (2002), and Tse and Tsui (2002) as special cases if B and G are assumed diagonal matrices (Conrad and Karanasos, 2010, p. 842).8 To obtain the k-period-ahead forecast of ht2, I define vt≡ut2−ht2 for Equation (1.7), so that ht2 has an autoregressive moving average (ARMA) presentation9  ht2=C+∑j=1s(Bj+Gj)ht−j2+∑j=1qBjvt−j, (1.8) where s = max(p,q). Provided that {Et(vt+i)=0}i=1∞, the k-period-ahead forecast of the structural variances, ht+k|t2, can then be derived as follows conditional on Ft [see also Lin (1997)]   ht+k|t2=C+∑j=1k*(Bj+Gj)ht+k−j|t2+∑j=ks(Bj+Gj)ht+k−j2+∑j=kqBjvt+k−j, (1.9) where k*=min(k−1,s), Bj = 0 if j > q and Gj = 0 if j > p. Owing to common influences in financial markets, Weber (2010a) further specifies an instantaneous common comovement among yt recovered by the dynamic conditional comovement assumption as   hijt=cijthithjt, for i ≠ j and i,j = 1, … ,N. cijt∈(−1,1) denotes the common comovement between the ith and jth variables due to common influences (see the discussion in footnote 2). Let Rt designate the time-varying common comovement matrix for yt, holding ones on its main diagonal and cijt as its off-diagonal elements. The structural volatility matrix is then expressed as   Ht=UtRtUt, (1.10) where Ut = diag(ht) is an N ×  N diagonal matrix with ht=(h1t,…,hNt)′ on its main diagonal, an N ×  1 vector of standard deviations of ɛt. Building on Tse and Tsui (2002), I define the time-varying conditional common comovement matrix Rt as10  Rt=(1−φ1−φ2)R¯+φ1Rt−1+φ2Ψt−1, (1.11) where R¯ is the unconditional comovement matrix as a sample comovement target, φ1 and φ2 are assumed to be non-negative with the constraint that φ1 +  φ2 ≤ 1, and Ψt−1 is an N × N symmetric matrix whose elements are functions of the lagged observations of yt. Hence, if Ψt−1 and R0 are well-defined correlation matrices (i.e., positive definite with unit diagonal elements), Rt will also be a well-defined correlation matrix. Following Tse and Tsui (2002), I denote Ψt={ψijt}, and consider the following specification for Ψt−1  ψij,t−1=∑l=1Mɛ˜i,t−lɛ˜j,t−l∑l=1Mɛ˜i,t−l2∑l=1Mɛ˜j,t−l2,  1≤i<j≤N, (1.12) where ɛ˜t=(ɛ˜1t,…,ɛ˜Nt)′=Ut−1ɛt. Thus, Ψt−1 is the sample correlation matrix of {ɛ˜t−1,…,ɛ˜t−M}. Furthermore, define Γt−1=(ɛ˜t−1,...,ɛ˜t−M)(ɛ˜t−1,...,ɛ˜t−M)′ as an N  × N matrix, and Λt−1 as a diagonal matrix where its ith diagonal element is (∑l=1Mɛ˜i,t−l2)1/2 for i = 1, … , N. Then, Equation (1.12) can be rewritten in a matrix form as   Ψt−1=Λt−1−1Γt−1Λt−1−1. (1.13) Note that Equations (1.12) and (1.13) ensure that the main diagonal elements of Ψt have values of 1. In practice, imposing M ≥ N is necessary for Ψt−1 to be positive definite. Following Tse and Tsui (2002), I set M = N in this paper. Similar to Equation (1.8), Rt specified in Equation (1.11) can also be rewritten in an ARMA presentation   Rt=(1−φ1−φ2)R¯+(φ1+φ2)Rt−1+φ2ϑt−1, (1.14) where ϑt=Ψt−Rt. Then, the k-period-ahead forecast, Rt + k|t, can be obtained from Equation (1.14) as follows:   Rt+k|t=(1−φ1−φ2)R¯+(φ1+φ2)Rt+k−1 (1.15) with Rt+1|t=(1−φ1−φ2)R¯+(φ1+φ2)Rt+φ2ϑt and {Et(ϑt+i)=0}i=1∞. Finally, from Equations (1.9) and (1.15), I have the k-period-ahead forecast, Ht+k|t, as   Ht+k|t=Ut+k|tRt+k|t Ut+k|t, (1.16) where Ut+k|t=diag(ht+k|t). And then, the following proposition gives the solution to ϒk. Proposition 1 The impulse response function of the structural conditional volatility based on the SDCC model can thus be derived as  ϒk=DN+[(Ut+k|tRt+k|t⊗IN+IN⊗Ut+k|tRt+k|t)(Ut+k|t⊗IN+IN⊗Ut+k|t)−1DNDN−ϖk+(Ut+k|t⊗Ut+k|t)Πk], (1.17)where ϖk=∂ht+k|t2/∂ut2′is an N  × N derivative matrix given by   ϖk=∑i=1k*(Bi+Gi)ϖk−i+Bk (1.18) with k*=min(k−1,s), Bi = 0 if i > q, Gi = 0 if i > p, and ϖ1 = B1. If k > q, then Bk = 0, and Πk=∂vecRt+k|t/∂ut2′ is an N2 × N derivative matrix given by Πk=(φ1+φ2)k−1φ2ΠΨt. ΠΨt=∂vecΨt/∂ut2′ is an N2 × N derivative matrix derived as ΠΨt=(Λt−1⊗Λt−1)ΠΓt−(Λt−1ΓtΛt−1⊗Λt−1+Λt−1⊗Λt−1ΓtΛt−1)ΠΛt.11 ΠΛt=∂vecΛt/∂ut2′=(Λt⊗IN+IN⊗Λt)−1 (Ut−1⊗Ut−1)DNDN− and ΠΓt=∂vecΓt/∂ut2′=(Ut−1⊗Ut−1)DNLn,N are N2 × N derivative matrices. Ln,N=∂vech(ɛtɛt′)/∂ut2′ is an n ×  N matrix with zero and one elements [see Lin (1997)]. Proof See the Online Supplementary Material for detailed proof. Remark 2 If considering a constant common comovement, such as Rt+k|t = R, then Πk = 0 in Equation (1.17). If the common comovement is not modeled in the system, so that Rt+k|t = IN and Πk = 0, then ϒk=DN−ϖk in Equation (1.17). The SDCC model can directly be estimated by maximizing the log-likelihood for a sample of T observations under the assumption of conditional normality   L(Θ)=∑t=1Tlog(lt(Θ))=−12∑t=1T(Nlog2π+log|Σt|+yt′Σt−1yt)=−12∑t=1T[Nlog2π+log|A−1HtA′−1|+yt′(A−1HtA′−1)−1yt]. (1.19) All the listed parameters are stacked in Θ=(A,C,B1,…,Bq,G1,…,Gp,φ1,φ2). 1.3 Asymptotic Inference Comte and Lieberman (2003) and Hafner and Preminger (2009) have proved the asymptotic theories for multivariate GARCH models, including the strong consistency of the quasi-maximum-likelihood estimator (QMLE) under mild regularity conditions, and its asymptotic normality when the initial state is either stationary or fixed. In this paper, the SDCC model is identified through heteroscedasticity and estimated with the assumption of conditional normality as in Equation (1.19). Based on Comte and Lieberman (2003) and Hafner and Preminger (2009), I further conduct analytical inference for the derived SVIRF using Delta method. To establish asymptotic normality for Qk, the following assumptions are made. A1. The SDCC model is identifiable. A2.  φ1,φ2 are positive with φ1+φ2≤1, and M ≥ N for the dynamic process of Ψt−1. A3. All roots of the characteristic polynomials, det(IN−∑i=1qBiλi−∑i=1pGiλi)=0, lie outside a unit circle. A4. Qk and ϒk are continuously differentiable functions of Θ. Assumption 2 ensures that Rt is well-behaved. If Assumption 3 holds, then the impulse responses of conditional volatilities to a shock die out eventually as the step goes to infinity. Under conditions listed by Hafner and Preminger (2009) and the assumptions of A1–A3, the QML estimator Θ^ that maximizes Equation (1.19) is assumed to be consistent and asymptotically normally distributed. Let vec(Θ)=(vec(A)′,C′,vec(B1)′,…,vec(Bq)′,vec(G1)′,…,vec(Gp)′,φ1,φ2)′ be the m × 1 vector of parameters in the SDCC model with m=[(1+q+p)N+1]N+2. The following theorem provides the asymptotic distribution of Qk. Theorem 1 Apply the Delta method with the asymptotic normality of the QMLE and the assumption of A4. The SVIRF, Qk, is thus asymptotically normally distributed as  T1/2(vec Q^k−vec Qk)∼N(0,ΩkJ−1VJ−1Ωk′), (1.20)where V=E[(∂loglt(Θ)/∂vecΘ)(∂loglt(Θ)/∂vec(Θ)′)] is the information matrix, J=−E [∂2log(lt(Θ))/∂vec Θ ∂vec(Θ)′] is the negative expectation of the Hessian, lt(Θ) is the likelihood function defined in Equation (1.19), and Ωk=∂vecQk/∂vec(Θ)′ is an nN × m Jacobian matrix with respect to vec(Θ). A consistent estimator of the covariance matrix for the SVIRF structural volatility impulse response function can be obtained in two steps. First, obtain the QMLE which is robust to the non-normal density as proposed by Bollerslev and Wooldridge (1992). Second, evaluate the Jacobian matrix at the QMLE values. Since typical numerical procedures that are used to compute the score and the Hessian are numerically unstable, Hafner and Herwartz (2008) provide analytical formulas for the score and Hessian for a variety of multivariate GARCH models including DCC models. Their Monte Carlo simulation study shows that employing analytical derivatives for inference is clearly preferable to numerical methods. In the context of this paper, this is also crucially important when one constructs confidence intervals for statistical inference. Therefore, I derive the analytical solutions for both V and J for the SDCC model (see the Online Supplementary Material for detailed derivation), and the following proposition further provides the analytical Jacobian matrix for Equation (1.20). Proposition 2 Based on Theorem 1, the Jacobian matrix for the SVIRF’s asymptotic distribution of the SDCC model can be derived as  Ωk=2{[diag(ɛ0)⊗DN+(A−1⊗A−1)DN]∂vecϒk/∂(vecΘ)′−(diag(ɛ0)ϒk′DN′⊗DN+)(IN⊗KNN⊗IN)(IN2⊗vecA−1+vecA−1⊗IN2)(A′−1⊗A−1)∂vecA/∂vec(Θ)′}, (1.21)where KNN is an N2 × N2 commutation matrix, such that KNNvec(X)=vec(X′), ∂vecA/∂vec(Θ)′ is an N2 × m matrix with zero and one elements,12 and ∂vecϒk/∂vec(Θ)′ is an nN ×  m matrix. Proof See the Online Supplementary Materialfor detailed proofs Remark 3 As a special case, the SDCC(1,1) model is a parsimonious and widely applied volatility model in the literature by choosing p = q = 1. This special case has ht+k|t2=C+(G+B)ht+k−12 with ht+1|t2=C+(G+B)ht2+Bvt and ϖk=(G+B)k−1B for Equation (1.17), and the analytical solution to ∂vecϒk/∂(vecΘ)′ in Equation (1.21) is provided in the Online Supplementary Material. Particularly, I apply this SDCC(1,1) model to the empirical study in the next section. 2 Empirical Application As an empirical illustration, this section estimates the proposed SVIRF for the U.S. commodity and stock markets. Over the last few decades, massive inflows into commodity investments rely on low correlation with traditional asset classes (i.e., stocks and bonds), positive comovement with inflation, and a tendency to backwardation in the futures curve, so as to allowing for portfolio diversification benefits (Gorton and Rouwenhorst, 2006; Chong and Miffre, 2010). The underlying rationale is that the price of commodities is driven by different fundamentals—such as weather conditions, supply constraints in the physical production, geopolitical events—which determine different price patterns and dynamics in respect to traditional assets (Baldi, Peri, and Vandone, 2016). However, commodity financialization has generated a gradual integration between commodity and financial markets, which, in turn, has risen levels of correlation, convergence of risk-adjusted returns, and volatility spillovers between markets, see for example, Tang and Xiong (2012), Silvennoinen and Thorp (2013), Adams and Gluck (2015), among others. Simon (2013) finds that conditional correlations between commodity and equity index returns moved higher over the sample period from 1991 to 2011, and sharply spiked during the financial panic that began in 2008. In addition, Baldi, Peri, and Vandone (2016) report that volatility spillovers increased significantly after the 2008 financial crisis, signaling a rising interconnection between financial and commodity markets. Using non-mean–variance spanning tests, Daskalaki and Skiadopoulos (2011) find that the increased financialization of commodities is impairing diversification benefits of commodities. Contributing to this recent challenge in asset allocation, I consider daily prices of the S&P 500 stock index and the S&P GSCI index (formerly known as the Goldman Sachs Commodity Index) taken from Yahoo.Finance and Financial Times, respectively.13 The S&P GSCI index is a benchmark for investment in commodity markets and as a measure of commodity performance over time.14Table 1 summarizes the logarithm returns of GSCI and S&P 500 market indexes for the sample period from December 17, 1992 to March 17, 2017 with 6087 daily observations. Both commodity and stock market returns are positive on average, while the stock market with an average return of 2.8 basis points and a standard deviation of 115.2 basis points outperforms the commodity market which has a lower average return of 1.4 basis points and a higher standard deviation of 136.5 basis points over the sample period. This sample performance is in line with Simon (2013) who assesses the conditional relationship between weekly commodity and equity indexes for the sample period from 1991 to 2011. Table 1 Data summary statistics Data statistics    GSCI      S&P 500  Mean (%)    0.014      0.028  Median (%)    0.030      0.050  Maximum (%)    10.42      10.96  Minimum (%)    −9.190      −9.470  Standard deviation (%)    1.365      1.152  Skewness    −0.126      −0.254  Excess kurtosis    3.506      8.738  ρ1    −0.030      −0.063  ρ2    0.003      −0.043  ρ3    0.012      0.005  Q2(26)    <0.001      <0.001  Cross-correlations  S&P500t−2  S&P500t−1  S&P500t  S&P500t+1  S&P500t+2  GSCIt  −2.6%  7.1%  20.9%  −3.9%  0.6%  Data statistics    GSCI      S&P 500  Mean (%)    0.014      0.028  Median (%)    0.030      0.050  Maximum (%)    10.42      10.96  Minimum (%)    −9.190      −9.470  Standard deviation (%)    1.365      1.152  Skewness    −0.126      −0.254  Excess kurtosis    3.506      8.738  ρ1    −0.030      −0.063  ρ2    0.003      −0.043  ρ3    0.012      0.005  Q2(26)    <0.001      <0.001  Cross-correlations  S&P500t−2  S&P500t−1  S&P500t  S&P500t+1  S&P500t+2  GSCIt  −2.6%  7.1%  20.9%  −3.9%  0.6%  Notes: The daily sample period is from December 17, 1992 to March 17, 2017 (6087 observations). p-values of the Ljung–Box test (Q2(26)) are reported for squared returns. Autocorrelation coefficients, ρ1, ρ2, and ρ3, are also reported. GSCI and S&P 500 are the commodity and stock returns computed from the S&P GSCI and S&P 500 price indexes, respectively. Table 1 Data summary statistics Data statistics    GSCI      S&P 500  Mean (%)    0.014      0.028  Median (%)    0.030      0.050  Maximum (%)    10.42      10.96  Minimum (%)    −9.190      −9.470  Standard deviation (%)    1.365      1.152  Skewness    −0.126      −0.254  Excess kurtosis    3.506      8.738  ρ1    −0.030      −0.063  ρ2    0.003      −0.043  ρ3    0.012      0.005  Q2(26)    <0.001      <0.001  Cross-correlations  S&P500t−2  S&P500t−1  S&P500t  S&P500t+1  S&P500t+2  GSCIt  −2.6%  7.1%  20.9%  −3.9%  0.6%  Data statistics    GSCI      S&P 500  Mean (%)    0.014      0.028  Median (%)    0.030      0.050  Maximum (%)    10.42      10.96  Minimum (%)    −9.190      −9.470  Standard deviation (%)    1.365      1.152  Skewness    −0.126      −0.254  Excess kurtosis    3.506      8.738  ρ1    −0.030      −0.063  ρ2    0.003      −0.043  ρ3    0.012      0.005  Q2(26)    <0.001      <0.001  Cross-correlations  S&P500t−2  S&P500t−1  S&P500t  S&P500t+1  S&P500t+2  GSCIt  −2.6%  7.1%  20.9%  −3.9%  0.6%  Notes: The daily sample period is from December 17, 1992 to March 17, 2017 (6087 observations). p-values of the Ljung–Box test (Q2(26)) are reported for squared returns. Autocorrelation coefficients, ρ1, ρ2, and ρ3, are also reported. GSCI and S&P 500 are the commodity and stock returns computed from the S&P GSCI and S&P 500 price indexes, respectively. The summary statistics in this table also show that the U.S. commodity and stock market returns are negatively skewed. The much higher excessive kurtosis from the stock market return implies fatter tails than those of the commodity return. The return autocorrelations are insignificant and quite small in magnitude, except that the first-order autocorrelation of S&P 500 returns is still small in magnitude but significant at 5% level. I thus filter the stock return for estimating Equation (1.1) by an AR(1) model. In addition, the p-values of the Ljung–Box test statistics (Q2(26)) demonstrate that the squared returns are highly persistent, suggesting that autoregressive specifications might be appropriate for modeling conditional volatilities of the commodity and stock market returns. The bottom panel shows sample cross-correlations of GSCI returns with S&P 500 returns. The unconditional contemporaneous correlation of GSCI and S&P 500 index returns is 20.9%, which is modest positive over the sample period considered. Nonetheless, the cross-serial correlations of the first and second lags and leads of the returns are either positive or negative and quite low. Furthermore, Figure 1 plots the time series of the asset prices and returns. The figure shows the spike of the GSCI price on July 3, 2008, and then the substantial drop during the financial panic, where the index reached the lowest price on February 18, 2009, when it lost roughly 70% of its peak value. The figure also shows that the S&P 500 price index peaked on October 9, 2007, before the commodity index, and then fell to its lowest price on March 9, 2009, during the financial crisis period. The comovement of the asset price drops can thus easily be observed from July 2008 to February 2009. The returns also exhibit the stylized fact of financial return clustering as widely documented in literature. Figure 1 View largeDownload slide Time series data plots. Daily prices of the S&P 500 stock index and the S&P GSCI index (formerly known as the Goldman Sachs Commodity Index) are taken from Yahoo.Finance and Financial Times, respectively. The sample period is from December 17, 1992, to March 17, 2017. The returns are computed in logarithm. Figure 1 View largeDownload slide Time series data plots. Daily prices of the S&P 500 stock index and the S&P GSCI index (formerly known as the Goldman Sachs Commodity Index) are taken from Yahoo.Finance and Financial Times, respectively. The sample period is from December 17, 1992, to March 17, 2017. The returns are computed in logarithm. 2.1 Model Estimation Results Table 2 reports the model estimation results for (GSCI, S&P 500)′. t-Statistics are computed using the derived analytical results of the information matrix and the Hessian (see the Online Supplementary Material), evaluated at the QMLE values. The results show that the off-diagonal elements of the structural coefficient matrix, A, are statistically significant, indicating the significant simultaneous transmission effects among these asset markets.15 In particular, the contemporaneous effects among the commodity and stock market returns are positive. In addition, the positive instantaneous effect of the stock market on the commodity market is much stronger than that of the commodity market on the stock market. Table 2 Model empirical estimation results Estimates and statistics  Parameter estimates  t-Statistics  A  (1.000−0.149−0.0591.000)  (∼−8.248−3.335∼)  C  (0.689,1.194)×10−6  (6.025, 7.784)  B  (0.0390.0260.0020.095)  (7.5144.0812.0978.519)  G  (0.957−0.0240.0030.888)  (31.13−6.1994.17748.07)  φ={φ1,φ2)  (0.984, 0.015)  (33.35, 5.992)  Joint significance test of all off-diagonal elements of A  H0: aij=0  <0.001  ∀i≠j and i,j = 1, … ,N    Joint significance test of all off-diagonal elements of B and G  H0: bij=gij=0  <0.001  ∀i ≠ j and i,j = 1, … ,N    Tests for volatility spillovers:      (1) GSCI does not spillover to S&P 500  H0:b21=g21=0  <0.001  (2) S&P 500 does not spillover to GSCI  H0:b12=g12=0  <0.001  Estimates and statistics  Parameter estimates  t-Statistics  A  (1.000−0.149−0.0591.000)  (∼−8.248−3.335∼)  C  (0.689,1.194)×10−6  (6.025, 7.784)  B  (0.0390.0260.0020.095)  (7.5144.0812.0978.519)  G  (0.957−0.0240.0030.888)  (31.13−6.1994.17748.07)  φ={φ1,φ2)  (0.984, 0.015)  (33.35, 5.992)  Joint significance test of all off-diagonal elements of A  H0: aij=0  <0.001  ∀i≠j and i,j = 1, … ,N    Joint significance test of all off-diagonal elements of B and G  H0: bij=gij=0  <0.001  ∀i ≠ j and i,j = 1, … ,N    Tests for volatility spillovers:      (1) GSCI does not spillover to S&P 500  H0:b21=g21=0  <0.001  (2) S&P 500 does not spillover to GSCI  H0:b12=g12=0  <0.001  Notes: In this table, Wald test statistics are used for joint significance tests, and the corresponding p-values are reported. t-Statistics are computed using the derived analytical results of the information matrix and the Hessian, evaluated at the QMLE values. Table 2 Model empirical estimation results Estimates and statistics  Parameter estimates  t-Statistics  A  (1.000−0.149−0.0591.000)  (∼−8.248−3.335∼)  C  (0.689,1.194)×10−6  (6.025, 7.784)  B  (0.0390.0260.0020.095)  (7.5144.0812.0978.519)  G  (0.957−0.0240.0030.888)  (31.13−6.1994.17748.07)  φ={φ1,φ2)  (0.984, 0.015)  (33.35, 5.992)  Joint significance test of all off-diagonal elements of A  H0: aij=0  <0.001  ∀i≠j and i,j = 1, … ,N    Joint significance test of all off-diagonal elements of B and G  H0: bij=gij=0  <0.001  ∀i ≠ j and i,j = 1, … ,N    Tests for volatility spillovers:      (1) GSCI does not spillover to S&P 500  H0:b21=g21=0  <0.001  (2) S&P 500 does not spillover to GSCI  H0:b12=g12=0  <0.001  Estimates and statistics  Parameter estimates  t-Statistics  A  (1.000−0.149−0.0591.000)  (∼−8.248−3.335∼)  C  (0.689,1.194)×10−6  (6.025, 7.784)  B  (0.0390.0260.0020.095)  (7.5144.0812.0978.519)  G  (0.957−0.0240.0030.888)  (31.13−6.1994.17748.07)  φ={φ1,φ2)  (0.984, 0.015)  (33.35, 5.992)  Joint significance test of all off-diagonal elements of A  H0: aij=0  <0.001  ∀i≠j and i,j = 1, … ,N    Joint significance test of all off-diagonal elements of B and G  H0: bij=gij=0  <0.001  ∀i ≠ j and i,j = 1, … ,N    Tests for volatility spillovers:      (1) GSCI does not spillover to S&P 500  H0:b21=g21=0  <0.001  (2) S&P 500 does not spillover to GSCI  H0:b12=g12=0  <0.001  Notes: In this table, Wald test statistics are used for joint significance tests, and the corresponding p-values are reported. t-Statistics are computed using the derived analytical results of the information matrix and the Hessian, evaluated at the QMLE values. The Wald tests in Table 2 support statistically significant ARCH and GARCH spillovers between the commodity and stock markets. The magnitudes of the volatility spillovers from the commodity market to the stock market are small, that is, b21 = 0.002 and g21 = 0.003, respectively, for the ARCH and GARCH spillovers. Note that the GARCH spillover from the stock market to the commodity market takes a negative value, g12=−0.024. Conrad and Weber (2013) show that a negative GARCH spillover is a necessary condition for ensuring that the effect of an own volatility innovation on conditional variance is more persistent than that of a foreign innovation. Furthermore, the common comovement between the commodity and stock markets caused by common influences, such as economic growth, monetary policy, and inflation, and so on, varies over time, as the estimates of φ1 and φ2 are highly significant. 2.2 Structural Volatility Impulse Responses This subsection discusses the structural volatility impulse responses computed for the U.S. commodity and stock markets using the SDCC model estimates. Specifically, six volatility impulse responses are obtained including the impulse responses of two conditional variances and one conditional covariance to a shock to each market. In particular, unity shocks are employed as the initial shocks (diag(ɛ0) = I), and the 95% confidence intervals are calculated for the structural volatility impulse responses using Theorem 1 and the analytical solutions derived in the Online Supplementary Material. Figure 2 plots the predicted future conditional variances which respond to a shock to GSCI. As expected, the commodity market positively responds to its own shock, implying an immediate increase in the commodity market instability. Observably, the GSCI variance response to its own shock is persistent and exhibits a slow-decaying speed.16 The figure also shows that a shock to GSCI quickly increases the instability of the U.S. stock market within a short period of time, and then the shock effect slowly decays to zero. Figure 2 View large Download slide Conditional variance impulse responses to a shock to GSCI. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 2 View large Download slide Conditional variance impulse responses to a shock to GSCI. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 3 plots the predicted future conditional variances which respond to a shock to S&P 500. As seen, a shock to S&P 500 immediately and significantly worsens the stabilities of both U.S. commodity and stock markets. Compared with the variance response of the S&P 500 to a shock to GSCI shown in Figure 2, the immediate variance response of GSCI to a shock to S&P 500 appears to be much stronger, that is, about four times higher in magnitude. In addition, the S&P 500 variance response to its own shock decays to zero much faster than the GSCI variance response to its own shock, mainly due to its persistent estimate, g^22=0.886, lower than g^11=0.957 for GSCI. Figure 3 View large Download slide Conditional variance impulse responses to a shock to S&P 500. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 3 View large Download slide Conditional variance impulse responses to a shock to S&P 500. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 4 plots the predicted future conditional covariance responding to the shocks to GSCI and S&P 500, respectively. The figure shows that the shock to either GSCI or S&P 500 would immediately increase the covariance between the U.S. commodity and stock markets. A shock to S&P 500 positively impacts on the covariance much stronger than a shock to GSCI. In addition, the positive effect of a shock to S&P 500 also lasts longer than that from a shock to GSCI. However, the shock effects of both GSCI and S&P 500 shortly reverse to be negative, and thus decrease the covariance level. The negative shock effects are very persistent and slowly decay to zero. Figure 4 View large Download slide Conditional covariance impulse responses to shocks to GSCI and S&P 500. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Figure 4 View large Download slide Conditional covariance impulse responses to shocks to GSCI and S&P 500. The red dashed lines are the 95% confidence intervals computed from the analytical results of the information matrix and the Hessian derived for the SDCC model in the Online Supplementary Material. Eventually, the sign change of the covariance response in Figure 4 seems to suggest a nonlinear relationship between the U.S. commodity and stock markets, which is crucial for explaining the dynamics of their conditional correlation and common comovement. This finding is consistent with Delatte and Lopez (2013) who have identified nonlinear dependence structures between equity and commodity returns using a copula approach. Moreover, the sign of the covariance response changing from positive to negative might also indicate a correlation mean-reverse process, which adjusts short-run deviations toward a long-term correlation equilibrium. In this regard, theoretical equilibrium frameworks should be developed in future research for an in-depth analysis. In the next subsection, I perform a more detailed empirical investigation for the market comovement, in which a potential structural break has found partly explaining the sign change of the covariance response to shocks to GSCI and S&P 500. 2.3 A Structural Break in Correlation and Common Comovement Figure 5 plots the common comovement obtained from Rt (the red dashed line) along with the reduced-form counterpart (the black solid line) calculated from Σt. The figure shows that prior to the recent financial crisis that began in 2008, the reduced-form correlation fluctuates in a range between −20% and 30% centered close to zero. However, both reduced-form correlation and common comovement experience a sharp increase during the recent financial crisis of 2007–2009. Notably, these comovements between the asset markets continuously remain at the higher level even 8 years (2009–2017) after the crisis, supporting the recent findings in, that is, Silvennoinen and Thorp (2013), Simon (2013), among others. Figure 5 View largeDownload slide Conditional correlation (the black solid line) and common comovement (the red dashed line). Gray areas indicate the NBER-dated business cycles, including the dot.com bubble bust period from April 2, 2001 to November 30, 2001 and the recent financial crisis period from January 2, 2008 to June 30, 2009. Figure 5 View largeDownload slide Conditional correlation (the black solid line) and common comovement (the red dashed line). Gray areas indicate the NBER-dated business cycles, including the dot.com bubble bust period from April 2, 2001 to November 30, 2001 and the recent financial crisis period from January 2, 2008 to June 30, 2009. As seen in Figure 5, the turning point occurred on August 26, 2008 when the market correlation reached the lowest negative value of −16.3% during the financial crisis period of 2007–2009. The market correlation, nonetheless, quickly became positive on September 30, 2008 right after Lehman Brothers Bankruptcy on September 15, 2008 when the global panic was triggered. The correlation reached a much higher level at 54.7% on June 17, 2009. And, this relatively high correlation level remained until December 30, 2010 before it was adjusted to be lower around 30%. Importantly, this mean-reverse adjustment that can repeatedly be observed over the time period of 2009–2017 might be explained by the sign change of the covariance response discussed in Figure 4. The time-varying correlation between the asset markets is vital for asset allocation decisions. To this end, Table 3 reports the averaged comovements over the full and subsample periods, including (i) the prior-crisis period from February 1, 1995 to December 31, 2007, (ii) the during-crisis period from January 2, 2008 to June 30, 2009, and (iii) the after-crisis period from July 1, 2009 to March 17, 2017. Table 3 Correlations and comovements between the asset markets Sample periods  Reduced-form correlations (%)  Common comovements (%)  Full sample period  15.5  −3.4  Prior-crisis  2.9  −16.9  During-crisis  17.4  −1.4  After-crisis  39.7  22.4  Sample periods  Reduced-form correlations (%)  Common comovements (%)  Full sample period  15.5  −3.4  Prior-crisis  2.9  −16.9  During-crisis  17.4  −1.4  After-crisis  39.7  22.4  Notes: Entries in this table are the averaged comovements on the full and subsample periods. Full sample period is December 17. 1992–March 17, 2017. Prior-, during-, and after-crisis periods are December 17, 1995–December 31, 2007, January 2, 2008–June 30, 2009, and July 1, 2009–March 17, 2017, respectively. Table 3 Correlations and comovements between the asset markets Sample periods  Reduced-form correlations (%)  Common comovements (%)  Full sample period  15.5  −3.4  Prior-crisis  2.9  −16.9  During-crisis  17.4  −1.4  After-crisis  39.7  22.4  Sample periods  Reduced-form correlations (%)  Common comovements (%)  Full sample period  15.5  −3.4  Prior-crisis  2.9  −16.9  During-crisis  17.4  −1.4  After-crisis  39.7  22.4  Notes: Entries in this table are the averaged comovements on the full and subsample periods. Full sample period is December 17. 1992–March 17, 2017. Prior-, during-, and after-crisis periods are December 17, 1995–December 31, 2007, January 2, 2008–June 30, 2009, and July 1, 2009–March 17, 2017, respectively. Table 3 shows that, over the full sample period, the reduced-form correlation is positive with a modest magnitude (15.5%), while the common comovement appears to be negative with a relatively small magnitude (−3.4%). For the prior-crisis period, the correlation is positive but low and close to zero on average. However, during the crisis period, the average correlation increases to 17.4% from 2.9%, and further rises to 39.7% during the after-crisis period. Remarkably, the average common comovement has also changed to be positive (22.4% on average after the crisis period), in contrast to the negative comovements, −16.9% and −1.4%, respectively, from the prior- and during-crisis periods. 2.3.1 The common comovement between the U.S. commodity and stock markets In Figure 5, the common comovement is mostly negative during the prior-crisis period, indicating that the U.S. commodity and stock markets move in opposite directions due to the different reactions to common influence, such as changes in economic fundamentals. For instance, the inflationary shocks cause commodity prices to rise. Nonetheless, the equity–inflation relation is most likely non-linear. Low-but-positive inflation levels appear optimal for real earnings growth and equity returns, whereas deflation or higher inflation typically reduces real earnings growth potential and possibly requires higher equity risk premiums (Ilmanen, 2003). On the other hand, a positive common comovement might occur during the after-crisis period due to monetary policy easing in response to the economic great recession, which tends to boost the performance of both asset classes. Recall that the common comovement defined in this paper captures common influences in the asset markets. The reduced-form correlation is thus a joint of the common comovement and the idiosyncratic correlation through the contemporaneous coefficient matrix, A. In this regard, the difference between the reduced-form correlation and the common comovement should to some extent reflect the proportion of the reduced-form correlation contributed by idiosyncratic effects. In the context of this paper, idiosyncratic effects are generally attributed to market-specific changes, such as weather and geopolitical conditions, the impact of increased speculative traders in commodity markets who view commodities as “financial assets” rather than “real assets” (Buyuksahin and Robe, 2014; Haase, Zimmermann, and Zimmermann, 2016; Dimpfl, Flad, and Jung, 2017; Oglend and Kleppe, 2017), and so on. Figure 6 plots this difference computed from the results in Figure 5. The figure shows that prior to the recent financial crisis idiosyncratic changes positively contribute about 20–30% correlation between the U.S. commodity and stock markets to the reduced-form correlation, which is non-negligible. During the economic recession period, the idiosyncratic correlation is, nonetheless, reduced to about 15%. This sharp downward-trend is because both asset markets strongly react to the U.S. economy-wide adverse shocks to, that is, growth and unemployment. However, for the last 3 years, the idiosyncratic correlation has gradually been adjusted toward the prior-crisis level. This adjustment in the idiosyncratic correlation can still be explained by the sign change of the covariance response discussed in Figure 4, which implies a correlation mean-reverse process. Figure 6 View largeDownload slide The difference between the reduced-form correlation and the common comovement. Gray areas indicate the NBER-dated business cycles, including the dot.com bubble bust period from April 2, 2001 to November 30, 2001 and the recent financial crisis period from January 2, 2008 to June 30, 2009. Figure 6 View largeDownload slide The difference between the reduced-form correlation and the common comovement. Gray areas indicate the NBER-dated business cycles, including the dot.com bubble bust period from April 2, 2001 to November 30, 2001 and the recent financial crisis period from January 2, 2008 to June 30, 2009. 2.3.2 Structural volatility impulse responses before and after the structural break Both Figures 5 and 6 indicate a potential structural break in the relationship between the U.S. commodity and stock markets during the period of the recent financial crisis. Statistically, November 7, 2008 is detected in this paper as a structural break date.17 This statistical evidence raises the concern about the robustness of the volatility impulse responses estimated from the full sample period. Therefore, I reestimate the SDCC model for the subsample periods before and after the detected structural break date. Figure 7 plots the subsample conditional covariance impulse responses.18 The figure shows that the results from the after-break period are very similar to but less persistent than those obtained from the full sample period. However, for the before-break period, the impulse responses of the conditional covariance are very different from those in Figure 4. For instance, before the structural break, both shocks to GSCI and S&P 500 immediately reduce the level of the conditional covariance between the U.S. commodity and stock markets. The shock originated from the stock market has much stronger negative effect on the conditional covariance than the one from the commodity market, but this covariance reduction effect decays to zero much faster than those estimated from the full and after-break sample periods. In fact, this change in the covariance impulse responses of before- and after-break periods raises strategical cautions for asset allocation decisions. Figure 7 View largeDownload slide Subsample conditional covariance impulse responses to shocks to GSCI and S&P 500. The conditional covariance impulse responses in (a) and (b) are estimated from the periods before and after the break date on November 7, 2008, respectively. Figure 7 View largeDownload slide Subsample conditional covariance impulse responses to shocks to GSCI and S&P 500. The conditional covariance impulse responses in (a) and (b) are estimated from the periods before and after the break date on November 7, 2008, respectively. 3 Conclusions This paper derives an explicit function to describe over time the impulse responses of predicted future volatilities to independent shocks, which are identified through time-varying heteroscedasticity. The corresponding information matrix and the Hessian of the specified structural volatility model are analytically derived for statistical inference. The proposed SVIRF is further illustrated for the U.S. commodity and stock markets. Empirical results show that shocks to the asset markets positively increase their covariance, but after a short period of time the shock effects reverse to negatively reduce the level of the covariance. Eventually, this sign change in the conditional covariance response implies that: (i) the correlation between the U.S. commodity and stock markets is nonlinear due to different reactions to common influences, that is, changes in growth, inflation, monetary policy, and so on; and (ii) the correlation mean-reverse process exists and adjusts short-term deviations toward a long-run correlation equilibrium. Importantly, this mean-reverse process has been found very useful for explaining correlation dynamics over time, especially after the financial panic that began in 2008. Moreover, a significant structural break has been detected in the correlation and common comovement of the asset markets. Both the correlation and the common comovement have positively increased after the structural break date of November 7, 2008, which challenges the alleged diversification benefits of including commodities in portfolios. Supplementary Data Supplementary data are available at Journal of Financial Econometrics online. Footnotes 1 The April 2014 Global Financial Stability Report by International Monetary Fund assesses the challenging transitions that the global financial system is currently undergoing on the path to greater stability, but are far from complete and normal. Since the recent financial crisis of 2007–2009, financial stability has been considered as the regulatory goal to achieve by many international organizations, such as International Monetary Fund, Bank of International Settlement, Federal Reserve Bank, the Department of the U.S. Treasury (Office of Financial Stability), and central banks, etc. 2 Conceptually, this paper corrects the terminology of fundamental correlation used in Weber (2010a) as common comovement being more accurate. In general, two different types of relations may exist. One is the correlation which describes the direct influence of a change in one variable on other variables, whereas the other might occur when variables, which are uncorrelated, react to common influences so as to cause a comovement among them. Therefore, I refer to the latter relation as common comovement among uncorrelated variables due to common influences, while the former is referred to idiosyncratic correlation as the simultaneous transmission effects are the responses to idiosyncratic shocks. 3 Recently, Lanne, Meitz, and Saikkonen (2017) identify SVAR models with non-Gaussian errors. They show that the Gaussian case is an exception in that a SVAR model whose error vector consists of independent non-Gaussian components is, without any additional restrictions, identified and leads to essentially unique impulse responses. While their approach is quite general, it does not cover identification through heteroscedasticity proposed by Rigobon (2003) and extended by Weber (2010a). 4 DN+ is an N(N + 1)/2 × N2 Moore–Penrose inverse, which transforms vec(X) into vech(X) as DN+vec(X)=vech(X). DN is an N2 × N(N + 1)/2 duplication matrix, which transforms a symmetric matrix X from vech(X) into vec(X) as DNvech(X) = vec(X). ⊗ is the Kronecker product. vec as an operator that transforms a matrix X into a vector by stacking the columns of the matrix one underneath the other, and vech as an operator that eliminates all supradiagonal elements of a symmetric matrix X from vecX. 5 It is desirable to avoid imposing exclusion restrictions since they are difficult to be justified for financial assets, especially for highly correlated financial data (Rigobon, 2003). 6 See, for example, Magnus and Neudecker (2007, Ch. 5, Theorems 8 and 12), among others. 7 Conrad and Karanasos (2010) show that this model is identified in the sense of Jeantheau (1998, Definition 3.3). And, Corollary 4 of Conrad and Karanasos (2010) suggests that for N  =  2 at least one of the two GARCH spillover parameters can take a negative value, and this restriction is further relaxed for higher dimensions. 8 In the multivariate GARCH literature, BEKK is generally considered more flexible than DCC, while DCC is more parsimonious than BEKK. The framework proposed in this paper can readily be extended to BEKK. However, in this paper, I focus on deriving the SVIRF from the structural DCC model for the following reasons. First, in the terminology of Conrad and Karanasos (2010) and Conrad and Weber (2013), the unrestricted and extended DCC (UEDCC) specification of Equation (1.7) has generated a higher level of flexibility in modeling volatility transmissions than conventional diagonal DCC models. Second, SDCC allows for directly modeling common comovements among included variables, which reveals insightful implications in the empirical application of this paper, while BEKK is used indirectly to forecast conditional correlations (Caporin and McAleer, 2012, p. 737). Third, prior to deriving structural volatility impulse responses, the identification of BEKK in a structural system must be studied, which is not the focus of this paper. By contrast, the structural DCC model has already been studied and identified by Weber (2010a), which can directly be applied in this paper to derive structural volatility impulse responses. 9 vt is a martingale difference sequence, that is, E(vt) = 0 and cov(vt,vt−j)=0 for j ≥ 1. Nonetheless, {vt} in general is not an i.i.d. sequence. See, for example, Lin (1997) and Tsay (2010, §3.5, p. 132), among others. 10 This paper is in favor of the DCC specification of Tse and Tsui (2002), different from Weber (2010a) who uses Engle (2002). The following reasons are considered. In Engle (2002), Rt involves a transformation to ensure that its main diagonal elements have values of 1. This transformation makes the derivations of SVIRFs and their asymptotic distributions more complicated than necessary. In addition, the DCC model of Engle (2002) has an issue of inconsistency (Caporin and McAleer, 2012; Aielli, 2013). 11 Since the main diagonal elements in both Rt+k|t and Ψt take the value of 1, Πk=∂vecRt+k|t/∂ut2′ and ΠΨt=∂vecΨt/∂ut2′ have the value of 0 for the rows of N(i−1)+i for i = 1, … ,N. 12 Since the main diagonal elements in A is normalized to take the value of 1, ∂vecA/∂(vecΘ)′ has the value of zero for the rows of N(i−1)+i for i = 1, … ,N. 13 The S&P GSCI index data are available at https://markets.ft.com/data/indices/tearsheet/historical?s= GNX:IOM (Accessed 22 September 2017). 14 The S&P GSCI contains as many commodities as possible, with rules excluding certain commodities to maintain liquidity and investability in the underlying futures markets. The index currently comprises 24 commodities from all commodity sectors—energy products, industrial metals, agricultural products, livestock products, and precious metals. The S&P GSCI contains a much higher exposure to energy (the weight is 78.65%) than other commodity price indices such as the Dow Jones-UBS Commodity Index (the energy subgroup weight is 36.69%). The wide range of constituent commodities provides the S&P GSCI with a high level of diversification, across subsectors and within each subsector. 15 The Wald test rejects the null hypothesis of joint zero off-diagonal elements in A at all conventional confidence levels. 16 This high persistence feature of the volatility impulse responses might be caused by the high persistence of conditional variances as widely documented in finance literature for high frequency data when the values of b11  +  g11 and/or b22  +  g22 approach to one. In this paper, b^11+g^11=0.996 and b^22+g^22=0.983. 17 Particularly, I conduct the tests for structural breaks with unknown breakpoints based on Andrews (1993), Andrews and Ploberger (1994), and Bai and Perron (1998, 2003). The observations from Figure 5 and 6 suggest that the tests with one break during the recent financial crisis period can be considered. In other cases, multiple breaks can certainly be tested as well. An exogenous break date, such as August 26, 2008 or September 30, 2008 when the lowest or highest correlation is observed, is not automatically considered, as these dates do not necessarily imply the time when the structural break occurs in the correlation between the asset markets. In fact, any time point between these two dates is possibly the structural break date. To this end, I consider an endogenous break date tested with an unknown breakpoint. 18 Basically, the subsample impulse responses of the conditional variances to shocks to GSCI and S&P 500 are very similar to those obtained from the full sample period, except that they are more persistent for the before-break period but less persistent for the after-break period. 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Journal of Financial Econometrics – Oxford University Press

Published: Apr 1, 2018

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