3.1 Data source and a first look at the data

We obtain the daily excess returns of the broad (16 major individual) commodity futures market S&P GSCI indices between December 31, 1999 and June 14, 2021 from the Datastream database. The individual daily time-series of commodity futures market indices includes metals commodities of the S&P GSCI commodity futures indices (gold, silver, copper, precious metals, and industrial metals), agricultural commodities (agriculture, corn, cocoa, cotton, soybeans, sugar, and wheat), and energy commodities (energy, crude oil, petroleum, heating oil, natural gas, gas oil). Footnote ² The daily time-series VIX based on the S&P500 index options is our proxy for stock market volatility. The daily interest term spread is the difference between the 10-year constant U.S. government bond yield and the 3-month U.S. Treasury Bill rate. We also draw on monthly data of the three time-series and the U.S. industrial production index for the robustness analysis. In our estimation, we demean and standardize each variable to make each shock effect comparable in the analysis over time.

Figure 1 plots the point estimates of the residual standard deviations of VIX and the commodity futures market index in the model specified in Section 3.3 from January 10, 2000 to June 14, 2021. The figure indicates substantial variation in volatility. Following the Global Financial Crisis in 2008-09, there is a significant rise in the volatility of stock and commodity prices. The significant rise in the volatility of stock market prices is associated with a relatively moderate rise in the volatility of commodity prices during the COVID-19 pandemic in 2020. Figure 1 illustrates that the properties of the exogeneous shocks have changed over time in the commodity market. The change in the volatility implies that the transmission mechanism of stock market shocks on the commodity prices is likely to change over time. The understanding typically involves the investigation on the inter-relationships between the stock market volatility and commodity prices based on multivariate models where the transmission mechanism and the variances of the exogenous shocks can both potentially change over time (see the time-varying effects of monetary policy by Primiceri (Reference Primiceri2005) and Koop et al. (Reference Koop, Leon-Gonzalez and Strachan2009) and oil prices by Baumeister and Peersman (Reference Baumeister and Peersman2013) on the U.S. economy respectively).

3.2 Time-varying risk premium of commodity prices

The literature shows that commodity prices are affected by various factors including the demand for commodities, exchange rates, financialization of commodities markets, futures markets, expected inflation, investors’ risk aversion, supply of commodities, and interest rates (e.g., Gruber, Reference Gruber2011; Gruber and Vigfusson, Reference Gruber and Vigfusson2018). Pindyck (Reference Pindyck1993) and Gospodinov and Ng (Reference Gospodinov and Ng2013) emphasize that interest rates are related to the opportunity cost of buying and holding inventories. Decreases in interest rates lower the opportunity cost and attract investors to buy and hold commodities in search of yields. Gospodinov and Jamali (Reference Gospodinov and Jamali2018) show that monetary policy uncertainty is a key factor in determining the commodity risk premium. The monetary policy uncertainty is related to investors’ time-varying risk aversion in the commodity market, which differs from the economic policy/financial uncertainty in Bakas and Triantafyllou (Reference Bakas and Triantafyllou2018) and Kang et al. (Reference Kang, Ratti and Vespignani2020, Reference Kang, Ratti and Vespignani2021).

We relate stock market volatility and commodity prices through the risk aversion channel following Gospodinov and Ng (Reference Gospodinov and Ng2013) and Gospodinov and Jamali (Reference Gospodinov and Jamali2018) who discuss the time-varying risk premium that compensates for the risk of fluctuations in the commodity future spot prices in a non-arbitrage model based on the theory of normal backwardation. Define $S_{jt}$ and $F_{j,n}$ as the spot and futures prices for a contract delivery at time $(t+n),$ the futures basis ( $F_{j,n}-S_{jt})$ is then comprised of risk premium $(\Psi _{jt,n})$ and expected price change ( $E_{t}S_{jt+n}-S_{jt})$ in buying and holding the commodity,

(1) \begin{equation}F_{j,n}-S_{jt}=E_{t}S_{jt+n}-S_{jt}-\Psi _{jt,n},\end{equation}

that is the time-varying risk premium expressed as $\Psi _{jt,n}=E_{t}S_{jt+n}-F_{j,n}$ , the difference between the expected spot price at time $(t+n)$ and the futures prices of the commodity for delivery at time $(t+n)$ . A positive $\Psi _{jt,n}$ indicates that the futures price is backwardated at a discount. We utilize the daily excess returns of the broad commodity futures market S&P GSCI indices $(y_{t})$ as our proxy for the risk premium $(\Psi _{jt,n})$ and parameterize it in terms of observables such as,

(2) \begin{equation}\mathrm{y}_{t}={\unicode[Arial]{x03B1}} _{t}+{\unicode[Arial]{x03B2}} _{t}\mathrm{XIV}_{t}+{\unicode[Arial]{x03B8}} _{t}{\unicode[Arial]{x03C9}} _{t},\end{equation}

where the daily time-series $(\mathrm{XIV}_{t})$ based on S&P500 index options is decomposed into risk aversion $(s_{t}^{r})$ and expected stock market volatility $(s_{t}^{v})$ followed by Bekaert et al. (Reference Bekaert, Hoerova and Lo Duca2013), and ${\unicode[Arial]{x03C9}} _{t}$ the major determinants of commodity prices (interest rate $(\mathrm{i}_{t})$ using daily data and the additional variables including the U.S. industrial production index $(\mathrm{ip}_{t})$ and the U.S. consumer price index ( $({\unicode[Arial]{x03C0}} _{t})$ using monthly data), in search of the source of commodity prices and stock market volatility dynamics.

3.3 The time-varying structural-form VAR model

We relate the time-varying risk premium $(\Psi _{jt,n})$ to the multivariate model with time-varying parameters, which differs from current literature that normally parameterizes the time-varying risk aversion as models with constant parameters (e.g., Gospodinov and Ng (Reference Gospodinov and Ng2013), Gospoding and Jamali (Reference Gospodinov and Jamali2018), among many others), in the sense that the risk aversion varies over time as the market’s ability to bear risk is greater in good times but less so in bad times (Cochrane and Campbell, Reference Campbell and Cochrane1999; Cochrane, Reference Cochrane2017). Our multivariate model starts from the time-varying reduced-form VAR to the time-varying structural-form VAR specification. It follows Primiceri (Reference Primiceri2005) and Del Negro and Primiceri (Reference Del Negro and Primiceri2015) to begin with a state space model in which the sources of time variation include both the regression coefficients and the elements of variance-covariance matrix. The measurement equation in the state space system is

(3) \begin{equation}y_{t}=z_{t}\beta _{t}+\varepsilon _{t},\end{equation}

and the state equation is

(4) \begin{equation}\beta _{t+1}=\beta _{t}+k_{1t}\eta _{t},\end{equation}

where $y_{t}=(r_{t},s_{t},c_{t})$ is a $m\times 1(m=3)$ vector of endogenous variables including daily interest rate spread $r_{t}$ (i.e., the difference between the 10-year U.S. government bond yield and the 3-month U.S. T-Bill rate), daily stock market volatility $s_{t}$ , and the daily excess returns of the S&P GSCI indices on commodity futures prices $c_{t}$ .

We choose to place the interest rate spread $r_{t}$ first and financial variables $s_{t}$ and $c_{t}$ last in the VAR ordering selection by assuming that the shock to the financial market has no immediate effect on the monetary policy within a day. This is a standard assumption used in the literature by Bernanke et al. (Reference Bernanke, Boivin and Eliasz2005), Bekaert et al. (Reference Bekaert, Hoerova and Lo Duca2013) and Bakas and Triantafyllou (Reference Bakas and Triantafyllou2018) among many others, in the sense that the literature argues that monetary policy shocks measured by the innovations in the interest rates have a contemporary effect on all the fast-moving variables, the stock and commodity prices in our model, that are highly sensitive to the monetary policy news. With regards to the financial market block, we assume that the shock to the commodity prices in one day has no immediate effect on the broad stock market volatility accumulated over a long time (30 days for the VIX measure). These contemporaneous exclusion restrictions allow us to use standard Cholesky decomposition in the estimation of the variance-covariance matrix of the error term $\varepsilon _{t}$ to identify the structural-form VAR shocks, based on the estimation of the reduced-form VAR model (3).

In the state space model (3) - (4), $z_{t}=(c_{t},y_{t-1},\ldots y_{t-p})$ stands for a $m\times (p+1)$ matrix of data on explanatory variables where $p$ denotes the appropriate lag length of the endogenous variables,Footnote ³ $\beta _{t}=(\beta _{0,t},\beta _{1,t},\ldots, \beta _{p,t})^{\prime}$ the regression coefficients, and $\varepsilon _{t}\sim N(0,H_{t})$ the error term in the measurement Equation (1) and $\eta _{t}\sim N(0,Q)$ the error term in the state Equation (4). Define $H_{t}=A_{t}^{-1}\Sigma _{t}\Sigma^{\prime}_{t}(A_{t}^{-1})^{\prime}$ , where $\Sigma _{t}$ is a diagonal matrix, $A_{t}$ a lower triangular matrix with diagonal elements equal to one. Let the natural logarithm of the elements of the diagonal matrix $\Sigma _{t}$ be $\mathit{\ln } \sigma _{t}=(\mathit{\ln } \sigma _{1t},\ldots, \mathit{\ln } \sigma _{mt})^{\prime}$ allowed to evolve as follows:

(5) \begin{equation}\mathit{\ln } \sigma _{t+1}=\mathit{\ln } \sigma _{t}+k_{2t}\mu _{t},\end{equation}

where $\mu _{t}$ is $N(0,W)$ and independent over $t$ . We then stack the non-zero and non-one elements of the lower triangular matrix $A_{t}$ by rows into a $m(m-1)/2$ vector as $a_{t}=(a^{\prime}_{21,t},a^{\prime}_{31,t},a^{\prime}_{32,t},\cdots, a^{\prime}_{m(m-1),t})^{\prime}$ . The evolution of elements in $A_{t}$ is assumed to be a driftless random walk,

(6) \begin{equation}a_{t+1}=a_{t}+k_{3t}\varsigma _{t},\end{equation}

where $\varsigma _{t}$ is $N(0,S)$ and independent over $t$ . We assume that $\eta _{t}$ , $\mu _{t}$ and $\varsigma _{t}$ are independent over $t$ and independent of $\varepsilon _{t}$ , and $Q, W\text{ and }S$ are positive definite matrices.

Following Koop et al. (Reference Koop, Leon-Gonzalez and Strachan2009) in the state Equations (4), (5) and (6), the latent variables $k_{1t}, k_{2t}$ and $k_{3t}$ are assumed to be independent both contemporaneously as well as at all lags and leads and to take on the value of 1 if there is a change in the corresponding state errors and 0 otherwise. The probabilities of $k_{jt}, j=1,2,3,$ being zero or one are assigned with a Beta prior distribution, $p_{j}\sim Beta(1,1)$ and are subsequently updated from the data likelihood. The prior belief assumes the number of breaks through a non-informative choice at an even break with $E(p_{j})=0.5$ and $std(p_{j})=0.29$ , rather than the traditional hard restriction on the break points based on the economic theory. The mixture innovation specification is regarded as a special form of the Bayesian model stochastic search method for the model selection restrictions on the VAR models based on the data providing us with information on changes of parameters in the model in George and McCulloch (Reference George and McCulloch1997), George et al. (Reference George, Sun and Ni2008), Jochmann et al. (Reference Jochmann, Koop and Strachan2010), and Jochmann et al. (Reference Jochmann, Koop, Leon-Gonzalez and Strachan2013).

We follow Koop et al. (Reference Koop, Leon-Gonzalez and Strachan2009) and Kang et al. (Reference Kang, Ratti and Yoon2015a) to use similar prior distributions of unknown parameters and take the training sample of the first six years of the data to calibrate the key priors of hyperparameters at time $t=0\,:\,\beta _{0}\sim N(\hat{\beta }_{0},m(p+1)\times \hat{V}_{\beta })$ , $\ln (\sigma _{0})\sim N(\ln (\hat{\sigma }_{0}),I_{m})$ , and $a_{0}\sim N(\hat{a}_{0},m(m-1)\times \hat{V}_{a})$ . We utilize Wishart distribution priors for the variance-covariance matrices: $Q^{-1}\sim W(v_{Q},V_{Q}^{-1}),$ where $v_{Q}=m(p+1)+1$ and $V_{Q}=0.05\times m(p+1)\times I_{m(p+1)}$ , $W^{-1}\sim W(v_{w},V_{w}^{-1})$ , where $v_{w}=m+1$ and $V_{Q}=0.0001\times m\times I_{m}$ , and $S^{-1}\sim W(v_{s},V_{s}^{-1}),$ where $v_{s}=m(m-1)+1$ and $V_{s}=0.01\times m(m-1)\times I_{m(m-1)}$ . The calibration of $\hat{\beta }_{0}$ and $\hat{V}_{\beta }$ is obtained from the conditional Maximum Likelihood Estimates of the time-invariant version of Equation (3). The specification of $\hat{\sigma }_{0}$ , $\hat{a}_{0}$ and $\hat{V}_{a}$ is drawn from the decomposition of time-invariant error variance-covariance matrix $H=A^{-1} \Sigma \Sigma^{\prime}(A^{-1})^{\prime}$ .

We then run the Markov chain Monte Carlo algorithm executed 12,000 times with the first 10,000 draws discarded as burn-in iterates, based on the joint posterior density, $p(\beta ^{T},\sigma ^{T},a^{T},Q,W,S|y^{T})$ , obtained from the combination of the prior distribution and the likelihood function of the sample. Finally, the resulting estimates are used to construct the structural-form VAR representation of the model, $y_{t}=z_{t}\beta _{t}+A_{t}^{-1}\Sigma _{t}u_{t}$ , where $\varepsilon _{t}=A_{t}^{-1}\Sigma _{t}u_{t}$ .

4.1 Evidence on the time-varying parameters

Our mixture innovation specification in the state space system allows us to obtain the probabilities of a break at each point in time that presents evidence on how the parameters of model have changed over time. Table 1 shows the average probabilities of the break of the three parameters $\beta _{t}$ , $\mathit{\log } \sigma _{t}$ and $a_{t}$ for the broad/individual commodity price over time. The transition probabilities of the three parameter posteriors are $E(p_{1}|data)=0.817$ , $E(p_{2}|data)=0.997$ , and $E(p_{3}|data)=0.129$ for the regression coefficients $\beta _{t}$ , the elements of the diagonal error variance $\delta _{t}$ , and the non-zero and non-one elements of the off-diagonal error covariance $a_{t}$ , respectively, for the broad commodity price. It indicates the gradual evolution of parameters in the model, even though we use their priors that are non-informative, rather than the abrupt breaks of conventional structural break models argued (see Primiceri (Reference Primiceri2005) for monetary policy and Baumeister and Peersman (Reference Baumeister and Peersman2013) for oil prices). The evidence of parameter change is greatest for regression coefficients $\beta _{t}$ and error variances $\sigma _{t}$ and is appreciable for $a_{t}$ the covariance of commodity prices and stock market volatility (expected a break to occur about once per week). The results confirm that exogenous shocks generated particularly by the variance in the error variance-covariance matrix change over time and are of predominant importance in the commodity prices. The evolution of the regression coefficients and the covariance elements in the variance-covariance matrix is appreciable enough that should not be neglected, in the sense that the transmission mechanism is not merely the properties of the exogenous shocks that have changed in the commodity market over time.

Table 1. Transition probability of regression coefficients and error variance and covariance elements

Notes: The posteriors for the transition probabilities E(P₁|Data) for the regression coefficients β_t, E(P₂|Data) the elements of the diagonal error variance δ_t, and E(P₃|Data) the non-zero and non-one elements of the off-diagonal error covariance α_t respectively.

The results are very similar to those using the sectoral/individual commodity prices in Table 1. Note that the evidence of changes in the regression coefficients $\beta _{t}$ is relatively larger for the prices of petroleum, heating oil, gas oil, and crude oil that have relatively higher trading activities with the stock market. The result is in line with the findings in the literature showing the changes in the transmission mechanism by investigating the historical decomposition of oil and stock dynamics over time using the standard VAR model with time-invariant parameters in Kilian and Park (Reference Kilian and Park2009) for example. Our methodology allows the distinction by selecting restrictions for VAR models using the data information on changes of parameters in the model. In contrast, changes in the posterior probabilities of regression coefficients $\beta _{t}$ are relatively smaller for the prices of copper, gold, silver, and the previous metals. It implies that the changes in the transmission mechanism generated particularly by stock volatility on the precious metal prices are relatively smaller over time, in the fact that investing in precious metals is conventionally regarded as a risk management tool in the time of stock market turbulence. The evidence of changes in the covariance $a_{t}$ is appreciable in the agriculture sector, this being an expected break to occur about once per trading week in the contemporaneous co-movement between stock market volatility and cocoa, cotton, soybeans, sugar, and corn prices, respectively. The evidence of parameter change is consistently large, greater than 0.97 across industries for the error variances $\sigma _{t}$ , confirming that exogenous shocks generated by the variance are significantly changed over time and are of predominant importance in the fluctuation of commodity prices.

The substantial variation in the heteroskedastic VAR specification is consistent with major changes in the shocks in stock and commodity market over time (see, for example, Kang et al. Reference Kang, Ratti and Yoon2015a). It confirms that the transmission mechanism (shown via regression coefficients in particular) is not constant over time and the way the exogenous shocks (shown via variances in particular) are generated changes over time in the dynamics of stock market volatility and commodity prices.

4.2 Impulse response functions of stock market volatility shocks on commodity prices

In this subsection, we investigate whether the changes of the parameters in the model have important implications (i.e., the transmission mechanism) for the effect of stock market volatility on the commodity price through the analysis of impulse response functions at each point of time. We calculate the effect of a shock of size one to the structure errors, $\varepsilon _{t}=A_{t}^{-1}\Sigma _{t}u_{t}$ , in Equation (1). All the endogenous variables are demeaned and standardized to make each shock effect comparable in the analysis over time.

Figure 2 shows that the posterior median of the impulse response functions of the broad commodity prices to stock market shocks along with the 16^th and 84^th percentiles of the posterior median. Specifically, the impulse responses of the commodity prices to one point increase in stock prices are negative but small before 2005, fluctuate significantly thereafter over the period of 2005–2007, and then plummet to the nadir during the 2008–2009 Global Financial Crisis (GFC). The negative effects are persistently large for more than two years after the GFC, close to zeros between 2013–2017, built up in the 2018 Trump Tariff, and then increased dramatically since the start of COVID-19 pandemic in 2020. The evolution of the impulse response functions presents that the transmission mechanism has changed in the commodity market; that is the way exogeneous stock market shocks affect the commodity prices varies significantly over past two decades.

Figure 2. Responses of Commodity Price to Stock Volatility Shock.

Notes: The figure shows the posterior median, 16-th and 84-th percentiles of the impulse responses of commodity prices to stock market volatility shocks from January 10, 2000 to June 14, 2021.

Overall, the values of the posterior medians present that stock market shocks have a systematically negative effect on commodity prices over time. Note that each credible interval contains zero individually does not necessarily imply that there is no interesting difference in the impulse response functions jointly over time (Koop et al. Reference Koop, Leon-Gonzalez and Strachan2009). The evolution of impulse response functions is negative on average, with the most significantly negative effects being in October/November 2008 during the GFC. The negative effect is nonsignificant in March 2020 during the beginning of the COVID-19 shock. It indicates that the effect of stock market shocks associated with financial uncertainty is relatively more significant on the commodity prices than that associated with nonfinancial uncertainty. The impact is stronger in energy commodities such as crude oil, gas oil, heating oil, and petroleum compared to the agricultural and precious metals markets.Footnote ⁴ Similar results are also seen in Bakas and Triantafyllou (Reference Bakas and Triantafyllou2018) and Kang et al. (Reference Kang, Ratti and Vespignani2021)when investigating the effect of financial/nonfinancial uncertainty on commodity prices using the monthly sample data before the COVID-19 pandemics in 2020. The hedging strategy of longing agricultural and precious metals commodities and shorting energy commodities is beneficial, for example, for a stock market shock associated with financial uncertainty, vice versa on a stock market shock associated with nonfinancial uncertainty. This hedging strategy is conventionally regarded as a risk management tool in the time of energy market turbulence (e.g., Kang et al. Reference Kang, Ratti and Vespignani2021).

The evolution of the median of the cumulative responses of commodity prices in 3-day (short-term responses), 12-day and 24-day (long-term responses) are shown between January 2000 and June 2021 in Figure 3. First, the correlation between short-term and long-term responses to stock market volatility shocks are high, in that their upward/downward sloping trends are similar at each point in time. Second, the response of commodity prices to stock market shocks is often gradual, in the sense that it takes time for the commodity price response to reach the long-term response level following a stock market volatility shock. It is often the case that the commodity price responses reach approximately the same level after 24 days, in the sense that commodity investors have time to restructure their portfolios to hedge the risk associated with the stock market shocks.

Figure 3. Responses of Commodity Prices to Stock Volatility in 3, 12, and 24 Days.

Notes: The figure shows the posterior median of the impulse responses of commodity prices to stock market volatility shocks in 3-, 12--, and 24-days from January 10, 2000 to June 14, 2021.

6.1 Evidence on the time-varying parameters using monthly data

We draw on the monthly data of the three time-series of interest rate spread $(r_{t})$ , VIX index $(VIX_{t})$ , and the commodity futures market S&P GSCI index $(c_{t})$ as in the daily data analysis and the monthly U.S. industrial production index $(ip_{t})$ and the monthly U.S. consumer price index $(\pi _{t})$ between December in 1999 and June in 2021 for the robustness analysis based on our state space model using five variables, $y_{t}=(ip_{t},\pi _{t},r_{t},s_{t},c_{t})$ . We place the U.S. industrial production in first place in the order in the model by assuming that the industrial production within a month does not respond contemporaneously to innovations in the inflation, interest rate, stock market volatility or commodity prices, given the sluggishness of real aggregate economic activity. We also assume that the U.S. aggregate price level responds to innovations in the interest rate, stock market volatility, and commodity prices with, at least, a one-month delay. The posterior mean of transition probability is $E(p_{1}|data)=0.584$ , $E(p_{2}|data)=0.948$ , and $E(p_{3}|data)=0.527$ for the regression coefficients $\beta _{t}$ , the elements of the diagonal error variance $\delta _{t}$ , and the non-zero and non-one elements of the off-diagonal error covariance $a_{t}$ , respectively, for the monthly broad commodity price. Figure 6 shows the posterior median of commodity price responses to stock market volatility shocks in 18 months in November 2008 and in March 2020. The result confirms that there is a high probability of changes in all parameters in the model at each point of time, while shocks to stock market volatility may cause long-lasting negative effects on the commodity price.

Figure 6. Posterior median of commodity price responses to stock in 2008.11 and 2020.3.

Notes: The figure shows the posterior median of the impulse responses of commodity prices to stock market volatility shocks in 18 months in November 2008 and in March 2020.

We then compare the results using alternative measures of economic uncertainty related to the VIX index $(VIX_{t})$ , including economic policy uncertainty proposed by Baker et al. (Reference Baker, Bloom and Davis2016) and unobserved macroeconomic uncertainty based on Jurado et al., (2015), which have been studied thoroughly in the recent literature. While the similar result shows the high probability of changes in the parameters in the model using alternative measures of economic uncertainty, Figure 7 shows that the negative effects of economic policy uncertainty shocks on the commodity prices are short-lived followed by a strong recovery in 6 months. In contrast, Figure 8 illustrates that shocks to unobserved macroeconomic uncertainty may cause long-lasting negative effects on the commodity price, which are stronger in November 2008. Our results are in line with the findings in Bakas and Triantafyllou (Reference Bakas and Triantafyllou2018) who show that latent uncertainty shocks have the most significant impact on the commodity prices when compared to observable economic policy uncertainty.

Figure 7. Posterior median of commodity price responses to economic policy uncertainty shocks in 2008.11 and 2020.3.

Notes: The figure shows the posterior median of the impulse responses of commodity prices to the economic policy uncertainty shocks in 18 months in November 2008 and in March 2020.

Figure 8. Posterior median of commodity price responses to unobserved macroeconomic uncertainty shocks in 2008.11 and 2020.3.

Notes: The figure shows the posterior median of the impulse responses of commodity prices to the unobserved macroeconomic shocks in 18 months in November 2008 and in March 2020.

6.2 International evidence

In the second robustness check, we draw on the daily major world stock market volatility indices including the Euro Stoxx 50 volatility index in Europe (VSTOXX) and the Japanese volatility index in Asian (VXJ) between December 31, 1999 and June 14, 2021. We run the time-varying state space system (3) - (6) by replacing the U.S. stock market volatility index (VIX) with the major world stock market volatility index respectively. In Europe, the posterior mean of transition probability is $E(p_{1}|data)=0.772$ , $E(p_{2}|data)=0.996$ , and $E(p_{3}|data)=0.131$ for the regression coefficients $\beta _{t}$ , the elements of the diagonal error variance $\delta _{t}$ , and the non-zero and non-one elements of the off-diagonal error covariance $a_{t}$ , respectively, for the daily broad commodity price. In Asia, the posterior mean of transition probability is $E(p_{1}|data)=0.774$ , $E(p_{2}|data)=0.995$ , and $E(p_{3}|data)=0.145$ for $\beta _{t}$ , $\delta _{t}$ , and $a_{t}$ respectively. These results are very similar to those obtained using the U.S. VIX index, confirming that the transmission mechanism is not constant over time and the way the exogenous shocks are generated changes over time in the dynamics of stock market volatility and commodity prices.