1 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: Elucidating the Relationship among Volatility Index, US Dollar Index and Oil Price John Wei-Shan Hu* and Hsin-Yi Chang** Financial tsunamis have caused enormous disruption globally during the past decade. Particularly, the subprime mortgage loan crisis during 27 to 29 and the European debt crisis that began from 2 to 2, both significantly increased the volatility index (VIX). This study examines the cross causality and long-term equilibrium relationships among the VIX, US Dollar Index (USDX) and oil price from 24 to 2. This investigation also divides the full sample period into three sub-periods: 24 to 26, 27 to 29, and 2 to 2. Empirical results indicate that long-term equilibrium relationship among VIX, USDX and oil price exists for both the full sample period and the first sub-period. Hence, the VECM model was employed for the three parameters for full sample period and the first sub-period; while VAR model, impulse response function and forecast error variance decomposition are required for VIX, USDX and oil price the three parameters for the second and third sub-periods. This study finds that, VIX is significantly and negatively related with USDX and oil price throughout the full sample period. Meanwhile, only VIX and oil price are significantly and negatively related during the first sub-period, while oil price and USDX have a significantly and mutually causal relationship. This investigation finds that VIX affects USDX and oil price during the second sub-period. However, VIX only affects USDX during the third sub-period, suggesting it can serve as a reference instrument for investors engaged in arbitraging or hedging. This study also finds that each variable is most strongly affected by itself, and the forecast error variance decomposition of oil price arises from all three variables. Empirical results suggest spill-over effects from VIX and oil price, while co-movement occurs in response to large change in USDX. Empirical findings also show that all three parameters converge rapidly soon after major changes occur in each of them, indicating market efficiency. This study concludes that VIX, USDX and oil price are sensitive to increasing market uncertainty. JEL Codes: G, G4 *Dr. John Wei-Shan Hu, Dep t of Business Administration and Finance, Chung Yuan Christian University, 2 Zhong Bei Rd., Zhongli District, Tao Yuen City, Taiwan 3223, **Ms. Hsin Yi Chang, MBA graduate and Statistics Assistant, Dep t of Business Administration, Chung Yuan Christian University, 2 Zhong Bei Rd., Zhongli District, Tao Yuen City, Taiwan 3223.,
2 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: Introduction For the past decade, financial tsunamis occurred in succession (namely, the subprime mortgage loan crisis, bankruptcy of Lehman Brothers, and the European debt crisis), causing great unease globally. Consequently, investors are anxiously seeking hedging or arbitraging instrument. The volatility index (VIX) measures implied volatility for the U.S. market and was provided by the Chicago Board of Options Exchange (CBOE) in 993. VIX is a key measure of market expectations of near-term volatility conveyed by the S&P5 index option prices. Generally, VIX exceeding 4 suggests investors unease regarding future stock index trends; meanwhile, VIX smaller than 5 implies stable stock index trends. Additionally, international oil prices are important influence on global markets. Since crude oil is a traded commodity, foreign exchange (FX) rate is an essential reference for crude oil trading nations. Zhang et al.(28) argued that, in the long-term, the US exchange rate is a key influence on the price of crude oil. Since most previous studies examined the relationship between oil price (OIL) and FX rates or the correlation between VIX and stock index returns, this study follows a different direction and attempts to examine the long-term equilibrium relationship among the VIX, OIL and the US dollar index (USDX) to provide a reference for the hedging or arbitrating strategy of investors, and to examine the causal relationship among VIX, USDX and OIL. 2. Literature Review This study classifies previous literature into two categories: () Articles on the relationship between VIX and various stock indexes or commodities; (2) Studies on the relationship between OIL and FX rate. The following articles dealt with the correlation between VIX and various stock indexes or commodities: Copeland and Copeland (999), Whaley (2), Giot (25), Qadan and Cohen (2), Sari et al. (2) and Williams (2) all found that the VIX is significantly and negatively related to stock indexes or various commodities. Furthermore, the following studies dealed with the relationship between OIL and FX rates: Sadorsky (2), Yousefi and Wirjanto (23), Wirjanto and Yousefi (25), Zhang et al. (28), Lizardo and Mollick (2), Wang et al. (2) and Beckmann and Czudaj (22) all found a significant relationship between OIL and FX rates.
3 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: The Methodology and Model This study examines the cross causality and long-term equilibrium relationship among VIX, USDX, and OIL. This investigation uses the prices of West Texas Intermediate (WTI) Crude Oil to represent OIL because WTI is used as a benchmark in oil pricing. USDX measures the value of the United States dollar relative to a basket of foreign currencies. USDX was launched in March, 973, soon after the dismantling of the Bretton Woods system. The sample period runs from January, 24 to December 3, 2. There are 2,87 daily data available after deleting days on which data for any of the three variables was missing. Figure (a) shows that the VIX increased markedly from the end of 28 because of the shock of the financial tsunami, and the VIX peaked at 8.86 on November 2, 28. The OIL also increased rapidly from 24 to 28, and peaked at US$42.8 per barrel in early 28. However, the USDX declined from a peak of in 25 to a low of 7.2 in 28, and then fluctuated significantly. Figure. The trend of VIX, USD and Oil Price from Jan., 2 to Dec.3, First subperiod (a) VIX Second sub-period Third subperiod (b) USDX (c) OIL This study further divides the full sample period into three sub-periods to examine the reaction of a major shock resulting from the VIX. The first sub-period is defined as the VIX stable period, and runs from January, 24 to December 3, 26, and it contains 782 daily data. The second sub-period is labeled the VIX rising period, and runs from January, 27 to December 3, 29, coinciding with the U.S. subprime mortgage loan crisis causing the bankruptcies of Lehman Brothers and multiple investment banks and insurance companies. The second sub-period includes 784 daily data. The third sub-period runs from January, 2 to December 3, 2, when the VIX rapidly increased again because of the European debt crisis. The third sub-period contains 52 daily data. The data sources were CBOE and the Thomas Datastream. Before implementing the co-integration test, this study examines the stationarity of the time series variables and the integrated order of variance using the Augmented Dickey-Fuller (ADF) test. The ADF test model is as follows: Y t = α + β t + γy t + δ Y t + + δ p Y t p+ + ε t, ()
4 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: where Y t denotes the first-order difference of the Y logarithmic series; α is a constant, β is the coefficient for a time trend t, and p is the lag order of the autoregressive process that makes the residual white-noises. By including lags of the order p, the ADF formulation permits a high-order autoregressive process. Notably, ε t represents white-noise in the null hypothesis H : γ =. Failure to reject the null hypothesis implies a unit root in the event of a regime shift such as an oil shock. This study then employs the maximum likelihood estimation (MLE) proposed by Johansen (988) to examine whether co-integration (long-term equilibrium relationship) exists among variables, and to determine the number of co-integration vector groups: q t = t + ε t, (2) Equation (2) is then rewritten using the first-order difference function as q t = t + t q + t, (3) q where = ; =, and I is an identity matrix. Equation (3) denotes a VAR model with first-order difference plus an error correction item t q, where represents the short-term dynamic information, and the matrix reflects the long-term relevant information. According to Persaran and Shin (988), the co-integration relationship must be included in the VAR model (i.e., the vector error correction model, VECM) to estimate the dynamic price relationship among variables. Based on the co-integration Eqn. (3), a VECM examines the short-term adjustment as follows: q X t = X t + X t q + ε t, (4) where X t q is an error correction term; = αβ, which is the coefficient matrix of the adjustment speed of error correction. If α differs significantly from zero, then the variable which deviates from equilibrium in the short-term will immediately move toward long-term equilibrium. Furthermore, if α >, the parameter was under-estimated in the short-term and will be upward corrected in t+; otherwise, it will be downward corrected in t+. Although there are two main methods (i.e., the trace and maximum eigenvalue tests) to examine variable co-integration, this study uses the maximum eigenvalue test because Lûtkepohl et. al (2) found the powers of the corresponding trace and maximum eigenvalue tests to be very similar. The null hypothesis of the maximum eigenvalue test is as follows: H : There are r groups of co-integration vectors; H : There are r + groups of co-integration vectors. The maximum eigenvalue test statistic is λ max r, r + = T ln( λ r+ ), (5)
5 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: where λ max represents the statistic value of the Johansen maximum eigenvalue. This study then employs the Vector Autoregressive (VAR) model to capture the linear interdependence among multiple time series. This investigations uses two time series variables, =,, to illustrate the simplified VAR model, as follows: p p t = a o + α t + β t + ε xt, (6) p p t = b o + γ t + δ t + ε yt, (7) where a o and b o are intercepts, α & β are the coefficients of the lags of t and t of Eqn. (6); γ & δ are the coefficients of the lags of t and t of Eqn. (7); and ε xt and ε yt are error terms. Generally, optimal lag period selection of the VAR model is important; this study uses the Schwartz Bayesian Information Criterion (SBC) rule which has the lowest SBC value: SBC = T ln σ ε 2 + P ln T, where P is the number of parameters to be estimated; T is the number of observation; σ ε 2 is the error variance. The Granger causality test is then used to determine whether a time series Y is caused by X, in which the forecasts are linear and based on the information in series t and t. If no long-term equilibrium (co-integration) relationship exists between either VIX and OIL or USDX, a study on short-term interactions is necessary. This study applies the Granger causality test based on the VAR model in equations (6) and (7). The impulse response function (IRF) describes the reaction of the dynamic system in response to external change involving an endogenous variable as a function of time. After obtaining the VAR() model, the IRF is X t = π o + π X t + ε t. (9) Equation (9) can be transformed to obtain the following moving average form: X t = μ + ε t + Ψ ε t + Ψ ε t 2 + Ψ ε t ε t. () This investigation then partially differentiates Eqn. () using the lag error term to derive X t ε t = Ψ, () Specifically, matrix Ψ includes four impulse responses: Ψ = [ Ψ Ψ 2 Ψ 2 Ψ 22 ] = [ X t ε xt Y t ε xt X t ε yt Y t ], (2) ε yt where Ψ is the extent of the impact of external shock of one lag term of X t ( ε xt on X t ; Ψ 2 is that of external shock of one lag term of X t on Y t ; Ψ 2 is that of (8)
6 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: external shock of one lag term of Y t on X t ; and Ψ 22 lag term of Y t ( q yt on Y t. is that of external shock of one Equation (2) shows that other variables can be included in IRF to obtain cross-impacts. Furthermore, the influence of the speed adjustment can be assessed when one standard error shock of a parameter on the reaction of itself and other variables at t+. Restated, the sign of Ψ can indicate the reaction direction. According to Sari et al., (2), the generalized forecast - error variance decomposition (FEVD) demonstrates the extent to which the variance of a particular variable can be explained by a shock to itself and another variable. First, this study derives the expectation value of X t as follows: Ε X t = Φ + Φ + Φ Φ n + Φ n X t. (3) The forecast error of the n-th term is X t+n Ε X t+n = ε t + Φ ε t + + Φ n ε t n, (4) where Ε X t+n represents the possible forecast error of the n-th term when forecasting the t+n-th term. The variance matrix of the n-th term forecast error can be observed as Ω f = var ε t + Φ Φ 2n, where Ω f = Ω x Ω y. The forecast error variance of all the terms before the n-th term can be expressed using the linear function combination of σ y 2 +σ x 2 as follows: Ω x = Ω β σ x 2 + Ω α σ y 2. Equation (6) can be rewritten as Ω β σ x 2 Ω x + Ω 2 ασ y = %. Ω x Equation (7) demonstrates that the variance of each variable can be expressed as the sum of all the variances, and that can be used to assess the degree to which explanatory power of a specific variable contributes to itself and to other variables. (5) (6) (7) 4. The Findings This investigation applies the ADF test for the full sample period and three sub-periods. Table indicate that most original data are non-stationary. After applying the first-order difference all the data become stationary.
7 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: Variables Original Table : ADF test result: The full sample period and three sub-periods Full Sample Period st sub-period 2 nd sub-period 3 rd sub-period p-value p-value p-value p-value p-value p-value p-value p-value First-order Difference Original First-order Difference Original First-order Difference Original First-order Difference LVIX. **. **. **. **.24. **.6 *. ** LUSD.24. **.24. **.45. **.54. ** LOIL.22. **.22. **.57. **.2. ** Note: *denotes %; **represents 5% significance levels After employing the maximum eigenvalue test of Johansen (988) to examine the existence of the long-term equilibrium relationship for all variables, this study finds that, during the full sample period and the first sub-period, the VIX, USDX and OIL have at least one co-integration relationship (Table 2). A VECM test is then performed and Table 3 summarizes the empirical result. However, Table 2 also shows that, during the second and third sub-periods, the VIX, USDX and OIL have no co-integration relationship for the second and third sub-periods. This study thus uses the VAR to analyze all three variables and examine the impulse response and variance decomposition for the second and third sub-periods. Table 2. Co-integration test: The full sample period and three sub-periods Full Sample Period st sub-period 2 nd sub-period 3 rd sub-period Eigen Max E.V. Eigen Max E.V. Eigen Max E.V. Eigen Max E.V. -value stat. -value stat. -value stat. -value stat. None ** ** At most At most Note: **denotes 5% significance level Regarding the adjustment speed of error correction, Table 3 shows that the error correction significantly and negatively affects VIX and OIL during the full sample period. However, the adjustment speed coefficients of the VIX and OIL are minimal, suggesting that it takes extended adjustment for error correction to bring the VIX and OIL to equilibrium. Meanwhile, Table 3 demonstrates that the error correction significantly and negatively affects VIX during the first sub-period. However, the absolute value of adjustment speed of the VIX is greater than that of the full sample period, implying that it will take less time during the first sub-period than the full sample period to return to equilibrium.
8 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: Table 3: The Adjustment Speed for Error Correction (a) Full Sample Period (b) The st sub-period VIX USDX OIL VIX USDX OIL Adjustment speed t value -3.33*** *** *** Table 4 shows that OIL affects the USDX, while VIX affects OIL for the full sample period and the second sub-period. Furthermore, USDX affects OIL for the first and second sub-periods. This study also finds that VIX affects USDX for all three sub-periods, but does not influence USDX for the full sample period. The empirical findings demonstrate that neither USDX nor OIL influence VIX except for the third sub-period. Table 4. Causality among VIX USDX and OIL: Full Sample Period and three sub-periods Full Sample Period st sub-period 2 nd sub-period 3 rd sub-period F p- F p- F p- F p- Statistics value Statistics value Statistics value Statistics value USDX does not cause OIL ** **.5.7 OIL does not cause USDX ** ** **..99 VIX does not cause OIL ** **.9.76 OIL does not cause VIX * VIX does not cause USDX * ** 5.6. ** USDX does not cause VIX * Notes: * denotes %; **shows 5%; ***represents % significance levels. Since VIX, USDX and OIL do not have co-integration relationship for the second and third sub-periods, the impulse response function and forecast error variance decomposition tests are then examined for the second and third sub-periods. Figures 2(a) and 3(a) show that % influence on VIX resulting from VIX itself in the first term, then changes from being positive to negative from the second term, and thereafter gradually converges. This investigation also finds some influences on VIX arising from OIL and USDX from the second term. Figures 2 (b) and 3(b) denote that a shock of USDX affecting the reaction of USDX itself is the most significant among all three variables in the first term. However, a shock of USDX arising from VIX is also significant in the first term. Figures 2(c) and 3(c) indicate that a shock of OIL on the effect of itself is the most significant among all three variables in the first term, then decreases in the second term. However, the shocks to OIL arising from USDX and VIX are also significant during the first term. Additionally, Figures 2 and 3 show that all three parameters rapidly converge shortly after their dramatic change for the second and third sub-periods.
9 6 Proceedings of 47th Annual American Business 4 Research Conference July 25, Sheraton LaGuardia East 2 Hotel, New York, USA, 4 ISBN: Figure 2. Impulse Response of -2 Three Parameters to Cholesky -2 std. dev. innovations - The second sub-period Response of DLVIX to Cholesky Figure 3. Impulse Response of Three parameters to Cholesky std. dev. innovation. 6 Response of DLUSDX to Cholesky - The third sub-period 3 Response of DLOIL to Cholesky Response of DLUSDX to Cholesky Response of DLOIL to Cholesky Response of DLVIX to Cholesky Response of DLUSDX to Cholesky 2 One (a) S.D. DLVIX Innovations One (b) S.D. DLUSDX Innovations (c) DLOIL (a) DLVIX (b) DLUSDX (c) DLOIL 2 Response of DLVIX to Cholesky Response of DLOIL to Cholesky DLVIX One S.D. DLUSDX Innovations DLOIL Response of DLUSDX to Cholesky Response of DLOIL to Cholesky The Response forecast of DLUSDX error variance to Cholesky decomposition (FEVD) analyzes the influence of each - structure 5 shock on the endogenous variables, including the parameter itself and other 4 variables. Table 5 lists the explanatory power on the shocks to each variable 3 for the second and third sub-periods as follows: 2 () Regarding the FEVD of VIX, 5 the only explanatory power on the shocks to VIX in the first term arises from VIX itself (%). Although the explanatory power arising from VIX decreases slightly from the second term, it converges from the third term. (2) Regarding the FEVD of USDX, the most significantly explanatory power on the shocks to USDX at the first term arises from USDX itself and decreases Response of DLOIL to Cholesky from the second term to the third term, then converges from the fourth term. The second significant explanatory power at the first term arises from VIX and increases from the second to the third term, and then converges from the fourth term. (3) Regarding the FEVD of OIL, for both the second and third sub-periods, the Response of DLOIL to Cholesky most significantly explanatory power on the shocks of OIL in the first term arises from the OIL itself, which then decreases and converges from the fourth term. However, for the second sub-period, the second most significant influence
10 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: on OIL shocks is USDX, and converges from the third term; while for the third sub-period, the second significantly explanatory power at the first term arises from VIX and converges from the third term. Table 5. Variance decomposition:the second and third sub-periods 2 nd Sub-period 3 rd Sub-period Parameter Terms Terms Terms Terms DLVIX DLUSDX DLOIL Summary and Conclusions The conclusions are summarized as follows:. The co-integration test indicates that VIX, USDX and oil price have long-term equilibrium relationships for both the full sample period and the first sub-period. 2. The Granger causality test shows that, during the full sample period, oil price affects USDX and VIX affects oil price. However, VIX and USDX have no causal relationship. During the first sub-period, USDX and oil price exhibit mutual causality. During the second sub-period, USDX affects oil price; while VIX affects USDX and oil price. 3. The impulse response function and the FEVD demonstrate that each variable is most significantly affected by itself. All variables rapidly converge after being shocked, suggesting an efficient market. Additionally, this study finds that, for both the second and third sub-periods, VIX has high independence. The FEVD of USDX arises from itself and VIX. However, that of oil price arises from all three variables. 4. The error correction result shows that VIX, USDX and oil price are significantly and negatively related throughout the full sample period, suggesting that the oil price rises as the US dollar devalues; demand for oil increases, with decreasing VIX, decreasing oil price. However, VIX and oil price are significantly and negatively related during the first sub-period, implying that the prospect for the global economy are promising, increasing demand for oil, and thus increasing oil price. This study also finds that, it takes a longer time to achieve equilibrium in error correction for the full sample period than that for the first sub-period. This investigation concludes that VIX, USDX and oil price are sensitive to increasing
11 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: market uncertainty. Investors may observe the change in VIX to determine whether they should invest or hedge USDX or oil price. End Notes The authors would like to thank Ted Knoy for his excellent editorial assistance, and the precious remarks on the first draft provided by Chungfang Ho Chang and Catherina Y. F. Ku at the 88th WEA Conference at Seattle, Washington. Reference Beckmann, J and Robert C (22). Oil Price and US Dollar Exchange Rate Dynamics. Working paper, April pp-2. Copeland, MM, and Copeland, EC 999. Market Timing: Style and Size Rotation Using the VIX. Financial Analysts Journal, Vol.55, No.2, pp Giot, P 25. Relationships Between Implied Volatility Indexes and Stock Index Returns. The Journal of Portfolio Management, Vol. 3, pp.92-. Johansen, S 988, Statistical Analysis of Co-integration Vectors. Journal of Economics and Dynamics and Control, Vol. 2, Nos. 2-3, pp Lûtkepohl, H, Saikkonen, P and Trenkler, C 2, Maximum Eigenvalue versus Trace Tests for the Co-integrating Rank of a VAR Process. The Econometric Journal, Vol. 4, No. 2, December, pp Lizardo, RA. and Mollick, AV 2, Oil Price Fluctuations and U.S. Dollar Exchange Rates, Energy Economics, Vol. 32, No.2, March pp Pesaran, HH. and Shin, Y 993, Generalized Impulse Response Analysis in Linear Multivariate Models, Economic Letters, Vol. 58, pp Phillips, PC. 998, Impulse Response and Forecast Error Variance and Asymptotics in nonstationarity VARs, Journal of Econometrics, Vol. 83, pp.2-56 Qadan, M & Cohen G 2, Is It Profitable to Invest According to the VIX Fear Index? Journal of Modern Accounting and Auditing, Vol.7, pp Sadorsky, P 2, The Empirical Relationship between Energy Futures Prices and Exchange Rates, Energy Economics, Vol. 22, No. 2, pp Sari, R, U, Soytas U and Hacigasanoglu E 2, Do Global Risk Perceptions Influence World Oil Prices? Energy Economics, Vol. 33, pp Wang, ML, Wang CP and Huang TY 2, Relationships among Oil Price, Gold Price, Exchange Rate and International Stock Markets, International Research Journal of Finance and Economics, Vol. 47, pp Whaley, RE. 2, The Investor Fear Gauge, The Journal of Portfolio Management, Vol. 26, pp.2-7. Williams, B 2, Using the VIX to Time Markets, Futures Magazine. Wirjanto, TS, and Yousefi, A 25, A Stylized Exchange Rate Pass-through Model of Crude Oil Price Formation, OPEC Review, Vol. 29, No. 3, January, pp
12 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: Yousefi, A and Wirjanto TS 23, Exchange Rate of the US Dollar and the J Curve: the Case of Oil Exporting Countries, Energy Economics, Vol. 25, No. 6, November, pp Zhang, YJ, Fan, Y, Tsai, HT, and Wei YM 28, Spillover Effect of US Dollar Exchange Rate on Oil Prices,Journal of Policy Modeling, Vol. 3, No. 6, pp