Nonlinear Granger Causality

Baek and Brock (1992) developed a nonparametric test for potential nonlinear causality among time series. The test uncovers any remaining nonlinear causal relationship after the liner causal effect has been accounted for. To detect the nonlinear Granger causality from VRP to realized volatility, this study adopts the modified version of the Baek and Brock (1992)’s nonlinear Granger causality test, proposed by Hiemstra and Jones (1994). This modified test is based on the nonparametric estimators of temporal relations within and across time series. It relaxes the assumption of independent and identical distribution in each time series in Baek and Brock (1992), allowing each series to have weak temporal dependence. To determine whether nonlinear causality exist between given time series, we implement the modified Baek and Brock test to the residuals from Granger causality equation (10). Appendix B provides a detailed description of the modified Baek and Brock test used to detect the nonlinear causal relationship.

Assuming that Xt and Yt are strictly stationary and weakly dependent and satisfy the mixing conditions as specified in Denker and Keller (1983). If Y_t does not strictly Granger cause Xt, then the test statistic for nonlinear Granger causality, G, is asymptotically normally distributed. A rejection of the null hypothesis of Granger noncausality indicates that there exists nonlinear causality between the two time series. The statistic G is given as

1( , , ) 3( , ) 1 2

( ) (0, ( , , , )),

2( , , ) 4( , )

G m Lx Ly d G m Lx d a

G N m Lx Ly d

G Lx Ly d⁺ G Lx d⁺ nσ

= − ∼ (11)

where G1, G2, G3, and G4 are joint probabilities; m is the lead length; Lx and Ly are,

respectively, the lag lengths of X and Y; d is the distance measure; n = T + 1 – m – max(Lx, Ly); and σ²(m,Lx,Ly,d) is the asymptotic variance of the modified Baek and Brock test statistic.²⁸

3. DATA

The study collects two sets of data: the minute-by-minute Taiwan stock index data obtained from the Taiwan Economic Journal (TEJ) database for January 1, 2002 through December 31, 2009; and the minute-by-minute Taiwan VIX index data provided by the Taiwan Futures Exchange (TAIFEX) for December 18, 2006 through December 31, 2009.²⁹ The TAIFEX constructs its VIX index based on the European-style Taiwan index option (TXO) using the same approach as the CBOE new VIX index. This study retrieved, from TEJ database, the three month time deposit of the postal saving system for the risk-free interest rate.

For every 5 minutes, the daily risk-neutral volatility (IV) from the VIX index is computed using Equation (8) and the expected realized volatility (RV_E) is estimated using the VAR model in Equation (9). The 5-minute frequency VRP is obtained using equation (7) by subtracting RV_E from IV.

Table 8 presents the results of parameter estimation for the realized volatility forecast based on the vector autoregressive model in Equation (9). The 5-minute rolling window procedure generates 41,195 estimations for each parameter during the sample period. Table 2 reports the average. The mean coefficients for CV², the first elements in the M₁, M₅, and M22 matrices, are significantly positive, indicating a strong own persistence in the continuous variance over time. There are dynamically asymmetric dependencies between

28 Based on the Monte Carlo simulations, Hiemstra and Jones (1993) find that the modified test is not only robust to nuisance-parameter problems but also has good finite sample size and power properties.

29 The TAIFEX began releasing data on the VIX on December 18, 2006.

CV², nJV², and pJV². For instance, CV² is lagged to nJV² as shown by the significant estimates for the second elements in the M₁ and M5 matrices; and nJV² is lagged to CV² as shown by the significant estimates for the first elements in the M5 matrices in coefficients for nJV². Therefore, we include all three volatility components in forecasting the realized volatility.

Table 9 Parameter Estimates for the VAR Model in Equation (9)

CV² nJV² pJV²

Notes. This table presents the estimating results of parameters for realized volatility forecast every 5 minutes based on the vector autoregressive model in Equation (9):

5 22 estimations with 41,195 observations during the sample period. The realized variance is decomposed into three variance components, including continuous variance (CV²), negative jump variance (nJV²), and positive jump variance (pJV²). Based on the three variance components of realized variance, the expected realized volatility every 5 minutes is estimated using a vector autoregressive model for a three-dimensional vector.

Coeff. and t are the estimated parameters of regression and t-value of parameter test, respectively. In Equation (9), M₀ is a vector of the intercept term. M₁, M₅, and M₂₂ are matrices for the regression coefficients, in which the first column, second column, and third column in each matrix correspond to the parameters of the three variance components (CV², nJV², pJV²), respectively. All coefficients are multiplied by 100. ***, **, and * indicate that the t-values are significant at the 0.01, 0.05, and 0.10 level, respectively.

Figure 2 exhibits the plot of the 5-minute frequency time-series for the daily risk- neutral volatility (IV) and the expected realized volatility (RV_E). A visual inspection shows

that both the IV and RVE closely track each other. The relationship holds even during the period of financial crisis in 2008 when the market experienced large fluctuation in volatility.

Over the sample period, the daily risk-neutral volatility is slightly above the expected realized volatility by 15 basis points, indicating a positive VRP. This positive VRP is consistent with most other studies (Bakshi and Kapadia, 2003; Bollerslev and Todorov, 2011; Todorov, 2010).

Figure 2. Time Series Plots of the Daily Risk-Neutral Volatility and Expected Realized Volatility. This figure depicts the time-series relation between daily risk-neutral volatility (IV) and expected realized volatility (RV_E) at five minute intervals. The time period is from December 18, 2006 to December 31, 2009, inclusive. The IV is computed from the VIX index using Equation (8); the RV_E is estimated by using the vector autoregressive model in Equation (9).

Figure 2 also shows that the size of VRP changes over time and has a mean reversion tendency. It is consistent with the findings in Todorov (2010) that VRP increases significantly after large market moves and reverts to its long-term mean. Facing a mean-reversion trading opportunity, volatility traders engage trade based on the swing in the VRP rather than the size of the VRP. The deviation from the median VRP thus can better reflect the trading opportunities of volatility traders.³⁰ We therefore use the absolute deviations from the median, denoted by |dVRP|, as a proxy of volatility trading profit.

Figure 3 shows that the |dVRP| fluctuates over time and is more volatile during 2008. Any

30 We thank the referee for this insightful suggestion.

noticeable variation indicates a profitable opportunity for volatility traders.

Figure 3. Time Series Plots of Absolute Deviations from Median Volatility Risk Premium. This figure depicts the 5-minute time-series of the absolute deviations from the median of volatility risk premium (|dVRP|). |dVRP|_m is the average of the |dVRP| during the sample period. VRP is defined as the daily risk-neutral volatility less expected realized volatility, where the risk-neutral volatility is computed from the VIX index and the expected realized volatility is estimated by using the vector autoregressive model in Equation (9).

Table 10 provides summary statistics of the 5-minute time series for |dVRP|, VRP, IV, RVE, RV, and the three components of realized volatility (CV, nJV, and pJV).³¹ Note that both |dVRP| and VRP are positive in mean, indicating that volatility sellers, on average, may acquire profits about 4.8% annualized volatility spread. The augmented Dickey–Fuller (ADF) unit root tests significant reject the hypothesis of one unit root for every individual series, indicating that these variables are stationary.

31 For RV, this study first obtains 5-minute index return series while treating the entire overnight period (clock time 13:30 to 9:00 next day) as one interval. The realized volatility over a day is defined as the variation of returns in any window that contains 55 consecutive intervals (including 54 5-minute returns and one overnight return). This study then estimates return variation within the window for a daily volatility estimate, using the Equation (A1) in Appendix A. By rolling the window forward at 5-minute intervals, the RV estimation is obtained every 5 minutes. For example, at t=1, realized volatility RV₁ is calculated using 5-minute returns from 9:00 to 9:00 next calendar day. At t=2, we roll the window 5 minutes forward, calculating the realized volatility RV₂ using 5-minute returns from 9:05 current day to 9:05 next calendar day, and so forth. This will produce a time series of ‘daily’ RV every 5 minutes, where a ‘day’ is defined as any consecutive 55 intervals that does not necessarily begin at 9:00 am. This process is similar to the approaches used in Andersen et al.

(2003), Clements, Galvao, and Kim (2008), and Wright and Zhou (2009) to measure the monthly, quarterly, or yearly realized volatility with rolling window approach for every day. The data constructed above are used to

Table 10 Summary Statistics

Volatility components

|dVRP| VRP IV RV_E RV

CV nJV pJV

Mean 0.0030 0.0015 0.0181 0.0166 0.0153 0.0121 0.0029 0.0034 Med 0.0020 0.0023 0.0179 0.0153 0.0129 0.0112 0.0000 0.0000 Min 0.0000 –0.0536 0.0072 0.0002 0.0030 0.0022 0.0000 0.0000 Max 0.0559 0.0210 0.0379 0.0808 0.0796 0.0796 0.0691 0.0599 Std 0.0035 0.0046 0.0056 0.0081 0.0098 0.0060 0.0085 0.0074

Skew 3.98 –2.33 0.46 1.58 2.35 2.26 4.01 3.46

Kurt 33.91 16.84 3.19 7.33 10.78 17.65 21.96 19.05 ADF –283.84 –206.85 –20.56 –61.94 –350.29 –140.47 –722.13 –784.69 Notes. This table presents summary statistics for absolute deviations from the median of volatility risk premium (|dVRP|), volatility risk premium (VRP), risk-neutral volatility (IV), expected realized volatility (RV_E), realized volatility (RV), and three volatility components of realized volatility. The volatility components decomposed from the realized volatility are continuous volatility (CV), negative jump volatility (nJV), and positive jump volatility (pJV). The data cover the period from December 18, 2006 to December 31, 2009. All series are computed in 5-minute frequency. The VRP is defined as IV less RV_E, where the IV is directly computed from VIX index and the RV_E is estimated by a vector autoregressive model. ADF is the augmented Dickey–Fuller unit root test.

4. EMPIRICAL RESULTS