Linear Granger Test Results

The univariate OLS regressions in Table 12 indicate that VRP and RV are influenced by the lag-one terms of each other. Next, the linear Granger causality is used to test whether there is bidirectional causal relation between VRP and RV. The linear Granger causality test incorporates the own and other lag term beyond a one period lag. Thus, it is able to account for the autocorrelation in dependent variables. The dynamic relation between |dVRP| and RV is explored using the Granger causality test by specifying Z= [|dVRP| RV] in Equation (10). The appropriate lag lengths in this model are set to be four, according to the Akaike information criterion (AIC). If VRP Granger causes RV, then the past values of VRP should contain information that helps predict RV. In other words, the Granger causality helps test whether the VRP feedback effect exists.

In Panel A of Table 13, the results of the linear Granger causality test are summarized by presenting the t-tests on the sum of the estimated coefficients, b, which represents the

cumulative effect of lagged RVs on VRP, as well as the chi-square test for the jointly zero hypothesis of all the lag coefficients. For the |dVRP|, the null hypothesis of linear Granger non-causality from the realized volatility (RV) is strongly rejected by the significant summed coefficients of all lagged RVs (summed ˆβ^=0.0074, t-value = 13.47) and by the significant chi-square statistic (χ²=16.59, p-value <0.01). The finding that realized volatility positively Granger causes VRP suggests that the market prices of options reflect volatility risk. This is consistent with the findings of Bakshi and Kapadia (2003), Bollen and Whaley (2004), and Gârleanu et al. (2009) that the VRP compensates liquidity suppliers of options for bearing the volatility risk.

Panel B: Linear Causal Tests between |dVRP| and nJV, pJV, and CV

b ^0.3434

Notes. This table presents the sum b of all the lag coefficients for variable in first column and its t-values (in parentheses), and the chi-square statistics χ² for linear Granger causality test and its p-value (in brackets). Both test statistics are used to detect the linear Granger causality. The sum b indicates the cumulative effect of lagged |dVRP| on realized volatility (RV) and its volatility components, and vice versa. The χ² statistics test the null hypothesis that all the lag coefficients of column variable are jointly zero. A rejection of the null

hypothesis indicates that column variable (x) Granger causes row variable (y). Panel A reports the results of linear causal tests between |dVRP| and RV, and Panel B presents the results of pairwise linear causal tests between |dVRP| and three volatility components of realized volatility. The t-statistic is calculated as t= b/σ_b, where b= + + +a₁ a₂ ... a_nis the given sum of coefficients and n is number of lags on the independent variable. are calculated using the Newey-West (1987) autocorrelation and heteroskedasticity consistent covariance matrix. |dVRP| is absolute deviations from the median of volatility risk premium. nJV, pJV, and CV denote negative jump volatility, positive jump volatility, and continuous volatility, 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.

More importantly, linear Granger causality also supports the existence of the feedback effect from VRP to RV. In Panel A of Table 13, the null hypothesis of linear Granger non-causality from |dVRP| to RV is rejected by the significant summed coefficients of all lagged |dVRP|s (summed ˆβ=0.0103 and t-statistic 3.04) and the significant chi-square statistic (χ2=14.10, p-value <0.01). The finding supports the VRP feedback effect that higher VRP positively impacts future realized volatility. Frey and Stremme (1997), Gennotte and Leland (1990), Schoenbucher and Wilmott (2000), and Sircar and Papanicolaou (1998) suggest that dynamic hedging leads to greater market volatility. It is likely that the dynamic hedging induced by volatility trading that seeks to neutralize unanticipated price changes also affects the subsequent market volatility, resulting in the feedback effect found in Table 13.

To further examine the causal relation between |dVRP| and the three components of realized volatility (CV, nJV, and pJV), this study sets Z = [|dVRP| nJV pJV CV] in Equation (10) and performs the pairwise Granger-causality tests for |dVRP| versus each volatility component. The third column in Panel B shows the test whether individual volatility component Granger causes |dVRP|. The results show a significant causal relationship from CV to |dVRP| and from nJV to |dVRP| with t-values of 3.02 and 1.97, respectively.³³ This indicates that both continuous volatility and jump volatility due to large price declines

33 Based on the Akaike information criterion (AIC), this study includes 8 lag variables while performing this

enlarge VRP (Bollerslev and Todorov 2011; Eraker et al. 2003; Todorov 2010), whereas volatility due to large price increases has less of an effect on the VRP. This finding is consistent with the OLS results in Table 11 that the CV and nJV play more important roles than pJV in explaining the VRP changes. The OLS and linear Granger causality together suggest an asymmetric effect of positive jump and negative jump on the VRP.

In the first row of Panel B, where the test is presented for the feedback effect, the linear Granger causality is only found from |dVRP| to CV but not from |dVRP| to nJV or from

|dVRP| to pJV. One limit of the test in Table 13 is that the traditional Granger causality model aims to test for linear dependence. Thus, it has less power to detect nonlinear causal relations (Baek and Brock 1992; Hemstra and Jones 1994). If the impact from lagged

|dVRP| to any component of jump volatility is nonlinear, the traditional approach may fail to uncover the feedback effect.

4.3. Nonlinear Granger Test Results

In this section, a more general form of the Granger causality test is provided. The modified Baek and Brock test, which allows nonlinear dependence in both |dVRP| and components of realized volatility, is adopted to examine the VRP feedback effect. The test statistic is specified in Equation (11). In this test, values for the lead length m, the lag lengths Lx and Ly, and the distance measure d need to be selected. Unlike linear causality testing, no approaches exist for choosing optimal values for lag lengths and distance measure. Following Hiemstra and Jones (1994), this study sets the lead length at m=1, Lx=Ly, and a common distance measure of d=1.5σ, where σ denotes the standard deviation of the time series. The results for lag lengths from 1 to 8 are presented for the robustness analysis.

Table 14 reports the results of the modified Baek and Brock test applied to the estimated residuals of linear Granger causality model for |dVRP| and RV. The nonlinear

tests indicate stronger feedback effect than that shown previously by the linear test. The null hypothesis of no nonlinear Granger causality from RV to |dVRP| is strongly rejected at 1%

significance level in every specification. The null hypothesis of no nonlinear Granger causality from |dVRP| to RV is also rejected. Results of nonlinear Granger tests again support that the causality between |dVRP| and RV is bidirectional. This bidirectional nonlinear relation holds for all the common lag lengths used in constructing the test. It suggests that the duration of the predictability of |dVRP| for RV is equivalent to that RV for

|dVRP|. This nonlinear impact from lagged |dVRP| to current RV provides stronger evidence to the VRP feedback effect.

Table 14 Results of Nonlinear Granger Causality Test

Ho: RV Does Not Cause |dVRP| Ho: |dVRP| Does Not Cause RV

Notes. This table reports the results of the modified Baek and Brock nonlinear Granger causality tests applied to the vector autoregression residuals corresponding to absolute deviations from the median volatility risk premium (|dVRP|) and realized volatility (RV). Lx=Ly indicates the lag lengths of the residuals used in the test.

In all cases, the tests are applied to unconditionally standardized series, the lead length, m, is set to 1, and the distance measure, d, is set to 1.5. Stat. and t, respectively, denote the test statistic in Equation (11) and its t-value. Under the null hypothesis of nonlinear Granger noncausality, the test statistic is asymptotically distributed N(0,1). ***, **, and * indicate that the t-values are significant at the 0.01, 0.05, and 0.10 level, respectively.

Table 15 examines the pairwise nonlinear Granger causality between |dVRP| and each of the three volatility components of RV. The volatility components are found to be significantly Granger cause VRP, as shown by the significant t-values in the left part of the panel. This finding holds for every lag-length selection in every volatility component.

The feedback effects from VRP to volatility components are almost as pronounced as

the impact from volatility components to VRP. In the last two columns of Panels A and B, the hypothesis of no nonlinear Granger causality from |dVRP| to CV and from |dVRP| to nJV is rejected in every case, clearly showing evidence for the VRP feedback effect. Only the nonlinear Granger causality from |dVRP| to pJV, reported in the last two columns of Panel C, is somehow weaker. The VRP Granger causes the positive volatility jump component for models with lags up to lag 4.

In summary, the modified Baek and Brock test reports significant VRP feedback effect for all volatility components. The VRP feedback effect is stronger for continuous volatility (with greater coefficient and higher significance) and lower on jump volatility. The feedback effect is asymmetrical such that negative jump volatility (nJV) responds more than positive jump volatility (pJV) to the changes in volatility risk premium.

Table15 Results of Pairwise Nonlinear Granger Causality Test Panel A: Nonlinear Causal Relation between |dVRP| and CV

Ho: CV Does Not Cause |dVRP| Ho: |dVRP| Does Not Cause CV

Panel B: Nonlinear Causal Relation between |dVRP| and nJV

Ho: nJV Does Not Cause |dVRP| Ho: |dVRP| Does Not Cause nJV

Lx=Ly Stat. t Stat. t

Panel C: Nonlinear Causal Relation between |dVRP| and pJV

Ho: pJV Does Not Cause |dVRP| Ho: |dVRP| Does Not Cause pJV

Lx=Ly Stat. t Stat. t

1 0.0008 4.32^*** 0.0004 1.92^*

2 0.0021 7.05^*** 0.0014 4.61^***

3 0.0022 6.84^*** 0.0011 3.52^***

Note. This table reports the results of the pairwise nonlinear Granger causality tests between |dVRP| and CV, nJV, and pJV. They are reported in Panel A, Panel B, and Panel C, respectively. |dVRP| indicates absolute deviations from the median volatility risk premium; CV is continuous volatility; nJV is negative jump volatility; pJV is positive jump volatility. Lx=Ly indicates the lag lengths of the residuals used in the test. In all cases, the tests are applied to unconditionally standardized series, the lead length, m, is set to 1, and the distance measure, d, is set to 1.5. Stat. and t respectively denote the test statistic in Equation (11) and the t-value of test statistic. Under the null hypothesis of nonlinear Granger noncausality, the test statistic is asymptotically distributed N(0,1). ***, **, and * indicate that the t-values are significant at the 0.01, 0.05, and 0.10 level, respectively.

The explanations for the ranking of the VRP on the three volatility components are provided as follows. First, the higher VRP would lead to a measurable increase in jump volatility only if the dynamic-hedging transactions results in substantial price changes (a sudden shift in realized volatility is often associated with radical changes in price). This occurs in the scenario described in Gennotte and Leland (1990), that is, the dynamic-hedging transactions substantially alter the expectations and liquidity supply of other uninformed market participants. The consequence is that a relatively small amount of hedging would drive significant price and volatility change. This scenario, of course, is not commonly observed in the market, which is consistent with our results that the VRP feedback effect is less significant for jump volatility than continuous volatility.

Second, the results of asymmetric VRP feedback effect show that high VRP is more likely to be followed by a negative jump than a positive jump. This is consistent with prior evidence that investors are more sensitive to a large market decline than a large increase in return and are willing to pay more to hedge the potential decline than the possible increase.

That is why the volatility risk premium widens more prior to a negative jump than a positive jump.³⁴

34 Pan (2002) and Bollerslev and Todorov (2011) provide evidence for the asymmetric responses to upward versus downward jumps. They find that such asymmetry in the fear of jump risk leads to a larger premium in the

The asymmetric VRP feedback effect could be used to infer the strategies of volatility traders. The tendency of a negative jump after a large VRP implies that volatility traders tend to engage in hedge transactions that involve shorting spot assets or futures contracts, so that their hedging leads to negative jumps subsequently. Based on Chaput and Ederington (2005), popular volatility trading strategies that require delta hedge using short spot/futures positions include 1. long volatility by buying calls, 2. short volatility by selling puts, and 3.

straddles. Chang, Hsieh, and Wang (2010) show that strategy 1. and 2. are the most frequently used volatility trading strategies in Taiwan. Since the sample in this study spans the period of global financial tsunami, a period characterized by high volatility and substantial market decline, there should be more opportunities for long volatility than for short volatility. It is therefore speculated that the long volatility by buying calls (strategy 1.) would be the most likely approach for volatility trading. The delta hedging of such strategy creates downward pressure on the underlying asset price and leads to subsequent widening in the VRP.