5. Empirical Results
5.6 The GARCH Model
≤0
wt with 75.43 observations and the second regime i.e., extreme regime, occurs while wt >0.307617 with 24.57% observations. In the first regime, the error-correction phenomenon is only significant in the futures prices (ΔF) equation, which indicates that there exists mean reversion only in the futures price. However, as contrary to the first regime, the error-correction effect is not significant both in the futures (ΔF) and spot (ΔS) equations, which means that there does not have any long-run equilibrium relationship between futures market and spot market in the extreme regime. In other words, the two series behave like a random walk and free from the cointegration constraint (Tsay, 1998) probability due to the decrease of market depth at the best-quoted prices (Harris(1994, 1997) and Furfine (2003)) after the reduction of tick size. Equilibrium relationship between futures market and spot market in the extreme regime becomes insignificant. The decrease of arbitrage trade numbers might result from stronger co-movement between the two financial markets discovered in Section 5.4 after reduction of tick size. Similarly, for the short-term dynamic coefficients, the result seems consistent with earlier findings. The futures price tends to lead the spot price after the reduction of tick size.
5.6 The GARCH Model
We further examine the volatility alteration across the two sub-periods following Section 5.1 by constructing a GARCH mode in Equation (23), (24), (25), and (26). Before employing the GARCH model, we implement ARCH test in advance to investigate whether the futures and spot returns exist quadratic autocorrelation in the residuals. The ARCH test is conducted as following. First, we implement an OLS regression on the futures and spot returns with the constant as independent variable, and draw out the the residuals,
ε
ˆ . Second, we square the tresiduals and run the autoregression on q own lags to test whether ARCH-effects exists in the residuals:
t correlation and get the test statistic, TR , which is the chi-square distribution with q 2 degree of freedom,
χ
2( ) q
. Finally, the null and alternative hypotheses areTherefore, we test if any coefficient of the autoregression for q lags in Equation (22) significantly different from zero meaning the ARCH-effect exists in the residuals.
The chi-square test statistic is 315.4757 and 381.6646 for the futures and spot returns, respectively before the reduction of tick size, and 480.1283 and 502.4160 for the futures and spot returns, respectively after the reduction of tick size. The results above significantly reject the null hypothesis at 1% level expressing ARCH-effect in the residuals for the futures and spot returns in both sample periods. Thus, the conduction of following GARCH model is justified by the significant ARCH test results in both sample periods.
We construct the following GARCH Equation (23), (24), (25), and (26) to investigate the return and conditional volatility alteration before and after the reduction of tick size.
t
1 variance, , is composed of the lagged squared errors in the return process and the lagged conditional variance, . The variable stands for the futures market lagged volatility and the variable stands for the spot market lagged volatility. is the event dummy taking the value one for the period before the reduction of tick size and the value zero for the period after the reduction of tick size.
t futures return (spot return) on present futures return (spot return) and the coefficient,
3 ,
αi , measures the effect of last period spot return (futures return). The coefficient,
4 ,
αi , measures whether the return exists structural change after introduction of new tick size. In the conditional variance Equation (25) and (26), coefficient, βi,1, measures the lagged squared error effect on this period conditional variance.
Coefficient, βi,2, captures last period conditional variance impact on this period conditional variance. Coefficient, βi,3, catches return volatility alteration across the two sample period.
From the result of Table 8, we can find last period futures and spot returns both have significant effect on the current period futures and spot returns at 1% level. By the dummy variables in the two mean equations, we find futures and spot returns do not significantly differ in the two periods at any common significance level. From the result of variance equation in the Table 8, we can find the lagged squared error and lagged conditional volatility both have significant effect on the current period conditional volatility for both futures and spot markets. By the dummy variables in the variance equations, we can find the second sample period volatility significantly decrease at 1% level for both futures and spot markets. This result is consistent with
Section 5.1. We will further use this conditional volatility in the variance equations to implement the empirical research in Section 5.7.
5.7 Conditioning Mispricing Errors on Volatility
In Section 5.1, we find MPE significantly decreases at 1% level in the second sample period; meanwhile, futures and spot volatility also significantly decreases at 1% level in the same sample period discovered in Section 5.1 and 5.6. Arbitrageurs can trigger arbitrage requiring smaller MPE, because the smaller volatility reduces the execution risk for them. Hence, we cannot figure out the smaller MPE in the second sample period originated from smaller volatility or reduced tick size. In order to identify the reason that results in the smaller MPE in the second sample period, we implement the methodology of Jones and Lipson (2001) that conditions the mispricing errors on volatility as introduced in Section 4.7.
The first regression of Table 9 shows the results of Equation (20) that conditions the data on the 30-minute volatility of the futures returns immediately preceding every trade following Henker and Martens (2005). Equation (20) also conditions on the GARCH conditional volatility of futures returns, whose result is shown in the second regression of Table 9. We choose the futures volatility over the volatility of the underlying stocks because the bid-ask bounce in futures prices is regarded to be a less serious problem than the serial correlation in index returns (Henker and Martens (2005)). The table demonstrates the empirical result in the first sample period. We can confirm the futures volatility has the significantly positive relation to the MPE at 1% level in both regressions of Table 9, because the higher volatility could result in higher timing risk and tracking error risk for arbitrageurs’
position. The GARCH conditional futures volatility in the second regression has the similar result to the 30-minute futures volatility in the first regression. The indicator
variables of buy program and sell program both reveal significantly negative relation to the MPE at 1% level, because the trigger of arbitrage trades would shrink the MPE.
The high R-square in both regressions of Table 9, 0.7268 and 0.7389 respectively, means the independent variables have high explanatory power to the MPE in the first sample period.
We employ Equation (21) that uses the estimated coefficients in Equation (20) to calculate MPE prediction errors under new regime. While controlling for the change of volatility, buy program, and sell program effects, we further examine whether MPE shrinks after the reduction of tick size. The null hypothesis of the test is
H
0 :e
ˆipre,t −reduction =e
ˆipost,t −reduction, and the corresponding alternative hypothesis isreduction post
t i reduction pre
t
i
e
e
H
1: ˆ, − > ˆ, − . The bar denotes the average residuals over the two sample periods, respectively.In order to implement above mean residual difference test, we should fulfill residual volatility difference test above all4. The test statistic conditioning on 30-minute futures volatility is 2.8727, and that conditioning on GARCH conditional futures volatility is 0.2161. Clearly, the residual volatility is statistically significant different across two sample periods in both conditioning cases. Then, we further
4 s1 denotes the residual volatility before the reduction of tick size, s2 indicates the residual volatility after the reduction of tick size, and n1 and n2 are their respective number of observations. Test statistics 2
2 2 1
S
F
=S
is F(n1-1, n2-1) distribution.employ the mean residual difference test5 to explore whether MPE decreases after the reduction of tick size, while conditioning on the change for volatility and buy and sell program effects. For conditioning on 30-minute futures volatility and GARCH futures volatility, the test statistics are 121.5890 and 115.6704, respectively. In both cases, the null hypothesis is rejected in favor of the alternative hypothesis at any conventional significance level. The results means the reduction of tick size can lower the mispricing error and improve the pricing efficiency.
However, we use the arbitrage data drawn from the second regime of threshold VECM model instead to run the OLS regression of Equation (20) and Equation(21) again. We find the mispricing error increases and pricing efficiency deteriorates for the arbitrage data after the reduction of tick size. Because the market depth might reduce after the reduction of tick size, the error-correction terms of the second regime of threshold VECM model are insignificant which means long-term equilibrium relationship of futures and spot does not exist after the reduction of tick size. Hence, mispricing error of arbitrage data does not reduce for the arbitrage data after the reduction of tick size.