5. Empirical Results
5.2 Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) Unit Root Test.20
precluding the cash dividend effect, we adjust the spot index price by subtracting the present value of the cash dividends that will be paid during the remaining life of the corresponding futures contract, because the futures prices drop about the present value of that dividends in advance as Equation (2). These futures and adjusted index prices will be employed to implement the following empirical research. The log-prices of the futures and the adjusted index are used to fulfill our sequentially empirical analysis. The returns of futures and adjusted index are calculated by taking the difference in the log-price.
Stationary time series react to the shock transitorily and return to the long-term equilibrium with the shock passing by. Conversely, nonstationary time series have permanent effects with the passage of shock. We employ the two unit root tests examined for stationarity.
1. Augmented Dickey-Fuller (ADF) Test
∑
= Δ − +where the null hypothesis, H0: δ=0, represents nonstationarity, and the alternative hypothesis, H1: δ≠0, interprets stationary. The model constructed here contains a drift term, α and a time trend, t. We employ Schwarz Bayesian Information Criterion (SBIC)3 to choose the optimal lag length based on the parsimony principle.
2. Phillips-Perron (PP) Tests
3 SBIC(p)=N log(SSR)+p log(N), where SSR is the residual sum of squares, N is the sample size, and p is the total number of parameters.
t nonstationarity, and the alternative hypothesis, H1≠H0, is on behalf of stationarity.
Table 3 addresses the results of the two unit root tests examined for stationarity.
The ADF results and PP results are similar, both of which fail to reject the null hypothesis of unit root tests for each price series in level data, but reject it in first difference data at 1% significance level. The results suggest all the data series are integrated of order one, I(1). These results indicate that the futures price and spot price are integrated in the first difference, integrated of order one, I(1), before and after the introduction of new tick size regulation, and thus verify the fulfillment of the cointegration test.
5.3 Johansen Cointegration Test
Given that the two price series are integrated of the same order one, I(1), this study builds two Johansen multivariate cointegration tests to judge whether the price series are cointegrated. We set a 2×1 vector xt=[Ft
S
t]′ where Ft stands for futures price and St stands for spot in our study. If there exists a vector β (β≠0) that makes linear combination of two price series, β′xt, reduce the integrated order to stationarity, we can say the two price series exist cointegration relationship andβis the so-called cointegration vector.The reduced form error correction model formulates the test hypothesis as follows The test hypothesis is formulated as the restriction for the reduced rank of Π:
β α ′
=
Π for the reduced form error correction model:
t representing the adjustment speed of the parameter and cointegrating vector, respectively. denotes short-term relation of . is the error correction term. The rank of or the number of non-zero eigenvalue of
i
x
t−Δ
x
tx
tΠ Π determines the
number of cointegration vector. Johansen proposed two likelihood ratio statistics to test the number of cointegration vector.
Two test statistics for cointegration under the Johansen approach are formulated as follows:
1. Trace Statistic
r
2. Maximum Eigenvalue Statistic1
where r is the number of cointegrating vectors under the null hypothesis, T is the sample size and is the estimated value for the ith ordered eigenvalue from the Π matrix. The null hypothesis for joint test statistics,
λ
ˆitrace
λ
, is that the number of cointegrating vectors is less than or equal to r against the unspecified or general alternative one that the number of cointegrating vectors is more than r. Maximum Eigenvalue Statistic,λ
max, conducts separate tests on each eigenvalue. The null hypothesis that the number of cointegrating vectors r is against an alternative that the number of cointegrating vectors r+1.The result of the cointegration test reported in Table 4 demonstrates the rejection the null hypothesis of no cointegration at the 5% significance level in the both sample period and the rejection of the null hypothesis of only one cointegration relationship at the 5% significance level before the reduction of tick size. This evidence advocates that a stationary linear combination exists between the futures and spot prices in the both pre-reduction and post-reduction period. Thus, the futures and spot markets are cointegrated and have long-term equilibrium relationship.
5.4 Linear Vector Error Correction Model (VECM)
Since the futures and spot price are cointegrated, there must exists an error correction term in which so-called vector error correction model (VECM) constructing the dynamic system dominating the joint evolution of the futures and spot prices over time. From Equation (6), we can know that the error correction terms, )wt(
β
, illustrate the long-term equilibrium dynamics between the price series, and the dynamic coefficients of the lagged price series which capture the short-run dynamics resulted from market imperfections. Similarly, we employ Schwarz Bayesian Information Criterion (SBIC) to determine the optimal lag length d applied not only to the linear VECM but also to the threshold VECM. Eventually, the optimal lag length is selected for four and three for the pre- and post-reduction of tick size, respectively. The results of the linear VECM estimation are proposed in Table 5.For the pre-reduction of tick size period, we find the futures market have an obvious lead over the spot market. In Panel A of Table 5, all the coefficients of the lagged futures prices (ΔF) in the spot equation (ΔS) are statistically significant at the 1% level, but no for lagged spot prices (ΔS) in the futures equation (ΔF). These results indicate that the futures market leads the spot market. In addition, the
coefficients of the error correction terms for futures prices (ΔF) and spot prices (ΔS) equations are not statistically significant, so we cannot confirm that long-run co-movement exists between these two financial markets in the pre-reduction of tick size period.
For the post-reduction of tick size period, we discover appearance of bi-direction relationship between two markets. In Panel B of Table 5, all the lagged futures prices (ΔF) in the spot equation (ΔS) are significant at 1% level; whereas, we detect only one significant impact of lag 1 spot price on futures at 1% level in the post-reduction period as opposed to that in the pre-reduction period, i.e., a feedback relation. Besides, the coefficient of the error correction terms is only significant in the futures equation, suggesting that futures prices are inclined to adjust significantly as the prices deviate from long-run equilibrium took place. The z statistics of the coefficient of the error correction term for both price series after the reduction of tick size are more significant than that before. Evidently, this result imply that the reduction of tick size, lowering the spread cost, makes the long-run co-movement extent between these two financial markets turns stronger. This can be confirmed by the higher and closer to the unity Pearson’s correlation coefficient between the two market, which is described in Section 5.1. The cointegrating vector, 1.03697, in the post-reduction period is closer to unity than that, 1.03886, in the pre-reduction period.
It implies the two price series approximate to each other stronger in the second sample period. This result is caused by the lower transaction costs after the reduction of tick size, which reduces the obstacles for the two prices to return to long-run equilibrium.