Intraday price information flows between the CSI300 and futures market: an application of wavelet analysis

Abstract

This study investigates linear and nonlinear price information flows between the Chinese Stock Index 300 (CSI300) and futures market using high-frequency data and their wavelet transformed series for three regimes for which stock short-selling restrictions in China are different. Empirical results generally indicate information feedback between these two markets regardless of assumptions of linear and nonlinear causality and regimes for original series and wavelet transformed data at different scales.

Appendices

Appendix 1: The nonlinear Granger causality test

Consider two strictly stationary and weakly dependent time series, $\{R_{1,t} : t=1,\ldots ,T\}$ and $\{R_{2,t}: t=1,\ldots ,T\}$. Let the m-length lead vector of $R_{1,t}$ be $R_{1,t}^{m}$, and the $l_{R_{1}}-$length and $l_{R_{2}}-$length lag vectors of $R_{1,t}$ and $R_{2,t}$ be $R_{1,t-l_{R_{1}}}^{l_{R_{1}}}$ and $R_{2,t-l_{R_{2}}}^{l_{R_{2}}}$, respectively. For given values of $m\ge 1$, $l_{R_{1}}\ge 1$, and $l_{R_{2}}\ge 1$, and an arbitrarily small constant $\varepsilon >0$, $R_{2,t}$ does not strictly nonlinearly Granger cause $R_{1,t}$ if:

$$\begin{aligned}&\Pr \left( ||R_{1,t}^{m}-R_{1,s}^{m}||<\varepsilon \mid ||R_{1,t-l_{R_{1}}}^{l_{R_{1}}}-R_{1,s-l_{R_{1}}}^{l_{R_{1}}}||<\varepsilon \text {, }||R_{2,t-l_{R_{2}}}^{l_{R_{2}}}-R_{2,s-l_{R_{2}}}^{l_{R_{2}}}||< \varepsilon \right) \nonumber \\&\quad =\Pr \left( ||R_{1,t}^{m}-R_{1,s}^{m}||<\varepsilon \mid ||R_{1,t-l_{R_{1}}}^{l_{R_{1}}}-R_{1,s-l_{R_{1}}}^{l_{R_{1}}}||<\varepsilon \right) \text {,} \end{aligned}$$

(4)

where $\Pr$ stands for probability, $||\cdot ||$ stands for maximum norm, and $s,t=\max (l_{R_{1}},l_{R_{2}})+1,\ldots ,T-m+1$. The left-hand side of Eq. (4) is the conditional probability that two arbitrary m-length lead vectors of $R_{1,t}$ are within a distance $\varepsilon$ of each other, given that the corresponding two $l_{R_{1}}$-length lag vectors of $R_{1,t}$ and two $l_{R_{2}}-$length lag vectors of $R_{2,t}$ are within a distance $\varepsilon$ of each other. The right-hand side of Eq. (4) is the conditional probability that two arbitrary $m-$length lead vectors of $R_{1,t}$ are within a distance $\epsilon$ of each other, given that the corresponding two $l_{R_{1}}$-length lag vectors of $R_{1,t}$ are within a distance $\varepsilon$ of each other. Equation (4) states that if $R_{2,t}$ does not strictly nonlinearly Granger cause $R_{1,t}$, then adding lagged values of $R_{2,t}$ does not improve the prediction power of $R_{1,t}$ from only $R_{1,t}$ lagged values. By representing the conditional probabilities in Eq. (4) in terms of the corresponding ratios of joint probabilities, we have:

$$\begin{aligned} \frac{C1\left( m+l_{R_{1}},l_{R_{2}},\varepsilon \right) }{C2\left( l_{R_{1}},l_{R_{2}},\varepsilon \right) } =\frac{C3\left( m+l_{R_{1}},\varepsilon \right) }{C4\left( l_{R_{1}},\varepsilon \right) }\text {,} \end{aligned}$$

(5)

where

$$\begin{aligned}&C1\left( m+l_{R_{1}},l_{R_{2}},\varepsilon \right) \nonumber \\&\quad \equiv \Pr \left( ||R_{1,t-l_{R_{1}}}^{m+l_{R_{1}}}-R_{1,s-l_{R_{1}}}^{m+l_{R_{1}}}||< \varepsilon \text {, } ||R_{2,t-l_{R_{2}}}^{l_{R_{2}}}-R_{2,s-l_{R_{2}}}^{l_{R_{2}}}||<\varepsilon \right) \text {,} \end{aligned}$$

(6)

$$\begin{aligned}&C2\left( l_{R_{1}},l_{R_{2}},\varepsilon \right) \nonumber \\&\quad \equiv \Pr \left( ||R_{1,t-l_{R_{1}}}^{l_{R_{1}}}-R_{1,s-l_{R_{1}}}^{l_{R_{1}}}||<\varepsilon \text {, }||R_{2,t-l_{R_{2}}}^{l_{R_{2}}}-R_{2,s-l_{R_{2}}}^{l_{R_{2}}}||< \varepsilon \right) \text {,} \end{aligned}$$

(7)

$$\begin{aligned} C3\left( m+l_{R_{1}},\varepsilon \right)\equiv & {} \Pr \left( ||R_{1,t-l_{R_{1}}}^{m+l_{R_{1}}}-R_{1,s-l_{R_{1}}}^{m+l_{R_{1}}}||< \varepsilon \right) \text {,} \end{aligned}$$

(8)

$$\begin{aligned} C4\left( l_{R_{1}},\varepsilon \right)\equiv & {} \Pr \left( ||R_{1,t-l_{R_{1}}}^{l_{R_{1}}}-R_{1,s-l_{R_{1}}}^{l_{R_{1}}}||<\varepsilon \right) \text {.} \end{aligned}$$

(9)

The correlation–integral estimators of $C_{j}^{\prime }s$ in Eqs. (6–9) are:

$$\begin{aligned} C1\left( m+l_{R_{1}},l_{R_{2}},\varepsilon ,n\right)\equiv & {} \frac{2}{n(n-1)} \sum _{t<s} \sum _{s}I\left( R_{1,t-l_{R_{1}}}^{m+l_{R_{1}}},R_{1,s-l_{R_{1}}}^{m+l_{R_{1}}}, \varepsilon \right) \nonumber \\&\times \, I\left( R_{2,t-l_{R_{2}}}^{l_{R_{2}}},R_{2,s-l_{R_{2}}}^{l_{R_{2}}},\varepsilon \right) \text {,} \end{aligned}$$

(10)

$$\begin{aligned} C2\left( l_{R_{1}},l_{R_{2}},\varepsilon ,n\right)\equiv & {} \frac{2}{n(n-1)} \sum _{t<s} \sum _{s}I\left( R_{1,t-l_{R_{1}}}^{l_{R_{1}}},R_{1,s-l_{R_{1}}}^{l_{R_{1}}}, \varepsilon \right) \nonumber \\&\times I\left( R_{2,t-l_{R_{2}}}^{l_{R_{2}}},R_{2,s-l_{R_{2}}}^{l_{R_{2}}},\varepsilon \right) \text {,} \end{aligned}$$

(11)

$$\begin{aligned} C3\left( m+l_{R_{1}},\varepsilon ,n\right)\equiv & {} \frac{2}{n(n-1)}\sum _{t<s} \sum _{s}I\left( R_{1,t-l_{R_{1}}}^{m+l_{R_{1}}},R_{1,s-l_{R_{1}}}^{m+l_{R_{1}}}, \varepsilon \right) \text {,} \end{aligned}$$

(12)

$$\begin{aligned} C4\left( l_{R_{1}},\varepsilon ,n\right)\equiv & {} \frac{2}{n(n-1)}\sum _{t<s} \sum _{s}I\left( R_{1,t-l_{R_{1}}}^{l_{R_{1}}},R_{1,s-l_{R_{1}}}^{l_{R_{1}}}, \varepsilon \right) \text {,} \end{aligned}$$

(13)

where $n=T+1-m-\max (l_{R_{1}},l_{R_{2}})$, and $I(\cdot )$ denotes a kernel that equals 1 when both vectors are within the maximum-norm distance $\varepsilon$ of each other, and 0 otherwise. Using the joint probability estimators given in Eqs. (10–13), the strict nonlinear Granger noncausality condition in Eq. (4) can be tested as follows. For $m\ge 1$, $l_{R_{1}}\ge 1$, and $l_{R_{2}}\ge 1$, and an arbitrarily small constant $\varepsilon >0$, we have:

$$\begin{aligned} \sqrt{n}\left[ \frac{C1\left( m+l_{R_{1}},l_{R_{2}},\varepsilon ,n\right) }{ C2\left( l_{R_{1}},l_{R_{2}},\varepsilon ,n\right) }-\frac{C3\left( m+l_{R_{1}},\varepsilon ,n\right) }{ C4\left( l_{R_{1}},\varepsilon ,n\right) }\right] \sim N\left( 0,\sigma ^{2}\left( m,l_{R_{1}},l_{R_{2}},\varepsilon \right) \right) \text {,} \end{aligned}$$

(14)

where $\sigma ^{2}(m,l_{R_{1}},l_{R_{2}},\varepsilon )$ is the asymptotic variance of the modified Baek and Brock test statistic.¹⁶ Under strict stationarity of $R_{1,t}$ and $R_{2,t}$, Eq. (4) is actually a statement about the invariant distribution of the $(l_{R_{1}}+l_{R_{2}}+m)-$dimensional vector $\left( R_{1,t-l_{R_{1}}}^{l_{R_{1}}},R_{2,t-l_{R_{2}}}^{l_{R_{2}}},R_{1,t}^{m}\right)$. If we let $m=l_{R_{1}}=l_{R_{2}}=1$, Eq. (4) can be represented as ratios of joint distributions of $(R_{1,t},R_{2,t},R_{1,t+1})$ :

$$\begin{aligned} \frac{f_{r_{1,t},r_{2,t},r_{1,t+1}}(R_{1,t},R_{2,t},R_{1,t+1})}{ f_{r_{1,t},r_{2,t}}(R_{1,t},R_{2,t})}=\frac{ f_{r_{1,t},r_{1,t+1}}(R_{1,t},R_{1,t+1})}{f_{r_{1,t}}(R_{1,t})}\text {.} \end{aligned}$$

(15)

The major drawback of Hiemstra and Jones (1994) test is that it tends to reject too often under the null of no nonlinear Granger causality, especially for small values of $\varepsilon$ (Diks and Panchenko 2006). A modified test statistic is introduced to address this issue by Diks and Panchenko (2006). Their restated null hypothesis is:

$$\begin{aligned} q\equiv & {} E\left[ f_{r_{1,t},r_{2,t},r_{1,t+1}} (R_{1,t},R_{2,t},R_{1,t+1})f_{r_{1,t}}(R_{1,t}) \right. \nonumber \\&\left. -f_{r_{1,t},r_{2,t}}(R_{1,t},R_{2,t})f_{r_{1,t},r_{1,t+1}}(R_{1,t},R_{1,t+1}) \right] =0\text {.} \end{aligned}$$

(16)

And the modified test statistic is:

$$\begin{aligned} T_{n}(\varepsilon )= & {} \frac{n-1}{n(n-2)}\sum _{i}^{n}\left[ \hat{f}_{r_{1,t},r_{2,t},r_{1,t+1}}(r_{1,it},r_{2,it},r_{1,it+1})\hat{f}_{r_{1,t}}(r_{1,it}) \right. \nonumber \\&\left. -\hat{f}_{r_{1,t},r_{2,t}}(r_{1,it},r_{2,it})\hat{f}_{r_{1,t},r_{1,t+1}}(r_{1,it},r_{1,it+1})\right] \text {.} \end{aligned}$$

(17)

The local density estimator of each $d_{z}-$variate random vector Z at $z_{i}$ is expressed as:

$$\begin{aligned} \hat{f}_{z}(z_{i})=\frac{(2\varepsilon )^{-d_{z}}}{n-1}\sum _{j,j\ne i}I(z_{i},z_{j},\varepsilon )\text { for }z_{i}=R_{1,it},R_{2,it},R_{1,it+1} \text {.} \end{aligned}$$

(18)

Diks and Panchenko (2006) showed that, for $l_{R_{1}}=l_{R_{2}}=1$, if the sequence of bandwidth values is determined by $\varepsilon _{n}=Cn^{-\beta }$ for any $C>0$ and $\beta \in (1/4,1/3)$, $T_{n}(\varepsilon )$ converges to a standard normal distribution:

$$\begin{aligned} \sqrt{n}\frac{\left[ T_{n}(\varepsilon )-q\right] }{S_{n}}\overset{D}{ \longrightarrow }N(0,1)\text {,} \end{aligned}$$

(19)

where $S_{n}$ is the estimated standard error of $T_{n}(\cdot )$.

Appendix 2: Wavelet analysis

Consider a time series $\{f(t)\}_{t=0}^{T-1}$ of length T. Its wavelet decomposition is represented as:

$$\begin{aligned} f(t)\approx & {} \sum _{k}s_{J,k}\phi _{J,k}(t) +\sum _{k}d_{J,k} \psi _{J,k}(t)+ \sum _{k}d_{J-1,k} \psi _{J-1,k}(t) \nonumber \\&+\cdots +\sum _{k}d_{1,k}\psi _{1,k}(t)\text {,} \end{aligned}$$

(20)

where J, an integer no greater than $\log _{2}T$, is the number of scales, k ranges from 1 to the number of coefficients in the specific component, $s_{J,k}, d_{J,k}, \ldots , d_{1,k}$ are wavelet transform coefficients, and $\phi _{J,k}(t)$, $\psi _{J,k}(t), \ldots , \psi _{1,k}(t)$ are approximating wavelet functions. Specifically, $\phi _{J,k}(t)$, the father wavelet, captures the low frequency underlying smooth part of the time series and characterizes the long scale trend, and $\psi _{J,k}(t), \ldots , \psi _{1,k}(t)$, the mother wavelets, represent the high-frequency parts and describe increasing finer scale deviations from the smooth trend. Given the family of wavelets, scaling coefficients $s_{J,k}=\int \phi _{J,k}(t)f(t)\mathrm{d}t$ and detail coefficients $d_{J,k}=\int \psi _{J,k}(t)f(t)\mathrm{d}t, \ldots , d_{1,k}=\int \psi _{1,k}(t)f(t)\mathrm{d}t$. Accordingly, $s_{J,k}$ captures the trend and $d_{J,k}, \ldots , d_{1,k}$ capture the scale deviations. Their magnitude is a measure of the contribution of the corresponding wavelet function to the series. A specific coefficient set is named a crystal, in which coefficients in level $j\in [1,J]$ are associated with scale $[2^{j-1},2^{j}]$. With these coefficients, f(t) is replicated in multi-resolution decomposition as:

$$\begin{aligned} f(t)=S_{J,k}+D_{J,k}+\cdots +D_{1,k}\text {,} \end{aligned}$$

(21)

where $S_{J,k}=\sum _{k}s_{J,k}\phi _{J,k}(t)$, $D_{J,k}=\sum _{k}d_{J,k}\psi _{J,k}(t), \ldots , D_{1,k}=\sum _{k}d_{1,k}\psi _{1,k}(t)$, and they are in order of increasing finer scales.

To obtain the wavelet transform coefficients with the classical discrete wavelet transform (DWT), a wavelet filter and a scaling filter are used. A wavelet filter, $\{h_{l}\}_{l=0}^{L-1}$ where L is its width and is an even integer, must satisfy three conditions listed in Eqs. (22)–(24).

$$\begin{aligned}&\text {The filter coefficients sum to zero: }\sum _{l=0}^{L-1}h_{l}=0\text {.} \end{aligned}$$

(22)

$$\begin{aligned}&\text {The filter coefficients have unit energy: }\sum _{l=0}^{L-1}h_{l}^{2}=1 \text {.} \end{aligned}$$

(23)

$$\begin{aligned}&\text {The filter coefficients are orthogonal to even shifts: } \sum _{l=0}^{L-1}h_{l}h_{l+2n}=0, \nonumber \\&\quad \text {where }n\text { is any nonzero integer. } \end{aligned}$$

(24)

A scaling filter, $\{g_{l}\}_{l=0}^{L-1}$, is related to the wavelet filter through a quadrature mirror filter relationship:

$$\begin{aligned} g_{l}=(-1)^{l+1}h_{L-1-l}\text { for }l=0,\ldots ,L-1\text {.} \end{aligned}$$

(25)

The scaling filter follows the same orthonormality condition as the wavelet filter, i.e., unit energy and orthogonality to even shifts, but instead of summing to zero, it satisfies:

$$\begin{aligned} \sum _{l=0}^{L-1}g_{l}=\sqrt{2}\text {.} \end{aligned}$$

(26)

Corresponding to the father and mother wavelet aforementioned, the scaling filter is referred to in the literature as the father wavelet filter and the wavelet filter as the mother wavelet filter. Let $s_{0,t}=f(t)$ be the zeroth-level scaling coefficients. Starting with level $j=1$ and recursively continuing on for levels $j=2,\ldots ,J$, the DWT detail coefficients are calculated as:

$$\begin{aligned} d_{j,t}=\sum _{l=0}^{L-1}h_{l}s_{j-1,(2t+1-l)\mod \text { } N_{j-1}}\text {, }t=0,\ldots ,T_{j-1}\text {,} \end{aligned}$$

(27)

and the DWT scaling coefficients as:

$$\begin{aligned} s_{j,t}=\sum _{l=0}^{L-1}g_{l}s_{(j-1,2t+1-l)\mod \text { } N_{j-1}}\text {, }t=0,\ldots ,T_{j-1}\text {,} \end{aligned}$$

(28)

where $T_{j}=\lfloor \frac{T}{2^{j}}\rfloor$ is the largest integer that is smaller than or equal to $\frac{T}{2^{j}}$.

The classical DWT exhibits translation in time because it applies nonzero phase filters. As a result, the details at different time scales may not contain the same number of observations and not line up in time with the original time series. More information regarding the limitations of the DWT could be found in Walden (2001). In this study, the maximal overlap discrete wavelet transform (MODWT) is adopted. It basically provides all functions of the DWT, such as multi-resolution analysis (MRA) decomposition and variance analysis, and has several advantages. Percival and Walden (2000) indicated that (1) while the DWT of level J restricts the sample size to be an integer multiple of $2^{J}$, the MODWT of level J is well defined for any sample T. (2) As the DWT, the MODWT can be used to perform MRA. But in contrast to the DWT, the details $D_{J,k} \ldots , D_{1,k}$ and smooth $S_{J,k}$ of this MRA are shift invariant in the sense that circularly shifting the time series by any amount will circularly shift by a corresponding amount each detail and smooth. (3) Different from the DWT, the MODWT details and smooths are associated with zero phase filters, making it easy to meaningfully line up features in MRA with the original time series. (4) As is true for the DWT, the MODWT can be used to conduct variance analysis based on the wavelet and scaling coefficients.

To preserve energy in the MODWT, rescaled versions of the wavelet and scaling filter used by the DWT are defined as:

$$\begin{aligned} \tilde{h}_{l}=\frac{h_{l}}{\sqrt{2}}\text { and }\tilde{g}_{l}=\frac{g_{l}}{ \sqrt{2}}\text {.} \end{aligned}$$

(29)

Let $\tilde{s}_{0,t}=f(t)$ be the zeroth-level scaling coefficients. The MODWT detail coefficients are calculated as:

$$\begin{aligned} \tilde{d}_{j,t}=\sum _{l=0}^{L-1}\tilde{h}_{l}\tilde{s}_{j-1,(t-2^{j-1}l)\mod \text { }T}\text {,} \end{aligned}$$

(30)

and the MODWT scaling coefficients as:

$$\begin{aligned} \tilde{s}_{j,t}=\sum _{l=0}^{L-1}\tilde{g}_{l}\tilde{s}_{j-1,(t-2^{j-1}l)\mod \text { }T}\text {,} \end{aligned}$$

(31)

where $t=0,\ldots ,T-1$. It is also possible to obtain $\tilde{d}_{j,t}$ and $\tilde{s}_{j,t}$ from the time series directly through:

$$\begin{aligned} \tilde{d}_{j,t}=\sum _{l=0}^{L_{j}-1}\tilde{h}_{j,l}f_{t-l\mod \text { }T}\text { and }\tilde{s}_{j,t}=\sum _{l=0}^{L_{j}-1}\tilde{g}_{j,l}f_{t-l\mod \text { }T}\text {,} \end{aligned}$$

(32)

where $\tilde{h}_{j,l}$ and $\tilde{g}_{j,l}$ are called the level j equivalent MODWT wavelet and scaling filter, respectively. Both $\tilde{h}_{j,l}$ and $\tilde{g}_{j,l}$ have a width of $L_{j}=(2^{j}-1)(L-1)+1$. Obviously, when $j=1$, $L_{1}=L$, $\tilde{h}_{1,l}=\tilde{h}_{l}$, and $\tilde{g}_{1,l}=\tilde{g}_{l}$. The equivalent MODWT filters are generally of interest for theoretical developments and can be obtained from just the basic filters, $\tilde{h}_{l}$ and $\tilde{g}_{l}$ (Alzahrani et al. 2014; Percival and Walden 2000).

Following many studies in economics and finance (e.g., Almasri and Shukur 2003; Alzahrani et al. 2014; Ramsey and Lampart 1998a, b), the Daubechies (1992) least asymmetric wavelet filter is used in the current study. It has appealing regularity characteristics. In particular, it has the approximate linear phase and exhibits near symmetry about the filter midpoint that allows the scaling and detail coefficients to line up at all levels with the original time series.

Technically, a Daubechies (1992) scaling filter, $\{g_{l}\}_{l=0}^{L-1}$, is any one that has a squared gain function given by:

$$\begin{aligned} \mathscr {G}^{(D)}(p)=2\cos ^{L}(\pi p)\sum _{l=0}^{\frac{L}{2}-1}\left( \begin{array}{c} \frac{L}{2}-1+l \\ l \end{array} \right) \sin ^{2l}(\pi p)\text {,} \end{aligned}$$

(33)

where p is the passband, $\left( \begin{array}{c} \frac{L}{2}-1+l \\ l \end{array} \right) \equiv \frac{(\frac{L}{2}-1+l)!}{l!(\frac{L}{2}-1)!}$, and $\sin ^{0}(\pi p)=1$ for all p (including $p=0$). The associated wavelet filter, $\{h_{l}\}_{l=0}^{L-1}$, following the quadrature mirror filter relationship in Eq. (25) has a squared gain function given by:

$$\begin{aligned} \mathscr {H}^{(D)}(p)=2\sin ^{L}(\pi p)\sum _{l=0}^{\frac{L}{2}-1}\left( \begin{array}{c} \frac{L}{2}-1+l \\ l \end{array} \right) \cos ^{2l}(\pi p)\text {.} \end{aligned}$$

(34)

Given L, the scaling filter is not uniquely determined by the squared gain function $\mathscr {G}^{(D)}(\cdot )$. And for $L\geqslant 6$, possible scaling filters differ in nontrivial ways. The Daubechies (1992) least asymmetric wavelet filter incorporates an additional criterion for the unique identification of the scaling filter. In particular, it selects the scaling filter whose transfer function, $G(p)=(\mathscr {G}^{(D)}(p))^{\frac{1 }{2}}e^{i\theta ^{(G)}(p)}$, is the one for which the phase function, $\theta ^{(G)}(\cdot )$, is as close as possible to that of a linear phase filter. To be more specific, we calculate

$$\begin{aligned} \rho _{\tilde{\nu }}(\{g_{l}\})=\underset{-1/2\le p\le 1/2}{\max }|\theta ^{(G)}(p)-2\pi p\tilde{\nu }| \end{aligned}$$

(35)

for all possible scaling filters with a squared gain function $\mathscr {G}^{(D)}(\cdot )$ and for a given shift $\tilde{\nu }$. For a given $\{g_{l}\}_{l=0}^{L-1}$, let $\nu$ be the shift that minimizes $\rho _{ \tilde{\nu }}(\{g_{l}\})$. The Daubechies (1992) least asymmetric wavelet filter selects the scaling filter such that $\rho _{\nu }(\{g_{l}\})$ is as small as possible. We refer to the Daubechies (1992) least asymmetric wavelet filter of width L as LA(L). More technical details could be found in Daubechies (1992) and Percival and Walden (2000).

Following previous studies (e.g., Almasri and Shukur 2003; Alzahrani et al. 2014; In and Kim 2006a, b), the filter width $L=8$ is used in the current study. It is short enough to limit the number of boundary conditions and is long enough to generate coefficients that are approximately uncorrelated between scales (Alzahrani et al. 2014). Plots of the scaling and wavelet filter based on LA(8) can be found in Percival and Walden (2000).