Information flow and price discovery dynamics

Abstract

Non-homogeneous and time-varying information flow that affects the price discovery processes within and across markets is a common occurrence in reality but is often neglected in the literature of price discovery. To analyze such information flow within and across markets, we propose a new strategy with a new dynamic price discovery measure. We use this strategy to test the efficient home market hypothesis and the sector effect hypothesis based on the intraday data of the 115 stocks cross-listed and traded in the Canadian and U.S. stock markets. We find that the Canadian stock market is more efficient in price discovery for the Canadian stocks cross-listed in the U.S. stock market. A higher trading volume in the Canadian market makes price discovery in that market more efficient. The Canadian stock market is more efficient in price discovery for stocks in the basic materials sector but not in the technology and financial sectors. The NYSE Alternext is more efficient for junior stocks while the NASDAQ is more efficient for technology stocks.

APPENDIX A Stylized Measure of \(D_{i,j,t}\)

In this appendix, we show an stylized example for the price discovery measure $D_{i,j,t}$. For equation (A-1), which is reproduced from equation (6) in the text,

$$\begin{aligned} \varDelta \mathbf {p}_t&= {\varvec{\Psi }}(L) \mathbf {G}^{-1}\mathbf {G} \mathbf {A}^{-1}\mathbf {B}\mathbf {v}_t \\&= \left[ \begin{array}{cc} \mathbf {D}_1(L)&\mathbf {D}_2(L) \end{array}\right] \left[ \begin{array}{c} \alpha _{\perp }' \\ \beta ' \end{array}\right] \mathbf {A}^{-1}\mathbf {B}\mathbf {v}_t \\&= \mathbf {D}_1(L) \alpha _{\perp }'\mathbf {A}^{-1}\mathbf {B}\mathbf {v}_t +\mathbf {D}_2(L) \beta '\mathbf {A}^{-1}\mathbf {B}\mathbf {v}_t \\&= \underbrace{\mathbf {D}_1(1) \alpha _{\perp }'\mathbf {A}^{-1}\mathbf {B} \mathbf {v}_t}_{\mathrm{long}\text {-}\mathrm{run}\ \mathrm{impact}\ \mathrm{denoted}\ \mathrm{as}\ {\varvec{\Phi }} \mathbf {v}_t} + \underbrace{[\mathbf {D}_1(L) \alpha _{\perp }' \mathbf {A}^{-1}\mathbf {B}-\mathbf {D}_1(1) \alpha _{\perp }'\mathbf {A}^{-1} \mathbf {B} +\mathbf {D}_2(L) \beta '\mathbf {A}^{-1}\mathbf {B}] \mathbf {v}_t}_{\mathrm{pricing}\ \mathrm{error}\ \mathrm{denoted}\ \mathrm{as}\ {\varvec{\Phi }}^*(L) \mathbf {v}_t \text { with } {\varvec{\Phi }}^*(1)\mathbf {v}_t = \mathbf {0}}. \end{aligned}$$

(A-1)

We assume the number of lags $L =1$ and then we have

$$\begin{aligned} {\varvec{\Psi }} = \left[ \begin{array}{cccc} \psi _{11} &{} \psi _{12} &{} \psi _{13} &{} \psi _{14} \\ \psi _{21} &{} \psi _{22} &{} \psi _{23} &{} \psi _{24} \\ \psi _{31} &{} \psi _{32} &{} \psi _{33} &{} \psi _{34} \\ \psi _{41} &{} \psi _{42} &{} \psi _{43} &{} \psi _{44} \end{array}\right] . \end{aligned}$$

(A-2)

For equation (A-1),

$$\begin{aligned} \mathbf {G} = \left[ \begin{array}{c} \alpha _{\perp }' \\ \beta ' \end{array}\right] , \end{aligned}$$

(A-3)

where

$$\begin{aligned} \beta = [1, -1, 0, 0]'. \end{aligned}$$

(A-4)

We need to define $\alpha _{\perp }'$. Recall $\alpha = [\alpha _{cn}, \alpha _{us},\alpha _{idcn}, \alpha _{idus}]'$ and, empirically, $\alpha _{cn}<0$, $\alpha _{us}>0$, $\alpha _{idcn}=0$ and $\alpha _{idus}=0$. Therefore, we have

$$\begin{aligned} \alpha _{\perp }' =\left[ \begin{array}{cccc} -\alpha _{us} &{} -\alpha _{cn} &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ \end{array}\right] \end{aligned}$$

(A-5)

such that $\alpha _{\perp }' \alpha = 0$. Substituting equations (A-4) and (A-5) into equation (A-3) yields

$$\begin{aligned} \mathbf {G} = \left[ \begin{array}{cccc} -\alpha _{us} &{} -\alpha _{cn} &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 1 &{}-1 &{} 0 &{} 0 \end{array}\right] , \end{aligned}$$

(A-6)

We select a stylized $\mathbf {A}$ and $\mathbf {B}$:

$$\begin{aligned} \mathbf {A} = \left[ \begin{array}{cccc} 1 &{} 0 &{} A_{cn,idcn} &{} A_{cn,idus} \\ A_{us,cn} &{} 1 &{} A_{us,idcn} &{} A_{us,idus} \\ 0 &{} 0 &{} 1 &{} A_{idcn,idus} \\ 0 &{} 0 &{} 0 &{} 1 \end{array}\right] \end{aligned}$$

(A-7)

and

$$\begin{aligned} \mathbf {B} = \mathbf {I}_4. \end{aligned}$$

(A-8)

The i.i.d. structural shocks are given by

$$\begin{aligned} \mathbf {v} = \left[ \begin{array}{cccc} v_{cn} \\ v_{us} \\ v_{idcn} \\ v_{idus} \end{array}\right] . \end{aligned}$$

(A-9)

Then we obtain

$$\begin{aligned} \mathbf {D}_1 = \left[ \begin{array}{cccc} -\psi _{11}/(\alpha _{cn}+\alpha _{us})-\psi _{12}/(\alpha _{cn}+\alpha _{us}) &{} \psi _{13} &{} \psi _{14} \\ -\psi _{21}/(\alpha _{cn}+\alpha _{us})-\psi _{22}/(\alpha _{cn}+\alpha _{us}) &{} \psi _{23} &{} \psi _{24} \\ -\psi _{31}/(\alpha _{cn}+\alpha _{us})-\psi _{32}/(\alpha _{cn}+\alpha _{us}) &{} \psi _{33} &{} \psi _{34} \\ -\psi _{41}/(\alpha _{cn}+\alpha _{us})-\psi _{42}/(\alpha _{cn}+\alpha _{us}) &{} \psi _{43} &{} \psi _{44} \end{array}\right] , \end{aligned}$$

(A-10)

$$\begin{aligned} \mathbf {D}_2 = \left[ \begin{array}{cccc} \alpha _{cn}\psi _{11}/(\alpha _{cn}+\alpha _{us})-(\alpha _{us}\psi _{12})/ (\alpha _{cn}+\alpha _{us}) \\ \alpha _{cn}\psi _{21}/(\alpha _{cn}+\alpha _{us})-(\alpha _{us}\psi _{22})/ (\alpha _{cn}+\alpha _{us}) \\ \alpha _{cn}\psi _{31}/(\alpha _{cn}+\alpha _{us})-(\alpha _{us}\psi _{32})/ (\alpha _{cn}+\alpha _{us}) \\ \alpha _{cn}\psi _{41}/(\alpha _{cn}+\alpha _{us})-(\alpha _{us}\psi _{42})/ (\alpha _{cn}+\alpha _{us}) \end{array}\right] . \end{aligned}$$

(A-11)

The permanent shocks in this stylized example are given by

$$\begin{aligned} \mathbf {D}_1 \alpha _{\bot }'\mathbf {A}^{-1}\mathbf {Bv} =[permanent_1, permanent_2, permanent_3, permanent_4]'. \end{aligned}$$

(A-12)

Please note that each permanent shock is a weighted average of the i.i.d. structural shocks, $v_{cn}$, $v_{us}$, $v_{idcn}$, and $v_{idus}$. as shown below:

$$\begin{aligned} Permanent_1&= -{{a}_{{ cn}}}\left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{11}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) {{v}_{{ us}}}\\&\quad +\, \left( -\left( {{A}_{{ cn,idcn}}}{{A}_{{ idcn,idus}}} -{{A}_{{ cn,idus}}}\right) {{a}_{{ us}}}\left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}-\frac{{{\psi }_{11}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) \right. \\&\quad -\,\left( -{{A}_{{ us,idus}}}+{{A}_{{ idcn,idus}}} {{A}_{{ us,idcn}}}-{{A}_{{ cn,idcn}}}{{A}_{{ idcn,idus}}} {{A}_{{ us,cn}}} +{{A}_{{ cn,idus}}}{{A}_{{ us,cn}}}\right) {{a}_{{ cn}}}\\&\qquad \left. \left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{11}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) -{{\psi }_{13}}\,{{A}_{{ idcn,idus}}} +{{\psi }_{14}}\right) {{v}_{{ idus}}}\\&\quad +\,\left( {{A}_{{ cn,idcn}}}{{a}_{{ us}}} \left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{11}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) \right. \\&\quad \left. -\,\left( {{A}_{{ cn,idcn}}}{{A}_{{ us,cn}}} -{{A}_{{ us,idcn}}}\right) {{a}_{{ cn}}} \left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{11}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) +{{\psi }_{13}}\right) {{v}_{{ idcn}}}\\&\quad +\,\left( {{A}_{{ us,cn}}}{{a}_{{ cn}}} \left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{11}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) -{{a}_{{ us}}}\left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{11}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) \right) {{v}_{{ cn}}} \end{aligned}$$

(A-13)

$$\begin{aligned} Permanent_2&= -{{a}_{{ cn}}}\left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{21}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) {{v}_{{ us}}}\\&\quad +\,\left( -\left( {{A}_{{ cn,idcn}}}{{A}_{{ idcn,idus}}} -{{A}_{{ cn,idus}}}\right) {{a}_{{ us}}} \left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{21}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) \right. \\&\quad -\,\left( -{{A}_{{ us,idus}}}+{{A}_{{ idcn,idus}}} {{A}_{{ us,idcn}}}-{{A}_{{ cn,idcn}}}{{A}_{{ idcn,idus}}} {{A}_{{ us,cn}}}+{{A}_{{ cn,idus}}}{{A}_{{ us,cn}}}\right) {{a}_{{ cn}}}\\&\quad \left. \left( -\,\frac{{{\psi }_{22}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{21}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) -{{\psi }_{23}}\,{{A}_{{ idcn,idus}}} +{{\psi }_{24}}\right) {{v}_{{ idus}}}\\&\quad +\,\left( {{A}_{{ cn,idcn}}}{{a}_{{ us}}} \left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} \frac{{{\psi }_{21}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) \right. \\&\quad \left. -\,\left( {{A}_{{ cn,idcn}}}{{A}_{{ us,cn}}} -{{A}_{{ us,idcn}}}\right) {{a}_{{ cn}}} \left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{21}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) +{{\psi }_{23}}\right) {{v}_{{ idcn}}}\\&\quad +\,\left( {{A}_{{ us,cn}}}{{a}_{{ cn}}} \left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{21}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) -{{a}_{{ us}}}\left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{21}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) \right) {{v}_{{ cn}}} \end{aligned}$$

(A-14)

$$\begin{aligned} Permanent_3&= -\,{{a}_{{ cn}}}\left( -\frac{{{\psi }_{32}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{31}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) {{v}_{{ us}}}\\&\quad +\,\left( -\left( {{A}_{{ cn,idcn}}}{{A}_{{ idcn,idus}}} -{{A}_{{ cn,idus}}}\right) {{a}_{{ us}}} \left( -\frac{{{\psi }_{32}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{31}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) \right. \\&\quad -\,\left( -{{A}_{{ us,idus}}}+{{A}_{{ idcn,idus}}} {{A}_{{ us,idcn}}}-{{A}_{{ cn,idcn}}}{{A}_{{ idcn,idus}}} {{A}_{{ us,cn}}}+{{A}_{{ cn,idus}}}{{A}_{{ us,cn}}}\right) {{a}_{{ cn}}}\\&\quad \left. \left( -\,\frac{{{\psi }_{32}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{31}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) -{{\psi }_{33}}\,{{A}_{{ idcn,idus}}} +{{\psi }_{34}}\right) {{v}_{{ idus}}}\\&\quad +\,\left( {{A}_{{ cn,idcn}}}{{a}_{{ us}}} \left( -\frac{{{\psi }_{32}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{31}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) \right. \\&\quad \left. -\,\left( {{A}_{{ cn,idcn}}}{{A}_{{ us,cn}}} -{{A}_{{ us,idcn}}}\right) {{a}_{{ cn}}} \left( -\frac{{{\psi }_{32}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{31}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) +{{\psi }_{33}}\right) {{v}_{{ idcn}}}\\&\quad +\,\left( {{A}_{{ us,cn}}}{{a}_{{ cn}}} \left( -\frac{{{\psi }_{32}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{31}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) -{{a}_{{ us}}}\left( -\frac{{{\psi }_{32}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{31}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) \right) {{v}_{{ cn}}} \end{aligned}$$

(A-15)

$$\begin{aligned} Permanent_4&=-\,{{a}_{{ cn}}}\left( -\frac{{{\psi }_{42}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{41}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) {{v}_{{ us}}}\\&\quad +\,\left( -\left( {{A}_{{ cn,idcn}}}{{A}_{{ idcn,idus}}} -{{A}_{{ cn,idus}}}\right) {{a}_{{ us}}} \left( -\frac{{{\psi }_{42}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{41}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) \right. \\&\quad -\,\left( -{{A}_{{ us,idus}}}+{{A}_{{ idcn,idus}}} {{A}_{{ us,idcn}}}-{{A}_{{ cn,idcn}}}{{A}_{{ idcn,idus}}} {{A}_{{ us,cn}}}+{{A}_{{ cn,idus}}}{{A}_{{ us,cn}}}\right) {{a}_{{ cn}}}\\&\quad \left. \left( -\frac{{{\psi }_{42}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{41}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) -{{\psi }_{43}}\,{{A}_{{ idcn,idus}}} +{{\psi }_{44}}\right) {{v}_{{ idus}}}\\&\quad +\,\left( {{A}_{{ cn,idcn}}}{{a}_{{ us}}} \left( -\frac{{{\psi }_{42}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{41}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) \right. \\&\left. -\left( {{A}_{{ cn,idcn}}}{{A}_{{ us,cn}}} -{{A}_{{ us,idcn}}}\right) {{a}_{{ cn}}} \left( -\frac{{{\psi }_{42}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{41}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) +{{\psi }_{43}}\right) {{v}_{{ idcn}}}\\&\quad +\,\left( {{A}_{{ us,cn}}}{{a}_{{ cn}}} \left( -\frac{{{\psi }_{42}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{41}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) -{{a}_{{ us}}}\left( -\frac{{{\psi }_{42}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{41}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) \right) {{v}_{{ cn}}} \end{aligned}$$

(A-16)

Under the condition $L=1$, the transitory shocks are given by

$$\begin{aligned} \mathbf {D}_1\beta '\mathbf {A}^{-1}\mathbf {Bv}&= [transitory_1,transitory_2, transitory_3,transitory_4]' \\&= [0,0,0,0]'. \end{aligned}$$

(A-17)

Based on the permanent-transitory decomposition given in equation (A-1) as in equation (6) in the text, we propose a new dynamic price discovery measure for innovations transmitting from market j to market i in period t, for $t=0,1,2,\ldots$,

$$\begin{aligned} D_{i,j,t} = \frac{{\varvec{\Phi }}_{i,j}+{\varvec{\Phi }}^*(t)_{i,j}}{{\varvec{\Phi }}_{i,j}}, \end{aligned}$$

(A-18)

where ${\varvec{\Phi }}_{i,j}$ (${\varvec{\Phi }}^*(t)_{i,j}$) is the i, j-th element of ${\varvec{\Phi }}$ (${\varvec{\Phi }}^*(t)$). This measure is the ratio of the sum of the total impact of innovations, including both the long-term impact and pricing errors, from market j to market i on the price to the long-term impact of innovations from market j to market i on the price in period t.

In this stylized example, all transitory shocks will not have any impact while all permanent shocks will have immediate impacts on ${\varvec{\Delta }} \mathbf {p}_t$, where $\mathbf {p}_t = [p_{cn}, p_{us}, p_{idcan},p_{idus}]'$. If we are interested in price discovery in $p_{cn}$ caused by $v_{us}$ at time 0, we could use the price discovery measure, $D_{i,j,t}$, as follows:

$$\begin{aligned} D_{cn,us,0}&= \frac{{\varvec{\Phi }}_{cn,us}+{\varvec{\Phi }}^*(0)_{cn,us}}{{\varvec{\Phi }}_{cn,us}} \\&= \frac{ -{{a}_{{ cn}}}\left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{11}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) + 0}{ -{{a}_{{ cn}}} \left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{11}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) } \\&= 1. \end{aligned}$$

(A-19)

If we are interested in price discovery in $p_{us}$ caused by $v_{cn}$ at time 0, we could use the price discovery measure, $D_{i,j,t}$, as follows:

$$\begin{aligned} D_{us,cn,0}&= \frac{{\varvec{\Phi }}_{us,cn}+{\varvec{\Phi }}^*(0)_{us,cn}}{{\varvec{\Phi }}_{us,cn}} \\&= \frac{\left( {{A}_{{ us,cn}}}{{a}_{{ cn}}} \left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{21}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) -{{a}_{{ us}}}\left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{21}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) \right) +0}{\left( {{A}_{{ us,cn}}} {{a}_{{ cn}}}\left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{21}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) -{{a}_{{ us}}} \left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{21}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) \right) } \\&= 1. \end{aligned}$$

(A-20)

When the market-specific shocks are the same or $v_{us} = v_{cn}$, we can compare the long-run role of price discovery by comparing ${\varvec{\Phi }}_{cn,us}$ with ${\varvec{\Phi }}_{us,cn}$ in the ratio form:

$$\begin{aligned} \frac{{\varvec{\Phi }}_{cn,us}}{{\varvec{\Phi }}_{us,cn}} =\frac{-{{a}_{{ cn}}}\left( -\frac{{{\psi }_{12}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{11}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) }{\left( {{A}_{{ us,cn}}}{{a}_{{ cn}}} \left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}} -\frac{{{\psi }_{21}}}{{{a}_{{ us}}}+{{a}_{{ cn}}}}\right) -{{a}_{{ us}}}\left( -\frac{{{\psi }_{22}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}-\frac{{{\psi }_{21}}}{{{a}_{{ us}}} +{{a}_{{ cn}}}}\right) \right) }. \end{aligned}$$

(A-21)

As the above expression shows, the information channel ($A_{us,cn}$), the error correction coefficients ($a_{cn}$ and $a_{us}$), and moving average coefficients ($\psi _{12}, \psi _{12}, \psi _{21}, \psi _{22}$) all play a role. When $L = 1,2,\ldots$, the transitory shocks will have some noticeable impact on prices but will vanish over time. As this stylized example shows, the dynamic price discovery measure, $D_{i,j,t}$, will recover more information than $\frac{|\alpha _{cn}|}{|\alpha _{us}|}$ does. The dynamic price discovery measure is affected by the generic moving average representation of reduced form residuals over L lags (${\varvec{\Psi }}(L)$), the decomposition of error correction terms and integration terms ($\mathbf {G}$), and the information channels in the markets ($\mathbf {A}$ and $\mathbf {B}$).