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.

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Notes

  1. 1.

    There is another literature on price discovery on dual class stocks (see Wang and Yang 2015). We do not study how to price dual class stocks that have different voting rights.

  2. 2.

    This approach is quite consistent with that of Gonzalo and Granger (1995).

  3. 3.

    Out of these 115 stocks, 106 are Canadian stocks and 9 are the U.S. stocks.

  4. 4.

    Due to the space limitation, we do not report the simulation design and results in this paper.

  5. 5.

    As Yan and Zivot (2010) note, this measure may not be unique.

  6. 6.

    Frijns and Schotman (2009) propose a structural model in tick time for the quote setting dynamics on NASDAQ. Otsubo (2014) proposes a structural model and three price discovery measures based on the structural model but the model is only suitable to non-overlapping markets.

  7. 7.

    For example, Yeh et al. (2016) show that stock sales timing and profitability of insiders are highly correlated.

  8. 8.

    The cointegration relations under consideration in this study have no deterministic components, which lie outside the cointegrating space in the sense of Johansen (1991).

  9. 9.

    See Chang (1998) for the role of insiders and Securities and Exchange Commission (2010) for the role of high frequency traders.

  10. 10.

    The simulation results are available upon request.

  11. 11.

    The asset pricing models suggest that the return of a particular stock in a market is priced in relation to the return of that market portfolio that contains this stock, among other factors, but not the other way around. Meantime, the law of one price also ensures the equilibrium between the two market prices of, therefore the resulting returns on, the same constituting stock. Hence, the returns on this stock in the two markets are jointly influenced by the two market indices.

  12. 12.

    Please see Wu et al. (2015).

  13. 13.

    As proved by Wu et al. (2015), since \(\alpha _{\perp }'\mathbf {A}^{-1}\mathbf {B}\) still contains some transitory components, \(\mathbf {D}_1(1) \alpha _{\perp }'\mathbf {A}^{-1}\mathbf {B}\ne \mathbf {D}_1(0) \alpha _{\perp }'\mathbf {A}^{-1}\mathbf {B}\) and (\(\mathbf {D}_1(0) \alpha _{\perp }'\mathbf {A}^{-1}\mathbf {B}-\mathbf {D}_1(1) \alpha _{\perp }'\mathbf {A}^{-1}\mathbf {B}\)) is a part of the initial pricing discrepancies. For the detailed proof, see Wu et al. (2015).

  14. 14.

    In the empirical research, we can specify a cutoff point T at the time lag when pricing discrepancies vanish.

  15. 15.

    Quebec’s official language is French while English and French are official languages in Canada.

  16. 16.

    The Canadian exchanges adopted decimal pricing on April 15, 1996. The NASDAQ completed decimal pricing on March 12, 2001. The NYSE completed decimal pricing on September 11, 2001.

  17. 17.

    The data information can be found at https://www.google.com/googlefinance/disclaimer/. R API programmed by the authors is used to extract the intraday data every 2 weeks from February 14, 2016 to July 28, 2017. Google terminated Google Finance in mid-November 2017. We use the TSX 60 ETF and S&P 500 ETF as the proxies for the market indices. For each actively traded stock, its 1-min intraday data cover 391 data points (for 390 min) in each trading day, 385 trading days (for 15 months) in the entire sample, and in each of the two stock markets. That is, for each actively traded stock, there are about 301,070 (\(=2 \times 391 \times 385\)) data points. Some stocks that are traded less frequently would have less data points over time across the two markets.

  18. 18.

    We select these frequencies (1-, 5-,10-,15-, 30-, and 65-min frequencies) as they will not omit the critical end of day trading in the whole day which has 390 trading min. These frequencies can be multiplied by an integer to match 390.

  19. 19.

    The R package BatchGetSymbol is used to extract the end-of-day data.

  20. 20.

    According to Heinkel and Kraus (1988), if a stock trades less than 20% of all trading days, it is considered a thinly traded stock. Using this benchmark in the intraday setting, if a stock trades less than 20 % (78 min) of a trading day (390 min) in at least one of the two stock markets, it is considered a thinly traded stock.

  21. 21.

    Chiang and Fong (2001), Brooks et al. (2003), Adams et al. (2004), and Hautsch et al. (2011) examine how equity market prices and returns react, respectively, to the futures and option markets, unanticipated events, inflation news, and macroeconomic news using minute-by-minute data. Ellul (2006) analyzes the price discovery process across markets around large trades using 5-min frequency data. Chaboud et al. (2014) study how algorithmic trading affect the pricing efficiency of foreign exchange using minute-by-minute data. Hatheway et al. (2017) and Menkveld et al. (2017) study the impact of dark venues on price discovery using 1-min data sampled from much higher frequency data.

  22. 22.

    The simulation results are available upon request.

  23. 23.

    For example, when we use the intraday data of the 60-min frequency, the data are sampled every 60 min starting from 9:30 till 360 min. We shift this sample from 9:30 EST to 9:31 EST, 9:32 EST, ..., 10:00 EST, respectively, and the last point from 15:30 EST to 15:31 EST, 15:32 EST, ..., 16:00 EST for 31 shifts, respectively.

  24. 24.

    We multiply time lag t by the intraday data frequency in minutes to convert a particular number of time lags t into the number of lags in minutes for the intraday data. For example, Table 5 provides the average and standard error of \(D_{i,j,t}\) at intraday data of 5-min frequency, so the column \(t=10\) in Table 5 corresponds to information transmission at lags of 50 min (\(=10 \times 5\) min).

  25. 25.

    The country difference observed here is based on the pairwise comparison across the two countries. The net effect of country differences may differ when all other factors are held constant as is the case in the panel data model discussed later.

  26. 26.

    The ticker symbol for Barrick Gold became GOLD on January 2, 2019 after Barrick and Randgold merged into one corporation still named as Barrick Gold.

  27. 27.

    We use \(\log (S)\) as this logarithmic transformation gives the panel model a better fit.

  28. 28.

    Among the 115 companies studied, 106 are Canadian and 9 are American. Therefore, our inference would focus on Canadian corporations and their stocks.

  29. 29.

    The 115 companies are classified into 8 sectors. There are 70 companies in the basic materials sector, 10 companies in the technology sector, 9 companies in the financial sector, and 26 companies in the other five sectors (consumer goods, heath care, industrial goods, services, and utilities). Eun and Sabherwal (2003) find that the sectors do not affect price discovery significantly. In this research, we reexamine price discovery efficiency across sectors.

  30. 30.

    Eun and Sabherwal (2003) use the trade-by-trade data while we use the minute-by-minute data. Therefore, we use the trading volume variation within a small window, say 1 min, to get a sense of medium trades. When the variation is small, this is a reflection of less large trades and/or less small trades but more medium trades.

  31. 31.

    A higher level of medium trades is represented by a smaller standard deviation of the minute-by-minute trading volumes.

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Acknowledgements

Lei Wu thanks the National Natural Science Foundation of China for support (Grant No. 71671). Kuan Xu thanks University of Toronto and Renmin University of China for providing visiting opportunities during the Winter and Summer of 2018. The authors wish to thank Dr. C.-F. Lee, Editor-in-Chief, Review of Quantitative Finance and Accounting, and two anonymous referees for helpful comments and suggestions. The authors also wish to thank John Galbraith, Yin-Feng Gau, Victoria Zinde-Walsh, Xue Wang, and other participants at the 23rd Conference on the Theory and Practices of Securities and Financial Markets at National Sun Yat-sen University, the Frontiers of Business Research in China 2016 International Symposium, and the Canadian Econometrics Study Group 2017 Meeting for helpful comments on earlier drafts of the paper. Any remaining errors are of our own.

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

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 ij-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 \(\mathbf {B}\)).

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Wu, L., Xu, K. & Meng, Q. Information flow and price discovery dynamics. Rev Quant Finan Acc (2020). https://doi.org/10.1007/s11156-020-00896-8

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Keywords

  • Financial markets
  • Price discovery
  • Information flow

JEL classification

  • C5
  • F3
  • G1