Information Share. Bernt Arne Ødegaard 29 May 2018

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1 Information Share Bernt Arne Ødegaard 29 May 2018 Contents 1 Information Share, or, measuring the importance of different markets Setup Information share of a market Information share in R 2 3 Example of IS calculation - NHY Literature Information Share, or, measuring the importance of different markets The Information Share concerns ways of measuing which market place is most important in price discovery. It is attributed to Hasbrouck (1995). It is an example of the usage of cointegration. Let us use the notation of Hasbrouck (1995) Setup A security is traded in two separate markets. The evolution of prices is supposed to follow p 1,t = p 1,t 1 + w t p 2,t = p 1,t 2 + ε t p 1,t and p 2,t can be both transaction prices or quotes. w t and ε t are zero mean iid disturbances. Here p 1,t is a random walk (integrated). The price in the second market follows the price in the first market with a delay. It is thefore also integrated (of order 1). The difference between the two prices is p 1,t p 2,t = p 1,t 1 + w t p 1,t 2 + ε t = (p 1,t 2 + w t 1 ) + w t p 1,t 2 + ε t = w t 1 + w t ε t Since these variables are mean zero noise, the difference between the two prices is stationary. The price difference in the two markets is therefore cointegrated. (A linear combination of integrated variables is stationary.) Two representations of the model: Vector moving average representation p 1,t = w t 1 For a textbook, see Hasbrouck (2008). 1

2 Common trends representation p 2,t = δp 1,t 2 + ε t ε t 1 = w t 2 + ε t ε t 1 ( t ) p 1,t = p 1,0 + w s s=1 ( t ) p 2,t = p 2,0 + w s + (w t w t 1 + ε t ) s=1 Price series sum of initial value, common random-walk term, and stationary term. Interpretation: random walk component of security price the implicit efficient price. Information share: attribution of the source of variation in the random-walk component to the innovations in the various markets. Estimable representation of the model: Error correction model p 1,t = w t p 2,t = (p 1,t 1 p 2,t 1 ) p 1,t 1 + ε t However, not an unique representation. General representation ( t ) p t = p 0 + ψ e s ı + Ψ (L)e t s=1 e t is a vector of disturbances with covariance matrix Ω. 1.2 Information share of a market Suppose p t represents prices in n different markets. The information share S j should measure the contribution of market j to the total variance of the random walk component. Unfortunately, when the covariance matrix Ω is not diagonal there is not a unique solution to this calculation. Instead, Hasbrouck (1995) shows that one can find upper and lower bounds on the information share. 2 Information share in R There is an R routine that implements estimation of information shares. pdshare, in the ifrogs library. It is not part of the CRAN library, the following is necessary to install it: install.packages("ifrogs", repos=" The short documentation of the routine estimating information share is shown in figure 1 2

3 Figure 1 Documentation R routine for Information Share pdshare package:ifrogs R Documentation Computes information share & component share weights Description: This function implements the two most commonly used techniques to measure price discovery in multiple markets. These are information share (Hasbrouck (1995)) and the component share approach (Booth et al. (1999), Chu et al. (1999), Harris et al. (2002)) which utilises the Gonzalo and Granger (1995) permanent-transitory decomposition. Usage: pdshare(x, override.lags = NULL, lag.max = 10) Arguments: x: Is a numeric matrix / data frame which has two columns with log prices of two markets. override.lags: Is an integer that specifies user-defined lags to be used for VECM estimation. Uses NULL as default. See Details. lag.max: Is an integer that specifies the maximum lag order to be used for selecting the number of lags in VARselect. Is used when override.lags is NULL. Uses 10 as default. Details: The function estimates the information share (IS) and component share weights for a two market case. This is done by first estimating the VECM model for two cointegrated price series using the functionality in package urca. The subsequent VAR and VMA representation are obtained using package vars. The number of lags to be used for VECM estimation can be pre-specified by the user in the argument override.lags. The default is NULL. If the argument is kept as NULL, then the number of lags are decided based on the AIC criterion using the function VARselect from package vars. The maximum number of lags to be used in VARselect can be specified using the argument lag.max. To achieve the lower and upper bound of IS using triangularization of covariance matrix, the function first computes IS for the supplied ordering. The resulting IS estimate maximises the share of Market 1 and minimises the share of Market 2. The results are saved in is.original.ordering. Subsequently, the ordering is reversed. The corresponding IS estimate maximises the share of Market 2 and minimises it for Market 1. Value: A list of the following five elements: is.original.ordering: Information shares of Market 1 & 2 under the supplied ordering is.reversed.ordering: Information shares of Market 2 & 1 under the reversed ordering component.share: Component share weights of Market 1 & 2 var.covar.matrix: Variance covariance matrix of the residuals lags.used: Number of lags used in VECM estimation 3

4 3 Example of IS calculation - NHY The stock NHY (Norsk Hydro) is traded on several exchanges. We are interested in how much of the price discovery happens in the main market (OSE). In the case of NHY, another market with significant trading is Chi-X. The basis for inference is contemporaneous observations of prices at two market places. The prices can be transanction prices, quotes, or midprices. In our example we collect five-second observations of the last transaction price (i.e. if there is no transactions, we keep the previous transaction prices. Table 1 shows the first few elements of the data we use. Note that the time used is the time recorded by Reuters, in London (UTC). By matching on London time, we have some confidence that traders in different places actually have seen the last transaction in the other market. Anyway, together with the last transaction price we also indicate the number of transactions within that five second interval. From that we observe that Oslo is the place with more transactions. Table 1 Observations of five second prices time;nhy.ol-price;nhy.ol-no trans;nhyo.chi-price;nhyo.chi-no_trans 2012-Dec-03 08:00:40;27.3;10;27.3; Dec-03 08:00:45;27.3;0;27.3; Dec-03 08:00:50;27.24;3;27.3; Dec-03 08:00:55;27.27;1;27.3; Dec-03 08:01:00;27.27;0;27.3; Dec-03 08:01:05;27.24;1;27.3; Dec-03 08:01:10;27.24;4;27.3; Dec-03 08:01:15;27.24;0;27.26; Dec-03 08:01:20;27.24;0;27.26; Dec-03 08:01:25;27.25;1;27.26; Dec-03 08:01:30;27.25;0;27.26; Dec-03 08:01:35;27.24;4;27.26; Dec-03 08:01:40;27.24;0;27.26; Dec-03 08:01:45;27.24;0;27.26; Dec-03 08:01:50;27.24;0;27.26; Dec-03 08:01:55;27.24;0;27.26; Dec-03 08:02:00;27.24;0;27.26; Dec-03 08:02:05;27.24;0;27.26; Dec-03 08:02:10;27.24;0;27.26; Dec-03 08:02:15;27.24;3;27.26; Dec-03 08:02:20;27.24;0;27.26; The first few observations of the input file with five second prices of NHY on 3 dec traded in Oslo, and NHYo.CHI NHY traded at Chi-X. Here NHY.OL is the code for NHY Let us now calculate the IR for the two exchanges trading of NHY on this day. Load library and Prepare the data: > infile <- "/home/bernt/data/2017/reuters_trade_data/2017_01_sampled_prices/five_second_prices_nhy_2 > library(ifrogs) > data <- read.table(file=infile,sep=";",header=true) > head(data) time NHY.OL.price NHY.OL.no.trans NHYo.CHI.price Dec-03 08:00: Dec-03 08:00: Dec-03 08:00: Dec-03 08:00: Dec-03 08:01: Dec-03 08:01:

5 NHYo.CHI.no_trans > data <- na.omit(data) > NHY.OL <- data$nhy.ol.price > NHYo.CHI <- data$nhyo.chi.price > > > logprices <- as.matrix(cbind(log(nhy.ol),log(nhyo.chi))) > colnames(logprices) <- c("lnnhy.ol","lnnhyo.chi") > head(logprices) lnnhy.ol lnnhyo.chi [1,] [2,] [3,] [4,] [5,] [6,]

6 Estimating IS without specifying lag structure: > is <- pdshare(logprices) > print(is) $is.original.ordering IS lnnhy.ol lnnhyo.chi $is.reversed.ordering IS lnnhyo.chi lnnhy.ol $component.share CS lnnhy.ol lnnhyo.chi $var.covar.matrix lnnhy.ol lnnhyo.chi lnnhy.ol e e-08 lnnhyo.chi e e-08 $lags.used [1] 2 The numbers we want to look at is the information share for OSE. The Hasbrouck estimation is based on permuting the ordering of exchanges, and calculating a minimum and a maximum of the information share for a given exchange. The results above says that the IS for trading NHY at the OSE is in the interval [0.71, 0.94], which is supported with the CS for the OSE of

7 Let us also show the estimation imposing a lag structure with 12 lags: > is <- pdshare(logprices,override.lags=12) > print(is) $is.original.ordering IS lnnhy.ol lnnhyo.chi $is.reversed.ordering IS lnnhyo.chi lnnhy.ol $component.share CS lnnhy.ol lnnhyo.chi $var.covar.matrix lnnhy.ol lnnhyo.chi lnnhy.ol e e-08 lnnhyo.chi e e-08 $lags.used [1] 12 7

8 3.1 Literature The Information Share literature starts with Hasbrouck (1995), which introduces his calculation of a minimum and maximum information share. He argues that there is a degree of choice here that can not be bypassed, and therefore end up calculating a range instead of a single numer. A related measure, component share, is introduced by Harris, McInish, Shoesmith, and Wood (1995), Booth, So, and Tse (1999) and Chu, liang Gideon Hsieh, and Tse (1999), as an application of results in Gonzalo and Granger (1995). The CS is also called the common factor component. The relative merits of the two measures keep being discussed. A special issue of the Journal of Financial Markets in 2002 tried to conclude on the relative merits, Lehmann (2002); Baillie, Booth, Tse, and Zabotina (2002); Harris, McInish, and Wood (2002a); Hasbrouck (2002); Harris, McInish, and Wood (2002b) According to the summary by Lehmann, the different approaches correspond to different error correction models, and willingness to make restrictions on covariance structueres. But both approaches are argued to carry information. So it does not hurt to check both the Information Share and Component Share. The discussion does not end there. E.g. Yan and Zivot (2010) revisit the IS/CS discussion, and argues that IS is better. References Richard T. Baillie, G. Geoffrey Booth, Yiuman Tse, and Tatyana Zabotina. Price discovery and common factor models. Journal of Financial Markets, 5(3): , Price Discovery. G Geoffrey Booth, Raymond W So, and Yiuman Tse. Price discovery in the German equity index derivatives markets. Journal of Futures Markets, 19(6): , Quentin C Chu, Wen liang Gideon Hsieh, and Yiuman Tse. Price discovery on the S&P 500 index markets: An analysis of spot index, index futures, and SPDRs. International Review of Financial Analysis, 8(1):21 34, Jesus Gonzalo and Clive W J Granger. Estimation of common long-memory components in cointegrated systems. Journal of Business and Economic Statistics, 13(1):27 35, F Harris, TH McInish, and R Wood. Security price adjustments across exchanges: An investigation of common factor components for dow stocks. Journal of Financial Markets, 5(3): , 2002a. Frank H deb Harris, Thomas H McInish, G Shoesmith, and Robert A Wood. Cointegration, error correction and price discovery on informationally-linked security markets. Journal of Financial and Quantitative Analysis, 30: , Frederick H.deB. Harris, Thomas H. McInish, and Robert A. Wood. Common factor components versus information shares: a reply. Journal of Financial Markets, 5(3): , 2002b. Price Discovery. Joel Hasbrouck. One security, many markets: Determining the contributions to price discovery. Journal of Finance, 50(4): , Joel Hasbrouck. Stalking the "efficient price" in market microstructure specifictions: an overview. Journal of Financial Markets, 5: , Joel Hasbrouck. Empirical Market Microstructure. Oxford University Press, Bruce N Lehmann. Some desiderata for the measurement of price discovery across markets. Journal of Financial Markets, 5(3): , Bingcheng Yan and Eric Zivot. A structural analysis of price discovery measures. Journal of Financial Markets, 13(10): 1 19,