Measures of contributions to price discovery: a comparison☆
Introduction
This note attempts to clarify the relation between two competing definitions of the contribution to price discovery in market microstructure models: (i) the information shares as defined in Hasbrouck (1995) and (ii) the common factor component weight of Gonzalo and Granger (1995), applied in the market microstructure literature by Booth et al. (1999), Chu et al. (1999) and Harris et al. (2000).
Let Xt be the vector of prices for the same security in n markets. Each individual price series xit is non-stationary, but because of long run arbitrage, the series will be cointegrated. The multivariate price process is given by the vector error correction modelwhere zt=α′Xt are the stationary error correction terms.
There are several ways to decompose the price vector in a permanent, I(1), component and a transitory, I(0), component. The traditional decomposition is the Stock and Watson (1988) decomposition where the permanent component is a random walk with serially uncorrelated increments. This decomposition works from the vector moving average representation of the modelwhich can be written asIf the vector Xt is cointegrated, the Granger represention theorem (Engle and Granger, 1987) states that C(1) satisfies α′C(1)=0 and C(1)γ=0. Thus, we may writewith α⊥′α=0 and θ′γ=0. The term is the common stochastic trend component. The common trend innovations θ′εt are serially uncorrelated by construction. Notice that this common trend is defined as a function of the innovations εt and therefore involves current as well as lagged values of Xt.
Gonzalo and Granger (1995) propose an alternative decomposition of Xt into permanent and transitory components, where the components are linear combinations of Xt alone, and do not involve lagged values of Xt:with ft=β′Xt and zt=α′Xt as before. As an additional identifying assumption, GG assume that there is no long run Granger causality from zt to ft. It turns out that this assumption implies for the permanent component A1ft=α⊥ft andwith γ⊥′γ=0 and hence β′γ=0. This definition of the common factor is different from the Stock-Watson definition because the changes in ft are serially correlated.
How are the decompositions related? An instructive way to look at this issue is by substituting the Stock-Watson decomposition of Xt into the Gonzalo-Granger definition of the common factors ft:where st is stationary. So, we see that the random walk part of the GG common factor ft is identical to the Stock-Watson common factor (this result is also stated in Proposition 5 of Gonzalo and Granger).1 Sharing the same random walk component is however not a very special property, because all linear combinations of market prices have the same random walk component.2
Section snippets
Market microstructure models
For the applications to market microstructure models, it is natural to assume that the prices share the same, scalar, common non-stationary component. Because of market microstructure frictions, however, there are temporary deviations from the equilibrium price, but these are transient (stationary). Hence, the cointegrating rank of the VECM is n−1, and γ is an n×(n−1) matrix. The n−1 dimensional vector of error correction terms, zt, can be defined in many ways but the simplest definition is
Examples
A few simple examples may clarify these results. In both examples there are two markets. The error correction term is the difference between the prices on each market, zt=X1t−X2t. For simplicity, there are no further lagged price effects, so Ak=0 for all k. The VECM then is
The first example concerns the one-way price discovery hypothesis. Under that hypothesis, only the second market error corrects to the price difference, for some strictly positive γ
Conclusion
This note showed that there is a very close relation between the two definitions of contribution to price discovery. The coefficients β of the GG common factors are just normalized elements of the vector θ that defines the Stock-Watson common stochastic trend (Hasbrouck's efficient price). The major difference between the two approaches is the role of the variance of the innovations. The GG definition only works with the ‘weight’ that the innovation of market i has in the increment of the
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2022, Journal of Economic Dynamics and ControlCitation Excerpt :Both the IS and CS measures build their fundamentals upon the modeling of price changes through VECMs, with the substantial difference that while the CS is defined only in terms of speeds of adjustment toward the common trend (i.e. markets with lower cointegration loadings rapidly adjust and are thus more informative), the IS measure is more concerned with variations in the prices and seeks to measure the amount of information generated by each market. Both approaches have their merits and limits which have been documented by comprehensive discussions in the literature (Baillie et al., 2002; De Jong, 2002; Harris et al., 2002a; 2002b; Hasbrouck, 2002b; Lehmann, 2002). The IS approach, compared to the CS one, has a richer specification since it considers the speed of adjustment together with the relative share of variance of the efficient price process accounted by each market.
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I thank Joel Hasbrouck and Bruce Lehmann (the editor) for their comments on an earlier draft of this note.