Price discovery and common factor models

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Abstract

If a financial asset is traded in more than one market, common factor models may be used to measure the contribution of these markets to the price discovery process. We examine the relationship between the Hasbrouck (J. Finance (50) (1995) 1175) and Gonzalo and Granger (J. Bus. Econ. Stat. 13 (1995) 27) common factor models. These two models complement each other and provide different views of the price discovery process between markets. The Gonzalo and Granger model focuses on the components of the common factor and the error correction process, while the Hasbrouck model considers each market's contribution to the variance of the innovations to the common factor. We show that the two models are directly related and provide similar results if the residuals are uncorrelated between markets. However, if substantive correlation exists, they typically provide different results. We illustrate these differences using analytic examples plus a real world example consisting of electronic communications networks (ECNs) and other Nasdaq market makers.

Introduction

A financial asset such as a stock often trades in multiple markets. Its price in any given market is determined (discovered) by news being gathered and interpreted in one or more of these markets. Because only the trading venue differs, intermarket arbitrage keeps the prices in the different markets from drifting apart.1 Expressed in econometric terms, the prices are cointegrated I(1) variables, which means that the price series share one or more common stochastic factors. If there is only one common factor (and this is often the case), we typically refer to it as the implicit efficient price. It is this price that is driven by news, making it the source of permanent movement in the prices of all markets.

Currently there are two popular common factor models that are used to investigate the mechanics of price discovery: Hasbrouck (1995) and Gonzalo and Granger (1995). Hereafter we refer to these models as information shares (IS) and permanent-transitory (PT), respectively. Both models use the vector error correction model (VECM) as their basis, and Hasbrouck (1996) points out that the VECM is consistent with several market microstructure models in the extant literature. Despite this initial similarity, the IS and PT models use different definitions of price discovery. Hasbrouck (1995) defines price discovery in terms of the variance of the innovations to the common factor. Thus the IS model measures each market's relative contribution to this variance. This contribution is dubbed the market's information share. Gonzalo and Granger (1995), however, are concerned with only the error correction process. This process involves only permanent (as opposed to transitory) shocks that result in a disequilibrium. In the price discovery context, disequilibria occur because markets process news at different rates. The PT model measures each market's contribution to the common factor, where the contribution is defined to be a function of the markets’ error correction coefficients.

To more clearly distinguish between the two approaches of measuring price discovery, consider the case a stock being traded in two markets with the prices in these two markets being not only cointegrated but also highly correlated. The high correlation suggests that the two prices move together most of the time. Nevertheless, correlation and cointegration are different statistical concepts. Assume that the first market's price responds to deviations from the second market's price described by the error correction term, but the second market does not respond to deviations from the first market. According to the PT model price discovery only occurs in the second market.2 In contrast, the IS metric suggests that both markets contribute to price discovery because of the high correlation between the two markets.

The above example suggests that there are two important and related price discovery questions to be answered. First, from a microstructure perspective, which measure of price discovery is more useful in helping us understand how markets work? Second, how are the IS and PT metrics related and under what conditions, if any, do they provide the same or similar answers? The purpose of this paper is to answer the second question and in doing so provide some clues to resolve the issues raised by the first question.

We show that the IS and PT models are directly related and both models’ results are primarily derived from the error correction vector in the VECM. They provide similar results if the VECM residuals are uncorrelated. However, if substantial correlation exists, the two models usually provide different results. This is a direct result of Hasbrouck (1995) incorporating contemporaneous correlation in his metric, but Gonzalo and Granger (1995) not doing so in constructing their measure. Hasbrouck (1995) handles this correlation by using Cholesky factorization, which requires that the prices be ordered. Because the IS results are order dependent, Hasbrouck (1995) suggests that different orders be used so that upper and lower information share bounds can be calculated. Unfortunately, the bounds are often very far apart. We provide evidence to support the use of the mean of the bounds to resolve the interpretational ambiguities.

The remainder of this paper is organized in the following fashion. In Section 2 we discuss the theoretical background of cointegration, error correction, and common factors. We show that the IS and PT models are related. Section 3 provides some empirics and is divided into two parts. The first part compares the results of the two models using three analytical examples. These examples are distinguished from each other by different error correction values and each example is examined using different contemporaneous correlation values. The second part revisits Huang (2000) and examines the two price discovery metrics for five groups of Nasdaq participants. We offer concluding remarks in Section 4.

Section snippets

Cointegration, error correction, and common factors

Consider two cointegrated I(1) price series, Yt=(y1t,y2t)′ with the differential being the error correction term, i.e., zt=βYt=y1ty2t, and, therefore, the cointegrating vector is β=(1,−1)′. Both the IS and PT models start from the estimation of the following VECM:ΔYt=αβ′Yt−1+j=1kAjΔYt−j+et,where α is error correction vector and et is a zero-mean vector of serially uncorrelated innovations with covariance matrix Ω such thatΩ=σ12ρσ1σ2ρσ1σ2σ22.σ1222) is the variance of e1t(e2t) and ρ is the

Some empirics

As indicated in the previous section, the IS model results typically depend on the ordering of the variables in the Cholesky factorization of the innovation covariance matrix. The first (last) variable in the ordering tends to have a higher (lower) information share, and this discrepancy may be very large if the series’ innovations are highly contemporaneously correlated. In this section we empirically explore this phenomenon and compare IS model information shares to PT model common factor

Concluding remarks

The common stochastic factor, or implicit efficient price, in cointegrated financial series has received substantial attention from the researchers continuously trying to refine their estimations of market quality and efficiency. We explore the relationship between the popular common-factor models developed by Hasbrouck (1995) and Gonzalo and Granger (1995).

Both models use the VECM as their starting point. The feature that distinguishes them from each other is that Hasbrouck (1995) decomposes

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We thank an anonymous referee, Bruce Lehmann (the editor), and Joel Hasbrouck for valuable comments. All errors are our own responsibility. The paper was completed when he and Zabasina were in Binghamton University.

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