Feb 16, 2001 - demands (except their own, of course) that keeps them from becoming more far-sighted. Likewise, the far-sightedness in market microstructure ...
Reconciling Competitive And Strategic Approaches To Price Discovery In Financial Markets: The Empirical Evidence
Elena Asparouhova‡ , Peter Bossaerts§ and Charles Plott¶
16 February 2001
‡ California
Institute of Technology
§ California
Institute of Technology and CEPR
¶ California
Institute of Technology
1
Reconciling Competitive And Strategic Approaches To Price Discovery In Financial Markets: The Empirical Evidence
Elena Asparouhova, Peter Bossaerts and Charles Plott
Abstract: Price dynamics are studied on the basis of more than 11,000 transactions from twelve large-scale financial markets experiments. Transaction price changes are strongly correlated with the excess demands that form the core of competitive general equilibrium theory. The evidence for market microstructure theory (which focuses on strategic interaction, appealing to game theory) is more qualified. On the one hand, more predictability is uncovered than expected by chance. In particular, imbalances in the book too often predict subsequent transaction price changes. On the other hand, the nature of the predictability changes in unstructured ways, rendering it essentially useless. Weighted average quotes in the book strongly correlate with excess demands, providing evidence for a link between (competitive) general equilibrium theory and (strategic) market microstructure theory. There is no inconsistency in the findings that transaction price changes correlate with excess demands yet not with information in the book (at least not in structured ways) while at the same time the book correlates with excess demands. Indeed, the findings can be reconciled because weighted average quotes in the book explain part of the variance of transaction price changes that is unaccounted for by excess demands.
Keywords: Price Discovery, Stability Analysis, Walrasian Tatonnement, Perfect Competition, Game Theory, Market Microstructure Theory, Experimental Finance, General Equilibrium Theory, Asset Pricing Theory.
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Reconciling Competitive And Strategic Approaches To Price Discovery In Financial Markets: The Empirical Evidence¶
Elena Asparouhova, Peter Bossaerts and Charles Plott
1
Introduction
Price discovery is taken here to refer to the process with which markets reach their general equilibrium. With the risk of oversimplifying, one can recognize two approaches to studying price discovery. The first approach assumes that the process is competitive and stylized, essentially with excess demands correlating with price changes. We will refer to it as the Walrasian model. Originally, it was a tool to study stability of general equilibrium (see, e.g., [10, 2]), but we will consider it to be a simple and convenient model of competitive price discovery. The Walrasian model uses information parsimoniously: agents’ demands change with price changes, and hence, only price changes need to be commonly observed. The Walrasian model can be criticized, however, in that it is myopic. Agents’ demand adjustments are reactive, without much foresight into future price and demand adjustments. Later on, this criticism will be qualified. More recently, a second approach to studying price discovery has become popular, known as market microstructure theory. This approach focuses on the strategic aspects of trading in the marketplace, and hence, uses game-theoretic arguments. At the core of its predictions is that trade can only take place at prices equal to expected future (equilibrium) values given the size and nature (buy/sell) of the transaction. See [8, 9]. As a consequence, transaction price changes are not predictable using past information on quotes and trades (barring inventory costs and/or risk ¶ Financial
support was provided by the National Science Foundation, the California Institute of Technology, and the R.G. Jenkins
Family Fund.
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aversion). Market microstructure theory is primarily forward-looking, unlike the Walrasian model. However, it has a major shortcoming. Markets are supposed to reach general equilibrium with a type of game (and corresponding Bayesian Nash equilibrium) that requires far more common knowledge than the very general equilibrium that is to be discovered. Market microstructure theory seems to merely push back the question of equilibration, to a level where complexity really becomes demanding. It will be argued here that the the approaches need not contradict each other. The Walrasian approach predicts that price changes are correlated with excess demands. The market microstructure approach predicts that transaction price changes are not predictable. To the extent that excess demands are unobservable to agents, the two approaches can be consistent. If so, the Walrasian model can hardly be called myopic; it is agents’ inability to observe excess demands (except their own, of course) that keeps them from becoming more far-sighted. Likewise, the far-sightedness in market microstructure theory must not be exaggerated: agents lack the common knowledge needed to accurately predict other agents’ strategies. This paper provides an empirical exploration of the relative value of the two approaches. Are (transaction) price changes correlated with excess demands as predicted by the Walrasian model? Can transaction price changes not be predicted using information in the order book as implied by market microstructure theory? Are the two approaches related in that there is correlation between excess demands and the information in the book? Can the two be reconciled because the prediction error of the Walrasian model correlates with information in the book? While the answer to the second question has to be qualified, this paper reports supportive evidence based on more than 11,000 transactions from twelve large-scale financial markets experiments. Our using experimental data can be justified quite simply by the fact that supplies and payoff distributions are known in an experimental context (being design features), making excess demand readily computable (up to a scaling factor). In the field, neither supplies nor payoff distributions are known accurately and the exercises we perform in this paper would not generate equally clear answers. The experiments involved large-scale (up to 70 subjects) experimental markets with three or four securities.
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Our findings have implications for the quality of a market, with which we understand its ability to quickly discover general equilibrium. As the correlation between orders in the book and excess demands increases, the speed with which markets are able to reach their general equilibrium is enhanced. This correlation can only be increased if traders are given incentives to quickly reveal their true demands. This also means that market microstructure is crucial for markets to discover their general equilibrium.
2
The Theory
As mentioned in the Introduction, we plan to contrast two theoretical approaches to modeling price discovery in financial markets. In the first approach, the oldest one, price discovery is modeled as a competitive Walrasian tatonnement, as if a Walrasian auctioneer calls out prices, investors formulate their demands given those prices, and the auctioneer adjusts prices as a function of the sign and size of the excess demands. In the second approach, price discovery is modeled as the outcome of a trading game, whereby traders strategically submit quotes and orders taking into account the conjectured actions and information of others. In what follows, we will distill both approaches into a number of simple yet general mathematical restrictions. These restrictions will be brought to the data later in the paper. In this section, we will also determine to what extent the two approaches can be consistent with each other. In particular, we will demonstrate that the two approaches are not necessarily mutually exclusive. It will obviously be important to to confirm that consistency is not merely a theoretical possibility, but that the data support both approaches as well. Before we present the two price discovery modeling strategies, however, we should elaborate on the general equilibrium environment we have in mind.
2.1
The General Equilibrium Environment
To simplify the analysis, we will focus on the most popular general equilibrium asset pricing model, namely, the Capital Asset Pricing Model (CAPM). While it relies on very specific assumptions about preferences, it has turned out to be 5
extremely useful in analyzing prices and risk sharing in the large-scale experimental financial markets on which the empirical study later in this paper relies (see [5, 6]). Moreover, the CAPM is very much the prototype of asset pricing theory: its main pricing implication (that expected excess return will be proportional to covariance with aggregate risk) and its main risk sharing implication (that all investors should hold the same portfolio(s) of risky securities) are shared with all other asset pricing models that have been brought to bear on empirical data. The general equilibrium formulation of the CAPM that is convenient for our purposes may be described as follows. There are two dates, with uncertainty about the state of nature at the terminal date. A single good is available for consumption at the terminal date; there is no consumption at the initial date. J + 1 assets (claims to state-dependent consumption at the terminal date) are traded at the initial date and yield claims at the terminal date. The first asset is riskless, yielding one unit of consumption independent of the state; all other assets are risky. Without loss, we assume asset payoffs are non-negative.1 The N investors in the market are endowed only with assets; write h0n ∈ R+ for investor n’s endowment of the riskless asset and zn0 ∈ RJ+ for investor n’s endowment of the risky assets. Investors trade off mean return against variance. We assume that investors hold common priors, so that they agree on the mean and variance of any portfolio. Write Dj (s) for the return of the j-th risky asset in state s. Let µ be the vector of expected payoffs of risky assets and ∆ = [cov (Dj , Dk )] be the covariance matrix. An investor who holds h units of the riskless asset and the vector z of risky assets will enjoy utility un (h, z) = h + [z · µ] −
bn [z · ∆z] 2
(1)
Since there is no consumption at the initial date, we may normalize so that the price of the risky asset is 1; write p for the vector of prices of risky assets. Given prices p, consumer n’s budget set consists of portfolios (h, z) that yield 1 In
general, the number of assets may be smaller than the number of states, so markets may be incomplete. In all the experiments,
markets were complete, however.
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non-negative consumption in each state and satisfy the budget constraint h + p · z ≤ h0n + p · zn0
(2)
From the first order conditions that characterize optima, investor n’s demand for risky assets given prices p is zn (p) =
1 −1 ∆ (µ − p) bn
(3)
In particular, all investors choose a linear combination of the riskless asset and the same portfolio of risky assets — a conclusion usually referred to as portfolio separation. As usual, an equilibrium consists of prices p for assets and portfolio choices hn , zn for each investor so that investors optimize in their budget sets and markets clear. Given the nature of demands (3), market clearing requires that the portfolio ∆−1 (µ − p) be a multiple of the market portfolio z 0 = equilibrium prices yields p=µ− It is convenient to write B =
1 N
P
1 bn
µX
1 bn
P
zn0 (the supply of risky assets). Solving for
¶−1 (∆z 0 )
for the mean inverse risk aversion and z¯ =
1 N
P
zn0 for the mean endowment
(mean market portfolio); with this notation, the pricing formula becomes z p = µ − B −1 ∆¯
(4)
A little analysis shows that the pricing formula (4) implies that the mean market portfolio z¯ (equivalently, the market portfolio z 0 ) is mean-variance optimal (i.e., among all portfolios with the same variance, it has the highest expected net return). This is not obvious, but is not surprising either. Investors demand mean-variance optimal portfolios, which can all be decomposed into the riskfree security and a single portfolio of risky securities. In equilibrium, aggregate demand for risky securities must equal supply, which is the market portfolio. Hence, in equilibrium, the market portfolio must be the single mean-variance optimal portfolio that is part of each investor’s demand. That is, the market portfolio must be mean-variance efficient. 7
Our central question will be: how would financial markets discover their CAPM equilibrium? The theoretical literature can be interpreted as providing two answers.
2.2
Competitive Walrasian Tatonnement
In the first approach, the price discovery is presumed to be competitive and investors are myopic (although we will have to qualify myopia later on). No trade takes place as long as equilibrium has not been reached yet (although this no-trade restriction is inconsequential in our setting of mean-variance preferences). Competitive price adjustment without intermediate trade has become known as Walrasian tatonnement. In the sixties and seventies, Walrasian tatonnement was used to study stability of general equilibrium.2 The central question was, if markets are slightly kicked off their equilibrium, will they revert back to equilibrium? To answer this question, a model of equilibration was necessary, and most often a simple Walrasian tatonnement process was chosen. The Walrasian model is very stylized. It supposes that an auctioneer calls out prices, and that there is no intermediate trade before excess demand reaches zero (tatonnement). In the context of the CAPM, tatonnement is irrelevant, as demand (for risky securities) does not depend on endowment. Even if trade were to take place at each tatonnement step, the results would not change. The Walrasian model is competitive: agents do not lie about their demands. Moreover, it is short-sighted: agents do not speculate about subsequent price changes. These aspects contrast with recent models of the microstructure (trade-by-trade process) of financial markets, which have invariably been based on game theory. It would be foolish to take the Walrasian model literally. We choose it mainly because of the simplicity of its core empirical prediction: that price changes are related to excess demand. We will model price adjustment in discrete time. This allows us to characterize the adjustment process in terms of solutions to differential equations. Let t denote time. In the prototype Walrasian adjustment process, price changes 2 See,
e.g., [2, 10]. For recent experimental evaluation of stability issues, [1].
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are proportional to excess demand: pt+1 − pt = λz e (pt ),
(5)
where z e denotes the vector of per capita excess demand and λ denotes a positive constant. z e is a function of prices pt : z e (pt )
=
N 1 X zn (pt ) − z¯ N n=1
= B∆−1 (µ − pt ) − z¯.
(6)
Notice that in the basic Walrasian model, there are no cross-security effects: the price of a security reacts only to its own excess demand (even if the excess demand of a security depends on the prices of all other securities). Let us generalize this and allow for cross-security effects: pt+1 − pt = Az e (pt ),
(7)
where A is an arbitrary matrix with strictly positive diagonal terms. The ensuing adjustment process is exponential. That is, it does not exhibit the damped cycling characteristic of many physical phenomena. Cycling may seem to be present, but really only reflects the multidimensional nature of the adjustment process (J > 1). In stability analysis, the off-diagonal terms of A are assumed zero (no cross-security effects). If nonzero off-diagonal terms are allowed, stability may not be a foregone conclusion. All depends on the relationship between entries of A and ∆. It is shown in [3] that the price adjustment process is stable if and only if the real parts of the eigenvalues of A∆−1 are all strictly positive. In the empirical analysis, we will verify these restrictions. We are going to take this Walrasian price discovery process as an approximation of the actual, noisy transaction price process. In particular, we posit the following stochastic difference equation for transaction price changes: pt+1 − pt = Az e (pt ) + ²t+1 ,
(8)
where the noise ²t+1 is assumed to be mean zero and uncorrelated with past information. Past information includes everything that is available to all investors (order flow, transaction volume and prices). Let It denote this publicly 9
available information. In addition, we assume that the noise ²t+1 is uncorrelated with the time-t excess demand as well. Note that investors in general will not known the excess demand (because they may not know the aggregate per-capita supply z¯ or because they may not know other investors’ risk aversion). That is, z e (pt ) is not in It . Altogether, we impose that: E[²t+1 |It ] =
0,
(9)
E[²t+1 z e (pt )|It ] =
0.
(10)
The model in (8), (9) and (10) is meant to capture the essence of a competitive price discovery process: price changes are correlated with excess demand, where excess demand is defined as in general equilibrium theory. Later in this paper, we bring this model to experimental data (where we know excess demand up to a constant scaling factor even if subjects themselves did not). This way, we will be able to determine to what extent the stylized Walrasian price discovery process has any empirical content.
2.3
Market Microstructure Theory
A substantially different view on price discovery emerges in market microstructure theory. The focus is on strategic interaction among a number of key players in the marketplace, in particular, those who are ready to accept orders on either side of the market (market makers). The analysis is based on game theory in general and Bayesian Nash equilibrium in particular. All models in this area differ from the Walrasian model in the detailed description of the environment. The outcomes are sensitive to the specifics of the environment. Unlike the Walrasian model, the game-theoretic models of market microstructure theory are populated with strategic agents who have foresight (try to predict future price changes) and who are ready to lie about their demand when it is in their interest. The strategic sophistication comes at a cost. The models suppose that there is already an enormous amount of structural knowledge that the equilibration process itself ought to reveal. For instance, the distribution of
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the eventual equilibrium values of securities are taken as given and commonly known, whereas the equilibrium values are in fact the very quantities to be discovered without agents knowing what their distribution is. Also, game theoretic models are equilibrium models themselves, so the issue of equilibrium price discovery is pushed just one layer back. It remains to be explained how markets manage to play according to the equilibrium of the equilibration game. Despite the enormous variety of market microstructure models, a central prediction about transaction price changes can be recognized. As [8] argued, when there are risk neutral market makers who face no inventory costs, transaction prices must be a martingale with respect to the publicly available information (which includes the order flow). In the notation of the previous section, E[pt+1 − pt |It ] = 0.
(11)
This is an easily verifiable restriction. The restriction demonstrates why simple autocorrelation-based estimates of the bid ask spread do not work (see [12]). In our empirical analysis, we will verify the restriction by projecting transaction price changes on information that we distill from activity in the order book. Because (11) is based on subtle game theoretic arguments, we have to be careful when testing it. In particular, the prediction is that a security’s transaction price changes are not predictable. That is, the change in price over intervals defined with respect to the security’s own trading process is unpredictable. In the empirical section, however, we will generally measure time as the count of all transactions, irrespective of the security. In our records, no security ever trades simultaneously with others. Following tradition, the other securities are then assigned a transaction price equal to the last recorded transaction price.3 In that case, however, transaction price changes may seem to be predictable even if (11) holds. To see this, assume that security A traded at time t and t + 2. Security B traded at time t + 1. Hence, pA,t+1 (the time-(t + 1) price of A) equals pt,A . The subsequent change, pA,t+2 − pA,t+1 , may then be predictable from information 3 Another
possibility would be to assign a transaction price equal to the average of the best bid and ask quotes. The latter is implemented
in the datasets issued by the Center for Research in Securities Prices (CRSP) of the University of Chicago.
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at t + 1 (It+1 ) because this information is revealed after the last transaction in A, which was at time t. That is, the predictability is not a rejection of (11), because It+1 post-dates pA,t+1 (which actually is pA,t ). Consequently, when testing (11), one will have to be careful to measure transaction price changes and changes in information in a security’s own transaction time. If we find no predictability, it will be interesting to determine whether predictability does obtain when transaction price changes and information are recorded over a higher frequency. In principle, predictability will emerge if transaction price changes are recorded over time defined by the count of all transactions (regardless of the security). If predictability is not observed at this higher frequency, one should seriously question the power of the test of (11): predictability at the frequency of a security’s own transactions may not have appeared because one’s predictors are not powerful enough.
2.4
Can The Two Approaches Be Reconciled?
At first, the two approaches seem inconsistent. One approach (the Walrasian tatonnement) assumes that investors are short-sighted and nonstrategic. The other one (market microstructure theory) assumes that investors are far-sighted and act strategically. But inspection of the key mathematical implications reveals that they are not inconsistent. Which at the same time means that the Walrasian approach does not necessarily assume that investors are short-sighted. If both make empirically valid predictions, this also means that investors’ strategic power is limited, presumably because they lack the structural (common) knowledge to exploit predictability of transaction price changes using excess demand. Contrast the Walrasian predictions in (8), (9) and (10) with the prediction of market microstructure theory, (11). Even if publicly available information It (order flow, activity in the book) is correlated (though not perfectly) with excess demand z e (pt ), and excess demand predicts transaction price changes pt+1 − pt , publicly available information It may itself be of no use to predict transaction price changes pt+1 − pt . Of course, this can only occur if publicly available information It is correlated with the error of the Walrasian model ²t+1 so as to cancel predictability through
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the excess demand z e (pt ). Publicly available information It and excess demand z e (pt ) need not be correlated. One hopes, however, that there is correlation. It would indeed be puzzling to discover that there is no correlation between publicly available information (activity in the order book), on the one hand, and the excess demand of general equilibrium theory. Excess demand is central to the determination of the equilibrium price relationships predicted by general equilibrium theory. If the book is uncorrelated with excess demand, what is related to? What else would strategic activity in the book be meant to discover but the general equilibrium? Therefore, it will be interesting to verify whether information in the book is indeed correlated with excess demand. And, if both Walrasian tatonnement and microstructure theory receive empirical support, whether information in the book explains the error of the Walrasian model. Since the error ²t+1 of the Walrasian model obviously keeps markets from reaching their general equilibrium (it is orthogonal to the key variable of general equilibrium theory – excess demand), evidence of high correlation between this error ²t+1 and information in the book should be interpreted as characterizing a low quality microstructure. Conversely, low correlation between the error ²t+1 and the book should be indicative of a high quality market. At the extreme, the book is perfectly correlated with excess demand. In that case, however, for (11) to be satisfied simultaneously, markets must clear instantaneously. That is, no transactions occur before general equilibrium is reached. If so, the market microstructure is of extreme quality. If securities markets are complete, this would also mean that only Pareto-improving trades will take place. In summary, here are the predictions one would like to test. A. (The Walrasian model.) Transaction price changes and excess demand as defined in general equilibrium theory are correlated; a security’s transaction price changes are positive correlated with its own excess demand. These predictions follow from (8), (9) and (10). B. (The market microstructure model.) Transaction price changes are uncorrelated with information in the book. This prediction follows from (11). Transaction price changes, however, have to be measured over the 13
frequency of a security’s own trading process. Discovery that transaction price changes are predictable over a higher frequency suggests that one’s predictor (say, a statistic based on information the book) is powerful. C. (Link between the market microstructure model and the Walrasian model.) There is correlation between information in the book and excess demand as defined in general equilibrium theory. D. (Consistency between the market microstructure model and the Walrasian model.) Information in the book correlates with the error of the Walrasian model, i.e., the book correlates with the error term of (8). If these predictions are confirmed, we are interested in two more issues. 1. (Stability.) Are there cross-security effects, i.e., does the excess demand of security j correlate with transaction price changes of security k (j 6= k) after accounting for own-excess-demand influences? Are these cross-security effects such that the Walrasian price discovery system may be unstable? The latter involves verifying that the real parts of the eigenvalues of the adjustment matrix (A) times the inverse of the covariance matrix of payoffs (∆−1 ) are all strictly positive. 2. (Quality of the market.) Is the correlation between the error term of the Walrasian model (see (8)) and information in the book related to features of a market’s microstructure (number of investors, number of securities, etc.)? We won’t be able to shed much light on the latter, because the experimental design parameters did not change sufficiently from one experiment to another. Let us turn to the empirical analysis.
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3
Description of data
To evaluate the predictive power of our equilibration models, we collected transaction price changes in a series of large-scale financial markets experiments run at Caltech over the last two years. While prices in these experiments generally moved in the direction predicted by asset pricing theory, most transactions clearly occurred before markets reached equilibrium. Experimental data allow us to determine when markets are not in equilibrium. In field studies, such a clean delineation is impossible, absent precise knowledge of the economic structure of the market (parameters of demand and supply). In fact, it has been tradition in empirical finance to just treat all observations as if they are snapshots of a dynamic equilibrium, ignoring the very premise underlying this paper, namely, that markets may at times be in disequilibrium. Moreover, the experimental setup allows us to estimate excess demand with little error. Estimation of excess demand in field data is far more difficult, absent knowledge of investors’ beliefs, a good model for attitudes towards risk, and the total supply of assets. The three-asset experiments are described in more detail in [5] (focusing on pricing), [6] (focusing on allocations) and [7] (linking prices and allocations). The Appendix also discusses evidence from 4 four-asset experiments. Except for the number of assets and the payoff matrix, the setup for all experiments was the same. A continuous open book trading system was used, implemented through the Caltech Marketscape web-based program.4 . There were either two or three risky securities A,B, (C), in addition to the riskless security (the “Note”), and limited cash. Payoffs (dividends) on the risky securities depend on a state variable. Three states (or four in the four-asset experiments) were possible and one was drawn at random, with a commonly known distribution. The number of subjects ran from less than 20 to over 60. 4 Instructions
In each of several consecutive periods the assets were allocated, mostly unevenly, across
and screens for the experiment we discuss here can be viewed at http://eeps2.caltech.edu/market-991116uclacit and use
identification number:1 and password:a to login as a viewer. As a viewer you will not have a payoff but you will be able to see the forms used. If you wish to interact with the software in a different context you can go to http://eeps.caltech.edu and go to the experiment and then demo links. This exercise will provide the reader with some understanding of how the software works.
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subjects. The markets were then opened for a fixed period of time. When the period closed the state was announced and earnings recorded. New allocations of the asset were distributed and a fresh period opened.
Subjects whose
earnings were sufficiently low were declared bankrupt and were prevented from participating in subsequent periods. Earnings ranged from nothing (the bankrupt) to over two hundred dollars. In all the three-security experiments, the payoff matrix remained the same, namely: X
Y
Z
Security A
170
370
150
Security B
160
190
250
Note
100
100
100
State
In the four-security experiments (except 001113)5 , the payoff matrix was as follows: State
W
X
Y
Z
Security A
30
190
500
200
Security B
100
270
300
130
Security C
200
210
90
180
Note
100
100
100
100
The remaining data and parameters for the experiments are displayed in Table 1. (As mentioned before, the data for the four-asset experiments are given in the Appendix.) 5 Only
three states could occur in the 001113 experiment. The payoff structure was: State
X
Y
Z
Security A
200
0
0
Security B
0
200
0
Security C
0
0
200
100
100
100
Note
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In at least two of the experiments (990407 and 991011), strong evidence of speculation was discovered. That is, some subjects apparently did not take the drawing of states, and hence, dividends, to be independent over time, and, consequently, attempted to guess the actual sequence of states. While this did affect the number of periods where equilibrium was observed, apparently it did not have much effect on the price dynamics, as we will discover shortly. That is, price changes continued to be strongly influenced by an aggregate excess demand that was optimal relative to the correct beliefs (independent drawing of states). It appears that speculation about the state to be drawn merely prolonged the equilibration process, the nature of the price dynamics still being determined by nonspeculative aggregate demand. In this sense, the experiments produced far more evidence in favor of general equilibrium theory through the price dynamics (prices were attracted by the equilibrium predicted in the theory) than by the number of periods that equilibrium was effectively reached. Of course, there are only a few periods in each experiments (six to eight), and therefore, little occasion for the markets to show that they could fully equilibrate. In contrast, there were often over one thousand transactions per experiment which could be used to evaluate the equilibration process. That is, there is far more information on equilibration than there is about equilibrium. In collecting the times series of transaction price changes, we deleted outliers, defined to be observations where one of the prices changed with more than 30 francs (these are almost invariably trading mistakes, or they straddle the end of a period).
4
Empirical Evidence
We will discuss the evidence on each of the predictions in turn.
4.1
The Walrasian Model
The Walrasian model predicts that excess demand correlates with subsequent transaction price changes. See (8), (9) and (10). A security’s own excess demand should correlate positively with subsequent transaction price changes. 17
Excess demand is a quantity (measured as number of assets) that is at the core of general equilibrium theory. It equals the theoretical optimal aggregate demand at last prices, minus the aggregate supply. Per capita aggregate supply varied substantially across experiments, but remained fairly constant within an experiment.6 Our using experimental data is fortunate here, because aggregate supply is a design element in the experiments, and hence, known to the experimenter (but only to the experimenter - subjects were not given any information about supplies). In the field, aggregate supply is virtually impossible to measure.7 In contrast, per-capita aggregate demand can only be measured up to a constant which needs to be estimated (the average risk tolerance B in (6)). We arbitrarily set the B equal to 103 . The results hardly display sensitivity to variations in B. In fact, the error in B is absorbed in the intercept that we add to the basic equation (8). To see ˆ denote one’s this, use (6) (definition of excess demand) and (8) (the Walrasian model) to derive the following. Let B estimate of B. Appeal to the fact that per-capita aggregate supply z¯ hardly varies during an experiment, so that we can take it to be constant. pt+1 − pt = Az e (pt ) + ²t+1 ¡ ¢ = A B∆−1 (µ − pt ) − z¯ + ²t+1 =
(
´ B ³ ˆ −1 B − 1)A¯ z + A B∆ (µ − pt ) − z¯ + ²t+1 . ˆ ˆ B B
Notice the effect of errors in the estimation of the average risk tolerance B. ˆ = B. Therefore, we should allow for an intercept in the estimation. • The intercept is nonzero unless B • The diagonal elements of the coefficient matrix remain positive if they were originally. 6 Within
the reported experiments, aggregate supply changed only as a result of subjects being excluded from trading because they ran
a negative balance two periods in a row. 7 This
fact led to the Roll critique of tests of the CAPM. See [11].
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• The properties of the error term are unaffected. Consequently, the fact that we cannot measure exactly the average risk tolerance B has little impact on our estimation. Contrast this with field data. If we had to compute the theoretical excess demand from field data, we would not only have to estimate B, but also the covariance matrix ∆ and the vector of mean payoffs µ. The latter are known in an experimental setting. (They are part of the design.) The estimation results are displayed in Table 2. The F -statistics prove that the relationship between estimated per capita aggregate excess demand and subsequent transaction price changes is very strong. The p-levels are uniformly below 1%. The relationship is very noisy, however, because the R2 s are generally low. Table 2 also reveals that the effect of own excess demand on a security’s price is uniformly strong and positive. Lack of autocorrelation of error verifies (9), which means that our model is well specified. The autocorrelation coefficient is small but significant at the 5% level in two cases. Still, we expect to see a number of rejections even if the model is well specified. The chance of finding at least two rejections at the 5% level is about 1 in 9. Overall, Table 2 provides strong support for the Walrasian price discovery model. As mentioned before, this support contrasts with the evidence from several experiments (especially, the four-asset cases) where prices almost never satisfy the equilibrium predictions exactly. In contrast with the basic Walrasian model (5), there is strong evidence of cross-security effects: most often, there is a significant relationship between a security’s excess demand and another security’s subsequent price change. This raises the possibility that the Walrasian price discovery system is unstable (it may cycles around equilibrium or move away from equilibrium). Table 3 lists the results of tests of the stability conditions. For the price discovery system to be stable, the real part of the eigenvalues of the adjustment matrix A times the inverse covariance matrix of payoffs ∆−1 . In no experiment do we find evidence of instability. Somehow, markets manage to generate dynamics that guarantee mean-reversion back to the general equilibrium.
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4.2
The Market Microstructure Model
To test the main prediction of the market microstructure model, (11), we have to project transaction price changes onto past public information. Public information includes any activity in the book. We will use a summary measure based on the five best bid and ask quotes. As mentioned before, one has to be careful to record transaction price changes at the frequency of a security’s own transactions. In Tables 2 and 3 (the Walrasian model), time was measured as the count of transactions in any security. Here, we will decrease the frequency. It is often suggested that the relative thickness of the two sides of the book provides useful information to gauge the direction of the markets. For instance, if there are far more bid quotes that are tightly packed together whereas ask quotes are fewer and far apart, then transaction price changes are likely to increase. Market microstructure theory contradicts such predictability: traders will react immediately, and orders will flow in to destroy this predictability before a transaction takes place. This suggests the following statistic to be used as predictor of future transaction price changes. Let nat,l denote the number of securities offered to be sold at the l best price as of time (transaction) t (l = 1, 2, ..., 5; limit orders are ranked first by price and then by time of entry, older orders having precedence over more recent ones). Let pat,l denote the corresponding asked price level. Likewise, let nbt,l denote the number of securities offered to be bought in the l best limit order as of t. Let pbt,l denote the corresponding bid price level. Construct weighted averages of the asked and bid quotes, respectively: qta =
5 X l=1
qtb
=
5 X l=1
P5
nat,l
a k=1 nt,k
P5
nbt,l
k=1
nbt,k
pat,l ,
pbt,l .
Now construct the weighted average of these average asked and bid quotes: P5 P5 b a l=1 nt,l l=1 nt,l o a q + qtb . pt = P5 P P P t 5 5 5 a + b a + b n n n n l=1 t,l l=1 t,l l=1 t,l l=1 t,l
(12)
Our predictor of the transaction price change equals po minus the last transaction price change, i.e., pot − pt .8 8 When
one side of the book is empty, pot would not be defined. In that case, we set pot − pt equal to its time series average. To avoid
20
We will refer to our predictor as the signal in the book or the weighted average quote. Admittedly, our predictor is crude and poor predictability may merely reflect lack of power instead of conformity with microstructure theory. To evaluate power, we will also present results from projections of transaction price changes at a higher frequency, namely the count of transactions in any security. As mentioned before, this means that our predictor will frequently use information that post-dates the last transaction of a security. If our predictor is powerful enough, it should be possible to predict transaction price changes. We will document later on that it does. For comparison with empirical microstructure studies on field data, we will first project transaction price changes on information in a security’s own book. Afterwards, we will study predictability including information in the book of other securities. The latter possibility has only rarely been considered in field studies. Table 4 displays the results when projecting a security’s transaction price change onto the signal in its own book only. In seven out of 16 cases, the predictability is significant at the 1% level. This is far higher than expected by chance. Note however, that the nature of the predictability changes in unstructured ways: in some experiments, only the price of one asset is predictable, and this asset is not always the same; in others, both asset prices are predictable. The reader is urged to inspect the tables in the Appendix for richer evidence on this phenomenon of unstructured predictability. Also, the intercept is nowhere significant. Overall, the findings would reject the prediction of microstructure theory, namely, that transaction prices should not be predictable from publicly available information (see (??)). But the rejection becomes unessential if one takes into account the unstructured nature of the predictability. The R2 s reveal that the predictability is marginal. It is interesting to inspect the sign of the slope coefficients. Our predictor is the weighted average of bid and ask quotes in the order book, with more weight given to the side of the book that has the least volume of orders. Consequently, when the book is thin on, say, the ask side, the weighted average of bid and ask quotes will be above the average of the best bid and ask. Evidently, this often predicts that outliers when one side of the book is very thin and the bid-ask spread is very wide, we set pot − pt equal to its time series average when it would otherwise be bigger than 50 francs (which corresponds roughly to 25% of the expected payoff).
21
transaction prices will increase. Vice versa if the bid side of the book is thin. The picture changes only slightly when including information from the book of other securities when predicting transaction price changes. Table 5 displays the results when projecting a security’s transaction price change onto the signals in its own book and that of the second security. Significant predictability is recovered in less cases (the F test obviously corrects for the fact that it is easier to find evidence of predictability with more predictors). Table 6 reveals that our predictors do not lack power. It displays the projections of the previous table at a higher frequency, namely, the count of all transactions. Predictability emerges in all but one case.
4.3
Link between the market microstructure model and the Walrasian model
Table 7 demonstrates that the signal in the book is correlated with excess demand. Instead of projecting the signal in the book (pot − pt in the previous subsection) onto excess demand, we project it onto the difference between the last transaction price and the price at which our estimate of excess demand equals zero (i.e., our estimate of the equilibrium price). The latter is referred to as “price signals” in the table. This variable facilitates interpretation of the signs of the slope coefficients. The projections are almost uniformly significant, and the R2 s are sometimes remarkably high (up to 18%). In addition, whenever the relation between the signal in the book and the difference between the last transaction price and the estimated equilibrium price is significant, its sign is as expected: the more transaction prices are below estimated equilibrium prices, the higher the price increase the book signals, if we interpret relative thinness of the book at one side as predicting that the transaction prices are likely to move in the direction of thinness (see also our discussion of Table 4).
22
4.4
Consistency between the market microstructure model and the Walrasian model
As mentioned before, correlation between the signal in the book and the signal from excess demand, support for the market microstructure model and for the Walrasian model, can reconciled only if the signal in the book at the same time explains part of the error of the Walrasian model. Table 8 documents that the signal in the book (weighted average quote) generally does correlate with the error of the Walrasian model. The support is not uniform, but it should be pointed out that the dependent variable is but an estimate (of the error of the Walrasian model), and hence, the projections may not be very powerful.
5
Conclusion
This paper provided evidence in support of both the competitive (Walrasian) model and the strategic (microstructuretheoretic) model of price discovery in experimental financial markets. The evidence for correlation between price changes and the excess demands that form the core of general equilibrium theory is overwhelming. Unlike the standard Walrasian model, however, there are very significant cross-security effects, whereby transaction price changes of one security are correlated with excess demands of other securities. At first, the evidence for the strategic model is less pronounced, because transaction prices apparently can be predicted with information in the order book more often than expected by chance. Still, this evidence is of little consequence when one observes that the nature of the predictability changes from one experiment to another. The latter is not unlike the evidence from field data. [4], for instance, documents that there is ample predictability in stock returns from 15 different countries (even after adjusting for data mining), but that the prediction models are generally useless out of sample. That some evidence against the market microstructure model emerges should not come as a surprise. In contrast with the competitive price discovery model, the market microstructure model is based on game theory. Standard
23
notions of equilibrium in games (Bayesian/Nash) are very demanding in terms of the amount of information that has to be common knowledge: number of agents, their preferences, their endowments, their level of rationality; while subjects in the experiments may have been able to guess these parameters, they were not explicitly revealed, which means that even if all subjects eventually were to know them, there would still be the question whether this information was common knowledge. The competitive model deals with information in a far more parsimonious way (in principle, only prices need to be observed). It is often claimed that the Walrasian model is myopic because agents do not foresee further excess demand adjustments, and hence, price adjustments. This criticism is well founded if agents indeed know all demands and supplies. Absent such information, the criticism is not necessarily founded. At the same time, the market microstructure model is said to be far-sighted, but one wonders to what extent this bites, since the amount of information that is common knowledge is very limited. Altogether, this means that the two approaches may not be contradictory. This paper discovers evidence in favor of both approaches. The data also reveal that the two are indeed closely related: the correlation between the structure of quotes in the order book and excess demands of general equilibrium theory is uniformly high. To make the link fully consistent, the paper also documents that the order book explains part of the error of the competitive model in predicting transaction price changes. The last finding brings up some new issues about the quality of a market. If incentives to submit orders can be designed such that the book is perfectly correlated with excess demand, no transactions will take place except at equilibrium prices. Consequently, there is a close relationship between the design of a market’s microstructure, on the one hand, and the ability of a market to reach its competitive general equilibrium, on the other hand.
24
References [1] Anderson, C.M., S. Granat, C. Plott and K. Shimomura (2000), “Global Instability in Experimental General Equilibrium: The Scarf Example,” Caltech working paper. [2] Arrow, K. and F. Hahn (1971), General Competitive Analysis, San Francisco: Holden-Day. [3] Asparouhova, E. and P. Bossaerts (2000), “Competitive Analysis Of Price Discovery In Financial Markets,” Caltech working paper. [4] Bossaerts, P. and P. Hillion (1999), “Implementing Statistical Criteria to Select Return Forecasting Models: What Do We Learn?” Review of Financial Studies 12, 405-28. [5] Bossaerts, P. and C. Plott [1999], “Basic Principles of Asset Pricing Theory: Evidence from Large-Scale Experimental Financial Markets,” Caltech working paper. [6] Bossaerts, P., C. Plott and W. Zame (2000), “Prices And Allocations In Financial Markets: Theory and Evidence,” Caltech working paper. [7] Bossaerts, P., C. Plott and W. Zame (2001), “Structural Econometric Tests Of General Equilibrium Theory On Data From Large-Scale Experimental Financial Markets,” Caltech working paper. [8] Glosten, L. and P. Milgrom (1985), “Bid, Ask, and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders,” Journal of Financial Economics 14, 71-100. [9] Kyle, A.P. (1985), “Continuous Auctions and Insider Trading,” Econometrica 53, 1315-35. [10] Negishi, T. (1962): “The Stability Of The Competitive Equilibrium. A Survey Article,” Econometrica 30, 635-70. [11] Roll, R. (1977), “A Critique Of The Asset Pricing Theory’s Tests, Part I: On The Past And Potential Testability Of The Theory,” Journal Of Financial Economics 4, 129-76. 25
[12] Roll, R. (1984), “A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market,” Journal of Finance 39, 1127-39. [13] Scarf, H. (1960): “Some Examples of Global Instability of the Competitive Equilibrium,” International Economic Review 1, 157-172.
26
Table 1: Experimental design data. Experiment
Subject
Signup
Category
Reward
(Number)
(franc)
Endowments A
B
Cash
Loan
Exchange
Repayment
Rate
(franc)
(franc)
$/franc
Notes
981007
30
0
4
4
0
400
1900
0.03
981116
23
0
5
4
0
400
2000
0.03
21
0
2
7
0
400
2000
0.03
8
0
5
4
0
400
2000
0.03
11
0
2
7
0
400
2000
0.03
22
175
9
1
0
400
2500
0.03
22
175
1
9
0
400
2400
0.04
33
175
5
4
0
400
2200
0.04
30
175
2
8
0
400
2310
0.04
22
175
5
4
0
400
2200
0.04
23
175
2
8
0
400
2310
0.04
8
0
5
4
0
400
2200
0.04
7
0
2
8
0
400
2310
0.04
8
0
5
4
0
400
2200
0.04
7
0
2
8
0
400
2310
0.04
990211
990407
991110
991111
000804
001005
27
Table 2: The Walrasian Model Experiment
Coefficientsa
Security Intercept
981007
A
B
981116
A
B
990211
A
B
990407
A
B
991110
A
B
991111
A
B
000804
A
B
001005
A
B
a OLS
R2
F -statisticb
Nc
ρd
13.6
614
-0.03
614
0.05
1053
−0.10∗∗
1053
0.00
466
-0.03
466
0.05
1385
-0.03
1385
−0.06∗
1318
-0.04
1318
-0.04
1190
-0.04
1190
-0.05
312
-0.11
312
-0.03
194
0.03
194
-0.10
Excess Demande A
B
1.568
0.800
-0.152
(0.340)
(0.153)
(0.057)
-0.602
-0.096
0.187
(0.168)
(0.076)
(0.028)
0.451
0.581
-0.057
(0.121)
(0.110)
(0.014)
-0.126
-0.063
0.023
(0.092)
(0.084)
(0.011)
0.666
0.671
-0.044
(0.413)
(0.181)
(0.038)
-0.934
-0.297
0.093
(0.278)
(0.122)
(0.026)
1.097
0.561
-0.043
(0.189)
(0.090)
(0.011)
0.371
0.130
0.011
(0.135)
(0.064)
(0.008)
0.568
0.993
-0.085
(0.136)
(0.121)
(0.013)
-0.045
0.240
-0.006
(0.110)
(0.098)
(0.011)
0.843
0.505
0.000
(0.164)
(0.087)
(0.014)
-0.047
0.039
0.058
(0.135)
(0.072)
(0.012)
1.701
0.689
-0.126
(0.525)
(0.219)
(0.045)
-0.514
0.034
0.182
(0.560)
(0.234)
(0.048)
0.584
0.835
-0.102
(0.760)
(0.305)
(0.056)
-1.592
-0.438
0.126
(0.740)
(0.297)
(0.055)
0.04
(< .01) 0.08
25.1 (< .01)
0.03
14.1 (< .01)
0.01
3.5 (.03)
0.03
7.8 (< .01)
0.03
6.5 (< .01)
0.03
19.8 (< .01)
0.01
6.5 (< .01)
0.05
34.9 (< .01)
0.01
8.8 (< .01)
0.03
18.1 (< .01)
0.02
14.0 (< .01)
0.03
5.3 (< .01)
0.08
13.1 (< .01)
0.04
8.6 (< .01)
0.03
6.2 (< .01)
projections of transaction price changes onto (i) an intercept, (ii) the estimated excess demands for the two risky securities
(A and B). See (8). Time advances whenever one of the three assets trades. Standard errors in parentheses. b p-level
in parentheses.
c Number
of observations.
d Autocorrelation e Estimated
of the error term;
∗
and
∗∗
indicate significance at the 5% and 1% level, respectively.
on the basis of a risk aversion parameter equal to 10−3 .
28
Table 3: The Walrasian Model: Stability Experiment
a For
Eigenvalues Of A∆−1 (∗10−4 )a First
Second
981007
1.33
0.80
981116
0.59
0.13
990211
0.59 + 0.19i
0.59 − 0.19i
990407
0.61
0.15
991110
1.05
0.11
991111
0.70
0.32
000804
1.31
0.77
001005
0.98
0.48
stability, the real parts of the eigenvalue of A∆−1 should all be strictly positive.
29
Table 4: The Market Microstructure Model I Experiment
Coefficientsa
Security Intercept
981007
A
B
981116
A
B
990211
A
B
990407
A
B
991110
A
B
991111
A
B
000804
A
B
001005
A
B
a OLS
R2
F -statisticb
Nc
1.6
182
Weighted Average Quoted
0.081
0.034
(0.294)
(0.026)
0.057
0.126
(0.131)
(0.038)
-0.049
0.032
(0.189)
(0.026)
0.024
0.233
(0.625)
(0.152)
0.056
0.030
(0.355)
(0.027)
0.145
0.033
(0.234)
(0.030)
0.036
0.033
(0.088)
(0.017)
0.105
0.030
(0.084)
(0.015)
-0.027
0.005
(0.158)
(0.028)
0.067
0.187
(0.123)
(0.027)
0.083
0.197
(0.118)
(0.034)
0.146
0.293
(0.109)
(0.040)
0.646
0.552
(0.449)
(0.109)
0.411
0.259
(0.257)
(0.060)
0.493
0.354
(1.261)
(0.106)
0.886
0.215
(1.140)
(0.101)
0.01
(.20) 0.06
10.6
181
(< .01) 0.00
1.5
335
(.22) 0.01
2.3
367
(.13) 0.01
1.2
147
(.27) 0.01
1.2
166
(.27) 0.01
3.6
503
(.06) 0.01
4.2
411
(.04) 0.00
0.0
435
(.85) 0.08
46.5
514
(< .01) 0.08
33.2
396
(< .01) 0.12
54.6
409
(< .01) 0.25
24.9
77
(< .01) 0.13
18.5
128
(< .01) 0.15
10.7
65
(< .01) 0.06
4.4
76
(.04)
projections of transaction price changes onto (i) an intercept, (ii) weighted average quotes in the order book. Time
advances after each trade in the asset whose price is the regressor. The null hypothesis is given in (11). Standard errors in parentheses. b p-level
in parentheses.
c Number
of observations.
d Difference
between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely
proportional to quantities.
30
Table 5: The Market Microstructure Model II Experiment
Coefficientsa
Security Intercept
981007
A
B
981116
A
B
990211
A
B
990407
A
B
991110
A
B
991111
A
B
000804
A
B
001005
A
B
a OLS
R2
F -statisticb
Nc
0.9
182
Weighted Average Quoted A
B
0.077
0.034
-0.016
(0.294)
(0.026)
(0.055)
0.026
-0.005
0.129
(0.149)
(0.012)
(0.039)
-0.031
0.029
-0.039
(0.190)
(0.026)
(0.042)
0.075
0.118
0.244
(0.624)
(0.079)
(0.151)
0.037
0.032
0.020
(0.360)
(0.028)
(0.069)
0.253
-0.033
0.031
(0.246)
(0.025)
(0.030)
0.035
0.036
-0.021
(0.088)
(0.017)
(0.023)
0.093
-0.026
0.031
(0.085)
(0.023)
(0.015)
-0.023
-0.002
0.048
(0.158)
(0.028)
(0.028)
0.147
0.051
0.183
(0.127)
(0.021)
(0.027)
0.101
0.198
0.019
(0.124)
(0.034)
(0.041)
0.119
-0.033
0.293
(0.110)
(0.023)
(0.040)
0.670
0.544
0.053
(0.450)
(0.110)
(0.102)
0.251
-0.065
0.242
(0.314)
(0.074)
(0.063)
0.459
0.354
-0.020
(1.273)
(0.106)
(0.102)
0.766
-0.068
0.221
(1.144)
(0.082)
(0.101)
0.01
(.53) 0.06
5.4
181
(.01) 0.01
1.2
335
(.31) 0.01
2.3
367
(.10) 0.01
0.6
147
(.53) 0.02
1.5
166
(.22) 0.01
2.2
503
(.11) 0.01
2.8
411
(.06) 0.01
1.5
435
(.23) 0.09
26.4
514
(< .01) 0.08
16.7
396
(< .01) 0.12
28.4
409
(< .01) 0.25
12.5
77
(< .01) 0.13
9.6
128
(< .01) 0.15
5.3
65
(.01) 0.06
2.5
76
(.09)
projections of transaction price changes onto (i) an intercept, (ii) weighted average quotes in the order book for the two
risky securities (A and B). Time advances after each trade in the asset whose price is the regressor. The null hypothesis is given in (11). Standard errors in parentheses. b p-level
in parentheses.
c Number
of observations.
d Difference
between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely
proportional to quantities.
31
Table 6: Sensitivity Of The Market Microstructure Model Experiment
Coefficientsa
Security Intercept
981007
A
B
981116
A
B
990211
A
B
990407
A
B
991110
A
B
991111
A
B
000804
A
B
001005
A
B
a OLS
R2
F -statisticb
Nc
1.6
614
Weighted Average Quoted A
B
0.026
0.013
-0.021
(0.094)
(0.008)
(0.019)
0.038
0.000
0.028
(0.047)
(0.004)
(0.010)
-0.029
0.018
0.015
(0.064)
(0.009)
(0.013)
0.005
0.007
0.037
(0.049)
(0.007)
(0.010)
0.018
0.015
0.012
(0.131)
(0.011)
(0.017)
0.052
0.002
0.056
(0.086)
(0.007)
(0.011)
0.033
0.042
-0.001
(0.035)
(0.009)
(0.008)
0.041
0.002
0.025
(0.025)
(0.006)
(0.006)
0.071
0.037
-0.003
(0.055)
(0.009)
(0.012)
0.018
-0.016
0.032
(0.044)
(0.007)
(0.009)
0.026
0.060
-0.005
(0.050)
(0.010)
(0.016)
0.078
-0.014
0.091
(0.041)
(0.008)
(0.013)
0.280
0.164
-0.009
(0.143)
(0.035)
(0.030)
0.274
0.036
0.170
(0.155)
(0.038)
(0.032)
0.492
0.052
0.015
(0.378)
(0.030)
(0.034)
0.122
0.008
0.105
(0.359)
(0.028)
(0.032)
0.01
(.20) 0.01
4.3
614
(.01) 0.01
2.7
1053
(.07) 0.01
7.5
1053
(< .01) 0.00
1.0
466
(.37) 0.06
13.7
466
(< .01) 0.02
12.2
1385
(< .01) 0.01
10.2
1385
(< .01) 0.01
7.6
1318
(< .01) 0.01
8.0
1318
(< .01) 0.03
17.2
1190
(< .01) 0.04
26.2
1190
(< .01) 0.07
11.2
312
(< .01) 0.08
13.7
312
(< .01) 0.02
1.7
194
(.19) 0.05
5.2
194
(.01)
projections of transaction price changes onto (i) an intercept, (ii) weighted average quotes in the order book for the two
risky securities (A and B). Time advances whenever one of the three assets trades, and hence, the regressors are not necessarily in subjects’ prior information set. The null hypothesis is given in (11). Standard errors in parentheses. b p-level
in parentheses.
c Number
of observations.
d Difference
between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely
proportional to quantities.
32
Table 7: The Link Between The Market Microstructure Model And The Walrasian Model Experiment
Coefficientsa
Security Intercept
981007
A
B
981116
A
B
990211
A
B
990407
A
B
991110
A
B
991111
A
B
000804
A
B
001005
A
B
a OLS
R2
F -statisticb
Price Signalsc A
B
-4.046
0.036
0.418
(1.673)
(0.074)
(0.192)
-6.073
0.054
-0.880
(0.663)
(0.029)
(0.076)
2.850
-0.353
0.051
(0.413)
(0.038)
(0.022)
-0.812
0.070
-0.032
(0.284)
(0.026)
(0.015)
0.797
-0.449
-0.687
(1.700)
(0.075)
(0.099)
-4.102
0.074
-0.199
(1.187)
(0.052)
(0.069)
2.287
-0.195
0.104
(0.581)
(0.029)
(0.023)
-0.772
-0.002
0.021
(0.656)
(0.033)
(0.026)
-4.070
-0.220
0.093
(0.900)
(0.035)
(0.014)
-5.766
-0.192
-0.056
(0.763)
(0.030)
(0.012)
-4.575
-0.240
-0.172
(0.474)
(0.026)
(0.031)
-4.047
-0.087
-0.236
(0.305)
(0.017)
(0.020)
-1.822
-0.080
0.039
(0.551)
(0.035)
(0.042)
-6.751
0.065
-0.549
(0.513)
(0.032)
(0.039)
-12.036
-0.599
0.268
(1.953)
(0.059)
(0.051)
-10.430
0.063
-0.365
(1.817)
(0.055)
(0.048)
0.01
2.5 (.09)
0.18
68.0 (< .01)
0.08
45.9 (< .01)
0.01
5.9 (< .01)
0.16
42.8 (< .01)
0.02
5.1 (.01)
0.08
56.5 (< .01)
0.00
0.5 (.63)
0.07
52.0 (< .01)
0.04
27.5 (< .01)
0.07
44.5 (< .01)
0.11
69.7 (< .01)
0.02
2.9 (.06)
0.38
96.4 (< .01)
0.35
51.6 (< .01)
0.25
32.3 (< .01)
projections of weighted average quotes in the order book onto (i) an intercept, (ii) the estimated price signals of the
Walrasian model for the two risky securities (A and B). Weighted average quotes are the differences between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely proportional to quantities. Time advances whenever one of the three assets trades. Standard errors in parentheses. b p-level
in parentheses.
c Difference
between last transaction prices and estimated equilibrium prices.
33
Table 8: Consistency Between The Market Microstructure Model And The Walrasian Model Experiment
Coefficientsa
Security Intercept
981007
A
B
981116
A
B
990211
A
B
990407
A
B
991110
A
B
991111
A
B
000804
A
B
001005
A
B
a OLS
R2
F -statisticb
Weighted Average Quotec A
B
0.065
0.012
-0.003
(0.092)
(0.008)
(0.019)
0.025
0.004
-0.000
(0.046)
(0.004)
(0.009)
0.003
0.006
0.020
(0.064)
(0.009)
(0.013)
0.005
0.009
0.035
(0.049)
(0.007)
(0.010)
-0.004
0.001
0.007
(0.129)
(0.011)
(0.017)
0.036
-0.003
0.049
(0.085)
(0.007)
(0.011)
0.026
0.030
-0.000
(0.035)
(0.008)
(0.008)
0.021
0.003
0.025
(0.025)
(0.006)
(0.006)
0.042
0.028
-0.019
(0.054)
(0.009)
(0.011)
-0.017
-0.013
0.025
(0.044)
(0.007)
(0.009)
0.008
0.044
-0.021
(0.050)
(0.010)
(0.015)
0.060
-0.017
0.070
(0.041)
(0.008)
(0.013)
0.270
0.152
0.022
(0.141)
(0.035)
(0.029)
0.144
0.028
0.071
(0.154)
(0.038)
(0.032)
0.015
0.004
0.009
(0.373)
(0.029)
(0.034)
0.111
0.033
0.066
(0.358)
(0.028)
(0.032)
0.00
1.1 (.33)
0.00
0.6 (.46)
0.00
1.4 (.26)
0.01
7.2 (< .01)
0.00
0.2 (0.92)
0.05
11.3 (< .01)
0.01
6.6 (< .01)
0.02
10.7 (< .01)
0.01
5.7 (< .01)
0.01
5.0 (.01)
0.02
10.1 (< .01)
0.03
16.7 (< .01)
0.06
9.6 (< .01)
0.02
2.6 (.08)
0.00
0.0 (.95)
0.03
2.8 (.06)
projections of the error of equilibration model (8) (see Table 2) onto (i) an intercept, (ii) the weighted average quotes in
the order book for the two risky securities (A and B). Time advances whenever one of the three assets trades. Standard errors in parentheses. b p-level
in parentheses.
c Difference
between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely
proportional to quantities.
34
Appendix: Results For Four-Asset Experiments Note: Table 9 is the equivalent of Table 1; Table 10 is the equivalent of Table 2; etc.
35
Table 9: Experimental design data (Four Asset Experiments) Experiment
991026
001030
001106
001113a
a Only
Subject
Signup
Category
Reward
(Number)
(franc)
Endowments A
B
C
Cash
Loan
Exchange
Repayment
Rate
(franc)
(franc)
$/franc
Notes
13
0
4
0
5
0
400
2075
0.04
16
0
0
6
5
0
400
2350
0.04
46
0
4
0
5
0
400
2075
0.04
22
0
0
6
5
0
400
2350
0.04
47
0
4
0
5
0
400
2075
0.04
23
0
0
6
5
0
400
2350
0.04
30
0
7
13
9
0
400
2200
0.04
29
0
8
11
12
0
400
2310
0.04
three states could occur in the 001113 experiment. The payoff structure was: State
X
Y
Z
Security A
200
0
0
Security B
0
200
0
Security C
0
0
200
100
100
100
Note
36
Table 10: The Walrasian Model (Four Asset Experiments) Experiment
Coefficientsa
Security
991026
A
B
C
001030
A
B
C
001106
A
B
C
001113f
A
B
a OLS
R2
F -statisticb
Nc
ρd
5.9
710
-0.05
710
-0.06
710
−0.08∗
982
-0.04
982
-0.04
982
0.08∗
1642
-0.02
1642
-0.04
1642
0.01
2498
−0.04∗
2498
−0.09∗∗
Excess Demande
Intercept A
B
C
3.767
1.918
0.838
-0.473
(1.160)
(0.544)
(0.266)
(0.129)
1.784
0.639
0.425
-0.123
(0.689)
(0.323)
(0.158)
(0.077)
-2.039
-0.914
-0.467
0.214
(0.988)
(0.463)
(0.226)
(0.110)
2.556
2.933
1.085
-0.775
(0.413)
(0.378)
(0.152)
(0.098)
0.466
0.026
0.115
0.020
(0.297)
(0.272)
(0.109)
(0.071)
-0.336
-0.223
-0.032
0.076
(0.471)
(0.431)
(0.173)
(0.112)
0.670
0.510
0.231
-0.120
(0.366)
(0.168)
(0.073)
(0.043)
0.649
0.113
0.146
-0.003
(0.329)
(0.151)
(0.066)
(0.039)
-1.283
-0.517
-0.203
0.138
(0.298)
(0.136)
(0.059)
(0.035)
-0.327
0.103
0.033
(0.109)
(0.031)
(0.025)
-0.441
0.142
0.282
(0.188)
(0.053)
(0.043)
0.02
(< .01) 0.03
7.6 (< .01)
0.02
4.5 (< .01)
0.06
21.6 (< .01)
0.02
6.7 (< .01)
0.01
2.8 (.04)
0.01
8.0 (< .01)
0.02
12.1 (< .01)
0.01
7.3 (< .01)
0.01
8.8 (< .01)
0.03
34.4 (< .01)
projections of transaction price changes onto (i) an intercept, (ii) the estimated excess demands for the three risky securities
(A, B and C). See (8). Time advances whenever one of the four assets trades. Standard errors in parentheses. b p-level
in parentheses.
c Number
of observations.
d Autocorrelation e Estimated f Since
of the error term;
∗
and
∗∗
indicate significance at the 5% and 1% level, respectively.
on the basis of subjects’ final holdings and last transaction prices.
there were four securities and three states in experiment 001113, the demand for one of the securities was redundant.
Excess demands for A and B were computed based on a per capita endowment of -3 of A, 1.5 of B and 21 Notes, ignoring security C. This endowment is equivalent to one of 7.5 of A, 12 of B, 10.5 of C and zero notes.
37
Table 11: The Walrasian Model: Stability (Four Asset Experiments) Experiment
a For
Eigenvalues Of A∆−1 (∗10−4 )a First
Second
Third
991026
0.80
0.60
0.19
001030
1.19
0.27 + 0.06i
0.27 − 0.06i
001106
0.39
0.22 + 0.15i
0.22 − 0.15i
001113
0.64
0.06
stability, the real parts of the eigenvalue of A∆−1 should all be strictly positive.
38
Table 12: The Market Microstructure Model I (Four Asset Experiments) Experiment
Coefficientsa
Security Intercept
991026
A
B
C
001006
A
B
C
001106
A
B
C
001113
A
B
C
a OLS
R2
F -statisticb
Nc
3.2
134
Weighted Average Quoted
1.554
0.060
(0.859)
(0.033)
0.302
0.041
(0.187)
(0.021)
0.361
0.044
(0.214)
(0.022)
0.795
0.057
(0.428)
(0.022)
0.122
0.165
(0.245)
(0.043)
0.959
0.162
(0.443)
(0.036)
0.221
0.013
(0.576)
(0.029)
0.255
0.061
(0.261)
(0.025)
-0.014
0.017
(0.136)
(0.009)
-0.308
0.043
(0.125)
(0.010)
-0.081
0.010
(0.135)
(0.010)
0.010
-0.001
(0.090)
(0.008)
0.02
(.02) 0.02
3.7
169
(.06) 0.02
4.0
213
(.05) 0.03
6.4
247
(.01) 0.06
14.6
214
(< .01) 0.07
20.1
289
(< .01) 0.00
0.2
359
(.66) 0.02
6.2
355
(.01) 0.01
3.9
526
(.05) 0.02
17.2
688
(< .01) 0.00
1.1
717
(.30) 0.00
0.0
696
(.89)
projections of transaction price changes onto (i) an intercept, (ii) weighted average quotes in the order book. Time
advances after each trade in the asset whose price is the regressor. The null hypothesis is given in (11). Standard errors in parentheses. b p-level
in parentheses.
c Number
of observations.
d Difference
between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely
proportional to quantities.
39
Table 13: The Market Microstructure Model II (Four Asset Experiments) Experiment
Coefficientsa
Security Intercept
991026
A
B
C
001030
A
B
C
001106
A
B
C
001113
A
B
C
a OLS
R2
F -statisticb
Nc
1.2
134
Weighted Average Quoted A
B
C
1.355
0.060
-0.011
-0.019
(0.988)
(0.037)
(0.045)
(0.058)
0.977
0.027
0.054
-0.013
(0.429)
(0.016)
(0.025)
(0.027)
-0.322
-0.032
0.025
0.050
(0.486)
(0.017)
(0.021)
(0.024)
0.617
0.058
-0.015
-0.009
(0.473)
(0.023)
(0.057)
(0.029)
0.406
0.017
0.162
0.006
(0.402)
(0.020)
(0.043)
(0.023)
1.268
0.012
0.076
0.160
(0.611)
(0.030)
(0.067)
(0.036)
0.102
0.022
-0.051
0.014
(0.584)
(0.030)
(0.047)
(0.029)
-0.113
-0.036
0.079
0.006
(0.312)
(0.017)
(0.026)
(0.017)
0.065
0.009
-0.011
0.016
(0.170)
(0.009)
(0.013)
(0.009)
-0.344
0.043
0.000
0.009
(0.130)
(0.010)
(0.008)
(0.008)
0.064
-0.018
0.011
-0.004
(0.171)
(0.014)
(0.010)
(0.011)
0.010
0.003
-0.006
-0.001
(0.118)
(0.009)
(0.007)
(0.008)
0.03
(.32) 0.04
2.3
169
(.08) 0.05
3.5
213
(.02) 0.03
2.2
247
(.09) 0.07
5.1
214
(< .01) 0.07
7.2
289
(< .01) 0.00
0.5
359
(.66) 0.03
3.6
355
(.01) 0.01
1.8
526
(.14) 0.03
6.1
688
(< .01) 0.0
1.0
717
(.39) 0.00
0.3
696
(.83)
projections of transaction price changes onto (i) an intercept, (ii) weighted average quotes in the order book for the
two risky securities (A, B and C). Time advances after each trade in the asset whose price is the regressor. The null hypothesis is given in (11). Standard errors in parentheses. b p-level
in parentheses.
c Number
of observations.
d Difference
between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely
proportional to quantities.
40
Table 14: Sensitivity Of The Market Microstructure Model (Four Asset Experiments) Experiment
Coefficientsa
Security Intercept
991126
A
B
C
001030
A
B
C
001106
A
B
C
001113
A
B
C
a OLS
R2
F -statisticb
Nc
1.5
710
Weighted Average Quoted A
B
C
0.408
0.015
0.004
-0.002
(0.196)
(0.007)
(0.009)
(0.010)
0.126
0.004
0.007
-0.007
(0.117)
(0.004)
(0.005)
(0.006)
-0.171
-0.015
0.000
0.026
(0.166)
(0.006)
(0.007)
(0.009)
0.282
0.022
0.002
-0.002
(0.130)
(0.006)
(0.014)
(0.007)
-0.039
-0.004
0.054
-0.012
(0.091)
(0.004)
(0.010)
(0.005)
0.120
0.005
-0.005
0.018
(0.145)
(0.007)
(0.016)
(0.008)
0.082
0.001
0.002
-0.002
(0.071)
(0.004)
(0.005)
(0.004)
0.117
-0.002
0.024
0.008
(0.063)
(0.003)
(0.004)
(0.003)
-0.033
-0.001
-0.001
0.006
(0.057)
(0.003)
(0.004)
(0.003)
-0.043
0.007
0.000
0.002
(0.031)
(0.002)
(0.002)
(0.002)
-0.035
-0.002
0.011
0.001
(0.055)
(0.004)
(0.003)
(0.003)
-0.008
-0.000
-0.002
-0.000
(0.032)
(0.003)
(0.002)
(0.002)
0.01
(.21) 0.00
0.8
710
(.49) 0.02
5.0
710
(< .01) 0.01
4.1
982
(.01) 0.03
11.1
982
(< .01) 0.01
1.7
982
(.16) 0.01
0.4
1642
(.80) 0.02
9.6
1642
(< .01) 0.00
1.8
1642
(.14) 0.00
3.0
2494
(.03) 0.00
3.6
2494
(.01) 0.00
0.3
2494
(.81)
projections of transaction price changes onto (i) an intercept, (ii) weighted average quotes in the order book for the two
risky securities (A, B and C). Time advances whenever one of the three assets trades, and hence, the regressors are not necessarily in subjects’ prior information set. The null hypothesis is given in (11). Standard errors in parentheses. b p-level
in parentheses.
c Number
of observations.
d Difference
between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely
proportional to quantities.
41
Table 15: The Link Between The Market Microstructure Model And The Walrasian Model (Four Asset Experiments) Experiment
Coefficientsa
Security Intercept
991126
A
B
C
001030
A
B
C
001106
A
B
C
001113
A
B
C
a OLS
R2
F -statisticb
Price Signalsc A
B
C
-31.289
-0.274
0.673
0.045
(6.366)
(0.072)
(0.135)
(0.068)
11.234
-0.022
-0.335
0.515
(5.562)
(0.063)
(0.118)
(0.060)
-17.959
-0.165
0.700
0.175
(4.594)
(0.052)
(0.098)
(0.049)
2.488
-0.324
-0.360
-0.017
(2.062)
(0.070)
(0.059)
(0.037)
1.245
0.208
-0.342
0.067
(0.852)
(0.029)
(0.024)
(0.015)
-7.072
-0.076
-0.020
-0.232
(1.753)
(0.059)
(0.050)
(0.031)
-11.940
0.056
-0.141
-0.017
(2.651)
(0.038)
(0.050)
(0.058)
-2.712
0.044
-0.130
-0.021
(1.732)
(0.025)
(0.032)
(0.038)
-27.490
0.102
0.430
-0.443
(2.502)
(0.036)
(0.047)
(0.055)
2.597
-0.085
0.129
(0.875)
(0.035)
(0.029)
-1.330
-0.024
0.207
(1.182)
(0.048)
(0.040)
9.259
0.094
-0.136
(1.088)
(0.044)
(0.037)
0.09
22.8 (< .01)
0.11
29.6 (< .01)
0.21
61.3 (< .01)
0.09
32.2 (< .01)
0.20
79.2 (< .01)
0.06
19.7 (< .01)
0.01
3.0 (.03)
0.01
6.0 (< .01)
0.09
52.3 (< .01)
0.03
34.0 (< .01)
0.02
26.6 (< .01)
0.02
25.1 (< .01)
projections of weighted average quotes in the order book onto (i) an intercept, (ii) the estimated price signals of the
Walrasian model for the two risky securities (A, B and C). Weighted average quotes are the differences between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely proportional to quantities. Time advances whenever one of the three assets trades. Standard errors in parentheses. b p-level c Last
in parentheses.
transaction prices minus estimated equilibrium prices.
42
Table 16: Consistency Between The Market Microstructure Model And The Walrasian Model (Four Asset Experiments) Experiment
Coefficientsa
Security Intercept
991116
A
B
C
001030
A
B
C
001106
A
B
C
001113
A
B
C
a OLS
R2
F -statisticb
Weighted Average Quotec A
B
C
0.322
0.012
0.003
-0.002
(0.194)
(0.007)
(0.009)
(0.010)
0.174
0.006
0.002
0.002
(0.115)
(0.004)
(0.005)
(0.006)
-0.089
-0.013
0.008
0.026
(0.163)
(0.006)
(0.007)
(0.009)
0.207
0.012
0.025
-0.004
(0.127)
(0.006)
(0.014)
(0.007)
-0.092
-0.007
0.035
-0.011
(0.091)
(0.004)
(0.010)
(0.005)
0.122
0.004
-0.003
0.013
(0.145)
(0.007)
(0.016)
(0.008)
0.030
0.002
0.002
-0.001
(0.070)
(0.004)
(0.005)
(0.003)
0.040
-0.002
0.023
0.011
(0.063)
(0.003)
(0.005)
(0.003)
-0.021
-0.001
-0.002
0.003
(0.057)
(0.003)
(0.004)
(0.003)
-0.060
0.007
0.000
0.002
(0.031)
(0.002)
(0.002)
(0.002)
-0.042
0.001
0.013
-0.001
(0.054)
(0.004)
(0.003)
(0.003)
0.011
-0.001
-0.002
0.000
(0.032)
(0.003)
(0.002)
(0.002)
0.00
1.1 (.36)
0.00
0.9 (.42)
0.03
6.8 (< .01)
0.01
2.9 (.04)
0.02
6.0 (< .01)
0.00
0.9 (.46)
0.00
0.3 (.77)
0.02
10.2 (< .01)
0.00
0.6 (.62)
0.00
3.0 (.03)
0.01
5.8 (< .01)
0.00
0.6 (.65)
projections of the error of equilibration model (8) (see Table 10) onto (i) an intercept, (ii) the weighted average quotes in
the order book for the two risky securities (A, B and C). Time advances whenever one of the three assets trades. Standard errors in parentheses. b p-level
in parentheses.
c Difference
between weighted average ask prices (five best) and weighted average bid prices (five bids); weights are inversely
proportional to quantities.
43
