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I 1 W. .4 - 11.1.1. 3 .HMmuriftzrs 3.93....41 1...! .. . . June... .3. n1. .‘ Ht“. .cl.-WNVHII.. ”In.“ .QIflq. I... 4' nBIlVOu ' II. 4333 .9, lhfi. J.MH.A 3. 4|! . .3 u‘v‘hr...‘ a... 113...)! 3.4.25.5}. v. 3)..P..v«-4._ 1.7-! .9 I... 1‘. 3&4.... . .r .1 b. o/swa’ MICHOG AN SYATE UN IVER 1 “Hill ll’llf’ll'.’.'.'lllllllllllll 1293 00599 5331 LIBRARY Michigan State University This is to certify that the dissertation entitled The Effect of Information on the Behaviors of Security Price and Trading Volume presented by Kwok Sang Tse has been accepted towards fulfillment of the requirements for Doctor of BhilasnphLdegree in Finance KJrc/SuxéA Major professor Kirt C. Butler Assistant Professor Date May 3* 1990 of Finance MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 “W‘ \ 91", '. PLACE IN RETURN BOX to move this checkout from your record. TO AVOID FINES return on or before date om. DATE DUE DATE DUE DATE DUE ll l j l . MSU Is Anjfl‘irmativo Action/Equal Opportunity Institution The Effect of Information on the Behaviors of Security Price and Trading volume BY Kwok Sang Tse A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Finance and Insurance ABSTRACT THE EFFECT OF INFORHATION ON THE BEHAVIORB OE SECURITY PRICE AND TRADING VOLUME BY Kwok Sang Tse The price-volume relationship and the effect of the information arrival process on price and volume have been extensively examined by many authors. However, as noted by Hal R. Varian}, little analytical work has been done on the effect of diverse beliefs arising from heterogeneous information on the behavior of price and trading volume. In particular, little theoretical work has been done on the relationship between the nature of information and the behavior of security price and trading volume. The objective of the three essays in this dissertation is to investigate theoretically and empirically the impact of information characteristics and heterogeneous beliefs on security price and trading volume. The first essay develops a theoretical model in a noisy rational expectations equilibrium framework incorporating heterogeneous information and diverse beliefs. The quality of information is characterized by individual investor's confidence and the variability of opinion across investors. The effects of these two characteristics of information on security price and volume are examined. It is found that when the market is confidence driven, large trading volume normally accompanies large price variability. When the market is consensus driven, price variability is accompanied by low trading volume. Also, caution needs to be exercised when attempting to use price and volume to measure information content. The second essay similarly develops a theoretical model relating security price and volume reaction to earnings announcements. A potentially asymmetric price-volume relationship emerges from the theoretical model depending on investor optimism or pessimism just prior to the announcement and the effect of the announcement on investor uncertainty. Empirical tests using daily CRSP returns, Media General's Trading Volume Tapes, Compustat, and Lynch, Jones and Ryan's Institutional Brokers Estimate System database are developed to examine the model. Empirical evidence is consistent with the asymmetric response of price and volume to good news and bad news announcements according to the theory. The third essay develops a statistical test for estimating the onset and duration of security price and trading volume responses to new information. It extends the analysis of Hillmer and Yu (1979) by allowing a dependent relationship between price and volume. The dependent relationship between price and volume is addressed by orthogonalizing one market attribute with respect to the other. However, the resulting statistical test may provide biased estimates of the onset and duration of market responses to new information (see Giliberto (1985)). A practical procedure for implementing the statistical test is then prescribed. The statistical test allowing dependence is compared to the Hillmer and Yu (1979) and Pincus (1983) tests in simulations of real world responses to information. 1. Varian, H. R. ”Differences of Opinion in Financial Markets.” Working Paper, University of Michigan (March 1988). DEDICATION This dissertation is dedicated to my wife, Hun Chee Chan, my son, Kyle, my daughter, Kellie, and my mother, Lam Heung. ii ACKNOWLEDGEMENTS Obviously, this dissertation could not have been completed without substantial assistance. Research resources and computer facilities provided by both Michigan State University and Indiana University proved quite valuable. Committee members Ann Kremer and Dale Domain will never know the true extent of my thanks for their help and comments. Most of all, I thank my committee chairman, Kirt Butler, for all his help and advice. I will never be able to repay him for his dedication to my work and career. iii List of Chapter Chapter Chapter Chapter Chapter Tables: One: TWO: Three: Four: Five: TABLE OF CONTENTS Introduction The Effect of Divergent Opinions on Security Prices and Trading Volume The Effect of Forecast Bias and Investor Disagreement on Security Prices and Trading Volume The Estimation of Market Speed of Adjustment Using Security Prices and Trading Volume Conclusions iv Page 40 90 127 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table LIST OF TABLES The Effect of Confidence and Consensus on Price Variability and Trading Volume Summary Summary Test of of Propositions 1 Through 6 of Empirical Tests of Asymmetry Hypothesis 1 Using Two-Day Geometric Mean Return as Dependent Variable Test of Hypothesis 2 Using Two-Day Geometric Mean Return as Dependent Variable Test of Hypothesis 3 Using Two-Day Geometric Mean Return as Dependent Variable Test of Hypothesis 4 Using Two-Day Geometric Mean Returns as Dependent Variable Test of Average Test of Average Test of Average Test of Average Test of Average Test of Average Hypothesis 1 Using Two-Day Excess Returns as Dependent Variable Hypothesis 2 Using Two-Day Excess Returns as Dependent Variable Hypothesis 3 Using Two-Day Excess Returns as Dependent Variable Hypothesis 4 Using Two-Day Excess Returns as Dependent Variable Hypothesis 5 Using Two-Day Trading Volume as Dependent Variable Hypothesis 6 Using Two-Day Trading Volume as Dependent Variable Shift in Means - Rt and Vt Independent Shift in Variances - Rt and Vt Independent Shift in Correlation Coefficient Shift in Mean Levels - R related to V t t Page 26 70 71 72 73 74 75 76 77 78 79 80 81 123 124 125 126 1 CHAPTER ONE: INTRODUCTION The relationship between stock prices and trading volume has interested practitioners and financial economists for many years. Price and volume are widely used by financial analysts as market sentiment indicators to gauge rallies and declines, to forecast bull and bear markets, and to predict market turning points. For example, a lot of technical analysts agree on these principles:1 1. A price rise accompanied by expanding volume is a normal market characteristic and has no implications so far as a potential trend reversal is concerned. 2. A rally which reaches a new (price) high on expanding volume but whose overall level of activity is lower than the previous rally is suspect and warns of a potential trend reversal. 3. A rally which develops on contracting volume is suspect and warns of a potential trend reversal. Technical analysts generally also believe that security return and volume are associated and that volume changes may presage price changes. Bernstein [1983] states that an investor can predict the movement of the stock market if he can predict volume.“ 1Martin J. Pring, ”Technical Analysis Explained" (New York: McGraw- Hill, 1985, p.149. 2Peter L. Bernstein, “The Volume Indicator, Refurbished, and Retained," Peter L. Bernstein, Inc., April 1, 1983. 2 To the financial economist, price-volume relationship has important implications for understanding the microstructure of financial markets. Several important works have been developed to explain the impact of the rate of information flow and the way information is disclosed on price-volume relation.3 Some researchers also believe that if price and volume are jointly determined, incorporating volume information into event studies will improve the power of test statisticsw‘ Price-volume relationship is also important for identifying the empirical price distributions of speculative assets including options and futures. It is a common belief that speculative prices follow either the stable Paretian distribution with infinite variances or a mixture of distributions with different conditional variances.5 'On the other hand, volume information could 3See for example Copeland, T. E. "A Model of Asset Trading under the Assumption of Sequential Information Arrival." Ihg_ggg;ngl_gfi Eingggg 31 (September 1976), 1149-1168. Morse, D. ”Asymmetrical Information in Securities Markets and Trading Volume.” Journal of Einggcigl and angtitative Analysis 15 (March 1980), 1129—1148. Jennings, R. H., L. T. Starks, and J. C. Fellingham. "An Equilibrium Model of Asset Trading with Sequential Information Arrival." lenrnal_2f_£inan9e. 36 (March 1981). 143-161. ‘Richardson, G, S. E. Sefcik, and R. Thompson. "A Test of Dividend Irrelevance Using Volume Reaction to a Change in Dividend Policy.” l2urnal.2f.£inangial_EQQEgmiea. 17 (Dec. 1986). 313-333. 5See, for example, Epps, T.W., and M. L. Epps. "The Stochastic Dependence of Security Price Changes and Transaction Volumes: Implications or the Mixture-of-Distributions Hypothesis." Econogetrica 44 (March 1976), 305-321. 3 provide a good proxy for the changing variance,‘5 providing useful insight concerning the behavior of price around the event day in event studies. Empirical studies in the accounting literature have used security prices and trading volume to measure the effects of new public information on financial markets.7 Traditionally, price changes are used to measure the effect of informativeness.a 'Trading volume, on the other hand, is employed by researchers as a measure of consensus among investors" or of the information content of an event.“ Tauchen, G. E., and M. Pitts. "The Price Variability-Volume Relationship on Speculative Markets.” Eggngmgtgigg 51 (March 1983), 485-505. 6Rogalski, R. J. ”The Dependence of Prices and Volume." The Review 2f_E£2n2m123_and_fitatistiss 36 (may 1978). 268-274. 7Imhoff, E. A. Jr., and G. J. Lobo. "Information Content of Analysts' Composite Forecast Revisions." Jggzngl 9f Aggogntigg Research 22, No. 26 Autumn 1984, 541-554. 8Atiase, R. K. 'Predisclosure Information, Firm Capitalization and Security Price Behavior Around Earnings Announcements.” Journal 9f Aseeunfins_fiesgarsh 23 (1985). 21-35. Beaver, U. H. ”The Information Content of Annual Earnings Announcements." Empirical Research in Accounting: Selected Studies. SUPPIe-ent to l2urna1.2f_A2222ntina_82§eersh 6 (1968). 67-92. °Bamber, L. S. "The Information Content of Annual Earnings Releases: A Trading Volume Approach." u a 0 tin e earc 24 (Spring 1986), 40-56. 1"’Lakonishok, J., and T. Vermaelen. "Tax-Induced Trading around Ex- Dividend Days." l2urnal.2f.£inangial_figgngmiss 16(Ju1y 1986). 287-319. Pincus, M. "Information Characteristics of Earnings Announcements and Stock Market Behavior." Jgumgl 9f Acgounting 8mm]; 21 (Spring 1983), 155-183. 4 The price-volume relationship and the effect of the information arrival process on price and volume have been extensively examined by many authors. However, as noted by Hal R. Varian,11 little analytical work has been done on the effect of diverse beliefs arising from heterogeneous information on the behavior of price and trading volume. In particular, little theoretical work has been done on the relationship between the nature of information and the behavior of security price and trading volume. The objective of this study is to investigate theoretically and empirically the impact of information characterisitics and heterogeneous beliefs on security price and trading volume. The remainder of this dissertation is organized in this manner. Chapter 2 develops a theoretical model in a noisy rational expectations equilibrium framework incorporating heterogeneous information and diverse beliefs. The quality of information is characterized by individual investor's confidence and the variability of opinion across investors. The effects of these two characteristics of information on security price and volume are examined. The conclusion is that when the market is confidence driven, large trading volume normally accompanies large price variability. When the market is consensus driven, price variability is 11Varian, H. R. ”Differences of Opinion in Financial Markets.” Working Paper, University of Michigan (March 1988). 5 accompanied by low trading volume. Also, caution needs to be taken when we try to use price and volume to measure information content. Chapter 3 employs earnings announcement as a source of information to study how security price and volume react to good news and bad news. In a framework similar to Chapter 2, a theoretical model relating earnings announcements to security price and volume reaction is first developed. Empirical tests using daily CRSP returns, Media General's Trading Volume Tapes, Compustat, and Lynch, Jones and Ryan's Institutional Brokers Estimate System database are then developed to examine the model. Empirical evidence is consistent with the asymmetric response of price and volume to good news and bad news announcement according to the theory. Chapter 4 attempts to develop a multivariate statistical technique using security price and trading volume to measure the adjustment speed of financial market to new information disclosure. The technique serves to detect the point in time when the market attributes begin to react to the news and the point in time when the reaction is over. Simulation studies are conducted to confirm its properties. 6 CHAPTER 2: THE EFFECT OF DIVERGENT OPINIONS ON SECURITY PRICES AND TRADING VOLUME Abstract This essay develops a single-period rational expectation model with noise and diverse beliefs to investigate the effect of investors' divergent opinions on the behavior of prices and trading volume. Information characteristics in terms of investor's confidence in his forecast and diversity of opinion are employed to analyze the effects of heterogeneous information on equilibrium price and trading volume. I. Introduction Security prices and trading volume are the two most widely reported financial variables by the news media. What kind of insight about investors' opinions in securities can we gain from prices and trading volume? The relationship between security prices and trading volume has interested practitioners and financial economists for many years. Price and volume are popularly used by financial analysts as market sentiment indicators to gauge rallies and declines, to forecast bull and bear markets, and to predict market turning points. To the financial economist, the price and volume relationship has important implications for understanding the microstructure of financial markets, for event studies, and for identifying the empirical price distributions of speculative assets including options and futures. Empirical studies in the accounting literature have used security prices and trading volume to measure the effects of new public information on financial markets. In general, a public disclosure can cause a precision effect and/or a consensus effect. Precision measures the gain of knowledge and consensus measures the extent of agreement among agents caused by the new information. Traditionally, price changes are used to measure the effect of informativeness (Beaver [1968] and Atiase [1985]). It has also been used as a measure of information content (Beaver, Lambert, and Ryan [1987]). Trading volume, on the other 8 hand, is employed by researchers as a measure of consensus among investors (Beaver [1968], Morse [1981], and Bamber [1987]) or of the information content of an event (Beaver [1968], Lakonishok and Vermaelen [1986], Morse [1981], Ro [1981], and Pincus [1933]). The objective of this essay is to investigate the effect of investors' divergent opinions induced by information disclosure on the behavior of prices and trading volume. A single-period rational expectation model with noise and diverse beliefs of the sort examined by Admati [1985], Varian [1987], and Diamond and Verrecchia [1981] is developed. Information characteristics in terms of precision and consensus will be formally defined and incorporated into the model in order to analyze the effects of heterogeneous information on equilibrium price and trading volume. The rest of the chapter is divided into five sections. Section II describes the market structure, the demand for the risky asset, and its price at equilibrium. Noise trading is allowed to exist in the economy. Section III defines the precision and the consensus effects induced among investors by information disclosure. The effect of precision and consensus on price variability and trading volume are analyzed in Sections IV and V. Section VI discusses the circumstances in which the separate effects of precision and consensus may be observed based on price variability and trading volume. In most 9 cases it is impossible to use either price or volume alone to measure information content. 11. Market Structure A two-asset single period model is developed in which investors have different endowments of wealth and identical initial beliefs. Investors may have diverse preferences, but all investors have a negative exponential utility function for wealth and all their preferences exhibit constant absolute risk tolerance. During the period, a different signal is observed by each investor, causing him to have different expectations regarding the final price of each asset. Another ingredient of this model is the existence of noise in the form of random supply and demand of the risky asset. IIA. Assumptions (A1) Population There are two groups of traders in the market. The first group is the diversely informed investors who trade to maximize their utilities subject to their budget constraints. This group is composed of I investors, indexed by i - 1,2,...,I. The second group is the liquidity traders (noise traders) who submit their demand orders according to their liquidity needs. In general, noise trading can exist in different forms. It might be caused by some trade of a 10 nonspeculative nature such as for life-cycle or liquidity reasons. It can be caused by some traders lacking perfect knowledge of the market structure, or by agents who do not know the realized aggregate endowment, as in Diamond and Verrecchia (1981). In a recent article, Trueman (1988) argues that noise trading can ensue from the incentive of the investment funds manager which is related to investors' perceptions of his ability. In this analysis, noise trading comes only from liquidity trading. Therefore, the supply per capita of the risky asset is assumed to be the realization of a random variable 2. Trades from this group are assumed to arrive at the market in a random fashion and constitute the exogenous noise in the economy. (A2) Assets There are only two assets in the economy: a riskless bond with known payoff and a risky asset with uncertain payoff U. The realizations of U are given by U. Both the risky and the riskless asset pay off in a single consumption good. The riskless bond serves as numeraire and each unit yields one unit of the consumption good. That is, the return to one unit of the riskless asset is unity. No consideration is given to time preference since it would only obscure the analysis. 11 (A3) Endowments Each informed investor i (i = 1,2...I) is endowed with risky asset D01 and riskless bond BM. Assume that the total endowment of the risky asset is held by the informed investors and the net holding of the risky asset by the liquidity traders is zero. Let 2, be the total per-capita supply of the risky asset. Then i oi o’ (1) (A4) Preferences Every investor i has a negative exponential utility function for wealth w of the consumption good given by: 01““) g -exp(-w/ri) (2) where investor i exhibits constant absolute risk tolerance r1. (A5) Information At the beginning of the period, every investor has the same prior beliefs about the risky asset's uncertain end- of-period payoff U which is believed to be normally distributed with mean M and variance V. During the period, each investor 1 receives information.Y} concerning the liquidating value U of the risky asset, Y1 = U + n + ci (3) 12 where n is the common noise normally distributed with mean 0 and variance N, and E, is the idosyncratic noise term normally distributed with mean zero and variance 8,. Note that n > 0 implies that investors as a whole are optimistic about the liquidating value of the risky asset, and n < 0 implies that they are pessimistic. It is assumed that U, n and E are independent of each other. Also the 2,,are independent across all investors, E[E,, §,]=- O for i f j. Following Admati [1985], we assume that the variance of E, is uniformly bounded, and that, (Ei71)/I = U + n almost surely. (4) As the idiosyncratic noise terms 3, are aggregated across investors, the law of large numbers causes them to converge almost surely to their mean of zero. Investors submit their buy and sell orders to the auctioneer based on the information they receive during the period. The liquidity traders submit orders randomly. The total per-capital supply of the risky asset net of liquidity trading is a random variable 2 with mean Zo,and variance approaching a. The assumption of liquidity trading implies that z is independent of 0, fl, and 2,. Through this exchange of assets, a new equilibrium price P for the risky asset is established. At the end of the period, every agent liquidates his holdings of the two assets and consumes them. Let D, and B, be investor i's holdings of the risky and the 13 riskless asset at the end of the transaction. The objective of each investor is to maximize his expected utility of terminal consumption at the end of the period, 1 E,[-exp(-r,' (0,6 + B,))1 (5) E,[.] is the expectation operator of investor i based on his own information. The terminal wealth D,U + B, is to be consumed at the end of the period. The budget constraint of investor i is given by, DiP + Bi = D 1p + 801. (6) O 118. Definition of Rational Expectations Equilibrium Following Admati [1985], and Diamond and Verrecchia [1981], the rational expectations equilibrium for the finite economy is defined as the price P and allocation functions D,(Y,, P,) and B,(Y,, P,) for all i = 1,2...I such that a) P is (U + n, 2) measurable: b) [Di(Yi' P), Bi(Yi' P)] e arg maxD§[-exp(ri-1(DiU+Bi))IYi] subject to DiP + B1 = DOiP + Boi; c) ziDi(Yi’ P) = 2 almost surely. 14 Note that conditional on Y, =Y,, U is normally distributed with mean and variance as follows: EIUIY. =Y.1 = M + 3.0!. - M) Var(UIY, =Y,) = Var(U) - fi,Var(U) , where B, = Var(U)/Var(Y,) = V/(V + N + 8,). The distribution of exp{-r,'1(D,U + B,)) conditional on Y, =Y, is lognormal. Direct computation yields Et-exp{-r.“(o.0 + 8.)}IY. =Y.1 = -exp{-r."}. Since an exponential function is strictly increasing in its exponent, the maximization of (5) becomes MaxD {r.“(D.E[UIY. =Y.1 + B.) - 1/2(r,'2)D,2Var(UIY, =Y.)}. i subject to B1 = Din + Boi - DiP and its unique solution is provided by the first order condition. After determining individual i's optimal demand for the risky asset, we can solve for the risky asset's equilibrium price by using the market clearing condition 15 (1). The results are summarized below: Lemma_l: (a) Investor i's demand for the risky asset conditional on Y, =Y, is given by: -1 D, = ri[M + V(V +2N + Si) (vi-1 M) - P] (7) v - v (v + N + 81’ (b) The equilibrium price of the risky asset is a linear function of the form: i = (1 - A)M + A(U + 5) - Bi (3) where A = {zi[(ri/012)V(V + N + si)'1]}B , (9) ai2 = Var[UIVi = Y1] = v - v2(v + N + Si)"1 (10) and B = [1/21(ri/oiz)]. (11) 2199:: See Appendix III. Definition of Information Characteristics The precision effect is measured by the variability of each agent's observed signal about the unknown value of the risky asset. Consensus on the other hand is a measure of the degree of agreement among different agents. It is 16 measured by the degree of dispersion of opinions among the agents. Suppose we have two financial analysts A and B forecasting the value of a stock at the end of the period. At the start of the period, before new information about the company is released, analyst A's forecast is $50 with high and low being $45 and $55, while analyst B's figure is $50 with high being $60 and low $40. After the information is announced, A revises his forecast to $55 per share and B to $65 per share. However, A's high and low are $57 and $53, while B's are $70 and $60. The smaller post-information high-low spreads for both A and B's forecast reflect the precision effect induced by the information. Both A's and B's uncertainty about the unknown value of the stock become less. However, the information has induced a weaker consensus between A and B about the expected value of the stock. In this study, the precision effect of information follows the standard definitions. The consensus effect between two agents i and j is usually defined as the correlation coefficient between their diverse opinions, Y, and Y5. This definition, however, is not very descriptive of the overall consensus in the economy. Therefore, this study develops a different and yet intuitive definition of consensus. (1) Precision of information for investor i, 9,, is defined by ln[1/Var(Yi)] = ln[1/(V + N + 31)] (12) 17 The precision effect for each agent is simply defined as the inverse of the variability of his signal since less variability means more precision. (2) Consensus among the investors on the liquidating value 0 of the risky asset induced by the observation of information, ¢¢, is defined by: 1n[1/(zisi)] (13) Consensus is defined as the inverse of the dispersion of each agent's opinion from the mean consensus opinion since less dispersion of opinion means more consensus. From equations (3) and the independence of idiosyncratic noise across investors, the dispersion of agent i's opinion from the average opinion is E(Yi - 2121/1)2 = E(E - ZiEi/I)2 = si. 1 The second equality is true because equation (3) implies that 2,3, converges to zero. Adding across all agents and taking the inverse yields the definition for consensus in equation (13). 18 IV. lean-Variance Analysis of the Change in Price In this section, we analyze the effects of precision e, and consensus (is, on the mean and variance of the price change induced by the new information. The change in price is the new equilibrium price minus the beginning price Po. Based on the equilibrium price given in Lemma 1b, the mean and the variance of the change in price conditional on the beginning price and the supply 2 are respectively given by E(APIPO, z = Z) = M - BZ — P0 (14) and Var(APIPO, z = Z) = A2(V + N). (15) To examine the comparative statics of the expected change in price and the variance of the AP in terms of the precision and consensus effects, we take the derivative of the mean and the variance with respect to each effect while holding the other constant. Some preliminary results prove useful. W: By allowing N and S, to vary with precision e, and consensus 4),, we have: (a) If d, is kept constant, then (1) dN/dei + dSi/dei = -(v + N + Si) for all i; .. I _ .. (11) 2j=1 de/dei — 0 for all 1, and (iii) dA/dei > 0, and dB/dei < 0 for all i. (1)) (ii) 2. 19 If e, is kept constant, then (1) dN/dpL + dSi/doL = 0 for all 1: i=1 dSi/doL = -(zsi); and (iii) dA/d¢L = dB/doL = o. 2192:: See Appendix. Inggzgm_1: If consensus.¢, about the risky asset's liquidating value is kept constant, then (a) (b) An increase (decrease) in information precision about the value of the risky asset for all agents will increase (decrease) the expected change in asset price. The effect of information precision on the variance of the price change is indeterminate. However, if S,‘< (V + N), then an increase in information precision will lead to an increase in price variability. If the agents are symmetrically informed (S, = S for all i), then an increase in information precision will also lead to an increase in price variability provided 8 < (V + N). The price variability will decrease if s > (V + N). Erggf_of_1he2rem_l: See AppendiX- 20 W: If every agent i's information precision e, is kept constant, then (a) A change in consensus has no effect on the expected change in price. (b) An increase (decrease) in consensus will increase (decrease) the variance of the change in price. RIQQI_QI_IDEQIQE_Z: (a) By taking the first derivative of the expected change in price given Po and z with respect to ¢,, the result follows directly from lemma 2(b)(iii). (b) The first derivative of Var(APIPo , z = 2) with respect to ¢, is Az(dN/d¢,) + 2A(V + N) (dA/dq). By lemma 2(b), dN/dcp, is positive and dA/dcp, is zero, hence the result. (q.e.d.) v. Vblume of Trade In this section, the trading volume consequences of the effects of precision and consensus about the value of the risky asset are analyzed. The results developed are based on the assumption that all agents are symmetrically informed about the value of the risky asset, that is, S, = SJ for all i and j. When the agents are asymmetrically informed, the effects of precision and consensus on the volume of trade become uncertain. 21 The overall trading volume after the information disclosure is given by: T = 1/2 siloi - D (16) 01' Since Y, is normally distributed, so is D, which is a linear function of Y,. Let X, be the net demand (I), - D0,) of agent i. From the expression for D“ it.can be shown that the expected value ((1,) and the variance (a',z) of X, given D0, are respectively given by: 2 -1 riBZ/[V V (V + N + Si) ] - Doi' and (17) 2 -1 81 + [V(V + N + Si) - A](V + N)) [v - v2(v + N + sifl]2 . 2 2 - (13) Since X, is normally distributed with mean u, and variance a',‘, the expected value of the absolute value of X, is given by 2E0,2¢,(0) - u,¢,<0)1 + u, (19) where ¢,(0) is the normal density function with mean u, and variance a',z evaluated at zero, and O,(0) is the corresponding cumulative normal distribution function. Therefore, aggregating the expected absolute net demand over 22 all agents yields the expression for the expected trading volume given the initial demands DM: E[T|Z. D0,] = z,[a',2¢(0) - u,o(0)1 (20) Note that the aggregate expected net demand 2g“ is zero. Before we can state the effects of precision and consensus on trading volume, we need the following preliminary results: Lemma_;: Assume that the agents are symmetrically informed. Then (a) If consensus is kept constant, du,/de, 0, and dc',z/de, > 0; (b) If precision is kept constant, du,/d 0. Similarly, by applying lemma 2b to the first derivative, dc',2/d¢,_ < 0 is obtained. (q.e.d.) We can now state the effects of information on the behavior of trading volume. Inggrgm_§ If the agents are symmetrically informed, an increase (decrease) in the precision of information about the value of the underlying risky asset will increase (decrease) the expected volume of trade. 2399:: The first derivative of the expected volume of trade E(T) with respect to 6 is given by: z,[(dE(T)/da',)(do',/de) + (dE(T)/du,>(du,/de)1 (22) NOte that dE(T)/da'i = z:i[(l«‘iz/a'i + O'i)¢(0) "' (I-‘iz/a'i)¢(0)]l which is in turn equal to 2,[a',¢(0)] > 0. By lemma 3a, du,/de is zero and da',/de is positive: hence dE(T)/de > 0. [q.e.d.] 24 Thggrgm_1: If the agents are symmetrically informed, an increase (decrease) in the consensus of information about the value of the underlying risky asset will decrease (increase) the expected volume of trade. Ezggfi: The steps are exactly the same as in the proof of theorem 3, except that the results of lemma 3b are used instead. [q.e.d.] VI. The Effect of Information on Price Variability and Trading Vblume The interaction of confidence and consensus and their effects on the risky asset's price variability and trading volume are summarized in this section. Let t6 denote an increase in confidence for all agents and t¢ an increase in consensus among investors. Similarly let #9 and J¢ denote a decrease in confidence and consensus respectively. Let "t6 » i¢" denote the event that the effect on price variability or trading volume of an increase in confidence dominates that of a decrease in consensus. In the following discussion, we assume that (1) V + N > S,, and (2) all agents are symmetrically informed. Now, consider the effect of "16 » ¢¢” on the behavior of the change in price and trading volume. By theorem 1, :9 will increase the price variability, but by theorem 2, ¢¢ will lead to a decrease in 25 price variability, ceteris paribus. Since the effect of Te dominates that of t¢, we expect to observe a ”moderate" increase in price variability. Assuming everything else constant, theorem 3 implies that te leads to an increase in trading volume, and theorem 4 indicates that i¢ will increase the trading volume as well. Since both 16 and ip exert an upward pressure on trading volume, we should observe an extraordinarily large volume. Similarly, if the information disclosure induces 16, but does not influence the consensus among the agents, then theorem 1 and theorem 3 imply that we should observe a 'large' price variability and a 'large' increase in trading volume. The effects on price variability and trading volume of various combinations of change in confidence and consensus are presented in Table 3.1. Table 3.1 The Effect of Confidence and Consensus on Price Variability and Trading Volume Price Variablity Trading Volume Degree of Degree of Degree of Degree of Increase Decrease Increase Decrease 1.16»¢¢ + +++ 2. 16 only + + + + 3.te»f¢ +++ + 4.t¢»$6 + --_ 5. to only + + - - 6.t¢»16 +++ - 7.$¢»19 - +++ 8. &¢ only — - + + 9.$¢»t6 --- + 10.16»1¢ - --- 11. #6 only - - - - 12.16»¢¢ --- - + + extraordinarily large increase; + + large increase: moderate increase: extraordinarily large decrease: large decrease: and moderate decrease. 27 The results in Table 3.1 enable us to draw inferences about the characteristics of new information based on observed changes in price variability and trading volume. Suppose we observe a negligible change in price variability and trading volume after a news announcement is released. In this case, either the news announcement did not contain new information or the effect of consensus is greater than the effect of confidence (cases 4 through 9 in Table 3.1). If we observe instead a decrease in price variability and an increase in trading volume, we can conjecture from cases 7 through 9 in the table that there is a decrease in consensus and an increase in confidence among the investors, and the consensus effect dominates the confidence effect. From the price variability and trading volume that we observe, we can still infer about the possible combinations of the confidence and the consensus effect induced by the information. Cases 2, 5, 8, and 11 represent those situations in which isolated effects of confidence and consensus are observed. In all other cases, it is extremely difficult to use either the trading volume or price variablity as a measure or proxy for the information content or consensus effect. 28 VII. Conclusion This essay develops a theoretical model in a noisy rational expectations equilibrium framework incorporating heterogeneous information and diverse beliefs. The quality of information is characterized by individual investor's confidence and the variability of opinion across investors. The effects of these two characteristics of information on security price and volume are examined. The conclusion is that when the market is confidence-driven, large trading volume normally accompanies large price variability. When the market is consensus-driven, price variability is accompanied by low trading volume. Also, caution needs to be taken when we try to use price and volume to measure information content. As stated in Theorem 1, the effect of investors' confidence on price variability is not clear. However, if the variance of the idiosyncratic noise is small relative to the variance of the unknown risky payoff and the common noise, then increase in confidence will lead to increase in price variability. 29 Appendix 2:22f_9f_Lemma_llali The first order condition of Man (r." - P1 21 .. =Z Var(UIYi) Solving for P gives - - r;E(UIY;) - l_e 1 Var(UlYi) [2 r1- ] 1 Var(UIYi) Denote the rightmost term on the right hand side as B. By substituting the expression for E[OIY,=£Y] into the price function above, we obtain P: 1-821— - riVar(U)/Var(Y) riVar(U)/Var(Y) - M + B 21 __1 yi - BZ Var(UIYi) Var(UIYi) 30 Substitute Y, = U + n + e, in to the price function. Rewrite the first term on the right hand side of the equation as (1 - A)M. If it is justified to write riVar (U)/Var(Y) .. B 2i -— 6i Var(UIYi) then we have P = (1 - A)M + A(U + n) - Bz. See Admati [1985]. £r22f.2f.Lemma.Zlal= (1) By the definition of 6,, we have 11 dei/dei 1 = d[ln(V + N + 31). mai -(dN/dei + dSi/d61)/(V + N + si) This implies that dN/d6, + dS,/d6, = (V + N + S,) . (ii) By keeping 4:, constant while changing 6,, we have 1 den/eai = o = d1n(1/zjsj)/dei = -(zsj)‘ zj(dsj/dei) which implies that z,(dS,/d6,) = 0. 31 (iii) First, show that dB/d6,‘< 0. From the expression of B in lemma 1b, we have dB/d6, given by: -Bzzi[[-riV2(dN/dei+ dSi/dei)/(V + N + Si)2] /[v - v2(v + N + si)'1]2] By the result of 2(a)(i), dB/d6,‘< 0. Next show that dA/d6, > 0. From the expression of A in lemma 1b, rewrite A as CB where C is E,[(r,/a,2)V(V + N + s,)"]. By the chain rule of differentiation and rearrangement, it can be shown that the first derivative of A with respect to 6, is given by: 2 2 -l 2 2 2 -1 2 2 -CB [Eiriv (V+N+Si) ]/(oi) ] + B[Eiriv (V+N+Si) ]/(oi) ] _ 2 _ -1 2 2 -1 2 2 - B [ [EiriV(V+N+Si) /ai ][ZiriV (V+N+Si) /(ci ) ] + (1/B)[zirivz(v+N+si)'1/(aiz)2]] By adding the term, A, to the expression above, and substracting A from the first term, the derivative can be rewritten as: 32 A + 82((Ei(ri/ciz))[EiriV2(V+N+Si)-1/(aiz)2] - (2,(r,/(a,2)2>[B,r,v2(V+N+s,)'1/(a,2)J} From the expression for of in lemma 1(b), it is easy to show that V2(V + N + S,)'1 -= V - 0,2. Then, we replace V2(V + N + S,)'1 with (V-a,z) in the expression above, and by cancelling terms, we obtain the following expression: A + th(2,(r,/(a,2)2)(z,r,) - (2,(r,/(o,2))21 _ 2 2 2 2 _ 2 2 where f, = r,/ (E,r,) and E,f, = 1. Since f, can be regarded as a density function, the second term can be considered as the variance of (l/o,) which is positive. Also, since A is positive, we conclude that dA/d6, is positive. Breef_2f_Lemma_21bl= (1) By the definition of ¢L, we have dei/d¢L = o = -(dN/d¢L + dSi/doL)/(V + N + Si) which implies the stated result. (ii) By differentiating (p, with respect to ¢,, we have __ __ _ 2 33 which implies 4(b)(ii). (iii) The first derivative of B with respect to ¢, is given by: -Bzz [-r V2(dN/d¢ + dS /d¢ )/(v + N + s )2] i i L i L i /[v - v2(v + N + si)'1]2] By 4(b)(i), we can conclude that dB/d¢,== 0. To show that the derivative of A with respect toi¢m is zero, rewrite A as CB as done in 4(a) (iii). Then dA/dcp, is given by: B (dC/d¢L) + C (dB/dch) = B zi[[v-v2(V+N+si)'1][-riV(dN/d¢L+ ds,/d¢L)/(V+N+S,>21 - [riV(V+N+Si)-1][V2(dN/d¢L+ dSi/d¢L)/(V+N+Si)2]] /[V-V2(V+N+Si)-1]2 + C(dB/doL) Again, by the result of 4(b)(i), we have dA/d¢,== 0. (q.e.d.) Brgef_gf_1heorem_l: (a) The expected change in price given.£5 and Z is given 34 by: E(APIPO, z = Z) = M - BZ - Po Taking the first derivative with respect to 6, while keeping o, constant yields: dE(.)/dei = -Z(dB/d61) By the result of lemma 2(a) (iii), we have dE(.)/d6, > 0. (b) The variance of the change in price given Po and z is Var(APIPo, z = Z) = A2(V + N). The derivative of the variance with respect to 6, is Az(dN/d6,) + 2A(V + N) (dA/d6,). By lemma 2a, (dN/d6i) = -dSi/d61 -(v + N + s ) In the proof for lemma 2a(iii), we have demonstrated that dA/d6,>0 is given by: 2 2 2 2 2 2 which can be rewritten as A + A' where A' > 0. Therefore, dVar(API .)/d6, can be simplified to: 2 A2(-dSi/d61) - A 31 + A2(V + N) + 2AA'(V + N) which can be either negative or positive, and the sign is therefore indeterminate. However, if S, < (V + N), then 35 dVar(API .)/d6, > o. If the agents are symmetrically informed, that is, S, = S for all i, then A' = 0, dS,/d6, = 0 by lemma 2a(ii), and dVar(API.)/d6,:> 0 given S,‘< (V + N). If s, > (v + N), then dVar(API.)/d6, < o. (q.e.d.) 36 References Admati, A. R. ”A Noisy Rational Expectations Equilibrium for Multi-Asset Securities Markets." EQQBQEQEEiQQ 53 (1985), 629-657. Atiase, R. K. "Predisclosure Information, Firm Capitalization and Security Price Behavior Around Earnings Announcements." I2urnal_9f_Aesgunting_Be§eersh 23 (1985), 21- -35. Bamber, L. S. ”The Information Content of Annual Earnings Releases: A Trading Volume Approach." Beeeezeh 24 (Spring 1986), 40-56. Bamber, L. S. "Unexpected Earnings, Firm Size, and Trading Volume Around Quarterly Earnings Announcements." AQQQEDEIDQ Beyiey 62 (1987), 510-532. Banks, D. W. ”Information Uncertainty and Trading Volume." financial_3exieu (February 1985). 83-94. Barry, C. B., and S. J. Brown. 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"An Empirical Investigation of NYSE Volume and Price Reactions to Announcement of Quarterly Earnings." l2urnal_2f_Asseuating_Beseersh 10 (Spring 1972). 113-128. 39 McNichols, M. ”A Comparison of the Skewness of Stock Return Distributions at Earnings and Non-Earnings Announcement Dates." I9urnal_9f_Assennting_and_nspnemies 10 (1988). 239- 273. Morgan, I. G. "Stock Prices and Heteroskedasticity." lpurnal_9f_nusine§§. 49 (Oct- 1976). 496-508- Morse, D. "Asymmetrical Information in Securities Markets and Trading Velune-" I9urnal_2f_Iinansial_and_nuantitatixe Ahelyeie 15 (March 1980), 1129-1148. . "Price and Trading Volume Reaction Surrounding Earnings Announcements: A Closer Examination." EQQI231_QI Assgunting_Be§earsh 19 (Autumn 1981). 374-383- Osborne, M. F. M. "Brownian Motion in the Stock Market." Qperatigns_Researsh. 7 (March-April 1959). 145-173- Pearce, D. E., and V. V. Roley. "Stock Prices and Economic News." learnel_pf_nusine§s 58. No. 1 (1985). 49-67- Pincus, M. 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"The Influence of Estimation Period News Events on Standardized Market Model Prediction Errors.” The_Acsgunting_Rexiew 63 (July 1933), 443-471. 40 Trueman, B. "A Theory of Noise Trading in Securities Markets." IQRIBQL_QI_EinQBQ§ 43 (March 1988), 83-95. Varian, H. R. ”Differences of Opinion in Financial Markets.” Working Paper, University of Michigan (March 1933). Verrecchia, R. E. "On the Relationship Between Volume Reaction and Consensus of Investors: Implications for Interpreting Tests of Information Content.” lehghe1_e£ Asspnnting_Be§earsh 19 (Spring 1981). 271-283- Verrecchia, R. E. "The Rapidity of Price Adjustments to Information.” I2urnal_9f_Asspunting_and_nepngmics 2 (1980). 63-92 e Westerfield, R. "The Distribution of Common Stock Price Changes: An Application of Transactions Time and Subordinated Stochastic Models." Jehghe1_efi_zihehe1e1_ehe Quantifatixe.hnalxs1§ 12 (DeC- 1977). 743-765- Winsen, J. K. "Investor Behavior and Information. " 292228 a] 2f_I1nans1al_and_Quantitatixe_Analxeis 11 (March 1976). 13- 37. Wood, R. A., T. H. McInish, and J. K. 0rd. "An Investigation of Transactions Data for NYSE Stocks." lgnrnal_9f_E1nanse. 60 (July 1985). 723-739- Ying, C. C. ”Stock Market Prices and Volumes of Sales." Esgnemetrisa. 34 (July 1966). 676-686- 41 CHAPTER THREE: THE EFFECT OF FORECAST DIAS AND INVESTOR DISAGREENENT ON SECURITY PRICES AND TRADING VOLUME Abstract In a two-asset competitive equilibrium model incorporating subjective prior beliefs, the response of share price to an earnings surprise is shown to be asymmetric if 1) investors are optimistic or pessimistic about the unknown future value of the risky asset, and 2) if positive and negative surprises have differential effects on uncertainty. Trading volume also reacts asymmetrically to positive and negative surprises when the effect of uncertainty is taken into consideration. These theoretical results are consistent with empirically observed relationships between share prices and trading volume. 42 THE EFFECT OF FORECAST BIAS AND INVESTOR DISAGREENENT ON SECURITY PRICES AND TRADING VOLUME 1. Introduction This essay develops a two-asset competitive equilibrium model with subjective prior beliefs to incorporate the effect of an earnings announcement on share price and trading volume. In this model, price response to the size of a positive or negative earnings surprise is symmetrical if investors' expected liquidating value of the risky asset is unbiased. Price response is asymmetric if 1) investors are optimistic or pessimistic about the unknown future value of the risky asset, and 2) if positive and negative surprises have differential effects on uncertainty. Trading volume also reacts asymmetrically to positive and negative surprises when the effect of uncertainty is taken into consideration. When the behavior of price and volume are examined jointly, an asymmetric price-volume relationship can exist even in a perfect market. The relationship between stock prices and trading volume has interested finance practitioners and financial economists for many years. Price and volume are widely used by financial analysts as market sentiment indicators to gauge rallies and declines, to forecast bull and bear markets, and to predict market turning points. To the financial economist, the price and volume relationship has 43 important implications for understanding the microstructure of financial markets, for event studies, and for identifying the empirical price distributions of speculative assets including options and futures. Beginning with Osborne [1959], the price-volume relationship has been studied from a variety of empirical perspectives. Price-volume studies have examined both equity and futures markets and have included price change intervals ranging from the individual transaction level (Wood, McInish and 0rd [1985]) to two months (Morgan [1976] and Rogalski [1978]). In an early article, Granger and Morgenstern [1963] studied the relationship between price indices and aggregate exchange volume using spectral analysis of weekly data and found no association. More recent articles have focused on individual securities. Karpoff [1987] surveys the empirical literature and categorizes empirically documented relationships between contemporaneous changes in the price and trading volume of individual stocks as follows. First, there is an association between absolute changes in price and volume. Crouch [1970], Westerfield [1977], Rogalski [1978], and Tauchen and Pitts [1983] find a positive association between absolute price changes and volume. Epps and Epps [1976] find a positive association between the variance of price change and volume. Clark [1973] and Harris [1983] find a positive association between squared price change and 44 volume. Second, these and other authors (Smirlock and Starks [1985] and Harris [1986]) find a positive relationship between price change and volume. Third, trading volume is higher when prices increase than when prices decrease (Ying [1966], Morgan [1976], Harris [1986], and Richardson, Sefcik and Thompson [1986]). Karpoff [1987] argues that these results could all be true if the price- volume relationship is asymmetric. In particular, the correlation between volume and positive price changes could be positive while the correlation could be negative and smaller in magnitude for negative price changes. This asymmetry could exist in markets in which short positions are more costly than long positions. In a dissenting paper, Wood, McInish and 0rd [1985] find evidence of an asymmetry in the opposite direction using trade-to-trade data. The price-volume relationship has been examined from different theoretical perspectives as well. Copeland [1976], Morse [1980], and Jennings, Starks and Fellingham [1981] model the price-volume relationship with a sequential information arrival process. Clark [1973], Epps and Epps [1976], Tauchen and Pitts [1983], and Harris [1983] develop equilibrium models for the stochastic dependence between transaction volume and changes in security price and employ the relationship in modeling the distribution of stock price changes. However, no theoretical model has addressed the 45 observed asymmetry in the relationship between stock price and trading volume. The next section describes the model of trade and the equilibrium conditions attained before and after an earnings announcement. The responses of price and volume to earningssurprises are developed as well as the relationship between price and volume responses. Section III develops empirical tests of the hypotheses in Section II and reports results for a sample of quarterly earnings announcements. Section IV summarizes and concludes the paper. II. A Two-Asset Competitive Equilibrium.Model Consider a simple two-period framework as depicted by the following time line. Expectation Earnings Y Formation Announcement Realized l 1 l 0 Pre- 1 Post- 2 Announcement Announcement I<--- Period ------ >I<-- Period ------- >I We assume that there are n investors in the economy. Noise trading by liquidity traders is allowed to exist in the economy but the liquidity traders as a group have no net holdings of either the riskless or the risky asset. There are two assets, one with an unknown payoff and one with a certain payoff with zero rate of return. The unknown or 46 risky security value at the end of the second period is a random variable (Y) which is initially (at t=0) believed to be normally distributed with mean M and variance V. Each investor i has a constant absolute risk tolerance utility function for wealth with coefficient of risk tolerance r, Ui(w) = -exp(-w/ri). Each investor maximizes his expected utility of end-of- period wealth E [-e (-r ‘1(D i + B ))] ixPii 1 subject to D.P + B. = D .P + B . where r1 a constant absolute risk tolerance Boi I initial holdings of the riskless asset Doi I initial holdings of the risky asset B21 I end-of-period holdings of the riskless asset D21 I end-of-period holdings of the risky asset Po 5 initial equilibrium price of the risky asset. Note that the right hand side of the budget constraint, DMP + B“, represents individual i's initial wealth conditional on the market value of his holdings of the risky asset. The equilibrium price of the risky asset has yet to be determined by the market auctioneer who functions to 47 match aggregate demand with aggregate supply. Once a marketprice is announced by the auctioneer, each investor will re-allocate his initial wealth between the riskless asset and the risky asset to the extent that his utility of wealth is maximized. Therefore BM and Ba may not be the same. To focus on the effect of earnings on Y, decompose investor i's signal concerning Y into Ioi = Y + éoi = daixoi + G01' (1’ The idiosyncratic error term.é, is assumed to be independent of Y and is normally distributed with mean zero and variance SM. Before an earnings announcement, each investor has an expectation Xm of future earnings X to be announced at time t = t5.and interprets the impact of this expectation on security price according to his earnings interpretation coefficient d“. Each investor also has an expectation GM of a random variable G independent of X which represents the impact of all other factors on the value of the risky asset. Investor i's signal or expectation of the value of the risky asset is then I, = d,,x,, 4- G0,. With subjective prior beliefs, each investor forms an expectation about Y EoiEY] = E“”101: I 011 = M + (v + $01) (doixoi 01 48 with conditional variance (uncertainty) V2 V ' g varEYIIOi= 101] = V ' (v + 501) ' 01 where EM[.] represents the expectation of investor 1 based on his information set at time t = 0. Investors then trade on their diverse beliefs during the first period. First, the market auctioneer announces a price P for the risky asset. Given P, each investor determines his/her optimal demand for the risky asset by maximizing E[-exp(-ri-1(D1Y + Bi))IIi = Ii] =W subject to DiP + B1 = D iP + Boi 0 0 where W0 is the initial wealth. We already know that conditional on I“ = I“, Y is normally distributed with mean and variance as follows: EIYIIO. = I...) = M + m1... - M) Var(YIIo, = I0:1.) = Var(Y) ' 31173112): where pi = Var(Y)/Var(Io,) = V/(V + 501) - 49 Since Y is normal, the distribution of exp(-r,'1(D,Y+ B,)) conditional on I, =- I, is lognormal. Direct computation with the substitution of B, with the budget constraint yields E[-exp{-ri-1(DiY + Bi))IIoi = 101] = E[-exp(-ri-1(DiY + W0 - DiP))IIoi = 101] = -exp{-ri-1(D1E[YII°i = 101] + No - DiP) + -2 2 _ 12 1/(2ri )Di Var(YIIoi - Ici)}' Since an exponential function is strictly increasing in its exponent, the maximization of the objective function becomes -1 _ - M3x[ri (D1E[Y|Ioi-Ioi] + W0 01?) i 2 2 - 1/(2ri- )Di Var(YIIoi = I0i’] The first order condition with respect to D, yields (l/r1)(E[YIIoi=Ioi] - p) - l/(riz)DiVar(Ylloi=Ioi) = o. ” Let x be normally distributed with mean u and with varipnce 0. Then exp[X] is lognormal with mean exp[p + 1/20']. 50 Individual i's optimal demand for the risky asset based on the price P is therefore equal to r,(E[YII, = I,]-P)/Var(YII, I 1,). Each individual submits his demand/supply order to the market auctioneer who will then match the aggregate demand with total supply and adjust the market price. This kind of iteration process continues until the market price announced by the auctioneer equates total demand to total supply. If the clearing price is P5,‘the equilibrium demand of investor i for the risky asset is given by Voi D01 = (4) After determining individual i's equilibrium demand for the risky asset, we can solve for the risky asset's equilibrium price Po by using the market clearing condition 2.0. = z where 20 is the supply of the risky asset being traded initially. Substituting equation (4) into the market clearing condition, we obtain 20 3 z:i{ri[EOi(Y) - Pol/v01} ‘ Bi[riEoi(Y)/V0i] ' Po‘ziri/Voi)° 51 Rewriting this equation for P,, gives 21 [riEoiEYJ/Voi] zo P° 8 zi‘ri/VOi) - 21(ri/Voi) (5) At the end of the first period (t = 1), the firm discloses actual earnings xf. The information content of the disclosed earnings may change investors' beliefs. We assume that the signal observed by investor 1 after the earnings announcement is represented by 111- Y + 611 = dliX + cli (6) Investor i's revised expectations are then 1,, -- d,,x' + 6,,. Each investor's interpretation of earnings after the announcement, d“, may differ from the pre-announcement earnings interpretation coefficient dm. The new error term en is assumed to be independent of Y and is normally distributed with mean zero and variance 8“. Based on the information content of xi, investors revise their expectations and their subjective posterior beliefs about Y such that - - - _1 * E [Y] = E[YII = I 1 = M + V(V + s ) (d x + c - M), 11 11 1i li 1i 11 (7) and v = Var[YII = I 1 = v - v2(v + s )'1 (3) ii 11 1i 1i 52 As a result, the equilibrium demand of investor i and the equilibrium price of the risky asset are respectively given by r[E [in-p] i 11 1 D 2 _______________ (9) 1i v11 and p = 21 [risli[Y]/v1i] - z; (10) 1 21(ri/vli) 21(ri/Vli) where 21 is the supply of the risky asset being traded during the post-announcement period. A. The Effect of an Earnings Surprise on security Price To simplify our analysis, assume that the variance of the idiosyncratic error is the same for all investors at a particular point in time (i.e. so, I S,, and S,, = S, for all 1). Equations (3) and (8) then imply that at a particular point in time investors' conditional uncertainty regarding the unknown liquidating value Y is constant across investors (i.e. V0, = V,) and V,, = V, for all 1). Then P0 and P, can be written as: P = 2 W E [Y] - (V Z )/2 r (11) 0 i i 01 0 0 i 1 P1 = Eiwi E11[Y] - (Vlzl)/Eiri , where wi = ri/(Eiri) (12) Note that Eiwi = 1. The change in price (P1 - Po) due to 53 the earnings announcement is given by: P1 ' P0 = zi"i(Eli[U] ' Eoi[U]) ’ (Vlzl ’ Vozo)/(2iri’ (13) This equation states that the change in price is affected by the weighted average of the change in expectation formations about the underlying value and the change in supply of the risky asset caused by the earnings announcement. The weight w, is the percentage of investor i's constant risk tolerance in the total risk tolerance of the economy. To decompose the price change equation further, insert the expressions for E,,U and EMU into equation (13) . Since S, =- S and s, = s for all i, replace V(V + S)’1 and V(V + s)'1 by A and a respectively. Equation (13) can be rewritten as P1 ' P0 = ziwi(Aldlix* ' AodOixoi) + ziwi(A1G11 ' AoGOi) + Eiwi(Ao - A1)M - (v121 - vozo)/(21ri) . (14) Traditionally, an earnings surprise is defined as the deviation of actual earnings from the mean consensus forecast, X. - 2:,w,x,,. By adding and subtracting E,w, (A,d,,x,,,) to the first term in equation (14) , the change in price due to an earnings announcement can be seen to be linearly related to the size of the earnings surprise. In order to isolate the effect of an earnings surprise, we further assume that all investors have the same risk 54 tolerance (r, =- r). The weight w, for each investor in equation (14) becomes 1/n. Let X0 be the simple average earnings forecast (E,Xo,/n) before the announcement. By adding and substracting A,d,,x,, in the first term, equation (14) becomes * p1 - p0 = A1d1(X - x0) + (Aldl- Aod1)Xo + (A161 - A060) + (Ao - A1)M - (V121 - VOZO)/(nr) (15) where do = E,d,,,/n, d1 = E,d,,/n, G, - (E,G,,)/n, and G0 = (E,Go,)/n. Equation (15) states that the change in price caused by an earnings announcement is determined by (i) the size of the earnings surprise (X'-)%), (11) the change in interpretation and uncertainty about the unknown stock value, (111) the revision of the growth forecast (G,-(%), and (iv) the change in supply of the risky asset being traded due to the earnings announcement. Equation (15) can be used to examine the symmetry of a price response to an earnings surprise. Assume that (i) the average forecast of growth does not change (G0 = G,) , and (ii) investors' pre- and post-announcement average interpretations of the impact of earnings on value does not change (do = d, = d), and (iii) the number of investors is large enough so that the last term in equation (15) is 55 negligible. After rearrangement, the price change equation becomes * p - p = A1d(X - x0) + (A1 - Ao)(dXo + G - M) . (16) 1 O 0 It is obvious from equation (16) that the magnitude of the price change is directly proportional to the size of the earnings surprise. The second term represents the pre- announcement consensus forecast revision of the underlying expected value of the risky asset. The term (dx0 + G0 - M) is a measure of market sentiment about the unknown value of the risky asset. If the term is zero, the market does not revise its expectation before announcement. If the term is positive (negative), the market is optimistic (pessimistic). Three testable hypotheses about the reaction of price to the earnings announcement can be established at this point. Erepgsitign_l If 1) the market consensus about the value of the risky asset does not change before the earnings announcement, or 2) the earnings announcement does not affect the degree of uncertainty among the investors about the asset's value, then the reaction of price to both negative and positive earnings surprises is symmetrical. 56 Under either condition, the second term in equation (16) is zero and the magnitude of share price response to the earnings announcement is linearly related to the magnitude of the earnings surprise according to P, - P0 = A,d(x' - X0) . W Suppose the market consensus about the value of the risky asset is revised upward before the earnings announcement such that (dx0 + Go - M) > 0. Then, 1) If the earnings announcement reduces the uncertainty about the value of the risky asset among investors (Ao < A,), then for the same size earnings surprise the absolute change in price is larger for a favorable surprise than for an unfavorable surprise. 2) If the earnings announcement induces more uncertainty among investors (A0 > A,) , then the absolute change in price is smaller for a favorable surprise than for an unfavorable surprise. 21911951119114 Suppose the market consensus about the value of the risky asset is revised downward before the earnings announcement such that (dx0 + G0 - M) < 0. 1) If the earnings announcement reduces the uncertainty about the value of the risky asset among investors (Ao < A,), then for the same size earnings surprise the 57 absolute change in price is smaller for a favorable surprise than for an unfavorable surprise. 2) If the earnings announcement induces more uncertainty among investors (A0 > A,) , the absolute change in price is larger for a favorable surprise than for an unfavorable surprise. Intuitively, if investors are optimistic about the value of the risky asset before the earnings announcement, a large negative earnings surprise is likely to cause more uncertainty among investors (A, < A,) . A large positive earnings surprise is more likely to confirm their beliefs and hence resolve their uncertainty to some extent (A,:> A,). Equation (16) implies that the price change is more responsive to a positive surprise than to a negative surprise. Similarly, if the investors are pessimistic, a large positive surprise is more likely to introduce more uncertainty into their beliefs. A large negative surprise will confirm their expectations. In this case, equation (16) implies that the price change is more responsive to negative earnings surprise than to positive surprise. III. Earnings Surprise and Trading Volume This section develops the effect of an earnings announcement on the behavior of the volume of trade. Conditional on the original equilibrium demand DM, investor 58 i's volume of trade induced by the earnings announcement is (Du - Du). Equations (4) and (9) of Section II yield the net trade of investor i in the risky asset induced by the earnings announcement: T1... -D,=ri_[E_1_i_[Y_]_:P_1]-r_iiEoim‘Po] 1 11 01 V11 V01 (17) Assuming the variance of the idiosyncratic error is constant across investors (i.e. S,,, = So and S,, = S,) yields * * Ti- [r1(h + A1(d11X + 311 - M) - [ziwi(u + A1(d11X + 311 - M)])]/V1 ' z1/ziri ' D01 * * ' [riA1[(dlix + Gli) ' Ei"i(d'11x + 61)] ]/V1 ' zl/ziri ' D01 (18) If all investors agreed on the value of the risky asset after an earnings announcement, no trades would be necessary to drive price to its new equilibrium level. Trade occurs after the earnings announcement solely because of the information content that causes diverse opinions (i.e. prior probabilities) among investors about the future activity of the firm. That is, trading volume changes because investors 59 interpret information differently and not because of the new information itself. This result is consistent with Varian (1935). Next, the effect of an earnings surprise on the behavior of trading volume is examined. By substituting equations (5) and (10) into equation (17), and assuming r,== r for all investors, 15 can be written as: T, -[r[(E,,[Y1-E,[YJ) - V,/(nr)J]/V, - - [r[(Eo,[Y1-E0[YJ) - Vo/(nr)]]/Vo =[(VOE,,[Y1- V,EO,[Y1) - (VOE,[Y1- v ,E o[Y1)]/(VOV,/r) (19) where E1 Y] = M1.[Y]/n andE EO[Y] = 2.1Eoi[Y]/n. The overall volume of trade is given by: T s EiITiI/Z =[r,l(voE,,EY1- v ,E0,EY1) - (voE,[Y1 - V,Eo[Y])|]/2(VOV,/r) (20) The functional form of T makes further analysis difficult. 60 To alleviate this problem, note that the numerator is the sample mean absolute deviation (MAD) of (VOE,[Y] - V,E,,[Y]) .13 E,[Y] and E,[Y] are normally distributed because they are expectations conditional on the information I0 and I, which are normally distributed. Therefore, there exists a one-to-one correspondence between the MAD and the variance. Since E,[Y] and E,[Y] are normally normally distributed, the numerator of equation (19) can be approximated by the variance of (VoE,[Y] - V,E0[Y]) . Then T can be approximated by a function 1 of the form: Var(VoE1[Y] - V1E0[Y]) ’ = 2(v0v1/r)' (21) By substituting equations (2) and (7) into the numerator of (21) and dropping the subscript i, we have: Var(VoE1[Y] - VlEo[§]) * = Var[(VoA1d1X - leodoxo) + (V01161 - VleGo) + (VOM - VIM - VoAlM - VlAOM)] 13. E,(Y) and E,(Y) represent the expectation of the unknown payoff Y conditional on the information set at t=0 and t=1 respectively. The expressions for EOCY) and E,(Y) are the same as equations (2) and (7) without the subscript i. In fact the expressions given by (2) and (7) are analogous to the outcomes of the ith experiment with E,(Y) and E,(Y). Therefore, the interpretation (d), earnings forecast (X5), and the growth factor (G0) are random variables. 61 * = Var[(VoAld1X - VledoXo) + (V0A1G1 - V1A0G0)] (22) As in the development of equation (16), assume that investors' pre- and post-announcement average interpretations of the impact of earnings on value do not change ((10 = d, = d) . Furthermore, to concentrate on the impact of an earnings announcement on trading volume, assume that G0 = G, = G. Adding and substracting V,,A,dx0 in the first term of (22) yields: * Var[V0A1d(X - x0) + d(voA1 - le0)x0 + (VOA1 - V1A0)G] (23) Now let x = i + e (24) where X is the mean consensus earnings forecast across all investors and the error term 6,, has zero mean and finite variance Var(eo) and is independent of X, d, and G. Substituting X0 into equation (23) and rearranging yields Var(VoE1[Y] — VlEo[§]) * - _ = Var[VoA1d(X - X) - V0A1d(eo) + d(voA1 - leo)x + d(voA1 - V1A0)60] + Var[(voA1 - V1A0)G] = (v A x*- v A02 )Var(d) + (V1A0)2E(d2)Var(e 0 1 1 O) 2 + (VOA - Vle) Var(G) 1 = (voA1(x* - X)+(VOA v1A0)2 12Var(d) + (V1A0)2E(d2)Var(6 1' o) + (VOA - V1A0)2Var(G) (25) 1 Therefore, from equation (21), 4 — - 7 = (r/(2VOV1)[[VOA1(X - X) + (VOA1 - V1A0)X]2Var(d) + (V1A0)2Var(eo)E(d2) + (VOA - V1A0)2Var(G) ] (26) 1 Observe from equation (26) that even when the earnings surprise is zero trade still occurs if the earnings announcement has an impact on investors' beliefs about the value of the risky asset. Based on equation (26), we can establish two hypotheses about the response of trading volume to an earnings announcement. Erppgsitign_i If the earnings announcement resolves uncertainty about the future value of the risky asset among the investors (i.e., Vo > V, and A0 < A,) , then the volume of trade associated 63 with a favorable earnings surprise [(Xf - X) > 0] is larger than that associated with an unfavorable earnings surprise [(X' - X) < 0] of the same magnitude. This proposition can be demonstrated by inspecting the first term in equation (26) . If V0 > V1 and A,, < A,, then every component in the first term is positive. If V,) < V, and A,, > A,, then (VOA, - V,A,,) is negative and so is the cross-product component in the first term. Therefore, the extent of a volume increase to a negative earnings surprise is less than that of a positive surprise. Similarly, the impact of an increase in uncertainty on trading volume is presented below: £r229212122£= If the earnings announcement induces more uncertainty about the future value of the risky asset among the investors, that is, Vo < V, and A,, > A,, then the size of the volume of trade associated with favorable earnings surprise is less than that associated with the unfavorable earnings surprise of the same magnitude. A summary of the five propositions is provided in the Table 3.1. 64 IV. Empirical Tests Price and volume responses to earnings announcements are categorized in Table 1 according to pre-announcement forecast revisions and changes in uncertainty after the announcement. According to equation (16), price response to an earnings announcement depends on the size and direction of forecast revision before the announcement and change in the uncertainty of investors after the announcement. Four testable hupotheses about price responses to earnings announcements arising from equation (16) are summarized as follow: Hi: If the market consensus about the value of the risky asset is revised upward before an earnings announcement and the announcement reduces uncertainty about the value of the risky asset among investors, then the absolute change in price is more responsive to a favorable earnings surprise than to an unfavorable surprise. B2: If the market consensus about the value of the risky asset is revised upward before an earnings announcement and the announcement induces more uncertainty about the value of the risky asset among investors, then the absolute change in price is less responsive to a 65 favorable earnings surprise than to an unfavorable surprise. H3: If the market consensus about the value of the risky asset is revised downward before an earnings announcement and the announcement reduces uncertainty about the value of the risky asset among investors, then the absolute change in price is less responsive to a favorable earnings surprise than to an unfavorable surprise. B4: If the market consensus about the value of the risky asset is revised downward before an earnings announcement and the announcement induces more uncertainty about the value of the risky asset among investors, then the absolute change in price is more responsive to a favorable earnings surprise than to an unfavorable surprise. According to equation (27), trading volume response depends on the change in uncertainty. Two testable hypotheses about trading volume arising from equation (27) are summarized below: as: If an earnings announcement resolves uncertainty about the value of the risky asset among investors, then the 66 volume of trade is more responsive to a favorable earnings surprise than to an unfavorable surprise. H6: If an earnings announcement increases uncertainty about the value of the risky asset, then the volume of trade is less responsive to a favorable earnings surprise than to an unfavorable surprise. A. Variables and Data This section discusses the measures and proxies adopted in empirically testing the hypotheses derived from equations (16) and (26). Tests of these hypotheses require: 1) an estimate of the market consensus about the future value of the risky asset, ii) a measure of uncertainty about the future value of the risky asset, iii) a measure of earnings surprise, and iv) security price and volume response to the earnings announcement. Lynch, Jones and Ryan's The§i§g§ioneT Beekeze Estimahe SYSLQT (I/B/E/S) is our source of earnings forecast data. The mean and standard deviation of financial analysts reporting quarterly estimates of earnings per share to I/B/E/S are adopted as proxies for investors' consensus expectation and the uncertainty of investors about the future value of the risky asset. Earnings announcement dates are retrieved from I/B/E/S and from the Wall Street Journal when not reported by I/B/E/S. 67 Earnings surprise is defined as actual earnings per share minus the adjusted mean consensus forecast standardized by price on the announcement day. The I/B/E/S adjustment factor (ADJFAC) is used to adjust consensus forecasts for stock splits, stock dividends, and new issues. Quarterly earnings per share are taken from Standard and Poor's Compustat database. The standard deviation of analyst forecasts is often used to standardize the level of earnings across companies. This is inappropriate here since the standard deviation of analyst forecasts is one of the independent variables under study. The sample period includes the third quarter of 1984 through the fourth quarter of 1987. The sampling criteria include: 1) each firm must have at least three analysts during the sample period, ii) every firm must have monthly forecasts reports at least three months before the earnings announcement date and one month after, and iii) firms must survive the sample period. Daily returns to the sample companies and to the value weighted market index are taken from CRSP. Trading volume data is supplied by Media General. 68 B. Empirical Design Consider the following diagram depicting the timing of financial analyst forecasts and quarterly earnings announcements. x: X Q(-1) Month(-2) Month(-1) Q(0) Month(l) Table 3.2 categorizes the six hypotheses according to i) change in the uncertainty of investor expectations (for price and volume response), and 11) change in mean consensus expectations (for price response). Change in uncertainty induced by the earnings announcement is measured by the difference between the standard deviation of analyst forecasts in the months before and after the announcement date (0,) - 0%)). Revision in the mean consensus opinion about the future value of the risky asset is measured by change in the mean analyst forecast in the month prior to the quarterly earnings announcement date from the mean consensus forecast three months prior to the announcement (X‘d - X‘fi). This design eliminates some earnings announcements from the sample because there are not enough analysts following that firm. A two-day window is employed to capture the reaction of price and volume to an announcement as well as any reaction 69 to information leakage about actual earnings immediately prior to the announcement. Price reaction to an earnings announcement is examined in hypotheses 1 through 4. Two proxies for price reaction are employed to see if results are sensitive to expected stock return assumptions. One is the two-day geometric mean return ((1+Ri-,) * (1+Rio) 1-1)" and the other is the two-day mean excess return over a value weighted market index “ER-1 + ERo)/2) = ((R-l-Rm-l)/2 + (Ra-Rmo)/2- The negative one subscript refers to the day before an earnings announcement and the zero subscript refers to the day of an announcement. The effect of an earnings announcement on trading volume is examined in Hypotheses 5 and 6. The two-day average trading volume (V-l + Vo)/2 is used to capture the volume response to an announcement as well as any information leakage immediately prior to an announcement. 70 Earnings announcements from the fourth quarter of 1984 through the second quarter of 1987 are categorized along these two dimensions. Observations in each group in Table 3.2 are then aggregated across all quarters. The econometric model designed to investigate asymmetric price and volume reaction to favorable and unfavorable earnings surprises is RESPONSEi = 60 + Bl * DUMMYi + 32 * SURPRISEi I + 33 * DUMMYi * SURPRISEi I + e. . 1 (27) Earnings surprise (SURPRISE) for the ith observation is defined as (Actual EPS - Mean Forecast * ADJFAC)/P,,. The dummy variable has a value of one if SURPRISE is positive and a value of zero if SURPRISE is negative. The response variable RESPONSE, is the two-day average return for Hypotheses 1 through 4 and the two-day average volume for Hypotheses 5 and 6. Equation (27) tests whether change in the dependent variable (price or trading volume) in response to a unit increase of favorable earnings surprise is different in absolute magnitude from change in response to a unit increase of unfavorable surprise. The model is equivalent to two separate regressions: 71 SURPRISE > 0 => DUMMY I 1 I RESPONSEi I = 30 + 32 + (31 + 33) I SURPRISEi I + ei SURPRISE < 0 => DUMMY = 0 SURPRISEi I + e.. I RESPONSEi I = 60 + £1 * 1 The coefficient 6, serves to capture any asymmetric effect. Significance tests on the coefficient fig.are one-tailed t- tests. The null hypothesis is listed in each cell of Table 2. VII. Empirical Results Tables 3.3 to 3.10 display the regression results for H1 through H4 based on the two-day geometric mean returns and the two-day average excess returns respectively. Tables 3.11 and 3.12 report the results for testing H5 and H6. As can be seen from the tables, the directions for the price and volume response to favorable and unfavorable earnings surprise are consistent with the implications of the theory. However, only H4 is significantly supported by the empirical evidence. VIII. Conclusions 72 This paper develops a two-asset competitive equilibrium model with heterogeneous expectations and constant absolute risk aversion to investigate the effect of an earnings announcement on the change in price and trading volume. Theoretically, price response to a good or bad news earnings announcement is symmetrical only if investors' expectation of the liquidating value of the risky asset is unbiased. The response is asymmetric if (1) investors have biased opinions about the unknown value of the asset, or (2) the positive and negative earnings surprises have differential effects on the uncertainty of investors' expectations. Trading volume also reacts differently to positive and negative earnings surprises when the effect of uncertainty induced by the earnings announcement is taken into consideration even if the investors' expectations are unbiased. An asymmetric relationship between the change in price and trading volume exists even if there is no differential transaction cost between short and long positions. The asymmetry can exist in either direction, depending on whether the earnings announcement causes more or less uncertainty about the value of the risky asset among the investors. The empirical tests weakly support the theory developed. 73 TABLE 3.1 SUMMARY OF PROPOSITIONS 1 THROUGH 6 PRE-ANNOUNCEMENT “ UNCERTAINTY AFTER EARNINGS ANNOUNCEMENT FORECAST REVISION u INCREASE DECREASE IAPI IAPI X X C x X 0 1. UPWARD -ES +ES -ES +ES VOL VOL X X 0 x x 0 IA?! IAPI X X X X . . 2. NO CHANGE -ES +ES -ES +ES VOL VOL X X . X X . liPI IAPI X X X . . X 3. DOWNWARD - -ES +ES -ES +ES VOL VOL X X X 74 TABLE 3.2 SUMMARY OF EMPIRICAL TESTS OF ASYMMETRY CHANGE IN UNCERTAINTY AROUND THE TIME OF AN EARNINGS ANNOUNCEMENT Reduced Increased 00 - a_1 00 - o_1 > 0 PRICE RESPONSE MEAN Upward H1: 63 5 H2: 83 2 0 CONSENSUS i _ i > 0 FORECAST —1 -3 REVISION BEFORE AN Downward H3: 63 2 H4: 33 S 0 EARNINGS i _ i < 0 ANNOUNCEMENT -1 -3 VOLUME RESPONSE H5: 33 5 H6: 63 2 0 75 Table 3.3 Test of Hypothesis 1 Using Two-Day Geometric Mean Return as Dependent Variable. Dependent Variable: RESPONSE Misses Sum of Mean Source DF Squares Square F Value Prob>F Model 3 0.00127 0.00042 3.508 0.0158 Error 286 0.03455 0.00012 C Total 289 0.03582 Root MSE 0.01099 R-square 0.0355 Dep Mean 0.01392 Adj R-sq 0.0254 C.V. 78.97483 a t te Parameter Standard T for H0: Variable DF Estimate Error Parameter-0 Prob >|T| INTERCEP 1 0.011785 0.00105458 11.175 0.0001 DUMMY 1 0.002449 0.00143717 1.704 0.0895 SURPRISE 1 0.058612 0.02566401 2.284 0.0231 INTER 1 0.033827 0.07290147 0.464 0.6430 76 Table 3.4 Test of Hypothesis 2 Using Two-Day Geometric Mean Return as Dependent Variable. Dependent Variable: RESPONSE Source Error 0 Total Root MSE Dep Mean C.V. Variable DF INTERCEP l DUMMY l SURPRISE 1 INTER l 322W Sum of Mean DF Squares Square F Value 3 0.00006 0.00002 0.198 272 0.02679 0.00010 275 0.02685 0.00994 R-square 0.0022 0.01238 Adj R-sq «0 0089 80.28807 amete t m Parameter Standard T for H0: Estimate Error Parameter-0 0.012664 0.00111440 11.364 -0.000128 0.00141221 -0.09l -0.002075 0.06193654 -0.033 -0.037370 0.08396224 -0.445 Prob>F Prob > |T| 77 Table 3.5 Test of Hypothesis 3 Using Two-Day Geometric Mean Return as Dependent Variable. Dependent Variable: RESPONSE Error C Total Root MSE Dep Mean C.V. Variable DF INTERCEP l DUMMY l SURPRISE 1 INTER 1 W29 Sum of Mean DF Squares Square F Value 3 0.00019 0.00006 0.463 600 0.08266 0.00014 603 0.08285 0.01174 R-square 0.0023 0.01238 Adj R-sq -0 0027 94.83586 W Parameter Standard T for H0: Estimate Error Parameter-0 0.012366 0.00067593 18.295 -0.000151 0.00103482 -0.146 0.009891 0.01190614 0.831 -0.024784 0.02614147 -0.948 Prob>F Prob > |T| 78 Table 3.6 Test of Hypothesis 4 Using Two-Day Geometric Mean Returns as Dependent Variable. Dependent Variable: RESPONSE 32211212411311.2222 Sum of Mean Source DF Squares Square F Value Prob>F Model 3 0.00289 0.00096 6.669 0.0002 Error 664 0.09591 0.00014 C Total 667 0.09880 Root MSE 0.01202 R-square 0.0293 Dep Mean 0.01243 Adj R-sq 0.0249 C.V. 96.67977 Winnie: Parameter Standard T for H0: Variable DF Estimate Error Parameter-0 Prob > |T| INTERCEP 1 0.012749 0.00061397 20.764 0.0001 DUMMY 1 -0.001970 0.00098353 -2.003 0.0455 SURPRISE 1 0.011784 0.00374932 3.143 0.0017 INTER 1 0.042791 0.02174958 1.967 0.0495 79 Table 3.7 Test of Hypothesis 1 Using Two-Day Average Excess Returns as Dependent Variable. Dependent Variable: RESPONSE Misuse Sum of Mean Source DF Squares Square F Value Prob>F Model 3 0.00102 0.00034 3.373 0.0189 Error 286 0.02874 0.00010 C Total 289 0.02975 Root MSE 0.01002 R-square 0.0342 Dep Mean 0.01241 Adj R-sq 0.0240 C.V. 80.77937 W Parameter Standard T for H0: Variable DF Estimate Error Parameter-0 Prob > |T| INTERCEP 1 0.011019 0.00096181 11.456 0.0001 DUMMY 1 0.001188 0.00131075 0.907 0.3654 SURPRISE 1 0.062726 0.02340645 2.680 0.0078 INTER 1 0.026740 0.06648863 0.402 0.6879 80 Table 3.8 Test of Hypothesis 2 Using Two-Day Average Excess Returns as Dependent Variable. Dependent Variable: RESPONSE Amalgam Sum of Mean Source DF Squares Square F Value Prob>F Model 3 0.00023 0.00008 0.978 0.4033 Error 272 0.02151 0.00008 0 Total 275 0.02174 Root MSE 0.00889 R-square 0.0107 Dep Mean 0.01078 Adj R-sq -0.0002 C.V. 82.51684 W Parameter Standard T for H0: Variable DF Estimate Error Parameter-0 Prob > |T| INTERCEP 1 0.010883 0.00099663 10.920 0.0001 DUMMY 1 -0.000738 0.00126273 -0.584 0.5595 SURPRISE 1 0.070576 0.05539110 1.274 0.2037 INTER 1 -0.039284 0.07445819 -0.528 0.5982 81. Table 3.9 Test of Hypothesis 3 Using Two-Day Average Excess Returns as Dependent Variable. Dependent Variable: RESPONSE Wm Sum of Mean Source DF Squares Square F Value Prob>F Model 3 0.00026 0.00009 0.697 0.5540 Error 600 0.07409 0.00012 C Total 603 0.07435 Root MSE 0.01112 R-square 0.0035 Dep Mean 0.01161 Adj R-sq -0.0015 C.V. 95.76569 W Parameter Standard T for H0: Variable DF Estimate Error Parameter-0 Prob > |T| INTERCEP 1 0.011578 0.00064648 17.909 0.0001 DUMMY 1 -0.000386 0.00098446 -0.392 0.6951 SURPRISE 1 0.015298 0.01252128 1.222 0.2223 INTER 1 -0.015362 0.02535855 -0.606 0.5449 82 Table 3.10 Test of Hypothesis 4 Using Two-Day Average Excess Returns as Dependent Variable. Dependent Variable: RESPONSE Wm Sum of Mean Source DF Squares Square F Value Prob>F Model 3 0.00196 0.00065 4.661 0.0031 Error 664 0.09295 0.00014 C Total 667 0.09491 Root MSE 0.01183 R-square 0.0206 Dep Mean 0.01177 Adj R-sq 0.0162 C.V. 100.51809 W Parameter Standard T for H0: Variable DF Estimate Error Parameter-0 Prob > |T| INTERCEP 1 0.012439 0.00060441 20.580 0.0001 DUMMY 1 -0.002510 0.00096822 -2.592 0.0098 SURPRISE 1 0.006305 0.00369095 1.708 0.0881 INTER 1 0.042868 0.02141096 2.002 0.0457 83 Table 3.11 Test of Hypothesis 5 Using Two-Day Average Trading Volume as Dependent Variable. Dependent Variable: VOLUME Source Root MSE Dep Mean C.V. Variable DF INTERCEP l DUMMY 1 SURPRISE 1 INTER 1 V ce Sum of Mean DF Squares Square F Value Prob>F 3 522053928 174017976 5.685 0.0007 891 27273812090 30610339 894 27795866018 5532.66112 R-square 0.0188 4331.24972 Adj R-sq 0.0155 127.73822 Winners: Parameter Standard T for H0: Estimate Error Parameter-0 Prob > |T| 3558.336157 270.42977348 13.158 0.0001 1222.684040 417.58652131 2.928 0.0035 8897.814088 4485.0699235 1.984 0.0476 19064 21937.461554 0.869 0.3851 84 Table 3.12 Test of Hypothesis 6 Using Two-Day Average Trading Volume as Dependent Variable. Dependent Variable: VOLUME Error C Total Root MSE Dep Mean C.V. Variable DF INTERCEP l DUMMY 1 SURPRISE 1 INTER 1 0 V c Sum of Mean DF Squares Square F Value 3 360909650 120303216 3.671 926 30346326034 32771410 929 30707235685 5724.63190 R-square 0.0118 4460.85645 Adj R-sq 0.0086 128.33033 221222221_321122122 Parameter Standard T for H0: Estimate Error Parameter-0 4095.785966 284.33120143 14.405 265.128447 403.90846281 0.656 29098 9299.0543781 3.129 ~18615 13282.507100 -1.401 Prob>F 0.0120 Prob > |T| 85 References Admati, A. R. "A Noisy Rational Expectations Equilibrium for Multi-Asset Securities Markets." EQQBQESEIIQQ 53 (1935), 629-657. Atiase, R. K. "Predisclosure Information, Firm Capitalization and Security Price Behavior Around Earnings Announcements." Tourne; of ecceuhe1ng Reseazeh 23 (1985), 21-35. Bamber, L. S. "The Information Content of Annual Earnings Releases: A Trading Volume Approach." geegha; ef Accounting Beeeegeh 24 (Spring 1986), 40-56. Bamber, L. S. 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"The Random Walk Hypothesis of Stock Market Behavior." 51192. 17 (Fasc. 1, 1964), 1-30. Granger, C. W. J., and O. Morgenstern. "Spectral Analysis of New York Stock Market Prices." Kylee, 16 (Fasc. 1, 1964), 1-27. Harris, L. "Cross-Security Tests of the Mixture of Distributions Hypothesis." Quantitati_2_32212212 21 (March 1986). 39-46- 87 Harris, L., and E. Gurel. ”Price and Volume Effects Associated with Changes in the S&P 500 List: New Evidence for the Existence of Price Pressures." IQDID21_Q£_E122DQ§. 41 (Sept. 1936), 315-329. Ho 8. Y., R. A. Schwartz, and D. K. Whitcomb. "The Trading Decision and Market Clearing Under Transaction Price Uncertainty-” The.l22:221.2f_£122222 40 (March 1985). 21- 42. Holthausen, R. W., and R. E. Verrecchia. "The Effect of Sequential Information Releases on the Variance of Price Changes in an Intertemporal Multi-Asset Market.” geezheT_e; 3222222129_82222r22 26 (Spring 1988). 82- -106- Imhoff, E. A. Jr., and G. J. Lobo. ”Information Content of Analysts' Composite Forecast Revisions." Qeurnel of AQQQHDIIDQ_BSSSQIQD 22, No. 2 Autumn 1984, 541-554. Jain, P. C., and G. Joh. "The Dependence between Hourly Prices and Trading Volume." Working Paper, The Wharton School, University of Pennsylvania (September 1986). James, C., and R. O. Edmister. "The Relation between Common Stock Returns Trading Activity and Market Value." QQQIDQI e£_£Tnehee 38 (September 1983), 1075-1086. Jennings, R. H. ”Unsystematic Security Price Movements, Management Earnings Forecasts, and Revisions in Consensus Analyst Earnings Forecasts-" l22r22l_2f_A222222122_32222r22 25, No. 1, Spring 1987, 90-110. Jennings, R. H., and C. Barry. ”Information Dissemination and Portfolio Choice." I22:nal.2f.£12222121_222 922221222122_32212212 18 (March 1983). 1-19- . ”On Information Dissemination and Equilibrium Asset Prices: A Note." r o ncia and 922221222122.hnalysis 19 (December 1984). 395-402- Jennings, R. H., L. T. Starks, and J. C. Fellingham. "An Equilibrium Model of Asset Trading with Sequential Information Arrival.“ lehzne1_eT_EThehee, 36 (March 1981), 143-161. Karpoff, J. M. "The Relation between Price Changes and Trading Volume: A Survey-" 12urnal_2f_£inan2ial_222 922221122122_Analysis 22 (March 1987). 109-126- Kiger, J. E. "An Empirical Investigation of NYSE Volume and Price Reactions to Announcement of Quarterly Earnings." I22r221.21.3222221122_82222222 10 (Spring 1972). 113-128- McNichols, M. ”A Comparison of the Skewness of Stock Return Distributions at Earnings and Non-Earnings Announcement Dates." 222r22l_2f_A222222122.222_222222122 10 (1988). 239- 273. Morgan, 1. G. "Stock Prices and Heteroskedasticity." I22rnal_2f_22212222. 49 (Oct- 1976). 496-508- Morse, D. ”Asymmetrical Information in Securities Markets and Trading Volume.” I22r22l_2f_I12222121_222_922221222192 AheTyeTe 15 (March 1980), 1129-1148. . "Price and Trading Volume Reaction Surrounding Earnings Announcements: A Closer Examination.” Jeezhe1_e; 3222222122_32222222 19 (Autumn 1981). 374-383- Osborne, M. F. M. "Brownian Motion in the Stock Market." 922:2212n2_82222222. 7 (March-April 1959). 145-173- Pearce, D. K., and V. V. Roley. "Stock Prices and Economic News.“ 12urnal_2f_22212222 58. NO- 1 (1985). 49-67- Pincus, M. ”Information Characteristics of Earnings Announcements and Stock Market Behavior." Journe; e: 3222222129_32222r2h 21 (Spring 1983). 155-183- Richardson, G, S. E. Sefcik, and R. Thompson. "A Test of Dividend Irrelevance Using Volume Reaction to a Change in Dividend Policy-” l22r221.21.21222212l_222222122 17 (Dee- 1986), 313- 333. Ro, B. T. ”The Disclosure of Replacement Cost Accounting Data and Its Effect on Transaction Volumes." The_Aeeehh;Thg BeyTey, 56 (January 1981), 70-84. Rogalski, R. J. "The Dependence of Prices and Volume." The B2xi22_2f_E22222122.222_2122122122 36 (may 1978). 268-274- Smirlock, M., and L. Starks. "An Empirical Analysis of the Stock Price-Volume Relationship.” Jeezhe1_efi_§ehhihg_ehg {Thehee 12 (September 1988), 31-41. Tauchen, G. E., and M. Pitts. "The Price Variability-Volume Relationship on Speculative Markets." EQQDQEQLIIQQ 51 (March 1983), 485-505. Thompson, R. B., C. Olsen, and J. R. Dietrich. "The Influence of Estimation Period News Events on Standardized Market Model Prediction Errors." Ih§_AQQQEDIIDQ_32212H 63 (July 1933), 443-471. Trueman, B. "A Theory of Noise Trading in Securities Markets." EQQIDQI_Q£_EIDQDQQ 43 (March 1988), 83-95. Varian, H. R. "Differences of Opinion in Financial Markets." Working Paper, University of Michigan (March 1933). Verrecchia, R. E. "On the Relationship Between Volume Reaction and Consensus of Investors: Implications for Interpreting Tests of Information Content.” Jeeghe1_ef 3222222129_82222222 19 (Spring 1981). 271-283. Verrecchia, R. E. "The Rapidity of Price Adjustments to Information." 222r2al_2f_A222222122.2nd.E22222122 2 (1980). 63-92. Westerfield, R. ”The Distribution of Common Stock Price Changes: An Application of Transactions Time and Subordinated Stochastic Models.” EBBID21_QI_EIDQDQIQI_QDQ 922221222122.hnalysis 12 (Dec- 1977). 743-765. Winsen, J. K. "Investor Behavior and Information." 2922221 21.21222212l_222.922211222122_Anal¥212 11 (March 1976). 13- 37. Wood, R. A., T. H. McInish, and J. K. Ord. "An Investigation of Transactions Data for NYSE Stocks." l2urnal_2f_zinan22. 60 (July 1985). 723-739- Ying, C. C. ”Stock Market Prices and Volumes of Sales." 322222222122. 34 (July 1966). 676-686. 90 CHAPTER FOUR: THE ESTIMATION OF MARKET SPEED OF ADJUSTMENT USING SECURITIES PRICES AND TRADING VOLUME ABSTRACT This essay develops a statistical test for estimating the onset and duration of security price and trading volume responses to new information. It extends the analysis of Hillmer and Yu (1979) by allowing a dependent relationship between security price and trading volume. The dependent relationship between price and volume is addressed by orthogonalizing one market attribute with respect to the other. The resulting statistical test provides biased estimates of the onset and duration of market responses to new information (see Giliberto (1985)). A practical procedure for implementing the statistical test is then prescribed. The statistical test allowing dependence is compared to the Hillmer and Yu (1979) and Pincus (1983) tests in simulations of real world responses to information. 91 CHAPTER FOUR: THE ESTIMATION OF MARKET SPEED OF ADJUSTMENT USING SECURITY PRICES AND TRADING VOLUME I. Introduction This paper develops a statistical procedure for estimating the onset and duration of security price and trading volume responses to new information. Study of the adjustment period is important for testing market efficiency and for understanding the way in which markets respond to new information such as earnings announcements. For example, Pincus (1983) examines the relationship between the duration of market (price and volume) adjustment and earnings predictability. He concludes that firms with harder to predict earnings streams have longer adjustment periods. Defeo (1986) finds that price adjustment duration depends upon firm size, reporting lag, and whether the announcement is of annual or quarterly earnings. Statistical techniques for identifying the time of a change in the mean (Hinkley (1970) and Lee and Heghinian (1977)) and variance (Wichern, Miller and Hen (1976)) of a time series have been proposed in the literature. The drawback of these methods, as Hillmer and Yu (1979) point out, is that they are very complicated and difficult to implement. Hillmer and Yu (1979) introduce a statistical technique that signals the point of time when the market begins to react to new information and the time when the 92 reaction stops. Change in the mean or variance of a market attribute signals the onset and duration of the adjustment period. Hillmer and Yu's technique allows only one market attribute to be analyzed at a time. However, it is very general in that it allows the market attribute to be price, volume of trade, frequency of trade, number of block trades, or any other attribute which responds to information. Pincus (1983) extends Hillmer and Yu's conceptualization of the adjustment period with a maximum likelihood procedure which incorporates price and trading volume. In the development of his MLE procedure, Pincus assumes that returns and volume are independent. Pincus goes on to observe that the variance of trading volume is constant over his time series and hence omits the variance of abnormal volume from his estimation procedure. These assumptions are inconsistent with empirical findings regarding changes in price and trading volume (e.g. Tauchen and Pitts (1983) and Harris (1986)). A shortcoming common to these estimation methods is that they examine a single market attribute at a time. Beaver (1968), Copeland (1976), and Tauchen and Pitts (1983) demonstrate that volume and price are jointly determined by the arrival of information. Morse (1981) similarly argues that price and volume be used together to measure the information content of an event. Pincus (1983) comes the closest to a general method for identifying the adjustment 93 period by including both price and volume in his maximum likelihood procedure. But by assuming price and volume are independent, he does not retain the full power of his test. This essay develops a statistical test for estimating the onset and duration of security price and trading volume responses to new information. It extends the analysis of Hillmer and Yu (1979) by allowing a dependent relationship between security price and trading volume. The dependent relationship between price and volume is addressed by orthogonalizing one market attribute with respect to the other. The resulting statistical test provides biased estimates of the onset and duration of market responses to new information (see Giliberto (1985)). A practical procedure for implementing the statistical test is then prescribed. The statistical test allowing dependence is compared to the Hillmer and Yu (1979) and Pincus (1983) tests in simulations of real world responses to information. In this essay, the technique of Hillmer and Yu is extended to incorporate both return and volume into estimation of the adjustment period. The technique explicitly allows for jointly dependent return and volume response to new information. This conceptualization of the adjustment period is discussed in Section II. Section III reformulates Hillmer and Yu's (1979) statistical test assuming independent price and volume changes. Section IV addresses the dependent relationship between price and 94 volume by orthogonalizing one market attribute with respect to the other. Note that the resulting statistical test provides biased estimates of the onset and duration of market responses to new information (see Giliberto (1985)). Section V discusses implementation of the statistical test in practice. Section VI demonstrates use of the test statistic with 4 illustrative simulations. II. Characterisation of the Adjustment Process In most previous empirical studies, the adjustment period is defined as the length of time around a public disclosure date during which the distribution of return is different from when there is no new information. This adjustment period is represented by the interval t,,2 - t,,, in Figure 1. However, volume may change before price as investors anticipate information release (see Figure 1). For instance, Morse (1981) contends that trading before the public disclosure of information may ensue from an increase in the differences in beliefs about the probability of different signals being released by the public announcement. These differences may be caused by asymmetric information among investors before the event date. Estimation of the adjustment period using return alone tends to underestimate ‘the length of the adjustment period and cause an upward bias .1n estimation of the speed of adjustment. We define the adjustment period as the total length of time that both 95 price and volume take to fully reflect new information. For example, in Figure 1 this period is given by (tR2 - tv,) . Although the process of information emission and interpretation is complex and unobservable, using both price and volume together may provide further insight into how the market reacts to information. III. Estimation of the Adjustment Period Empirical findings on the correlation between price changes and trading volume around the time of an earnings announcement are mixed. The conflicting evidence may arise from sample differences such as earnings predictability (Pincus (1983)), the magnitude of analysts' earnings forecast revisions (Jennings and Starks (1985)), firm size, reporting lags, or quarterly versus annual announcements (Defeo (1986)). Hillmer and Yu (1979) develop a statistical test for identifying the onset and duration of market response to information by assuming independence between market attributes including price and trading.volume. This section reformulates Hillmer and Yu (1979) according to the conceptualization of adjustment period in Section II. Consider the situation where return and volume are independent. Let V and R denote the level of trading volume and the rate of change of price, respectively. Suppose V and R are generated by the following stochastic processes: Rt I ”R + aRS(Rt) (l) Vt = pv + oVS(Vt) (2) where u and a are the respective means and standard deviations. Let S(R,) and S(V,) be independent and identically distributed normal random variables with mean zero and variance one. The covariance between S(R,) and S(V;) is zero by assumption. Figure 2 depicts the market reaction of either return or volume around a public disclosure date. Suppose information is released at date to. Either market attribute may have begun to respond at.tq prior to the announcement date. After the announcement, the market attributes (price and volume) continue to adjust until t, in order to fully reflect the effect of the new information. During the adjustment period [t“,t§], the mean level of volume usually increases. However, return could either increase or decrease, depending on whether the news is good news or bad news. We develop one-tailed test statistics for a bad news scenario and a good news scenario in the remainder of this section. A. Bad News Scenario When the information released is bad news, we would expect the price level to decrease and the trading volume to 97 increase. The partial sums of the deviation from the mean for Rt and V, over the interval [t,, k] can be expressed as k S = E (R -u 3 R1: t=1: t R) () S k sv - 2: (v -).). (4) k t'ts t v Since SR, and SV, both have independent increments over time, their behaviors follow a Wiener process. The expected values of SR, and SV, are zero under the hypothesis that the mean levels do not change. Their respective standard deviations are 0,,./t and ath which are functions of time. As k increases beyond t, in Figure 1, we would expect SR, to decrease and SV, to increase. In order to signal the beginning of the market reaction, we need to determine the crossing boundary (B(R,)=0 and B(V,)=0) for SR, and SV, such that, under the null hypothesis of no change in u, and fly, the probability of either SR, or SV, drifting beyond the boundary is less than some preset value a. That is, Pr[(SRjSB(Rj) U SVjZB(Vj)) for some j S kIconstant means]= a 98 Under the independence assumption, this probability can be simplified to pr[(snjss(pj); jSk] + pr[svj_>_3(vj); j s. k] - Pr[(SRjSB(Rj); jSk] * pr[(svjzs(vj); jSk] = o[3(Rj)/(oR/j)] + [1 - a[B(Vj)/(cvlj)1] - ¢[B(Rj)/(oR/j)] * [1 - otB/B(R,)/a,,2] * 9[[-B(R,)-(uR'-uR)t]/(0Rt1/ 2)]. (3) From (8) , the expected first passage time for SR, determined by using the moment generating method is B(R,)/(u,'-u,,) . Similarly, the expected first passage time for SV, is B(V,)/(uv'-uv) . An unbiased estimate for t, is . B(R, ) B(R T ) t1 = Min I TR - p R , Tv - V . (9) O- I... R ”R "V ”v Detecting the ending point, t,, of the adjustment period of the market attributes under the bad news scenario is similar to the detection of t,. Beyond t,, we would expect SR, to increase and SV, to decrease. Again, to 101 signal the end of the market reaction, we need to determine the set of crossing boundaries (B(R,) > 0, B(V,) < 0) for SR, and SV, such that, under the null hypothesis of no change in It, and M.,, the probability of eighe: SR, or SV, drifting beyond the boundary is less than some preset value a. Similar to the derivation of Equation (5) under the independence assumption, [1- ”B(R,)/(0,721] + a[B(vj)/(ov./j)] - [1 - ”B(R,)/(0373)] * ”B(V,)/(0,731)] = a. (10) Following (6) , we can solve for B, and fi,': -1 B(R,) (OR/1H (./(1-c)) _ -1 _ _ B(vj) - (av/in [1 7(1 an. (11) The following table presents the values of B(R,) and B(V,) at different levels of significance assuming no change in the processes R, and V,. a - 1% a = 5% a = 10% B(Rj) 2.57aR/j 1.960R/j 1.64aR/j B (vj) -2 . 57ov/j -1 . 96¢:ij -1 . 64(7ij 102 As in the derivation of equation (9), an unbiased estimate of t,, is given by B(RTR) B(vT ) t2=MaXITR- .- ,Tv- '_ I. (12) “R “R “v “v B. Good News Scenario Empirical evidence indicates that both the price level and trading volume should increase in response to unexpected good news. The partial sums, SR, and SV,, of the deviation from the mean for R, and V, over the interval [t,, k] are given by Equations (3) and (4) . SR, and SV, have independent increments and their behaviors follow a Wiener process. Under the null hypothesis of no change in the mean levels, the expected values of SR, and SV, are zero and their standard deviations are a,./t and av./t, respectively. As k increases beyond t,, we would expect SR, and SV, to increase as R, and V, reflect the new information. To detect the beginning of the market reaction at a (1-a) significance level, we need to determine the set of crossing boundaries (B(R,) > 0, B(V,) > 0) for SR, and SV, such, that under the null hypothesis, the probability of eigher SR, or SV, exiting the boundary is less than a. That is, Pr[(SRj 2B(Rj)) U (SVj 2B(Vj)) for some jSkI constantmeans] =a. 103 Under the independence assumption, this probability can be decomposed into pr[(SRj 2 B(Rj)): j s k] + pr[(svj 2 B(Vj)); j s k] - pr[(SRj 2 B(Rj)): j S k] * pr[(svj 2 B(Vj)); j s k] (1 - 9[B(Rj)/(0le)]) + (1 - “B(V,)/(awn) - (1 - i[B(Rj)/(0R./j)]) * (1 - “B(V,)/(awn) = a. (13) By normalizing B(R,) and B(V,) with 0,, and 0v such that B(R,)/0,, = - B(V,)/av, then 11801,) NOR/1)] = “B(V,)/(awn . (14) Substituting equation (14) into (13), we can solve for B(R,) (OR/ju'luu-an and B(V,) (aij)4-1[./(l-a)]. (15) The values of B(R,) and B(V,) assuming different values of a are: a = 1% a = 5% a = 10% B(Rj) 2.57oR/j 1.96OR/j 1.64aR/j B(Vj) 2.57avjj 1.96oV/j 1.64av/j 104 IV. Dependent Return and Trading volume Changes Assume that R, and V, are generated by the same processes as specified by (1) and (2) except that B(R,) and B(V,) are bivariate normal with correlation coefficient equal to r. Although the relaxed assumption is simple, determination of the joint distribution of the first passage time of two stochastically dependent variables is difficult. Also, there may not be an explicit solution for the expected first passage time of either SR, or SV, exiting the boundaries. While lower or upper bounds could be found, it does not help in estimating the reaction time.tq or t,. As an alternative route toward a solution, consider the following model based on the relaxed assumption Rt = bo + b1 * V + e t t (16) where 6, has zero mean. Since R, and V, are correlated, the variable V, explains part of the variation of R,. The constant and the residual term 6, capture that part of the variation in R, which is not explained by V,. Suppose there is a shift in the mean levels of R, and V, due to the arrival of new information. If the mean level of V, shifts before those components of R, which are independent of V,, then R, will also change according to the sign and the size of b,. However, the term (b0 + 6,) will not reflect the change of V,. If the mean level of R, shifts first, then b0 105 must change and hence the sum (bo + 6,) will reflect the shift. If the mean levels of R, and V, independently shift at the same time, then (b, + 6,) will not reflect the total change if r > 0. Now, consider shifts in the variances of R, and V,. If the change in variance of V, comes first, (b0 + 6,) will not capture the effect on R,. If the variance of R, shifts first, then it must come from a change in the variance of 6,. If both variances change independently at the same moment, then (bo + 6,) will not reflect the total change if r>0. Under this formulation, (b, + 6,) is an instrumental variable for R, that can capture the response time of R, with respect to that of V, while remaining uncorrelated with V,. We transform R, and V, into two new random variables, R,* and V,*, that are stochastically independent. In vector notation, I Rt* ] = I: Rt - th I = I “R - buv + ORS(Rt) - bOVS (Vt) :I where b is such that El: (Rt-bvt) "’ (Pa‘bl‘v) 1 [ (Vt'l‘v)] = 0- (17) A solution for b exists if the variance-covariance matrix between R, and V, is non-singular. From (14) , b = rOR/ov, which is simply the regression coefficient of R,* on V,*. 106 IQ* is the residual plus the constant term obtained by regressing R, against V,. It is easily shown that the transformed variables R,* and V,* are independent with means ((1,, - buv) and u, and variances (1-r2)o,,z and 0,2 respectively. The same analysis as in Section IIIA can then be repeated on R,* and V,*. Using the first passage time approach with T, and TV, the estimates for t, and t,“ are, respectively, . . JITR(1-r2)] 6R4'1[1-/(1-a)1 (JT )av6'1/(1-a) t1” "1“ [TR’ (pH-11R) + r(cR/cv)(uV'-uv)’ Tv' (uV'-uv) I and . Java-r2” ovs’lti-Ju-an (J'r) aRa’lJu-a) ‘2' 1.... [TV (I‘V'fllv) + r