Essays in financial markets

This dissertation studies how heterogeneous opinions affect financial market outcomes, including price informativeness and trading volume. The dissertation contains two chapters. In both chapters, theoretical models are developed and then supportive empirical evidence is provided. In the chapter ""Index Trading and Its Effects on the Underlying Assets'' (Chapter 2), I present a rational expectation model of index trading. The key finding is that the efficiency of each of the underlying stocks decreases as the proportion of index traders increases, while the efficiency of the index itself is unchanged. This result is achieved despite the fact that no arbitrage opportunities exist, i.e., the price of the basket (index) is the sum of its components. Using S&P 500 ETFs data, I show that the index contributes to price discovery in its underlying stocks. In addition, the regression analysis is consistent with the model predictions: index trading impairs efficiency of the component securities but does not have effects on the index itself. In the chapter ""News, Influence, and Evolution of Prices in Financial Markets'' (Chapter 2), we study the influence of published views on the evolution of prices by constructing a theoretical model and using empirical work to test the model. Our sequential trade model demonstrates how the influence of published views creates patterns in prices and volume. Still, a ""wisdom of the crowds'' effect emerges endogenously in our framework and helps expunge such shared errors from the price, thus setting the paper apart from the information cascades literature. We use the timing of earnings announcements to test our model, and find evidence consistent with the theoretical predictions. The magnitude of the empirical effects is then used to calibrate the model, and our calibration exercise suggests that patterns in the evolution of prices are affected more strongly by the extent of the influence of the published view than by its accuracy.

ESSAYS IN FINANCIAL MARKETS by Xiaodi Zhang A dissertation submitted to the faculty of The University of Utah in partial ful llment of the requirements for the degree of Doctor of Philosophy in Business Administration David Eccles School of Business The University of Utah August 2015 Copyright © Xiaodi Zhang 2015 All Rights Reserved T h e U n i v e r s i t y o f U t a h G r a d u a t e S c h o o l STATEMENT OF DISSERTATION APPROVAL The dissertation of Xiaodi Zhang has been approved by the following supervisory committee members: Shmuel Baruch , Chair 04/15/2015 Date Approved James Schallheim , Member 04/15/2015 Date Approved Michael Cooper , Member 04/15/2015 Date Approved Feng Zhang , Member 04/15/2015 Date Approved Jingyi Zhu , Member 04/15/2015 Date Approved and by William Hesterly , Chair/Dean of the Department/College/School of David Eccles School of Business and by David B. Kieda, Dean of The Graduate School. ABSTRACT This dissertation studies how heterogeneous opinions a ect nancial market outcomes, including price informativeness and trading volume. The dissertation contains two chapters. In both chapters, theoretical models are developed and then supportive empirical evidence is provided. In the chapter \Index Trading and Its E ects on the Underlying Assets" (Chapter 1), I present a rational expectation model of index trading. The key nding is that the e ciency of each of the underlying stocks decreases as the proportion of index traders increases, while the e ciency of the index itself is unchanged. This result is achieved despite the fact that no arbitrage opportunities exist, i.e., the price of the basket (index) is the sum of its components. Using S&P 500 ETFs data, I show that the index contributes to price discovery in its underlying stocks. In addition, the regression analysis is consistent with the model predictions: index trading impairs e ciency of the component securities but does not have e ects on the index itself. In the chapter \News, In uence, and Evolution of Prices in Financial Markets" (Chap- ter 2), we study the in uence of published views on the evolution of prices by constructing a theoretical model and using empirical work to test the model. Our sequential trade model demonstrates how the in uence of published views creates patterns in prices and volume. Still, a \wisdom of the crowds" e ect emerges endogenously in our framework and helps expunge such shared errors from the price, thus setting the paper apart from the information cascades literature. We use the timing of earnings announcements to test our model, and nd evidence consistent with the theoretical predictions. The magnitude of the empirical e ects is then used to calibrate the model, and our calibration exercise suggests that patterns in the evolution of prices are a ected more strongly by the extent of the in uence of the published view than by its accuracy. CONTENTS ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vi LIST OF TABLES: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii ACKNOWLEDGMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ix CHAPTERS 1. INDEX TRADING AND ITS EFFECTS ON THE UNDERLYING ASSETS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Personal Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Index Fund Traders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.4 Stock Pickers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.5 Equilibrium and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Empirical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Hasbrouck Information Share . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 Index Trading and the Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Sample and Variable Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.1 Hasbrouck Information Share . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.2 Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2. NEWS, INFLUENCE, AND EVOLUTION OF PRICES IN FINANCIAL MARKETS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1 The Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.2 Properties of the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Empirical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4.1 Price Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4.2 Return Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4.3 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 APPENDICES A. PROOFS IN CHAPTER 1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 64 B. PROOFS IN CHAPTER 2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 69 REFERENCES: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 76 v LIST OF FIGURES 1.1 Relation between Posterior Variance of Return and Proportion of Index Traders 20 2.1 Price Adjustment when the Published View Is Incorrect . . . . . . . . . . . . . . . . . . 50 2.2 Volume in the In uence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 LIST OF TABLES 1.1 Variables Related to Calculating the Posterior Probability . . . . . . . . . . . . . . . . . 21 1.2 S&P 500 Additions and Deletions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3 Summary Statistics: S&P 500 Component Stocks in the Whole Sample . . . . . . 22 1.4 Summary Statistics: S&P 500 Component Stocks within the Size Groups . . . . 23 1.5 Summary Statistics: S&P 500 Index ETFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6 Price Discovery from S&P 500 Index to Component Stocks . . . . . . . . . . . . . . . . 25 1.7 Price Discovery from Component Stocks to S&P 500 Index . . . . . . . . . . . . . . . . 25 1.8 Regression Results for Realized Volatility of Stocks in the Whole Sample . . . . 25 1.9 Regression Results for Realized Volatility of Stocks within the Size Groups . . . 26 1.10 Regression Results for Realized Volatility of the Index . . . . . . . . . . . . . . . . . . . . 26 2.1 Trader Type De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.2 Trader Type Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.3 Number of Earnings Announcements by Year and Earnings Surprise Category 53 2.4 Number of Earnings Announcement Pairs by Industry . . . . . . . . . . . . . . . . . . . . 54 2.5 Summary Statistics: Categories 1&2 Matching Using Industry Classi cation, Market Capitalization, and Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.6 Summary Statistics: Categories 1&2 Matching Using Market Capitalization and Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7 Summary Statistics: Categories 6&7 Matching Using Industry Classi cation, Market Capitalization, and Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.8 Summary Statistics: Categories 6&7 Matching Using Market Capitalization and Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.9 Initial Price Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.10 Return Patterns: Raw Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.11 Return Patterns: Returns in Excess of the S&P 500 . . . . . . . . . . . . . . . . . . . . . . 59 2.12 Return Patterns: Abnormal Returns Relative to Preannouncement Period . . . 60 2.13 Return Patterns with Alternative Initial Adjustment Period: Categories 1&2 . 61 2.14 Return Patterns with Alternative Initial Adjustment Period: Categories 6&7 . 62 2.15 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 B.1 Events and Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 B.2 Arrival Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 B.3 Arrival Probabilities Conditional on the Published View . . . . . . . . . . . . . . . . . . 75 B.4 Valuations Conditional on the Published View . . . . . . . . . . . . . . . . . . . . . . . . . . 75 viii ACKNOWLEDGMENTS I would like to express my deepest gratitude to my committee chair, Professor Shmuel Baruch, for his excellent guidance, caring and constant encouragement. His passion and creativity inspired me during my whole Ph.D. study. Without his persistent help, this dissertation would not have been possible. I would like to thank my committee member, Professor James Schallheim, who provided important insights in my research and patiently corrected my writing. I would like to thank my other committee members, Professor Michael Cooper, Professor Feng Zhang, and Professor Jingyi Zhu, for their kind support and help. I would also like to thank Professor Peter Bossaerts and Professor Hendrik Bessembinder for their helpful comments on my dissertation. In addition, a thank you to Professor Gideon Saar of Cornell University, who led me to the world of empirical market microstructure and motivated me as a role model. I also owe my sincere appreciation to my colleagues at the University of Utah, especially to Huan Cai, Al Carrion, Barbara Chambers, Wenhao Yang, and Yan(Julian) Zhang. Last but not least, I would like to thank my lovely wife, Aiting Gao, and my parents. They have been always standing by me, supporting me and encouraging me with their best wishes. CHAPTER 1 INDEX TRADING AND ITS EFFECTS ON THE UNDERLYING ASSETS 1.1 Introduction The index mutual funds and the index-based exchange-traded funds (ETFs) have become very attractive investment vehicles since their rst introduction. By the end of the year 2013, index funds and index ETFs managed total net assets of about $1.7 trillion and $1.6 trillion, respectively.1 By de nition, the index fund is used to replicate the performance of a particular market index. Prior studies mostly focus on the U.S. stock index and the index future and examine which one is dominant in price discovery, for example, Kawaller, Koch, and Koch (1987), Stoll and Whaley (1990), and Chan (1992). These papers nd that the futures market leads the cash market but present weaker evidence in the reverse direction. However, not many empirical studies examine the impact of index trading on its underlying assets. Yu (2005) investigates how the introduction of index ETFs a ects the e ciency of individual component securities. Qin and Singal (2013) study the relation between passive ownership and price e ciency using a sample of S&P 500 and non-S&P 500 stocks. In this paper, I try to ll this gap by developing a theoretical model of index trading and providing empirical evidence for the model's predictions. Traditionally, people consider the index fund a redundant asset since it can be easily replicated using its component securities. However, the environment of nancial markets has changed since the introduction of index ETFs and the advent of high-frequency and algorithmic trading. Recent studies by Yu (2005) and Jovanovic and Menkveld (2012) show that the index related security delivers information about its components. Hasbrouck (2003) also nds that the S&P 500 ETF contributes information about its sector ETFs. These ndings both convey a signal that index trading may contain some pieces of information on its underlying assets. When market participants with di erent views on asset values trade 1Investment Company FactBook 2014. 2 the index future or ETF, it becomes nonredundant. The existence of the index now a ects prices of its underlying assets because these prices will adjust to a new equilibrium. To study the e ects of index trading on its component stocks, I investigate a rational expectation model with two types of investors: index traders and stock pickers. In my model, the indexers only trade the market portfolio (i.e., index fund or index ETF) but the stock pickers can pick any component stock to trade.2 As in Bossaerts (1993), I do not model explicitly why indexers exist on the market. Investors might trade index because it is simple or because the execution of index incurs low costs. Given the large investment in index funds and index ETFs, it is a reasonable assumption that some of the investors on the market love to trade index. More importantly, to capture the idea that di erent investors may possess di erent information sets, each investor in my model is endowed with personal views on the prospects of individual assets. Investors form their personal views via related news papers, TV programs, or the Internet. In the model, I assume that personal views are more likely to be correct and errors among these views are independent. This notion of views is similar to the imprecise information in Admati and P eiderer (1986) but here the formation of views requires no costs. My model set-up is di erent from models with private information, where information is precise and only a small number of investors can access it. In my model, everyone holds personal views which are self-generated signals (i.e., a piece of imprecise information) and the accuracy of views is very low. The view and the signal (or the piece of information) are mathematically equivalent so I will use these terms interchangeably in the paper. In the equilibrium of my model, the price of market portfolio is equal to the sum of prices of individual stocks so there does not exist any arbitrage opportunity. The model predicts that individual stocks become less e cient when there are more index traders but the e ciency of the index itself stays constant. The e ciency in my model is de ned as in Kyle (1985), which is the posterior variance of return. The reason for the above implications is that index traders do not utilize their views on individual assets. Instead, they use the combined views, i.e., views of the market portfolio. In the equilibrium, prices of individual stocks are not fully revealing since they only re ect indexers' views about the market as a whole. In my model, indexers want to cope with the winner's curse problem when they 2In this paper, I use the term \market portfolio" and \index fund or ETF" interchangeably. The term \index fund" is used here to be consistent with the empirical work. 3 make their investment decisions. Thus they optimally extract information from individual stock prices. As a result, the price of market portfolio is always fully revealing. The model closely relates to prior theoretical work on the index and index-related securities. Subrahmanyam (1991) studies how the introduction of a security basket a ects market liquidity and the informativeness of its underlying assets and nds that the nal results depend on di erent model parameters. He also concludes that the basket security has no e ects on the variance of price changes in individual securities when the number of informed traders is constant. Gorton and Pennacchi (1993) develop a model to study how liquidity traders choose between a single security and a basket. Both models have informed traders possessing accurate information and require the existence of liquidity traders. Thus the price informativeness of securities depends on where these liquidity traders trade and the model parameters. My model is di erent from theirs in two respects. First, every trader in my model has views although they are imprecise. Second, liquidity traders do not exist in my model and the ine ciency all comes from trading in the market. Higher index trading leads to higher posterior variances of individual stock returns and causes more information loss on the underlying assets. I also examine the model predictions using real world data and in particular, I use S&P 500 ETFs as the index. These heavily traded ETFs provide me with a good setting to study information ow between the index and its underlying assets. Using the Hasbrouck (1995) information share method, I nd that price changes in the index contribute to about 25.69% in the e cient price variance of component stocks. This result contradicts the traditional cognition that the index security is a simple basket of its underlying assets. This approximate one quarter of price discovery in the component stocks provides signi cant evidence that index trading contributes some relevant information. In order to test the model's implications, I use realized volatility as an empirical proxy for the theoretical e ciency measure, posterior variance. I use realized volatility for the following reasons. First, it is an estimate of ex post variance. Second, it does not require speci c distributional assumptions. Third, many prior studies examine the relation between realized volatility and trading volume to provide insights on how traders process new infor- mation. For example, Jones, Kaul, and Lipson (1994) and Chan and Fong (2006) present evidence that trading volume explains almost all of the asset's daily realized volatility. Since my model relates index trading to posterior variance, the realized volatility ts the model well and provides very good context to test the model's predictions. Following 4 Andersen, Bollerslev, Diebold, and Ebens (2001) (ABDE (2001) hereafter), I construct daily realized volatility by summing up squares of high-frequency intraday returns. Speci cally, the 1-minute and 5-minute interval returns are used to construct the proxy. For individual assets, I measure the proportion of index traders as follows: I estimate daily trading volume generated from index ETFs for each component stock and divide it by that stock's total daily trading volume. For the index, I sum all daily trading volume of its underlying stocks and divide its own daily volume by the achieved sum. Using a sample period from October 1, 2013 to December 31, 2013, I regress the daily realized volatility on the proxy of index traders for the index and each component stock. Consistent with the model implications, I nd that proportion of indexers is positively and signi cantly related to realized volatility for component stocks but such relation does not exist in the index itself. This paper also relates to the literature on price changes in the case of index additions and deletions. Most of the studies in this literature argue that index additions and deletions involve no information. Therefore the price changes caused by indexing are due to downward sloping demand curves. For example, Shleifer (1986) and Lynch and Mendenhall (1997) present evidence consistent with this hypothesis. Two recent studies provide evidence for other explanations. Denis, McConnell, Ovtchinnikov, and Yu (2003) nd that additions to the S&P 500 index provide good information about companies' future prospects. Chen, Noronha, and Singal (2004) show that asymmetric price changes due to S&P 500 additions and deletions are caused by investors' awareness. Their nding is consistent with Merton (1987)'s model of market segmentation. My model together with its empirical evidence, however, sheds new light on this literature. Since index trading changes the information en- vironment of its component stocks, index additions and deletions are no longer information free so prices will adjust to a new equilibrium. This may explain the price changes due to index additions and deletions to the extent that index trading alters prices of its component stocks. My study contributes to the literature in at least two ways. First, I develop a rational expectation model to capture the idea that the index trading may contain relevant infor- mation about its underlying assets. In particular, index trading generates ine ciency in its components. Second, I empirically examine the impacts of index trading and present new evidence that it impairs the market quality of its underlying assets. The remainder of the paper is organized as follows. Section 1.2 studies a model with 5 heterogeneous agents and presents the model's prediction. Section 1.3 discusses the empiri- cal methodology and describes the sample. Section 1.4 presents the empirical results of the tests, and Section 1.5 concludes the paper. 1.2 The Model I consider a competitive economy with N risk averse traders, a fraction of them are index fund traders and 1− are stock pickers. All agents have a continuous and di erentiable concave utility function U. The economy contains three assets: a risk-free bond and two risky assets. The economy has one period and all agents consume at the end of the period. Each agent is endowed with 1~N share of the two risky assets' supplies and in addition, everyone has personal views on each asset's terminal value. Investors form their views by reading related newspapers, watching TV programs, or searching on websites or forums and their view formation incurs very low costs. In the model, I assume that each investor forms personal views with zero cost and also require that errors among views be independent. As I mentioned, each investor's view is equivalent to a piece of information and its accuracy is very low. 1.2.1 Assets The risk-free bond pays 1 with certainty at the end of the period and serves as the numeraire in this economy. The two risky assets are indexed by j, j = 1; 2. The price of risky asset j is denoted by Pj . The end-of-period value ~vj of risky asset j follows the binary distribution, equalling either vh or vl. I assume that the two risky assets are independent from each other and the supply for each is normalized to 1. The independent assumption is without loss of generality and simpli es the calculation. The prior distribution of each risky asset j is the following: ¢¨¨¦¨¨¤ phj = Prob(~vj = vh) = 1~2 plj = Prob(~vj = vl) = 1~2: In the economy, the market portfolio (or index fund) contains one share of risky asset 1 and one share of risky asset 2. That is to say, the terminal value ~ VI of the market portfolio is given by the linear combination ~v1 + ~v2. So ~ VI can take value VH = 2vh, VM = vh + vl, or VL = 2vl. According to the prior distribution of risky assets and the independence assumption between them, the prior distribution of the index fund is: 6 ¢¨¨¨¨¦¨¨¨¨¤ pHI = Prob( ~ VI = VH) = 1~4 pMI = Prob( ~ VI = VM) = 1~2 pLI = Prob( ~ VI = VL) = 1~4: 1.2.2 Personal Views Every investor, in the model, is endowed with personal views on terminal values of both risky assets. I assume that the relation between a view and the terminal value of asset j is given by the linear function: ~vn j = ~vj(1 − ~ nj ) + (vh + vl − ~vj)~ nj ; where j = 1; 2, n denotes the nth trader, and n = 1; 2; :::N . The error term ~ nj is a zero-one random variable, identically and independently distributed across all traders (both indexers and stock pickers) and the risky assets. With the above construction, a view of risky asset j also takes either vh or vl. In addition, I assume a trader's view is more likely to be correct than the prior belief implying that the error term equals zero with probability j > 1~2. In this paper, views are self-generated signals, which are costless and imprecise. Mathematically, a view and a signal (or a piece of information) are equivalent so I will use these two terms interchangeably. Given views on individual assets, a view on the market portfolio then directly follows: ~ V n I = ~vn 1 + ~vn 2 ; where n denotes the nth trader and V n I can also be one of the three values: VH, VM, or VL. 1.2.3 Index Fund Traders In my model, index fund traders or indexers are traders who love to trade the market portfolio. Therefore these traders will condition on their views of the market when making investment decisions. At the same time, these traders also want to cope with the winner's curse problem so they will optimally extract information from prices of individual assets. Then indexer i's information set includes his view of the market and prices of individual assets, i.e., {V i I = vi1 + vi2 ;P1;P2}, where i = 1; 2; :::; N. Each indexer maximizes his expected end-of-period utility function by allocating between the market portfolio and the risk-free bond conditioning on his information set. I denote indexer i's posterior distribution of the market as qiX I , where X corresponds to each possible 7 value in the state space {VH; VM; VL} respectively. Then the ith indexer's problem is written as: max iI E[U( ~W i;1)SV i I = vi1 + vi2 ;P1;P2] =Q X qiX I 􀀀 U(WX i;1) (1.1) subject to the budget constraint ~W i;1 = (Wi;0 − iI(P1 + P2)) + iI ~ VI ; (1.2) where iI is the demand for the index fund. Since the index fund comprises the asset 1 and 2 with share ratio 1 1, the indexer's demands for individual assets are i1 = iI and i2 = iI . 1.2.4 Stock Pickers Stock pickers allocate their wealth among the two risky assets and the risk-free bond to maximize their expected end-of-period utility. Stock pickers are indexed by s, s = 1; 2; :::(1− )N. Since these traders will choose the demand of two risky assets respectively, their state space e regarding the nal values of assets is {(v1 = vh; v2 = vh); (v1 = vh; v2 = vl); (v1 = vl; v2 = vh); (v1 = vl; v2 = vl)}. I index each event in the state space by e. The information set of stock picker s contains his own views and prices of the two risky assets, i.e., {vs 1; vs 2;P1;P2}. I let the posterior probability distribution of each event for the stock picker s be qs e and then his problem can be written as: max s1; s2 E[U( ~W s;1)Svs 1; vs 2;P1;P2] =Q e qs e 􀀀 U(We s;1) (1.3) subject to the budget constraint ~W s;1 = (Ws;0 − 2Q j=1 sjPj) + 2Q j=1 sj ~vj ; (1.4) where sj is the demand for asset j. 1.2.5 Equilibrium and Its Properties A rational expectation equilibrium in this economy is de ned as: (i) each indexer solves his maximization problem by allocating his wealth between the index fund and the risk- free bond, given his information set; (ii) each stock picker solves his full blown portfolio optimization problem, conditional on his information set; and (iii) prices P1 and P2 clear the relative markets. 8 The market clearing conditions are given by: ¢¨¨¦¨¨¤ P N i=1 i1 +P(1− )N s=1 s1 = 1 asset 1 P N i=1 i2 +P(1− )N s=1 s2 = 1 asset 2: (1.5) Market clearing conditions contain the demand from all traders, i.e., indexers and stock pickers. The trading of the index fund plays a role when stock pickers make their investment decisions. This is because the demands for asset 1 and 2 from the indexers contain their views on the index portfolio. These views of the market will be re ected in the prices of individual assets through the market clearing conditions. In addition, views contain imprecise but relevant information on terminal values of assets so the equilibrium prices will be di erent from those in the model without personal views. Since each trader's demands depend on his views of either individual assets or the market as a whole, equilibrium prices, formed from market clearing conditions (1.5), are functions of views from all traders. I denote Os = {(v1 1; v1 2); (v2 1; v2 2); :::; (v (1− )N 1 ; v (1− )N 2 )} the set of views of stock pickers and Oi = {V 1 I ; V 2 I ; :::; V N I } the set of views of indexers. Then the view set of all traders is given by O = {Os;Oi} and the prices can be written as P1(O) and P2(O). Given the above assumption about assets, I need the following lemmas to nd the economy's equilibrium prices. Lemma 1 Let X be a discrete random variable taking values {0; 1; 2; :::; k}. The unknown probability distribution of X is the vector p = {p0; p1; p2; :::; pk}, where pi = Prob(X = i) and Pki =0 pi = 1. In addition, I(X = i) is an indicator function, equalling 1 if X = i and 0 otherwise. Let {X1;X2; :::;XN} be a sample of N independent such random variables. Let Ni = PNn =1 I(Xn = i), where Pki =0Ni = N. Then ^p = {N0;N1;N2; :::;Nk} is su cient for p. Proof. See Appendix A.2. Using this lemma, I can nd su cient statistics for O, given v1 and v2, for indexers and stock pickers, respectively. Since indexers only care about the value of the index, in their view, the information set Os of stock pickers is equivalent to Os = {V 1 I ; V 2 I ; :::; V (1− )N I }. For this reason, the total information set O is reduced to O = {Os ;Oi} for indexers. So by Lemma 1, their su cient statistic for O given v1 and v2 is Ni = {NHs;NMs;NLs;NHi, NMi;NLi}, where NHs, NMs, and NLs correspond to the numbers of occurrence of VH, VM, and VL in Os and NHi, NMi, and NLi are the relative numbers in Oi. Stock pickers, on the 9 other hand, select their demands for asset 1 and asset 2 individually so they try to extract information on each risky asset from O. Then according to Lemma 1, for stock pickers, their su cient statistic for O conditional on v1 and v2 is given by Ns = {Nh1;Nl1;Nh2;Nl2;NHi, NMi;NLi}, where Nh1 and Nl1 are the numbers of occurrence of vh and vl for asset 1 in Os, and Nh2 and Nl2 denote the corresponding numbers for asset 2 in Os. I combine Ni with Ns to form a su cient statistic No = {Nh1;Nl1;Nh2;Nl2;NHs;NMs;NLs;NHi;NMi;NLi} of the information set O for all traders. Then, for indexers, No is equivalent to Ni which is a su cient statistic for the conditional joint probability mass function h(O SVI = v1 + v2); stock pickers consider No as Ns which is a su cient statistic for h(OSv1; v2). Together with the above analysis, I also need another lemma to solve the equilibrium problem. Before describing the next lemma, I de ne the conditional probabilities for the terminal values of risky assets and the index fund for later use. I let yxj be the conditional probability of a view of asset j on the value of asset j, where j = 1; 2, y = {vh; vl} is the value set of views, and x = {vh; vl} is the value set of risky assets. I denote Y XI the corresponding conditional probability for values of the index, where Y and X are value sets of views and the index, respectively, both equalling {VH; VM; VL}. I list all relevant variables in Table 1.1. Lemma 2 The conditional joint probability mass function Prob(V i I ;OSVI = v1 + v2) for indexers is equal to Prob(O SVI = v1 + v2), where O = {Os ;Oi}; and the conditional joint probability mass function Prob(vs 1; vs 2;OSv1; v2) for stock pickers is equal to Prob(OSv1; v2). Proof. See Appendix A.2. Lemma 2 simply says that for each trader, if he knows the information contained in O(or O ), his individual opinion provides no additional information regarding v1 and v2. In other words, all index traders have the same posterior belief of the market portfolio and all stock pickers have the same belief regarding individual assets. With the above two lemmas, I can characterize the equilibrium prices P 1 (O) and P 2 (O). By Lemma 1 and its subsequent analysis, given the realization of random variables, the information conveyed in the equilibrium price pair {P 1 (O);P 2 (O)} is equivalent to No. So the individual investor's information set is reduced to {V i I ;No} or {vs 1; vs 2;No}. Using Lemma 2, this is further reduced to Ni or Ns. With the relevant information in mind, each investor can calculate his posteriors regarding the terminal values. Then the equilibrium price pair is achieved by solving (1.1), (1.3), and (1.5). So by looking at the prices of two 10 risky assets, investors can infer No and then by conditioning on this inferred information, they form their demands such that the corresponding prices clear the asset markets. In this sense, the price pair achieved above constitutes the rational expectation equilibrium. After characterizing how to solve the equilibrium prices, I examine the model's implica- tions. I have the following proposition for the model with personal views: Proposition 1 The posterior variance of return for each risky asset j in the index increases with the proportion of index traders ; the posterior variance of the index return, however, is independent of . The above proposition actually considers how trading in the index a ects the price generation of its underlying assets. Traditionally, investors take the index fund as a redundant asset, i.e., a simple basket of underlying components which provides no addi- tional information. However, the development of index ETF and trading technologies has changed the equity market signi cantly. Hasbrouck (2003) nds that the S&P 500 ETF contributes to its sector ETFs' price discovery more than the other way round. Yu (2005) and Jovanovic and Menkveld (2012) also show that price changes of index related securities convey information about their components. My model shows that index trading makes the prices of its component assets less informative or e cient in terms of the posterior variance of their returns. The ine ciency arises since indexers do not utilize their views on individual assets. Instead, index traders trade on their views of the index portfolio. Therefore stock pickers cannot tell exactly the individual opinion for each asset in the cases of V i I = VM. For this reason, they have to consider all possible combinations of vi1 and vi2 that give V i I = VM and this makes the underlying assets less informative. The system of (1.1), (1.3), and (1.5) produces two complicated functions thus making the model less tractable therefore I resort to numerical solutions. Figure 1.1 gives a numerical example of the relations described in Proposition 1. In this example, I assume that each investor has a quadratic utility function: U( ~W ) = ~W − 1 2b ~W 2 = − 1 2b ( ~W − b)2 + b 2 ; (1.6) where b is large enough to avoid the problem of negative marginal utility. In addition, I let N = 100, vh = 11, and vl = 10. I also assume 1 = 2 for simpli cation and take the error precision as 0.60. I simulate the model's random variables based on di erent s (i.e., proportion of index traders). With the speci c realization of random variables and the 11 utility in (1.6), there always exists a unique equilibrium price pair such that vl B P 1 B vh and vl B P 2 B vh. Then the investor can always establishes a one-to-one correspondence between {P 1 ;P 2 } and the su cient statistic No. I calculate the equilibrium prices of two risky assets and the corresponding posterior variances of returns, and then plot posterior variances against di erent s in Figure 1.1. Figure 1.1 shows clearly that the posterior variance of each asset j increases with but the posterior variance of the index stays constant. My model is di erent from models in Subrahmanyam (1991) and Gorton and Pennacchi (1993) in that I do not have liquidity traders but instead have index traders. In their models, there exist some liquidity traders and it is these traders who provide camou age for the informed traders who possess very accurate information. The price informativeness depends on where the liquidity traders trade. Since the introduction of a basket security may change the investment decisions of liquidity traders, the price informativeness may change correspondingly. However, my model does not require liquidity traders but instead assumes existence of indexers with views. This nonfully revealing equilibrium due to index trading di erentiates my model from the traditional noisy rational expectation equilibrium which requires the existence of liquidity traders, noisy supply, or nontradable endowment shock. It is the trading of indexers (i.e., the constrained trading in the index) that generates the information loss in prices of component stocks. The more indexers exist on the market, the higher posterior variance each component stock has. This prediction is di erent from the conclusion in Subrahmanyam (1991) that the basket security has no e ects on the variability of price changes of its component securities. 1.3 Empirical Methodology My model allows me to study how index fund trading a ects its underlying assets using real world data. The component stock becomes less e cient in terms of its posterior variance when there are more indexers trading on the market. Empirically, I use realized volatility to measure the posterior variance. The realized volatility is an estimate of ex post variance, which is just the concept used in my model. Furthermore, unlike the stochastic or implied volatility, it does not depend on restrictive distributional assumptions. In addition, prior literature studies how market participants react and process new information by examining the relation between volatility and trading volume and in those studies, realized volatility is widely used. For the above reasons, I consider realized volatility a good t of my model. 12 In the test, I use S&P 500 ETF as the index since it is the most widely replicated index ETF on the market. This highly traded ETF also enables me to study the information ow between the index and its underlying assets by applying the information share approach in Hasbrouck (1995). The information share method can provide empirical evidence on whether the index contains some information about its component assets in short time horizons. 1.3.1 Hasbrouck Information Share The information share method in Hasbrouck (1995) is used to measure the contribution to price discovery from each market in two or more related markets. It assumes that there is a single random walk component, called the e cient price, which is common to prices from all markets. But this method can also be extended to include prices which may not cointegrate with each other. Within the context of studying information contribution between index and the underlying assets, there no longer exists a single random walk common to all prices. Speci cally, I construct the following multivariate price vector: P = [Midquote Indext Midquote Stockj;t] (1.7) for each component j in the index, where Midquote Indext represents the midquote (ask plus bid divided by two) of the index at time t and Midquote Stockj;t denotes the midquote of the stock j at time t. Then the vector error correction model (VECM) is written as: Pt = B1 Pt−1 + B2 Pt−2 + ::: + BM Pt−M + (zt−1 − ) + t; (1.8) where t is a 2×1 vector of zero-mean innovations with variance matrix , Bi is a 2×2 matrix of autoregressive coe cients corresponding to lag i of the price changes, and (zt−1− ) is the error correction term with = E(zt). The VECM also has a moving average representation: Pt = t + A1 t−1 + A2 t−2 + ; (1.9) where Ai matrices are moving average coe cients and the sum of all the moving average coe cients is denoted as A(1) = I + A1 + A2 + . With the above systems, I can nd the proportion of stock j's innovation variance that is attributed to the innovation of the index and also the other way round. Since A(1) accounts for the permanent impact innovations, then the total variance of permanent price changes for the lth component in P is ll = [A(1) A(1) ]ll, where is the covariance matrix of 13 the price innovations and ll is the lth diagonal element. If is diagonal, the information contribution of the kth component in P to the lth security is given by: Sk l = a2l k kk 2l l ; (1.10) where alk is the kth element in the lth row of the sum matrix A(1) and kk is the kth diagonal element in . If is not diagonal, the information share is not exactly identi ed. In this case, using the Cholesky factorization of one can determine the upper and lower bounds of the information share. 1.3.2 Index Trading and the Realized Volatility The main conclusions of my model are in Proposition 1 which says that the higher proportion of index traders a component stock contains, the higher posterior variance it has but the e ciency of the index itself is independent of such a proportion. In order to test these predictions empirically, I rst develop a variable that re ects the proportion of index trading for a component stock. I de ne my main variable as follows: Index Trading Portionj;t = Indextrading Dvolj;t Trading Dvolj;t ; (1.11) where Indextrading Dvolj;t is an estimate of the dollar volume generated by index trading for stock j on day t and Trading Dvolj;t is the total dollar volume for stock j on day t. For the index itself, the corresponding equation is written as: Index Trading PortionI;t = Index DvolI;t~Q j Trading Dvolj;t; (1.12) where Index Dvolt is the dollar volume of all ETFs and Pj Trading Dvolj;t is calculated by summing all the dollar volume of its component stocks on the trading day t. I then follow ABDE (2001) to construct the empirical proxy for posterior variance: realized volatility. ABDE (2001) shows that, under weak regularity conditions, the summed square of return within in nite small interval converges almost surely to the integrated latent volatility. Thus, by summing su ciently nely sampled high-frequency returns, an arbitrarily accurate measure of daily return volatility can be constructed. A practical estimator of the integrated volatility is then computed by summing up all the intraday 14 squared returns over many small intervals within a day. This estimator is the realized volatility and de ned as: 2 j;t(m) = Q k=1;::;m r2 j;t+k; (1.13) where j;t(m) is the realized volatility for stock j on day t by dividing the day into m intervals and rj;t+k is the natural logarithmic return of the kth interval. This approach is in line with the previous work by French, Schwert, and Stambaugh (1987) and Schwert (1989) who calculate the monthly realized stock volatilities using daily returns. The realized volatility is obtained for each component stock on every trading day. I also construct the realized volatility I;t(m) for the index itself using the same method. ABDE (2001) also mentions that, practically, it is impossible to construct an estimator free of measurement error due to the bid-ask bounce and the uneven spacing of the observed prices. Since I use the S&P 500 ETF as the index, its component stocks are highly liquid so nonsynchronous trading is not a problem. In order to reduce the noise generated by the bid-ask bounce, I use midquotes to calculate interval returns. With the above proxies, I can examine the relation between proportion of index traders and posterior variance. I regress the realized volatility on Index Trading Portion for the index and each component stock, controlling for daily dollar trading volume. The control variable is in line with Jones, Kaul, and Lipson (1994) and Chan and Fong (2006) who nd that trading volume explains almost all of the daily volatility. Speci cally, I evaluate e ects of the index trading by analysing the following equation: j;t(m) = 0 + 1Index Trading Portionj;t + 2log(Trading Dvolj;t) + j;t; (1.14) where Trading Dvolj;t is the daily dollar trading volume for stock j on day t, j;t(m) is de ned in equation (1.13), Index Trading Portionj;t is de ned in (1.11), and j;t is the error term. These variables are discussed in more detail in section 1.3.3. The above equation is estimated for each individual stock using daily observations. I also evaluate the corresponding equation for the index itself: I;t(m) = 0 + 1Index Trading PortionI;t + 2log(Index DvolI;t) + I;t: (1.15) where, in this case, volatility is the index's realized volatility I;t(m) and daily dollar volume is the index's trading volume Index DvolI;t. 15 1.3.3 Sample and Variable Construction The primary data sources used in this study are TAQ and CRSP. Following Hasbrouck (2003), I consider the sample period from October 1, 2013 to December 31, 2013, the most recent three-month period during which TAQ data are available on WRDS when I started the paper's empirical study. The index I use in this study is the S&P 500 value-weighted index ETF. Although my model applies to any index fund, there are several advantages of using this ETF. First, the S&P 500 index is one of the most commonly followed equity indices. At the end of 2013, mutual funds indexed to the S&P 500 alone account for 33% of all assets invested in the index funds (excluding ETFs).3 Second, it contains a very diverse set of companies and is a good representation of the U.S. equity market. Third, the S&P 500 index ETF is heavily traded. On April 9, 2013, the average daily volume of the SPDR (an S&P 500 value-weighted ETF) itself was 117 million shares, the highest volume among all ETFs.4 This highly liquid market provides me an ideal context to study the information transmission between the index and its underlying assets. In this paper, I only include three value-weighted ETFs in the sample: SPDR S&P 500 (SPY), iShares core S&P 500 (IVV), and Vanguard S&P 500 (VOO).5 During the sample period, there are several changes to the S&P 500 index. I list the corresponding companies and reasons for additions and deletions in Table 1.2.6 So after adding those companies already deleted from the index, in total I have 507 stocks for the sample period. I construct all prices used in the empirical study from TAQ database. Speci cally, on each day, I de ne the opening price as the midquote for the quote in TAQ with MODE=10 from the stock's primary listing market or the last regular midquote before 9:31 am if such a quote does not exist.7 I de ne the closing price as the midquote for the quote in TAQ with 3Investment Company FactBook 2014 4See Wiki: http://en.wikipedia.org/wiki/SPDR S%26P 500 Trust ETF 5I exclude the equal-weighted ETF, e.g., Guggenheim S&P 500, since the S&P 500 index is constructed based on market capitalization. 6See Wiki: http://en.wikipedia.org/wiki/List of S%26P 500 companies 7Following Boehmer, Saar, and Yu (2005), I apply certain criteria to screen the TAQ data. I keep quotes with TAQ's mode equal to 0, 1, 2, 3, 6, 10, 12, 23, 24, 25, or 26. I eliminate quotes with nonpositive ask or bid prices, or where the bid price is higher than the ask. I keep trades for which TAQ's CORR eld is equal 16 MODE=3 from the stock's primary listing market or the midquote prevailing at 4:00 pm if such a quote does not exist. In addition to the opening and closing prices, I also de ne the 1-minute (5-minute) midquote as the last regular midquote in every minute (5 minutes) from 9:30 am to 4:00 pm in each day. The 1-minute midquotes are used in estimating the Hasbrouck information share. I compute two realized volatilities using 1-minute and 5-minute midquotes, respectively. Following the literature, I calculate the natural logarithmic return for each 1-minute or 5-minute interval and sum up all these squared interval returns to get the daily volatility measures j;t(m) for each stock j and day t. When constructing volatilities I;t(m) for the index, I only use midquotes of SPY because this ETF is the most heavily traded among the three and often considered as a proxy for the market.8 To calculate my main variable Index Trading Portion, I need to estimate the index trading Indextrading Dvol and the total trading Trading Dvol for stock j on day t. For each day, I sum all dollar volume from three ETFs in TAQ as the Index DvolI;t and also construct Trading Dvolj;t for each stock and day from TAQ. Since the S&P 500 index follows the value-weighted rule, I estimate volume generated by the index trading as: Indextrading Dvolj;t = Pricej;t × Share Outstandingj;t PPricej;t × Share Outstandingj;t ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ value−weight of stock j on day t × Index DvolI;t ´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¶ daily volume of ETFs ; where Pricej;t is the average of the opening and closing prices and Share Outstandingj;t is the shares outstanding for the stock which is from CRSP. Then for each stock and each day, I calculate an Index Trading Portion as in equation (1.11) and winsorize this number at the top and bottom 0.5%.9 To calculate this proportion for the index, I divide Index DvolI;t by the total dollar volume of its component stocks on each trading day as in equation (1.12). I also construct the market capitalization MktCap, which equals the product of price and shares outstanding, for each stock and day. After constructing the above variables, I have in total 32195 stock-day observations and 64 observations for the index. During the sample to either zero or one, and for which the COND eld is either blank or equal to @, B, J, K, S, E, or F. I also exclude trades with nonpositive prices. 8SPY is heavily traded on multiple exchanges so the midquotes of SPY are constructed by using the National Best Bid and O er (NBBO). 9This construction certainly underestimates the index trading since I exclude the equal-weighted and levered ETFs and do not consider the volume from traditional index funds. 17 period, DELL's bid and ask quotes do not change too much so I drop this stock when conducting related tests. Thus I have in total 506 stocks and 32181 stock-day observations left. The summary statistics of relevant variables for all component stocks and the index are provided in Table 1.3 and Table 1.5, respectively. 1.4 Empirical Results 1.4.1 Hasbrouck Information Share Before analysing the empirical results of my model prediction, I rst discuss results of the Hasbrouck information share. In estimating the VECM, I use the 1-minute midquotes, thirty lags and include thirty moving average coe cients in the sum matrix A. For each component stock j in the S&P 500 index, I compute an estimate of the information share attributed to the price changes in the index for the whole three months by only including prices of the index ETFs and stock j in (1.7). I use the midquotes of SPY in the price vector (1.7) because this ETF is the most heavily traded among the three and often considered as a proxy for the market. This information share is calculated by assigning the index ETF precedence (i.e., placing the index rst in the price vector P in (1.7)) so the estimate is approximately the maximal information contribution from the index. Table 1.6 provides the summary statistics on price discovery from SPY to its 500 components. It shows that the mean and median information contribution from SPY to the underlying assets is 25.69% and 24.61%, respectively. This around one quarter of the price discovery delivers reasonable evidence that the index provides material information to its component stocks, at least in short time horizons. In addition, I rotate the price vector and put the stock midquote rst in P to estimate the maximal information share from stocks to the index. Table 1.7 presents the results and shows that the mean and median are 21.57% and 20.31%, respectively. This slightly lower information contribution from stocks to the index is somewhat surprising because traditionally investors would think there is more information ow from stocks to the index but not the other way round. Yu (2005) and Jovanovic and Menkveld (2012) also nd that the index related securities contribute to price discovery of individual assets. They mention that this is because price changes in the market index provide valuable hard information. In addition, Hasbrouck (2003) studies information ow between the S&P 500 ETF and its sector ETFs. He nds that the index has larger information contribution to its sector ETFs and his results are consistent with relatively low production of information at the sector level. With my ndings 18 in Table 1.6 and Table 1.7 and also the results from prior literature, it is reasonable to state that the index trading contains material information about its underlying assets. 1.4.2 Realized Volatility Does index trading impair the market quality of its underlying assets? In this subsection, I address this question by examining the relation between proportion of index trading and realized volatility. Speci cally I evaluate the main regression (1.14) for each component stock which controls daily dollar trading volume and present the regression results in Table 1.8. The coe cient in Table 1.8 is the average of coe cients from all the individual regressions and t-statistic and Adjusted R2 are computed in the same way. t-statistics are reported in parentheses. Using realized volatility as the dependent variable, I nd a positive relation between proportion of index trading and volatility for the component stocks. The results in Table 1.8 are statistically signi cant for both 1-minute and 5-minute volatility measures. This nding is consistent with the prediction in Proposition 1 and the pattern shown in Figure 1.1 for the underlying assets. Moreover, I also nd that dollar trading volume is signi cant in explaining the stock's daily volatility and this is consistent with prior literature. In my model, the two component assets are identical by assumption so the model produces no size e ect. In reality, however, rms are di erent in their capitalization. Therefore it is also interesting to see if size plays a role in the index trading. Thus I divide the 506 (excluding DELL) stocks into three size groups: large, medium, and small, based on their market capitalization at the beginning of the sample period. I provide the summary statistics of relevant variables within each size group in Table 1.4. The regression results are provided in Table 1.9. The empirical results within each size group are qualitatively similar to those in Table 1.8, which are calculated by using the total 506 stocks. The e ect of index trading is stronger in the large size group and it is con rmed by the larger proportion of index traders in this category. I also evaluate equation (1.15) for the S&P 500 ETFs and the results are presented in Table 1.10. Consistent with my model prediction, the proportion of index traders has no e ects on volatilities of the index. 1.5 Conclusions Index funds and index ETFs are both very popular investment vehicles among investors. These days, index ETFs have grown particularly quickly and they attract nearly twice the 19 ows of traditional index funds since 2007 in the U.S. market.10 By de nition, the index fund is redundant because it is easily replicated by the underlying securities. However, with the development of trading technology, index funds, especially index ETFs, may attract investors with di erent opinions on assets' payo s. Therefore the index may contain valuable pieces of information from these heterogeneous traders. As a consequence, the prices of its component securities will adjust to a new equilibrium to re ect these valuable opinions conveyed by the index trading. The purpose of this paper is to study the in uence of index trading on its component stocks. I rst develop a rational expectation model to capture the idea that the index contains valuable opinions on the assets' terminal values. In equilibrium, the index's price is equal to the sum of prices of its component stocks so there is no arbitrage opportunity. The model predicts that the e ciency of the underlying assets decreases with proportion of indexers but the e ciency of the index itself stays unchanged, where e ciency is de ned as the posterior variance of return. Unlike the models with private information, such as Subrahmanyam (1991) and Gorton and Pennacchi (1993), the ine ciency all comes from the index traders because they only trade the index that follows a predetermined rule, for example, the value-weighted rule. Then I test my model using real world data. Before examining the model implications, I con rm that the index does contribute to price discovery in its component stocks, at least in short horizons, using the S&P 500 ETF as the index. I use the realized volatility as a proxy for the posterior variance, the theoretical measure of price e ciency. I show that index trading is positively and signi cantly related to the volatility measures. On the other hand, such results do not exist in the index ETF. This di ers from prior works which focus on lead-lag e ects between the index future and the stock index. My empirical ndings also shed new light on the studies of price e ects when there are index additions or deletions. Overall, the empirical ndings are consistent with my model predictions that index trading makes its underlying assets less e cient but does not a ect the index itself. 10Investment Company FactBook 2014 20 Figure 1.1: Relation between Posterior Variance of Return and Proportion of Index Traders 21 Table 1.1: Variables Related to Calculating the Posterior Probability Variable Description pxj prior probability of risky asset j; x = {vh; vl} and j = 1; 2. pXI prior probability of index fund I; V = {VH; VM; VL}. qxj posterior probability of risky asset j; x = {vh; vl} and j = 1; 2. qXI posterior probability of index fund I; V = {VH; VM; VL}. yxj risky asset j's conditional probability of y given x; x = {vh; vl}, y = {vh; vl}, and j = 1; 2. Y XI index fund I's conditional probability of Y given X; X = {VH; VM; VL} and Y = {VH; VM; VL}. NHs number of occurrence of a view on index equalling VH in the set Os NMs number of occurrence of a view on index equalling VM in the set Os NLs number of occurrence of a view on index equalling VL in the set Os NHi number of occurrence of a view on index equalling VH in the set Oi NMi number of occurrence of a view on index equalling VM in the set Oi NLi number of occurrence of a view on index equalling VL in the set Oi Nhj number of occurrence of a view on asset j equalling vh in the set Os; j = 1; 2. Nlj number of occurrence of a view on asset j equalling vl in the set Os; j = 1; 2. 22 Table 1.2: S&P 500 Additions and Deletions Date Addition Deletion Reason 10/21/2013 Transocean DELL Inc. Dell acquired by private equity consortium 11/13/2013 Michael Kors NYSE Euronext ICE Exchange acquired NYSE Euronext 12/2/2013 Allegion J.C. Penney Allegion spun o by Ingersoll Rand 12/10/2013 General Growth Properties Inc. Molex Inc. MOLX acquired by Koch Industries 12/21/2013 Alliance Data Systems Abercrombie & Fitch Market capitalization changes 12/21/2013 Mohawk Industries JDS Uniphase Market capitalization changes 12/21/2013 Facebook Teradyne Market capitalization changes Data Source: http://en.wikipedia.org/wiki/List of S%26P 500 companies Table 1.3: Summary Statistics: S&P 500 Component Stocks in the Whole Sample Variable Mean Std 10% 25% 50% 75% 90% Realized Vol 1min 0.0115 0.0077 0.0070 0.0083 0.0101 0.0127 0.0165 Realized Vol 5min 0.0112 0.0067 0.0066 0.0079 0.0098 0.0126 0.0165 Index Trading Portion 0.2426 0.1586 0.0797 0.1299 0.2074 0.3157 0.4483 Trading Dvol ($millions) 157.16 249.45 27.63 48.06 90.23 172.18 336.16 MktCap ($millions) 32.42 50.35 5.84 8.96 15.78 32.25 71.68 23 Table 1.4: Summary Statistics: S&P 500 Component Stocks within the Size Groups Variable Mean Std 10% 25% 50% 75% 90% Large Group Realized Vol 1min 0.0104 0.0061 0.0065 0.0077 0.0093 0.0115 0.0145 Realized Vol 5min 0.0100 0.0054 0.0060 0.0073 0.0090 0.0113 0.0145 Index Trading Portion 0.3013 0.1580 0.1273 0.1870 0.2736 0.3851 0.5089 Trading Dvol ($millions) 300.54 369.75 81.82 122.95 198.08 347.31 593.88 MktCap ($millions) 73.37 70.50 25.99 32.05 47.85 82.54 157.33 Medium Group Realized Vol 1min 0.0115 0.0082 0.0073 0.0085 0.0102 0.0126 0.0159 Realized Vol 5min 0.0112 0.0067 0.0068 0.0081 0.0098 0.0124 0.016 Index Trading Portion 0.2304 0.1512 0.0831 0.1274 0.1965 0.2892 0.4131 Trading Dvol ($millions) 107.40 114.38 36.63 54.14 82.55 124.72 189.29 MktCap ($millions) 16.19 3.63 11.85 13.36 15.71 18.59 21.39 Small Group Realized Vol 1min 0.0127 0.0085 0.0075 0.0089 0.0110 0.0140 0.0188 Realized Vol 5min 0.0123 0.0075 0.0070 0.0085 0.0107 0.0140 0.0187 Index Trading Portion 0.1951 0.1475 0.0565 0.0966 0.1588 0.2479 0.3726 Trading Dvol ($millions) 61.63 60.95 17.93 27.30 44.85 74.92 120.97 MktCap ($millions) 7.19 2.22 4.17 5.36 7.30 8.92 10.08 24 Table 1.5: Summary Statistics: S&P 500 Index ETFs Variable Mean Std 10% 25% 50% 75% 90% Realized Vol 1min 0.0047 0.0015 0.003 0.0036 0.0044 0.0056 0.0065 Realized Vol 5min 0.0045 0.0017 0.0026 0.0032 0.0043 0.0055 0.0065 Index Trading Portion 0.2167 0.0519 0.1609 0.1741 0.2126 0.2539 0.2837 Index Dvol ($millions) 17,136.85 5,639.65 10,579.17 13,366.82 15,450.45 20,624.04 24,868.99 25 Table 1.6: Price Discovery from S&P 500 Index to Component Stocks Price Type Mean Median 1-Minute Midquote 25.69% 24.61% Table 1.7: Price Discovery from Component Stocks to S&P 500 Index Price Type Mean Median 1-Minute Midquote 21.57% 20.31% Table 1.8: Regression Results for Realized Volatility of Stocks in the Whole Sample Variable Realized Vol 1min Realized Vol 5min Index Trading Portion 0.0088** 0.0063** (2.34) (2.03) log(Trading Dvol) 0.0036*** 0.0044*** (5.2) (4.86) Adjusted R2 0.35 0.32 The coe cients, t-statistics, and R2s are averages of the corresponding numbers of total sample stocks. ***, **, and * represent signi cance at the 1%, 5%, and 10% levels, respectively. 26 Table 1.9: Regression Results for Realized Volatility of Stocks within the Size Groups Variable Realized Vol 1min Realized Vol 5min Large Group Index Trading Portion 0.0072*** 0.0075** (2.77) (2.34) log(Trading Dvol) 0.0039*** 0.0046*** (6.21) (5.77) Adjusted R2 0.41 0.37 Medium Group Index Trading Portion 0.0102** 0.0102* (2.19) (1.88) log(Trading Dvol) 0.0031*** 0.0038*** (4.90) (4.55) Adjusted R2 0.34 0.31 Small Group Index Trading Portion 0.0089** 0.0012* (2.05) (1.85) log(Trading Dvol) 0.0038*** 0.0047*** (4.48) (4.25) Adjusted R2 0.31 0.29 The coe cients, t-statistics, and R2s are averages of the corresponding numbers of stocks within each size group. ***, **, and * represent signi cance at the 1%, 5%, and 10% levels, respectively. Table 1.10: Regression Results for Realized Volatility of the Index Variable Realized Vol 1min Realized Vol 5min Index Trading Portion 0.0007 0.0016 (0.24) (0.52) log(Index Dvol) 0.0039*** 0.0043*** (7.75) (8.19) Adjusted R2 0.66 0.7 ***, **, and * represent signi cance at the 1%, 5%, and 10% levels, respectively. CHAPTER 2 NEWS, INFLUENCE, AND EVOLUTION OF PRICES IN FINANCIAL MARKETS 2.1 Introduction As social beings, we are in uenced by many elements in our environment. This in uence can cause us to behave in a manner that is suboptimal. The literature on information cascades (e.g., Banerjee (1992), Bikhchandani, Hirshleifer, and Welch (1992), and Welch (1992)) demonstrates how herding behavior can occur when agents follow others even when their private information suggests they shouldn't. A major source of in uence in the life of investors is the media, or more speci cally, opinions by experts and media pundits that are widely disseminated in various media forms that include both traditional outlets such as newspapers and television as well as newer forms such as Internet blogs and on-line forums. Such in uence has the potential to help prices better adjust to information, but also could introduce shared errors into prices and detract from their informational e ciency. Treynor (1987) claims that shared errors in published research are particularly important to asset prices. While the published opinion could be more accurate than the average accuracy of individuals' opinions, it replaces many independent estimates with a single number. Errors in the published opinion, therefore, will be re ected in the estimates of all investors it in uences. Treynor suggests that the impact of such published opinions on asset prices would depend on their accuracy as well as the extent of their in uence, or how persuasive they are. Our objective in this paper is to study the in uence of published opinions on the evolution of prices. Our model starts with an asset-speci c news event. After the event, rational investors form personal \views" about the true value of the asset. Investors do not have full con dence in their perceptions of the value of the asset, and therefore they use Bayes' rule to combine their views with the market view. This is similar in spirit to Black and Litterman's (1992) de nition of views as \feelings that some assets or currencies are overvalued or undervalued at current market prices." If errors in views are 28 not systematically biased, then, at least formally, there is no di erence between information and views. The interpretation of views, however, may be somewhat di erent. In a model with private information, the number of informed investors is often assumed to be small (sometimes just one), and information is considered accurate. Views, on the other hand, can be held by anyone, although the accuracy of personal views is considered to be marginal. Views can also give rise to the \wisdom of the crowds" e ect whereby the aggregate behavior of many people with limited (but independent) information can bring about an accurate outcome (see, e.g., Surowiecki (2004)). We assume that the analysis provided by experts and media pundits|the published view|is valuable in that it is more accurate than the average personal view. Still, the published view can also be incorrect, which is how shared errors are introduced into the price. In our sequential trade model, some investors are exposed to and in uenced by the published view. This in uence creates patterns in the trading process and the evolution of prices. The in uence of the published view increases the price impact of trades (i.e., the trading costs) during an initial price-adjustment period. The reason for this result is that investors who seek (and are in uenced by) the published view incorporate it into their trading, e ectively imposing adverse selection on other investors. These increased costs, which last for a (random) number of trading rounds, deter regular investors who are not in uenced by the published opinion, and they refrain from trading. In other words, their valuation is inside the spread when they arrive in the market, and hence they optimally choose not to trade. As prices adjust to the published view, uncertainty decreases and the spread becomes smaller. From a certain point on, regular investors start trading on their views, and trading volume increases because both in uenced and nonin uenced investors trade in the subsequent price-adjustment period, while only in uenced investors trade in the ini- tial price-adjustment period. This leads to changes in the evolution of prices during the subsequent price-adjustment period. In particular, the wisdom of the crowds e ect emerges as the independent personal views of all investors get impounded into prices via the trading process. It is this wisdom of the crowds e ect that ensures that prices converge to the true value, and also that shared error introduced by the published view is ultimately expunged from prices. This wisdom of the crowds e ect, which ultimately corrects prices, sets the model apart from the information cascade papers. In particular, once investors in information cascades 29 models start relying on common information rather than their own signals, they continue to do it inde nitely unless some exogenous in uence breaks that dependency. This exogenous in uence can be the release of public information or the introduction of a class of better- informed investors. In our model, the endogenous reduction in trading costs increases the pool of regular investors who trade the asset after some time passes, and the con uence of their marginally informative views delivers the wisdom of the crowds e ect and corrects prices. Of course, along the way, the in uence of published opinions changes the evolution of prices and creates volume patterns in the market. The more in uence is exerted by the published view, the more impact it has on the trading process and the resulting prices. Treynor (1987) states that the number of investors in uenced by the published opinion increases with the time elapsed since publication. We use this insight to empirically test the implications of our model. Our empirical tests use corporate earnings announcements as the information events. These are often associated with heightened media attention in traditional media outlets as well as in blogs, on-line discussion groups, and investor newsletters. Various experts and media pundits use this opportunity to voice their opinions on rms, which re ect their interpretations of the information in the earnings announce- ments. The passing of time following an earnings announcement provides opinion writers with more opportunities to distribute their views, and increases the likelihood that investors get exposed to and are in uenced by these views. Almost all earnings announcements take place either in the morning, before the market opens, or in the afternoon, after the main trading session on organized exchanges closes. Morning announcements are followed very quickly by the opening of the market, decreasing the likelihood that published opinions are produced and many investors are exposed to them before they submit their orders to trade at the opening of the regular trading session. For afternoon announcements, there is much more time for published opinions to be generated and in uence investors before the investors submit orders for the regular trading session on the following day.1 We therefore compare how prices evolve for morning versus afternoon earnings announcements to test the implications of our model on how the in uence of published opinions a ects the trading process. 1While investors could potentially trade after-hours (i.e., outside the 9:30 a.m.{4:00 p.m. regular trading session), the illiquid nature of trading outside the regular trading session deters many investors. Most investors, therefore, have a strong preference for trading during the regular trading session, providing published views on afternoon announcements with more opportunity to reach and in uence investors. 30 It is reasonable to assume that any published opinion following the earnings announce- ment would re ect or rely on the earnings surprise (i.e., how far the earnings number is from the consensus analysts' forecast prior to the announcement). Therefore, we use the earnings surprise as a proxy for the nature of the published view. To test the implications of our model, we match pairs of morning (less in uence) and afternoon (more in uence) announcements on the strength of the published view as well as various attributes of the stocks. The rst implication we test is that the initial price adjustment (or price impact) is larger when there is stronger in uence of the published view on investors. We look at both the absolute value of close(t − 1)-to-open(t) returns and at Amihud's price impact measure, and nd evidence consistent with our model. The most important implications of the model, however, involve the dynamic adjustment of prices. In the presence of in uence, positive published views would generate a large positive return during the initial price- adjustment period. As the wisdom of the crowds e ect emerges, however, the return over the subsequent price-adjustment period could exhibit one of two patterns: (i) a reversal, if the published view was incorrect and prices adjust downward, or (ii) a small positive return as the independent views of additional investors complete the adjustment of prices upwards. In either case, the di erence in return between the subsequent and initial price-adjustment periods should be negative. A similar logic suggests that for negative published views, this subsequent-minus-initial return di erence in return should be positive. We nd that indeed these return patterns are stronger in afternoon announcements than in morning announcements, in line with the predictions of the model. In addition to price patterns implications, the in uence model also yields a volume implication. In particular, some investors (who are not in uenced by the published view) optimally choose to refrain from trading during the initial price-adjustment period due to the higher trading costs. Therefore, there is lower volume initially as prices adjust to the published opinion, and volume increases when the wisdom of the crowds e ect emerges and all investors join the trading process during the subsequent price-adjustment period. We test this prediction, comparing afternoon to morning earnings announcements, and nd results consistent with the theory. Hence, our results suggest that in addition to the evolution of prices, published views also a ect the trading process itself. Besides the literature on in uence and information cascades that we mentioned at the beginning, our study is also related to the literature on the timing of the release of earnings 31 information (for example, during the trading day versus after the market closes, or during the week versus on Friday). Patell and Wolfson (1982) nd that bad news is more often released after the market closes and suggest a couple of possible explanations. First, managers could opportunistically release bad news after the market closes at a time of reduced media coverage and investor attention. Second, managers could release news after the market closes to allow a longer period for dissemination and evaluation of the news.2 Doyle and Magilke (2009) nd evidence consistent with the latter explanation, and they also provide evidence that more complex rms tend to release earnings after the market closes. Jiang, Likitapiwat, and McInish (2012) examine price discovery following earnings an- nouncements in after-hours trading and during the regular trading session. They observe that price adjustment to announcements made after the close is greater than to announce- ments made in the morning before the market opens, which is consistent with our rst nding. They do not observe a di erence in the e ciency of price discovery between announcements made in the morning and in the afternoon. Michaely, Rubin, and Vedrashko (2014) nd that the initial price impact (an hour after an announcement) is smaller for announcements made during the trading day compared with outside regular trading hours. They suggest that not all investors follow the market continuously, and hence announcing earnings outside the regular trading hours results in better price discovery because investors are given more time to evaluate the news.3 This is consistent with our nding that the initial price impact of afternoon announcements is greater than that of morning announcements. Our study suggests that this e ect can arise due to the in uence of published opinions on investors. The remainder of the paper is structured as follows. Section 2.2 presents the theoretical model. We describe the economy, characterize the equilibrium, and provide propositions on price impact and the dynamic evolution of prices and volume that describe how in uence a ects the trading process. These propositions also provide the basis for the empirical tests 2Gennotte and Trueman (1996) model the decision of a manager over the timing of his rm's earnings release. In their model, the price response is greater if the announcement is made during trading hours rather than after the market closes. The driving force behind this result is that there is greater likelihood of trades coming from informed traders during trading hours, while postponing trading to the following day increases the likelihood of additional noise trading. 3Michaely, Rubin, and Vedrashko (2014) also investigate the relation between corporate governance and the timing of earnings releases. They nd that rms with poor corporate governance are more likely to release the news during the trading day. 32 of the model. Section 2.3 puts forward the empirical methodology, describing the sample and the matching procedure, while Section 2.4 presents the empirical results of the tests. Section 2.5 concludes the paper. 2.2 The Model 2.2.1 The Economy We study a variant of the Glosten and Milgrom (1985) sequential trade model for a single asset with the interest rate set to zero. Before trade commences, the consensus is that the true value of the asset, denoted by ~v, is equally likely to be zero or one. Competitive and risk-neutral market makers set the ask and bid prices. Traders show up sequentially, and each trader has a personal view about whether the true value is zero or one. We call a view that the asset is worth one (zero) a positive (negative) view. The view of the n-th trader (i.e., the trader who arrives in period n) can be written as: ~vn = ~v(1 − ~ n) + (1 − ~v)~ n; where the error term, ~ n, takes the value zero if the view is correct and one otherwise. The probability that a personal view is correct, denoted by , is identical for all traders and satis es > 1~2. There are two classes of traders: knowledgeable investors and noise traders. The probability that a knowledgeable investor shows up at any given period is , and the probability that a noise trader arrives in the market is 1 − . Noise traders, irrespective of their views and for reasons that are exogenous to the model (e.g., risk sharing, liquidity needs), either buy or sell with equal probabilities. Knowledgeable investors, on the other hand, consider their personal views when making a decision whether to buy, sell, or abstain from trading. A fraction C 0 of the knowledgeable investors seek out the opinions of experts and media pundits. We call the consensus opinion propagated by these experts the \published" opinion. The fraction , therefore, represents the extent of in uence of this published opinion. The published opinion is also associated with a view about the asset's true value that we denote by ~ve and let ~ e denote the error associated with the view, i.e., ~ve = ~v(1 − ~ e) + (1 − ~v)~ e: The probability that the published expert view is correct, denoted by e, is strictly greater than , the probability that an investor's personal view is correct. However, the published 33 view can also be incorrect (with probability 1 − e), and this is important to the intuition discussed by Treynor (1987) regarding how such experts and media pundits can introduce errors into the price. We call knowledgeable investors \in uenced" if they seek and pay attention to the published opinion and \regular" if they do not. We assume that errors in views, the true value of the asset, the probabilistic selection model that governs the choice of traders, and the trading decision of noise traders are all independent. As in traditional sequential trade models (e.g., Glosten and Milgrom (1985) and Easley and O'hara (1992)), each trading period is long enough to accommodate at most one trade. When a trader arrives in period n, he has the option of buying one unit of the stock at the quoted ask price, selling one unit at the quoted bid price, or abstaining from making a trade. We let Hn denote the history of trading up to the n-th period. Given the ask and bid prices, the expected pro t of a regular investor is: ¢¨¨¨¨¦¨¨¨¨¤ E[~vSHn; ~vn] − askn if buys 0 if abstains bidn − E[~vSHn; ~vn] if sells (2.1) and the expected pro t of an in uenced investor is: ¢¨¨¨¨¦¨¨¨¨¤ E[~vSHn; ~vn; ~ve] − askn if buys 0 if abstains bidn − E[~vSHn; ~vn; ~ve] if sells: (2.2) An equilibrium is a bidn and askn quote (a mapping from Hn to R), such that (i) given the quoted prices, investors maximize their pro ts, and (ii) market makers quote the following regret-free prices: askn = E[~vSHn; buy] bidn = E[~vSHn; sell]: Given the realizations of all random variables in the model, we are interested in con- trasting the equilibrium outcomes in two environments: > 0 and = 0. We refer to the former as the \in uence" model, wherein a portion of investors is in uenced by experts and media pundits, while the latter is the \benchmark" model. In the benchmark model, all knowledgeable investors are regular investors. 2.2.2 Properties of the Equilibrium We can classify traders into eight types based on the source and realization of their views. Table 2.1 provides a taxonomy of the types, which we denote by = 1; :::; 8. For 34 example, type = 2 is an in uenced investor whose personal view is positive but is exposed to a negative published view. Types 3 and 4 represent regular investors who pay attention to their personal views only, while types 7 and 8 are the noise traders who buy and sell for exogenous reasons. We let ~ n denote the (random) type that arrives to the market in period n. Knowing which type is picked is informationally equivalent to the information that the trader knows. For example, knowing that ~ n = 1 is equivalent to knowing that both the investor's own view as well as the published view are positive (~vn = 1 and ~ve = 1). Proposition 2 There exists a Markovian equilibrium with three state variables.4 ph n P(~v = 1SHn; ~ve = 1) pmn P(~v = 1SHn) pl n P(~v = 1SHn; ~ve = 0) (2.4) For every nite n, and regardless of how the history of trading unfolds, the state variables remain in the open interval (0; 1) and the bid-ask spread remains strictly positive. In the benchmark model (i.e., = 0), ph n and pl n are redundant. Proposition 3 As n goes to in nity, pmn converges to ~v almost surely. Proofs of all the propositions are provided in Appendix B. In the Markovian equilibrium, regular investors interpret the history of trading in the exact same manner as market makers do, and then weigh in with their personal views. In contrast, in uenced investors interpret the trading history di erently. This is because in uenced investors know that some types of traders are not present in the market. For example, if ~ve = 1 (i.e., the published view is positive), then in uenced investors know that types ve and six are not present in the market. Table 2.2 shows the valuation each trader type attaches to the asset, and how they can be expressed as functions of the three state variables. This choice of the state variables enables us to write the valuations, which are equivalent to conditional probabilities because the asset value can only take the values zero or one, in an especially simple form in the Markovian equilibrium. 4The initial conditions of the state variables are given by ph 0 = e pm 0 = 1~2 pl 0 = 1 − e: (2.3) 35 Our goal is to use the model to investigate the patterns in prices and trading created by the in uence of experts and media pundits on investors. For that purpose, the next two propositions contrast the in uence model and the benchmark model. Proposition 4 When trading opens (i.e., at n = 1), the initial price impact of orders in the in uence model is larger than the initial price impact in the benchmark model. In a sequential trade model, the price impact is equivalent to the updating of beliefs about the asset value brought about by the order ow.5 Due to the greater precision of the beliefs that incorporate the published view, the initial price impact is larger in the in uence model. This, irrespective of whether the published view is correct or not, is the reason an incorrect published view impacts prices more than an incorrect personal view. The most interesting insights of the model come from the dynamic nature of trading and price adjustment that are investigated in the next proposition. Let v be the valuation of a trader of type (from Table 2.2). Proposition 5 Assume the probability that a personal view is correct is su ciently small (i.e., is close to 1/2). In the in uence model (i.e., when > 0), there exists an ~N C 1 such that for all n B ~N 1. ph n and pl n remain constant, and pmn = E[~veSHn]. 2. The bid-ask spread is larger than v3 − v4 (i.e., the ask (bid) is higher (lower) than the valuation of a regular investor with a positive (negative) view). 3. Regular investors abstain from trading, and in uenced investors trade in the direction of the published view regardless of their personal views. For n > ~N , we can show that for su ciently large n, D) The bid-ask spread is smaller than v3 − v4 (i.e., the ask (bid) is lower (higher) than the valuation of a regular investor with a positive (negative) view). E) All investors trade when they arrive in the market. In the benchmark model ( = 0), arriving knowledgeable investors never abstain from trading and always trade in the direction of their personal views. In particular, the bid-ask spread is always smaller than v3 − v4. 5The bid-ask spread in the model is simply the sum of the price impacts for buying and selling. 36 Trading costs|which in the sequential trade model are simply the price impacts|are higher when the market opens and in the early trading periods. Proposition 5 demonstrates the di erence in uence makes for the evolution of prices and volume in the market. These trading costs during that initial price-adjustment period (when n B ~N ) deter regular in- vestors from participating and they refrain from trading (i.e., their valuation is inside the spread when they arrive in the market and hence they optimally choose to refrain from trading). As prices adjust to the published view, uncertainty decreases and the spread narrows. After some time, regular investors start to trade on their views, and this leads to changes in price and volume patterns. First, volume increases due to the fact that all investors trade (previously only the in uenced investors traded). Second, the wisdom of the crowds e ect emerges as the independent views of all investors get impounded into the price, and ensures convergence to the true value (in Proposition 3). The convergence may still be slowed by the published view's impact on the valuations of the in uenced investors, but the wisdom of the crowds e ect eventually prevails. To provide a feel for the result, Figure 2.1 shows the evolution of prices when the published view is incorrect: it is negative while the true value of the asset is one. Panel A focuses on the rst 500 periods and Panel B continues the simulation up to period 3,000. The red line in the gure shows the simulated prices in the in uence model while the dotted blue line is from a simulation of the benchmark model. The parameters we use for both simulations are = 0:55, e = 0:9, and = 0:5. We take = 0:5 for the in uence model, while = 0 by de nition in the benchmark model. Panel A clearly shows the larger initial price impact of order ow in the in uence model compared with the benchmark model. Prices adjust downward rapidly towards the valuation implied by the published view. They then move around that level for a while, until the independent personal views of investors become more dominant and the wisdom of the crowds e ect brings prices up. Given that the published opinion is still re ected in the valuations of the in uenced investors, however, convergence to the true value appears slower in the in uence model, and only in panel B do we observe that prices converge to the true value. Our maintained assumption is that the analysis provided by experts and media pundits| the published view|is valuable in that it is more accurate than the average personal view. It is this di erence in accuracy that makes prices adjust very rapidly at the beginning of trading in the in uence model. Still, the gure demonstrates that the published view can slow down the eventual convergence of prices to full information values. When the 37 published view is correct, there is also a larger price impact in the in uence model relative to the benchmark model at the beginning, and it is followed by a somewhat muted price adjustment later on. Irrespective of whether a negative published view is correct or not, the initial return in the in uence model (i.e., the return from the prior expected value to the price at the market open or at any period n B ~N ) is very negative while the return over the subsequent price-adjustment period (i.e., from any n B ~N to the convergence of prices to the true value) is either positive as in Figure 2.1 or negative but of smaller magnitude. Figure 2.2 shows the ratio of trading volume in the in uence model to trading volume in the benchmark model in the rst 100 periods of the same simulation. Speci cally, we aggregate volume in 10-period buckets (1−10; 11−20; :::; 91−100), and present the in uence- to-benchmark ratio of trading volume in each bucket. In the benchmark model, all traders trade when they arrive in the market, and hence volume in each bucket is equal to ten. Consistent with Proposition 5, volume in the in uence model is lower than in the benchmark model during the initial price-adjustment period (n B ~N ) when regular investors abstain from trading, but it picks up after about 70 periods, from which point on volume is the same in the in uence and benchmark models.6 One of the interesting insights that our theory yields, therefore, is the dynamic way in which in uence of published views impacts market prices. In the model, trading by in uenced investors can introduce errors into the price. Regular investors, who at the beginning (optimally) choose to wait on the sideline, start trading after the information in the published view is impounded into the price. The trading on independent personal signals, which we call the wisdom of the crowds e ect, ultimately drives prices to the true value. This natural correction mechanism di erentiates the implications of our model from those of models in the information cascades literature (see, for example, Banerjee (1992), Bikhchandani, Hirshleifer, and Welch (1992), and Welch (1992)), where agents' reliance on common information (rather than their own signals) continues inde nitely unless some exogenous in uence is introduced (e.g., a public news announcement or a new agent type with more precise information). In our model, in contrast, the in uence of the published opinion eventually disappears and we observe the wisdom of the crowds emerging to impact the price path endogenously without the need for an exogenous intervention. 6How long regular investors abstain from trading depends on the parameters of the model and the speci c realization of the sequence of trader types who arrive in the market. 38 2.3 Empirical Methodology Our model enables us to examine the in uence of expert published opinions on nancial markets. The more persuasive these published opinions are, the greater the impact they have on the evolution of prices. Our empirical tests use corporate earnings announcements as identi able news events. These are often associated with much discussion in media outlets as well as Internet bulletin boards and investor newsletters. Various pundits and nancial newsletter writers use this information to update their audiences on their views with respect to the rms that announce the earnings numbers. Treynor (1987) notes that the number of investors persuaded by a piece of published analysis or opinion increases with the time elapsed since publication. This insight suggests that time elapsed following the earnings announcement gives more opportunity to opinion writers to distribute their views, and to investors to be exposed to these views and become persuaded. Earnings announcements tend to take place either in the morning, before the stock market opens, or in the afternoon, after the regular trading session on the organized exchange closes. For morning announcements, it is less likely that published views are produced and many investors are in uenced by these views before the regular trading session opens. We expect that the orders of investors in this case more likely re ect their own personal views. These morning announcements are therefore closer to our theoretical benchmark model (where = 0). In all likelihood, is positive for morning announcements but rather small, as published opinions could be issued before trading begins but reach only a small number of investors. Afternoon announcements are di erent in that there is an evening, night, and morning before the main trading session on organized exchanges opens, providing more time for published opinions to be generated and in uence investors. While there is \after-hours" trading in the U.S. in which investors can potentially trade on their views before the opening of the main trading session on the following day, volume in after-hours trading is low in general and markets are illiquid.7 Many investors, therefore, have a strong preference for trading during the regular trading session, providing more time for their views to be in uenced by published opinions. We take these afternoon announcements to represent the in uence model of our theory (where > 0). In other words, we use the insight from Treynor (1987) that the number of investors in uenced by a published opinion increases with time as the basis for our maintained assumption that 7Jiang, Likitapiwat, and McInish (2012), however, note that more after-hours trading occurs when an earnings announcement takes place than on nonannouncement days. 39 Afternoon > Morning.8 We therefore compare how prices evolve for afternoon versus morning earnings announcements to test the implications of our theory on the di erence between the in uence and benchmark models. We use the sample of corporate earnings announcements from Doyle and Magilke (2009). The sample period is 2000 through 2005, and the sample identi es both the date of the earnings announcement as well as whether the announcement occurs in the morning (before the market opens at 9:30 a.m.) or in the afternoon (after the market closes at 4:00 p.m.). There are 26,443 morning announcements and 23,893 afternoon announcements in the dataset. We merge the earnings announcement dataset with three additional data sources: CRSP, I/B/E/S, and TAQ. The merging of datasets results in the loss of some observations, and we are left with 25,008 morning and 21,848 afternoon announcements. We need a proxy for the published view, or the news content, of each announcement. It is reasonable to assume that any published opinion of pundits or newsletter writers at the time the announcement is made would rely on the earnings surprise (i.e., how far the earnings number is from the consensus analysts' forecast prior to the announcement). We therefore use a standardized earnings surprise measure (ES), de ned as the di erence between actual earnings and the mean consensus analysts' forecast from I/B/E/S divided by the price of the stock a month prior to the announcement, to sort the earnings announcements into seven categories.9 All positive ES announcements are sorted into three equal-sized categories, whereby category 1 contains the strongest positive earnings surprises and category 3 the weakest positive earnings surprises. Category 4 contains all earnings announcements with zero ES. All negative ES announcements are sorted into three equal-sized categories: category 5 contains the weakest negative earnings surprises and category 7 the strongest, most negative, earnings surprises. In the model we assume that the published view is either very good (ve = 1) or very bad (ve = 0). We therefore carry out the empirical work on categories 8Engelberg and Parsons (2011) show that local newspaper coverage of an earnings announcement increases trading from local retail investors in their sample of clients of a discount broker. Of course, only afternoon announcements from the previous day can make it into the newspaper, suggesting that the in uence of published opinions would in fact be greater for afternoon announcements than for morning announcements. 9It is likely that is positive to some extent in all earnings announcements, and hence we use ES as a proxy for the published view for both afternoon and morning announcements. We chose the terminology we use in the paper|comparing the in uence versus benchmark models|to simplify the exposition. All the implications of the model that we discuss carry through when the in uence model has two levels of such that Afternoon > Morning. 40 1 and 2, representing the positive published view in the model, and categories 6 and 7, representing the negative published view in the model.10 Table 2.3 provides information on the number of morning and afternoon earnings announcements in the entire sample as well as separately in ES categories 1, 2, 6, and 7. To properly compare the predictions of the benchmark and in uence models, we need to match pairs of morning and afternoon announcements to neutralize di erences in the strength of the published views as well as in other attributes of the stocks. Since we use ES to represent the published opinion, we control for the strength of the view by matching morning to afternoon announcements only within the same ES category (1, 2, 6, or 7). The rst general attribute of stocks that we match on is industry, and we classify each earnings announcement into 10 industries using the classi cation method developed by Kenneth French.11 This second step leaves us with 40 ES/Industry cells. Table 2.4 provides the number of earnings announcement pairs in each ES/Industry cell as well as in the entire sample. Outside of the \Other" classi cation, the largest number of pairs can be found in the Business Equipment classi cation that includes computers, software and electronic equipment. Consumer Durables as well as Telephone and Television Transmission are two of the smaller industries. Davies and Kim (2009) discuss the merits of various matching procedures, and their analysis suggests matching by market capitalization and price as the attributes of stocks. Therefore, within each ES/Industry cell, we match a morning earnings announcement with an afternoon earnings announcement (with replacement) by choosing the afternoon announcement that minimizes the distance function: Pricei − Pricej Pricei + Pricej 2 + MktCapi −MktCapj MktCapi +MktCapj 2 (2.5) where i denotes the morning announcement and j denotes an afternoon announcement. Table 2.5|Table 2.8 provide percentile summary statistics for the matched pairs. Table 2.5 and Table 2.7 present market capitalization, price, and ES of the morning and afternoon earnings announcements in the matched pairs of ES categories 1 and 2 (6 and 7). We observe a rather close matching on these three attributes. To see whether moving from four ES 10The results of the tests in Section 2.4 are similar in nature when we use just the extreme categories (1 and 7). 11The mapping of four-digit SIC codes into the 10 industries can be found on Kenneth French's web site: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data Library/det 10 ind port.html 41 categories to 40 ES/Industry cells a ects the quality of matching on market capitalization and price, we provide in Table 2.6 and Table 2.8 summary statistics for an alternative matching scheme in which we proceed directly to match on market capitalization and price within each ES category without controlling for industry. The matching on market capitalization in the 90th percentile is slightly better when we do not match on industry rst, but otherwise controlling for industry does not appear to impact the market capitalization and price matching. We therefore present in the paper the empirical analysis using the matching procedure that controls for ES, industry, market capitalization, and price.12 Throughout the empirical investigation, we test for di erences between afternoon an- nouncements (representing the in uence model) and morning announcements (representing the benchmark model) using pairs' tests. We report the mean and median of the paired di erences between the afternoon and morning announcements together with p-values from a pairs' t-test and a Wilcoxon signed-rank test against the two-sided hypothesis of zero di erences.13 2.4 Empirical Results 2.4.1 Price Impact The rst implication that we test empirically (from Proposition 4) is that the initial price adjustment, or price impact, is larger in the in uence model, where investors listen to and rely on the published opinion, than in the benchmark model. Time elapsed since the earnings announcement gives more opportunity for expert analysis and opinion to be distributed and for investors to be in uenced, and so the initial price adjustment should be larger in afternoon announcements than in morning announcements. The intuition behind this prediction of the model is that whether or not the published opinion is correct, the price impact is driven by the precision of the published view, which is greater than that of the average personal view of investors. For an earnings announcement on day t, we take the closing price on day t − 1 as the 12We carried out the empirical analysis presented in Tables 2.9 through 2.15 on the alternative matching scheme that does not control for industry, and found that the results were similar to those we present in the paper. The results without the industry control are available from the authors. 13Only a negligible number of pairs (e.g., eight pairs in category 6) had both afternoon and morning announcements sharing the same date. Similarly, there were only a very small number of pairs in each category that consisted of the same two stocks. Therefore, we did not carry out clustering of the errors by date or rm. 42 price that re ects the prior beliefs before the earnings announcement. The closing price is de ned as the midquote (ask plus bid divided by two) for the quote in the TAQ database with MODE=3 (Closing Quote) on day t − 1 from the market on which the stock is listed, or if such a quote does not exist, the midquote prevailing at 4:00 p.m.14 For the price that re ects the initial price impact after the information event and the dissemination of the published view we take the opening price on day t. The opening price is de ned as the midquote with MODE=10 (Opening Quote) from the market on which the stock is listed, or if such a quote does not exist, the rst quote after 9:30 a.m.. We test the magnitude of the initial price adjustment in two ways. Our rst test looks at the di erences between afternoon and morning announcements in the magnitude (or absolute value) of the close-to-open return, AbsRet. According to the model, greater in uence by the published view implies a larger initial price adjustment and hence the di erence in AbsRet between the in uence and benchmark models should be positive for all ES categories. For our second test, we divide AbsRet by dollar volume to create Amihud's measure of price impact.15 Due to after-hours trading and because we do not know the exact time of the afternoon earnings announcement, we need to make a choice as to the time from which we begin aggregating volume. We choose 6:00 p.m. based on evidence that most afternoon earnings announcements occur between the close of trading and 6:00 p.m. (see Jiang, Likitapiwat, and McInish (2012)), and hence the volume that accumulates from 6:00 p.m. on is likely to re ect order ow that arrives after the announcement. We include the volume at the open (the opening auction for NYSE stocks or the opening trade for NASDAQ stocks) for all earnings announcements. Since Amihud's Measure requires dividing by volume, we exclude announcements with very little volume in order to minimize the number of outliers.16 The rst line of Table 2.9 presents the results for AbsRet. We observe that the di erence 14Following Boehmer, Saar, and Yu (2005), we apply certain lters to the raw quote and trade data to minimize data errors. Speci cally, we keep only quotes with TAQs MODE eld equal to 0, 1, 2, 3, 6, 10, 12, 23, 24, 25, and 26. We eliminate quotes with nonpositive ask or bid prices, or where the bid price is higher than the ask price. We keep trades for which TAQs CORR eld is equal to either zero or one, and for which the COND eld is either blank or equal to B, J, K, S, or E. We also exclude trades with nonpositive prices. 15Following Amihud (2002), we multiply the measure by 106. 16Speci cally, we exclude an earnings announcement if the volume is less than $10,000. We used other methods for robustness (e.g., winsorizing the measure at 2%) and the results were not sensitive to the choice of method by which we exclude outliers. 43 between afternoon and morning announcements is positive and statistically di erent from zero both for positive announcements (categories 1 and 2) and negative announcements (categories 6 and 7). The second line of Table 2.9 presents the results for Amihud's Measure. Here as well, the results are positive and highly signi cant using both the t-test and the Wilcoxon signed-rank test, although the fact that the mean is much larger than the median suggests that the division by volume creates some outliers. Using both measures, therefore, we nd that the initial price adjustment is larger for afternoon announcements, which is consistent with the idea that more time allows the published opinion to reach and in uence more investors and hence increases the magnitude of the initial price impact. 2.4.2 Return Patterns After the initial adjustment of prices to the published view, regular investors who were on the sideline start trading, and trading on independent personal views brings about eventual convergence to the true value. If a negative published view is incorrect, as in Figure 2.1, we see fast initial price adjustment downward in the in uence model, and therefore the initial return is very negative. Once prices incorporate the published view and regular investors start trading, the wisdom of the crowds e ect emerges and we observe a reversal as prices adjust upward until they reach one (in Panel B), and hence the subsequent return is positive. RetChg, de ned as the return over the subsequent price-adjustment period minus the return over the initial price-adjustment period, is therefore very positive, indicating the reversal. In contrast, the benchmark model shows a slower initial adjustment and a continuation in the same general direction (upward) rather than a reversal. RetChg for the benchmark model would therefore be very small.17 Hence, the di erence between afternoon announcements (the in uence model) and morning announcements (the benchmark model) when the published view is incorrect should be positive in ES categories 6 and 7 (which represent negative published views). If the published opinion is correct, there is faster initial adjustment downward in the in uence model than in the benchmark model. The subsequent price adjustment to the true value in the in uence model is both small (because much of the price change occurred in the initial price-adjustment period) and slow (due to the tight posterior beliefs). Therefore, 17The exact magnitude of RetChg for the benchmark model depends on the length of time assumed for the initial price-adjustment period in the in uence model. In general, it can be slightly positive, zero, or slightly negative. 44 RetChg(Afternoon) would be positive due to the strictly concave curvature of the price adjustment, while RetChg(Morning) would be close to zero. Hence, the di erence between afternoon and morning announcements when the published view is correct should also be positive. Let Di RetChg(6&7) be de ned as: Di RetChg(6&7) = RetChg(Afternoon) − RetChg(Morning) > 0: (2.6) Since Di RetChg(6&7) would be positive in the in uence model irrespective of whether the published view is correct or not, this prediction lends itself nicely to testing using our data. For positive published views (ES categories 1 and 2), one can follow the same logic to arrive at the following relationship: Di RetChg(1&2) = RetChg(Afternoon) − RetChg(Morning) < 0: (2.7) To test these predictions, we need to de ne the return intervals for the initial and subsequent price-adjustment periods. We use several de nitions to ensure that our ndings are robust. Our main de nition for the initial price adjustment is the return from previous day close to the opening of the market subsequent to the announcement (the close-to-open return, as in Section 2.4.1). A somewhat more arbitrary decision concerns the point at which prices have already adjusted to the information and converged to the true value. Therefore, we use several return intervals to ensure that our results are not sensitive to the particular choice of an endpoint for the subsequent price adjustment. Speci cally, we compute four di erent returns: open-to-10:30 a.m., open-to-11:00 a.m., open-to-11:30 a.m., and open-to-12:00 p.m. using midquotes from the primary market at 10:30 a.m., 11:00 a.m., 11:30 a.m., and 12:00 p.m. to represent alternative points that are far enough into the subsequent price-adjustment period. Table 2.10 shows the mean and median of Di RetChg separately for positive and negative earnings surprises. For categories 6 and 7, Di RetChg is positive and statistically signi cant as predicted in Equation (2.6), while Di RetChg is negative and signi cant as in Equation (2.7) for categories 1 and 2. These results are therefore consistent with the predic- tions of the model pertaining to di erences in the evolution of prices between the in uence model (afternoon announcements) and the benchmark model (morning announcements). The magnitude of the e ect appears twice as large for negative earnings announcements, perhaps suggesting a more dramatic initial reaction to negative published views. The results in Table 2.10 re ect di erences in raw returns between afternoon and morn- ing announcements. However, the matched announcements in each pair do not generally 45 happen on the same day, and therefore it could be that they re ect di erences in market return alongside the e ect predicted by our theory. To separate them, we repeat the analysis using excess returns computed by subtracting from each return measure the return on the largest ETF that follows the S&P 500 index (SPY) as a proxy for the market return.18 The results of the tests using excess returns are provided in Table 2.11. A comparison of the two panels reveals that most of the e ect remains intact, and the numbers have similar magnitudes and statistical signi cance. Our third analysis seeks to ensure that our results are not driven by unknown rm attributes that may impact the return and that are not perfectly controlled for by our matching procedure. Speci cally, for each earnings announcement we calculate the close- to-open or open-to-endpoint returns in the same stock in each of the four weeks prior to the announcement on the same day of the week as the earnings announcement. We average these four return observations to get the \normal" return of the stock, and subtract that normal return from the raw return to get an abnormal return measure for the earnings announcement. Table 2.12 provides the results for the abnormal return measure, and we observe a signi cant negative Di RetChg for categories 1 and 2 and a signi cant positive Di RetChg for categories 6 and 7. The results from Table 2.10 to Table 2.12 are consistent with the prediction of our theory, and they appear very robust to both various endpoints that we consider for the subsequent price-adjustment period as well as to multiple de nitions of returns. One interesting feature of the in uence model that can be observed in the simulations in Figure 2.1 is that the initial price adjustment to the published view continues after the market opens for a short period of time. The reason for this pattern is that prices adjust to the published view through the trading of sequentially arriving investors, not instantaneously via some sort of mysterious coordination of beliefs. While the adjustment of prices on the rst trade is large, it takes up to 70 periods in the simulation for prices to adjust to the published view before emphasis shifts to the independent personal views and prices begin to re ect the wisdom of the crowds. If the price in the rst period in the gure is equivalent to the price at the opening of the regular trading session in the empirical analysis, Di RetChg should be somewhat smaller than if we had used an alternative price 18SPY is heavily traded on multiple trading venues that are not its primary market. Therefore, we compute returns for SPY using the midquote of the National Best Bid and O er (NBBO) that we construct using the quote information in TAQ. 46 after the opening of the market that re ects some additional trading. Our choice of the opening price for the tests in Table 2.10|Table 2.12 was meant to be conservative because it is di cult to ascertain when prices nish their initial adjustment to the published view and the focus shifts to the independent signals of investors.19 Still, it is instructive to see how this particular choice impacts the results. Therefore, we carry out additional tests using the midquote 30 minutes after the opening (at 10:00 a.m.) as an alternative de nition of the end of the initial price-adjustment period. Table 2.13 and Table 2.14 compare the results of the original and the alternative de nitions side by side. The rst two columns (of both panels) simply show the results from Table 2.10|Table 2.12, where the initial price-adjustment period is de ned as the close-to-open return and we use for the subsequent price-adjustment period the open-to- 10:30 a.m. return. The last two columns use the alternative de nition, whereby the initial price-adjustment period is close-to-10:00 a.m. and the subsequent price-adjustment period is 10:00 a.m.-to-10:30 a.m. Notice that both de nitions begin with the closing price on day t − 1 and end with the price at 10:30 a.m. on day t. The only di erence is the end of the initial price-adjustment period: opening of the market (left two columns) versus 10:00 a.m. (right two columns). The results in the tables are consistent with the price pattern we observe in Figure 2.1. In particular, we see that the mean and median magnitudes of Di RetChg in categories 6 and 7 are about 50% to 100% larger when we allow more time for the initial adjustment of prices. The mean magnitude is also twice as large for the positive earnings announcements in categories 1 and 2. While the medians are not larger in categories 1 and 2, all results appear to be more statistically signi cant when we use the midquote 30 minutes after the open to represent the initial adjustment of prices. These results are further consistent with our model, and demonstrate that our conclusions on the manner in which in uence a ects the evolution of prices are not very sensitive to the various choices we need to make when bringing the model to the data. 2.4.3 Volume One of the interesting insights that come out of our theory is the dynamic way in which the in uence of published opinions by experts and media pundits a ects investors. Investors 19Prices could also be adjusting in after-hours trading, in which case our use of the opening price in the empirical work is not necessarily equivalent to the rst period in the model. 47 who do not observe the published view are taken aback by the fast adjustment of prices (and the illiquidity associated with it) in the initial price-adjustment period and abstain from trading. Only after prices adjust to the published view does the market become liquid enough to facilitate trading by both types of investors (those who were in uenced by the published view and those who were not). As such, there is lower volume in the in uence model before prices adjust to the published view, and volume increases when the wisdom of the crowds e ect emerges. This volume pattern does not arise in the benchmark model. Testing this e ect is somewhat more nuanced than testing the return implications. When dealing with returns, the risk premium over a very short interval is not an important consideration relative to the movement of prices following news, and hence one can easily compare periods of varied lengths. With volume, on the other hand, increasing the length of a period over which we measure the e ect necessarily means that volume in the longer period will be at least as large as in the shorter period. With this caveat in mind, though, it could be still meaningful to compare the volume patterns in afternoon and morning announcements for a given de nition of initial and subsequent price-adjustment periods. We therefore de ne VolChg to be volume during the subsequent price-adjustment period minus volume during the initial price-adjustment period, and empirically test: Di VolChg = VolChg(Afternoon) − VolChg(Morning) > 0; (2.8) pooling together categories 1, 2, 6, and 7. As the volume period for the initial price- adjustment period we follow the same de nition as in Section 2.4.1: we accumulate volume starting from 6:00 p.m. on the previous day and up to and including the opening trade (or opening auction). For volume in the subsequent price-adjustment period we use two measures for robustness: (i) cumulative volume from 10:00 a.m. to 10:30 a.m. (VOL1), and (ii) cumulative volume from 10:30 a.m. to 11:00 a.m. (VOL2).20 We also use two alternative de nitions of volume. The rst de nition is turnover, which is the number of shares traded from TAQ divided by the number of shares outstanding from the CRSP database.21 The second de nition is dollar volume. 20Our conclusions are similar if we use periods later in the trading day. Also, the results hold if we run categories 1 and 2 separately from categories 6 and 7. 21Our use of turnover is motivated by Lo and Wang (2000), who discuss the theoretical advantages of using turnover as a measure of volume. 48 Table 2.15 shows that the means and medians of Di VolChg are positive and statistically signi cant for both VOL1 and VOL2 using either turnover or dollar volume.22 These results are consistent with the predictions of our theory, and in particular, with the idea that published views create not just price patterns but also patterns in the trading process itself. 2.5 Conclusions When news about a rm is made public, it is not the case that the information is simply announced to the world of investors and nothing further is ever said about it. Rather, the news is picked up by media pundits, newsletter writers, on-line discussion group leaders, and other experts who present their interpretations of the news. While it is likely that they are better than the average investor in terms of interpreting the news, their interpretations are de nitely not perfect. At the same time, they wield in uence and hence their opinion a ects the trading of other investors who listen to media reports or read the opinions in on-line newsletters and discussion groups. How does their in uence impact the evolution of prices? Our goal in this paper is to answer this question rst by constructing a theoretical model and second by providing some empirical work to test and calibrate the model. We note from the outset that the idea that published opinions could introduce a shared error into prices is not new. Treynor (1987) made the point that price e ciency in the market comes from the independent opinions of a large number of investors who err inde- pendently. Even if the accuracy of the published opinion is greater than the accuracy of the the average individual opinion, an error in the published opinion will be re ected in the trading of all investors who are in uenced by it, and hence would a ect prices much more than an error in one individual's opinion. Our paper's contribution is twofold. First, we model this intuition rigorously using a sequential trade framework, and investigate the consequences of this idea for the evolution of prices as well as for the trading process itself. Second, we take the model to the data to see if we can nd evidence consistent with the model, and then use the empirical estimates to get a sense of the magnitude of in uence in in U.S. equity markets. 22Jiang, Likitapiwat, and McInish (2012) note that volume numbers may be di erent across the two primary listing exchanges (NYSE and NASDAQ) due to di erences in market structure. If there is a systematic bias in exchange listing between rms that announce their earnings in the morning versus those that announce in the afternoon, our volume tests could be a ected. To ensure that this was not the case, we also carried out the tests only on pairs for which both the morning and afternoon announcements belong to rms that are listed on the same primary exchange. The results were similar. 49 Our model provides novel implications pertaining to how in uence matters for prices. We show that in uence will initially increase the price impact associated with news, but later could slow the adjustment of prices to the true value. In the presence of in uence, some investors choose to stay on the sideline during the initial price-adjustment period, reducing the intensity of trading. As prices adjust to the published view and trading costs decline, additional investors join the trading process and a wisdom of the crowds e ect emerges whereby the independent views of many investors contribute to the informational e ciency of prices. Their trading can correct the shared error that was introduced by the published opinion, but at the cost of slower convergence of prices to the true value compared with what would happen in an economy where a published view does not yield in uence. Unlike in information cascades papers, the arrival of investors in our model changes the terms of trade for subsequent investors. As such, the decision rule changes over time endogenously, and the error does not propagate forever. We do not need exogenous arrival of public information or investors with high-precision information that arrive late in the sequence of trading to counter the impact of the error (see, for example, Bikhchandani, Hirshleifer, andWelch (1992)). Rather, as arriving investors face evolving prices and trading costs over time, the population of investors who choose to trade changes, leading to the elimination of the error by letting the wisdom of the crowds take e ect. Following Treynor's observation that the number of in uenced investors increases with the time elapsed since the release of the published opinion, we use the timing of earnings announcements (or the interval of time between the announcement and the opening of the next regular trading session) to test our theory. We nd evidence consistent with the predictions of the model, and the results are robust to various choices we make in the empirical speci cations of the tests. We hope that future work will use the insights of our model to assess how the in uence of media impacts prices in other markets or under di erent circumstances. As the plurality of media forms and the prevalence of media in our lives increase over time, the in uence media exerts over nancial markets remains an important topic for study. 50 (a) Panel A: Price Path in the In uence and Benchmark Models up to Period 500 (b) Panel B: Price Path in the In uence and Benchmark Models up to Period 3,000 Figure 2.1: Price Adjustment when the Published View Is Incorrect 51 Figure 2.2: Volume in the In uence Model 52 Table 2.1: Trader Type De nitions Type ( ) ~ve ~vn Description 1 1 1 Positive published view; Positive personal view 2 1 0 Positive published view; Negative personal view 3 N/A 1 Positive personal view 4 N/A 0 Negative personal view 5 0 1 Negative published view; Positive personal view 6 0 0 Negative published view; Negative personal view 7 N/A N/A Noise Trader Buyer 8 N/A N/A Noise T