Measuring Sentiment and Herding in Financial Markets

Transcription

1 Measuring Sentiment and Herding in Financial Markets Soosung Hwang 1 Cass Business School Mark Salmon Warwick Business School April Corresponding author. Faculty of Finance, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ, UK. Tel) +44 (0) , Fax) +44 (0) , ) s.hwang@city.ac.uk. We would like to thank Gordon Gemmill, Andrew Karolyi, Colin Meyer, Roger Otten, Steve Satchell, Peter Schotman, Meir Statman, Franz Palm and seminar participants at the International Conference on the Econometrics of Financial Markets, the PACAP/FMA Finance Conference, University of New South Wales, the Bank of England, and Said Business School for their comments on earlier versions of this pape titled A New Measure of Herding and Empirical Evidence for the US, UK, and South Korean Stock Markets.

2 Abstract This study proposes a non-parametric measure of herding based on linear factor models and applies it to investigate the nature of slowing moving herd behaviour in the US, UK, and South Korean stock markets. We find evidence of herding towards the market portfolio when investors believe that they know where market is heading rather than when the market is in crises. Keyword Herding, Non-central Chi Square Distribution, Market Crises. JEL Code C12,C31,G12,G14

3 1 Introduction For the last couple of decades there have been an increasing interest on if and when investors decide to follow the observed decisions of others rather than their own belief. Many hypotheses have been proposed to explain herd behaviour and others have investigated the evidence of herding in financial markets. 1 However empirical evidence on herding is not conclusive. Most studies have failed to find evidence of herding except a few cases, i.e., herd behavior by market experts such as analysts and forecasters (see Hirshleifer and Teoh, 2001). The difficulty lies in the lack of methods that can differentiate changes in fundamentals or investors irrational behaviour. It is important to discriminate empirically between herding and common or correlated movements based on fundamentals, since the former potentially leads to market inefficiency whereas the latter simply reflects an efficient reallocation of assets. Several measures to investigate herd behaviour in financial markets already exist. However, it has not been easy to develop appropriate methods that discriminate between the two cases, because both motivations represent collective movements in the market towards some position or view and hence some class of assets. In this study we define herding slightly different from the conventional definition. We define herding as the behavior of investors in the market who follow the performance of factors such as the market portfolio, sectors, styles, 1 See Banerjee (1992), Bikhchandani, Hirshleifer, and Welch (1992), and Welch (1992) for information-based herding, Scharfstein and Stein (1990) reputation-based herding, and for compensation-based herding Brennan (1993), and Roll (1992). These studies on herd behaviour are closely related to the study of contagion, see Eichengreen, Mathieson, Chadha, Jansen, Kodres and Sharma (1998) for example. 1

4 or macroeconomic signals, to buy or sell individual assets at the same time and disregard the long-run risk-return relationship. Although our measure can be easily applied to other factors and thus we may have some interesting herding activity for these factors, we focus on herding towards the market portfolio in this study which has been a major topic in many previous studies. Our definition of herding is at the market-wide level similar to Christie and Huang (1995). 2 The existence of this type of herding suggests that individual assets are mispriced, and for practitioners, using styles to form hedge portfolios may not work when there is herding towards the styles. Hedging strategies could work well when there is adverse herding where factor sensitivities (betas) are dispersed. 3 Basedonthisdefinition, Hwang and Salmon (HS) (2004) propose a model which includes a measure that captures the extent of herding. The measure can be estimated by the cross-sectional variance of the factor sensitivities of individual assets of the market. In this paper, we extend the previous study of HS in several significant ways. We investigate the effects of sentiment on herding. We find that when sentiment works in a way that individual asset returns are expected to increase regardless of their systematic risks, herding increases. This could happen in both bull and bear markets. Therefore, our measure of herding has two sources; one from cross-sectional herding towards the market portfolio, 2 Throughout this paper we implicitly assume that herding should naturally be viewed in a relative sense rather than as an absolute and that no market can ever be completely free of some element of herding. Thus we argue that there is either more or less herding in a market at some particular time and herding is a matter of degree. It seems to us conceptually difficult if not impossible to rigorously define a statistic which could provide an absolute measure of herding. However, most herd measures that have been proposed, such as Lakonishok, Shleifer, and Vishny (1992), Wermers (1995), and Chang, Cheng and Khorana (2000), have apparently tried to identify herding in absolute terms. 3 The term adverse herding is consistent with disperse in Hirshleifer and Teoh (2001). 2

5 the other from the sentiment. The results suggest that our measure has some power to detect herd behaviour which develops over time with sentiment. Secondly, we propose a non-parametric method. This method should be more flexible since we do not assume any particular parametric dynamic process for herding. In addition, we develop a formal statistical framework to calculate the confidence level of the estimated herd measure. The herd measure we derive follows a non-central Chi-square distribution, which enables us to investigate if there is any significant difference in the estimated levels of herd behavior between two periods, or if the estimated level of herding changes over time significantly. Finally our study examines the dynamic nature of herd behavior over a longer time horizon in stock markets. Herding is generally perceived to be a phenomenon which arises rapidly and thus most studies of herding explicitly or implicitly assume that herding arises during very short time intervals. However, Summers and Porterba (1988) and Fama and French (1988) show that noise may be highly persistent and slow-moving over time. Shiller (2000) argues that if markets are not efficient at both macro and micro levels and if the conventional wisdom given by experts changes very slowly, then the short-run relationship may provide us with biased information about the level of stock prices. A dramatic case of the slow-moving noise is the bubble. A cycle of the bubble may not be completed within days, weeks or months. For example, bubbles such as the Tulip Bubble in seventeenth-century Holland, the real estate bubble in the late 1980s Japan, and the recent dot-com bubble were not formed within a short time period. It took years for these bubbles to develop and finally make a huge impact on society. 4 If this argument is correct, then we should find evidence for slow moving herd behavior, and this 4 See Shiller (2003) for example. 3

6 isonereasonwhyweusemonthlydatarather than high frequency data. We apply our non-parametric measure to the US, UK, and South Korean stock market and find that herding towards the market movements does move slowly, but is heavily affected by the crises. Contrary to the common belief that herding is significant when the market is in stress, we find that herding can be more apparent when market continues to rise or it becomes apparent that market goes down. Once a crisis appears herding toward the market portfolio becomes much weaker, as individuals become more concerned with fundamentals rather than overall market movements. We also show that the herd measure we propose is robust to business cycles and stock market movements. Our results confirm that herding occurs when investors expectations on the market is more homogeneous, or in other words when the direction towards which the market is heading is clear whether it is a bull or bear market. The non-parametric results in this paper are consistent with what Hwang and Salmon (2004) previously found using high frequency data. These results also suggest that herding is persistent and moves slowly over time like stock prices moving around the fundamental values as discussed in Shiller (1981, 2000, 2003). In the next section we briefly explain the concept of the herding and its measurement in the presence of market sentiment; we then develop a nonparametric method to estimate herd behavior based on the cross-sectional variance of the t-statistics of the OLS estimates of betas in a linear factor model. In section 3, we apply the new methods to the US, UK, and South Korean stock markets, and finally we offer some conclusions. 4

7 2 A Measure of Herding Thetypeofherdingweinvestigateisthebehaviorofinvestorswhofollow the performance of factors such as the market index (or market-wide movements) to buy or sell individual assets at the same time disregarding the equilibrium risk-return relationship. To investigate this type of herding we first discuss how individual betas are affected by herding and sentiment. We show that under some conditions sentiment has the same effects on betas; cross-sectional vaiance of betas decreases as herding and sentiment increases. Then measures of herding follow with test statistics. 2.1 Herding, Sentiment, and Betas HS show that in the presence of herding towards the market portfolio the equilibrium CAPM model can be presented in the following relationship; E b t (r it ) E t (r mt ) = βb imt = β imt h mt (β imt 1), (1) where β imt is the equilibrium beta of asset i at time t, and Et b (r it )andβ b imt are the market s biased conditional expectation on excess returns of asset i and its beta at time t, andh mt is a parameter that changes over time, h mt 1. This is a generalized model encompassing the equilibrium CAPM with h mt =0. Let us consider several cases in order to see how the herd measure affects individual asset prices given the evolution of the fundamentals (market returns). First of all, when h mt =1,β b imt =1foralli and the expected excess returns on the individual assets will be the same as that on the market portfolio regardless of their systematic risks. Thus h mt = 1 can be interpreted as the perfect herding toward the market portfolio. In general, when 5

8 0 <h mt < 1, herding exists in the market, and the degree of herding depends on the magnitude of h mt. In terms of betas, when 0 <h mt < 1, we have β imt >β b imt > 1foranequitywithβ imt > 1, while β imt <β b imt < 1for an equity with β imt < 1. Therefore when there is herding, individual betas are biased towards 1. We can also explain adverse herding by assuming h mt < 0. In this case an equity with β imt < 1 will be less sensitive to the movements of the market portfolio (i.e., β b imt <β imt < 1), while an equity with β imt > 1 will be more sensitive to movements of the market portfolio (i.e., β b imt >β imt > 1). Based on these arguments, HS propose a method to measure herd behaviour in the stock market. It is worth noting that E t (r mt ) is treated as given in this framework and thus h mt is conditional on market fundamentals. Therefore, the herd measure is not assumed to be affected by market-wide misspricing like bubbles, but is designed to capture cross-sectional herd behavior only within the market. We extend the model by allowing the expected returns of the market portfolio and individual assets to be biased by investors sentiment. Let δ mt and δ it represent sentiment on the market portfolio and asset i respectively. Then investors biased expectation in the presence of sentiment is sum of fundamentals and sentiment, Et s (r it ) = E t (r it )+δ it,and Et s (r mt ) = E t (r mt )+δ mt, where δ mt = E c (δ it ) and the superscript s represents bias due to the senti- 6

9 ment. Then we have β s imt = Es t (r it ) E s t (r mt ) (2) = E t(r it )+δ it E t (r mt )+δ mt = β imt + s it, 1+s mt where s mt = δmt E t (r mt and s ) it = δ it E t (r mt represent sentiment in the market ) portfolio and asset i relative to the expected market return. Positive values of s mt and s it are usually expected in bull markets, while negative market sentiment during bear markets. For example, the Japanese stock market bubble in the late 1980 s happened after long period of bull markets, and the US dot-com bubble happened during the bull markets of the 1990 s. On the other hand, it is hard to suggest any example that positive sentiment has existed during bear markets or negative sentiment has existed during bull markets. Equation (2) suggests that when δ it = β imt δ mt,wehaveβ s imt = β imt. That is, only when the sentiment affects the expected return of the individual asset through the equilibrium relationship, the beta in the presence of sentiment is equivalent to the equilibrium beta. However, when investors have a market-wide sentiment on the future perspective, it is hard to expect that the market-wide sentiment affects individual assets via the equilibrium relationship. When investors are over-confident on the outlook of markets with positive sentiment, a similar level of sentiment is likely to be expected for individual assets regardless of the equilibrium relationship. In an extreme case, when sentiment is the same for all assets in the market, s mt = s it > 0 for all i, β s imt moves towards 1; 1 >β s imt >β imt for assets with β imt < 1and 1 <β s imt <β imt for β imt > 1. Similarly when s mt = s it < 0, 1 <β imt <β s imt for assets with β imt > 1and1>β imt >β s imt for β imt < 1. 7

10 Therefore a reduction in Var c (β s imt) could be observed when there is a market wide positive sentiment s mt but idiosyncratic sentiments do not differ each other. In other words, the cross-sectional variance of β s imt Var c (β s imt) = 1 (1 + s mt ) 2 [Var c(β imt )+Var c (s it )] decreases when s mt increases and Var c (s it ) decreases for a given Var c (β imt ). The pattern - when there is a market wide sentiment the cross-sectional variance of betas decreases - is very similar to what we observe when there is herd behaviour in the market. The effects of sentiment on the systematic risk are similar to those of cross-sectional herding. Herd behaviour decreases large betas but increases small betas and so does the positive sentiment. In fact, sentiment under some conditions - s mt increases for a given Var c (s it )or Var c (s it ) decreases for a given s mt - is a herd behaviour, since it shows the behavior of investors who follow market movements to buy or sell individual assets at the same time disregarding the equilibrium risk-return relationship. Both herding towards the market portfolio and sentiment can be combined together as follows; E bs t (r it ) E s t (r mt ) = β bs imt (3) = β s imt h mt (β s imt 1) = β imt + s it 1+s mt h mt µ βimt + s it 1+s mt 1, where the superscript bs represents bias from herding and sentiment. Under 8

11 the assumption that β imt is not related with sentiment, we have " µβimt µ # 2 Var c (β bs + s it βimt + s it imt) = E c h mt s mt 1+s mt = (1 h mt) 2 (1 + s mt ) E 2 c (βimt 1) 2 + E c (sit s mt ) 2 = (1 h mt) 2 (1 + s mt ) 2 [Var c(β imt )+Var c (s it )] = Var c (β s imt)(1 h mt ) 2. (4) For a given level of Var c (β imt ) the left hand side of equation (4) decreases, ceteris paribus, when1)h mt increases, 2) s mt increases, and 3) Var c (s it ) decreases. 5 That is, we observe a decrease in Var c (β bs imt) whenthereisa cross-sectional herding, a positive market sentiment, or a lower idiosyncratic sentiments. 2.2 A Herd Measure and Non-parametric Test Methods In this section we use a simple market model to develop the herd measure. Similar explanation is possible in multifactor models such as Fama and French (1993), but explanation becomes more complicated. We show why the t-statistics of estimated betas are a better way of measuring herd behaviour than the estimates of betas. Following the previous section our definition of herding is Definition 1 The measure of herding towards the market portfolio is defined 5 We assume that Var c (β imt ) is constant. As argued in HS, we do not expect Var c (β imt ) to change significantly during a short time period though individual β imt s may change over time very slowly. With a large number of stocks, idiosyncratic movements in β imt s are expected to be cancelled out. 9

12 as H mt = 1 XN t β bs imt 1 2, (5) N t i=1 where N t is the number of stocks at time t. Herding towards the market portfolio therefore decreases with H mt. One major obstacle in calculating the herd measure is that β bs imt is unknown and needs to be estimated. It is quite well documented that betas are not constant but time-varying. (See Harvey (1989), Ferson and Harvey (1991, 1993), and Ferson and Korajczyk (1995) for example.) Several methods have been proposed to estimate time-varying betas by Gomes, Kogan and Zhang (2003), Santos and Veronesi (2004), and Ang and Chen (2005). Ang and Chen (2005) show that although a common procedure is to obtain OLS estimates of betas using rolling samples of 60 months, these betas are biased and inconsistent when betas vary over time and are correlated with time-varying market risk premia. In spite of these studies, in what follows, we use rolling windows to capture the time variation in betas. This simple method have been chosen for the following reasons. First, as in Jagannathan and Wang (1996) multifactor unconditional models could capture the same effects as a single factor conditional model. We estimate betas using multi-factor models in our empirical tests. Second we could avoid the parametric restrictions used in Ang and Chen (2005), where latent processes such as betas, market risk premia, and volatility are assumed to follow AR(1) processes. This could be significant restrictions unless these processes represent the true processes. Ghysels (1998) points out that it is difficult to obtain time-varying betas unless the true model is known. Third, betas are extremely persistent. For example the monthly autocorrelations of conditional betas reported by Gomes, Kogan and Zhang (2003) and Ang and Chen (2005) are 0.98 and 0.99 respectively. 10

13 Therefore choosing shorter windows (i.e., 24 months rather than 60 months) could reduce the problems of OLS estimates for the time-varying betas. In addition we can also easily evaluate any unfavourable effects of using long windows on the herd measure. Finally using the OLS estimates of betas we could investigate how the estimation errors affect the herd measure, which we discuss in this section. The state space model estimated with Gibbs sampling and Markov Chain Monte Carlo (MCMC) by Ang and Chen (2005) is not very helpful to evaluate how the estimation errors affect the herd measure (H mt ). We use a simple market model to show why herd measure based on the t- statistics of the OLS estimates of betas is better than that based on the OLS estimates of betas. The same results are obtained if factors in multi-factor models are authogonal to each other. Given τ (window size) observations, the market model is represented as r it = α bs it + β bs imtr mt + ε it, (6) where ε it is the idiosyncratic error which we assume ε it N(0,σ 2 iεt). The OLS estimator of β bs imt for asset i at time t, b bs imt, is then simply b bs imt = bσ 2 imt/bσ 2 mt, (7) Var(b bs imt) = bσ 2 iεt/bσ 2 mt, (8) where bσ 2 imt isthesamplecovariancebetweenr it and r mt, bσ 2 mt is the sample variance of r mt, and bσ 2 iεt isthesamplevarianceoftheolsresiduals. Using the OLS betas, we then estimate the herd measure as H O mt = 1 N t XN t i=1 b bs imt 1 2. (9) However, in general there could be some large but insignificant estimates of β b imt s which could dramatically increase H O mt and the variance of the es- 11

14 timated herd measure, Var(Hkt O ). More importantly, the significance of OLS betas may change over time, affecting Hmt O even without changes in β b imt. In addition, the OLS estimates in equations (7) and (8) have some undesirable properties. Suppose that the market model in (6) multiplied by δ. Then we have r it = α it + β bs imtr mt + ε it, (10) where rit = δr it, rmt = δr mt, α it = δα it,andε it = δε it,leavingb bs imt unchanged. Only when r it,r mt,andε it move at the same rate, the market model holds with the same beta, and the OLS estimator will not be affected. However, in general this is not always expected to happen. A similar econometric problem has been investigated for contagion effects during market crises. When the volatility of the country of origin of an international financial crisis increases dramatically, the volatility of the returns of its neighbour country or country-specific (idiosyncratic) risk may not move in proportion to that of the origin country, and the correlation coefficient between the two countries may not reflect the true relationship. 6 Similarly when r it, r mt,andε it do not move at the same rate, an important problem arises; Var(b bs imt) isaffected by heteroskedasticity in ε it. The heteroskedasticity could create significant difference between the measure in (5) we try to measure and the one in (9). To see the impact of the heteroskedasticity on our herd measure, we first see E c [b bs imt] =E c [β bs imt + η imt ]=1, 6 For example Forbes and Rigobon (2002) show that the correlation coefficient between the two countries increases during market crises when the volatility of idiosyncratic errors remains unchanged. Therefore they conclude that increased correlations between two countries may not necessarily be the evidence of contagion. However, Corsetti, Pericoli, and Sbracia (2003) point out that the assumption is not appropriate and suggest some evidence of contagion. 12

15 where η imt is the OLS estimation error, η imt N (0,σ 2 iεt/σ 2 mt). So the herd measureestimatedwiththeolsestimates,hmt, O is given by " # E[Hmt] O 1 XN t = E (b bs imt 1) 2 h since E 1 N t P Nt i=1 (βbs = E = E " " N t 1 N t 1 N t i=1 XN t i=1 XN t i=1 (β bs imt + η imt 1) 2 # (β bs imt 1) N t = H mt + 1 XN t σ 2 iεt, N t σ 2 mt i=1 XN t i=1 η 2 imt # (11) imt 1)η imt i =0. When r it, r mt,andε it move at the same P 1 rate, Nt N t i=1 σ2 iεt/σ 2 mt is constant over time and any movement in H mt can be captured by Hmt. O However, if either cross-sectional average of idiosyncratic P 1 variances (i.e., Nt N t i=1 σ2 iεt) or market variance (i.e., σ 2 mt) isheteroskedastic, then changes in Hmt O do not necessarily arise from herd behavior, but could come from the changes in the ratio of firm level variance against market variance. Campbell, Lettau, Malkiel, and Xu (2001) discuss some relationships between firm, industry, and market-level volatilities in the US market. What is interesting for our study is that the correlation coefficients between the firm and market-level monthly volatilities are high, i.e., 0.7 to 0.8. This result suggests that the difference between H O mt and H mt may not be significant, but we leave this for the empirical tests. To avoid this unfavourable property of Hmt, O we standardize b bs imt with its ownstandarddeviation. Thist statistic standardises η imt to be homoskedastic and the herd measure based on the t statistics, denoted by Hmt, isnot P 1 liable to any heteroskedastic behaviour of Nt σ 2 iεt N t i=1 or market volatility. σ 2 mt 13

16 Asimplewaytoseethisis b bs imt bσ iεt /bσ mt = bσ2 imt/bσ 2 mt bσ iεt /bσ mt (12) = bρbs imtbσ it, bσ iεt which consists of correlation coefficient between asset i s return and market return (ρ it ), asset i s return volatility (σ it ), and idiosyncratic volatility (σ iεt ). The difference between OLS estimates and t statistics is clear when equations (7) and (12) are compared. Equation (7) shows that OLS betas decreases when market volatility increases unless individual stock returns comove with the market returns. Using t statistics in (12) could reduce the influence the impact of changes in market volatility in particular during market crises. The implication of using t-statistics is that we now measure herding with OLS betas as well as their significance. Consider the following three cases in a world where there are only two stocks (numbers in parentheses are standard deviations); case 1) b bs imt =1.2 (0.1) and b bs jmt =0.8 (0.1), case 2) b bs imt =1.2 (0.2) and b bs jmt =0.8 (0.2), and case 3) b bs imt =1.1 (0.1) and b bs jmt =0.9 (0.1). Herd measure calcualted with estimated betas is not necessarily the same as that calculated with t statistics. Cases 1 and 2: There is no difference between cases 1 and 2 with Hmt. O However, the two stocks in case 1 are significantly different from 1 while those in case 2 are not. Thus ceteris paribus case 2 shows a high level of herding than case 1 with Hmt. Cases 1 and 3: Both Hmt O and Hmt show the same choice. Cases 2 and 3: We observe more herding in case 3 with Hmt, O while there is no difference between the two cases with Hmt. 14

17 Therefore H O mt and H mt give the same level of herding only when the estimation error is the same. Herd measure based on t statistics H mt concerns more on the probability that how estimated betas are distributed around their cross-sectional mean (i.e., one). This leads to our herd measure based on the following t statistic; b bs imt 1 t µdf; βbs bσ iεt /bσ mt imt 1 σ 2 iεt /σ mt, (13) where DF is the degrees of freedom and βbs parameter. imt Ec(βbs imt ) σ 2 iεt /σ mt is a non-centrality Definition 2 The measure of herding towards the market portfolio is defined as H mt = 1 N t XN t i=1 µ b bs 2 imt 1, (14) bσ iεt /bσ mt where b bs imt are the observed (OLS) estimates of betas for the market portfolio for stock i at time t, and bσ 2 iεt and bσ mt are defined in equations (7) and (8). Herding towards the market portfolio increases with decreasing H mt. For the measure defined as in (14) we have the following distributional result. ³ 0 Theorem 1 Let B mt = B1mt B2mt. BNmt, where B imt = bbs imt 1 bσ iεt /bσ mt. Then with the classical OLS assumption, µ B mt, Vmt N t N t N, δ mt N t 1 N t N t ³ 0 where δ mt = δ 1mt δ 2mt. δ Nmt,δ imt = βbs imt 1, and V σ 2 iεt /σ 2 mt is covariance (correlation) matrix of B mt. Then, the herd measure for the mt market 15

18 portfolio, follows H mt = 1 N t B 0 mtb mt (15) 1 N t χ 2 (R; δ R k )+c, where R istherankofv mt, δ R m = P R i=1 (δa i ) 2 /λ i,andc = P N i=r+1 (δa i ) 2, where δ A i is the ith element of the vector C 0 mtb mt, where C mt is the (N t N t ) matrix of eigenvectors of Vmt, i.e., Vmt = C mtλ mtc 0 mt, whereλ mt is the (N t N t ) diagonal matrix of eigenvalues. The eigenvalues are sorted in descending order and eigenvectors are sorted according to the sorted eigenvalues. Proof. See the Appendix. This measure can be calculated easily from any standard estimation program since they are given by the cross-sectional variance of the t statistics of the estimated coefficient on the market portfolio. Theorem 1 shows that the new measure of herding is distributed as 1/N t times the sum of non-central χ 2 distributions with degrees of freedom R and with non-centrality parameters δ R m and a constant. Therefore the variance of H mt is given by; Var[H mt] = 2 N 2 t R +2δ R m. (16) It is worth noting that the result with t-statistics depends on the assumption that the number of observations to estimate β bs imt is large and B kt is multivariate normal. For a small number of observations, the confidence level from the above theorem is smaller than it would be asymptotically and will reject the null hypothesis too frequently. For high frequency data which are characterized by non-normality, the statistics we propose in the Theorem could provide a narrower confidence level. However with a large number of N t the theorem is expected to work well asymptictically. In practice, the 16

19 non-centrality parameter would be replaced with the sample estimates. In addition, with a large number of stocks H mt follows non-central χ 2 distribution. In practice some components such as E c (β bs imt), Vmt, and δ mt are not known and we need to replace them with the sample estimates. 3 Empirical Tests One straight forward approach to testing herding in a market is to calculate the measure given in (15) and its confidence level as given in Theorem 1 for particular periods of interest. If there is any significant difference between two periods, we may conclude that one period shows relatively more herding behavior than the other. Many studies on herding and contagion take this view, especially by examining behavior around and during market crises. See Bikhchandani and Sharma (2000) for a survey of empirical studies. An alternative approach adopted below is to calculate the statistics recursively and in this way we can investigate in a more detailed manner whether the degree of herding has increased or decreased significantly over time. That is, using equation (14), we calculate the herd measure at time t given an appropriate choice of τ, andobtainconfidence intervals from equation (16). The same procedure is then repeated over time by advancing the start date by one period, i.e., t, t + 1,... These recursively generated test statistics in effect provide us with a sequence of hypothesis tests. That is, we can use the confidence level calculated at time t to test if the value of the test statistic at t + 1 changes significantly. In this way we can determine if the level of herding is significantly different over time. In the empirical study, we first present our results using the Center for Research in Security Prices (CRSP) data with Fama and French s (1993) 17

20 three factor model with momentum (four factor model). The results are further investigated with various different options. Then using UK and South Korean stock market data, we compare herd behaviour in different markets. These tests give us answers such if herding has happened, and if yes, when markets herd. 3.1 Herding in the US Market Data We use monthly data from July 1964 to December 2002 to investigate herding in the US stock market. Our sample period covers several crises as well as several bull and bear markets. We use the CRSP data which start with 1084 stocks at the beginning of our sample period and reach to 7197 stocks at November For excess market returns we use the CRSP value weighted market portfolio returns and 1 month treasury bills. For the other factors we use Fama-French s (1993) size (Small minus Big, SMB) and book-to-market (High Minus Low, HML), and momentum from Kenneth French s data library in addition to the escess market returns. As explained above, we need to choose an appropriate number of monthly observations, i.e., τ, to obtain the OLS estimate. As explained earlier we have chosen a shorter window, i.e., τ=24. But we also tried a range of values, i.e., τ=36, 48, and 60, and found that the results were effectively not different from one other (see the results in the next subsection). The procedure by which we calculated each herd measure is as follows. We use the first 24 observations up to June 1966 to obtain the OLS estimates of betas and their t statisticsforeachstockandthencalculatehmt and its test statistic for June We then add one observation at the end of the sample and drop 18

21 the first and so use the next 24 observations up to July 1966 to calculate the herd measure and its statistic for July 1966, and so on. An important issue in estimating betas is the nonsynchronous trading problem for illiquid stocks, which is first recognized by Fisher (1966). Scholes and Williams (1977) show that the OLS estimates of betas of infrequently traded stocks are negatively biased while those of frequently traded stocks are positively biased. They derive consistent estimators of the market model in (6). However, it is not clear how the consistent estimators work in multifactor models where factors such as size and value/growth are included as independent variables. Thus in our study we filter out small illiquid stocks by controlling the following three liquidity proxies; volatility, size, and turnover rate. 7 (Volatility) When the true betas are not known, the market-adjustreturn model of Campbell et al (2001) is useful; r it = r mt + ε it, (17) which is a restricted version of the market model in (6) with α bs it =0 and β bs imt =1. Here it is easy to show that σ it σ mt. Weremoveany stocks whose volatility is less than half of the market volatility, i.e., σ it < 0.5σ mt. 8 We find that there are less than 5 percent of stocks (19 percent in market value) whose volatility is less than half of the market volatility. 7 There are many studies on liquidity. Liquidity is a function of 1) the cost of liquidating a portfolio quickly, 2) the ability to sell without affecting prices, 3) the ability of prices to recover from shocks, and 4) costs associated with selling now, not waiting. See Kyle (1981) and Grossman and Miller (1998) for example. 8 Most stocks removed by this restriction have betas close to zero. 19

22 (Turnover Rate) Annual turnover rates on the NYSE from 1980 to 2000 rangefrom33to88percents(swan, 2002). In our study we remove stocks from our sample whose average monthly turnover rates (average over τ period) are less than 0.5 percent. The proportion of stocks removed by this process is less than 45 percent in both numbers and capitalisation values. (Size) We also remove small stocks whose market values are less than 0.01 percent of the total market capitalisation. The proportion of stocks removed by this process is less than 14 percent in values, but as large as 88 percent in numbers. These filtering method leaves number of stocks ranging from 697 to 1286 for our sample period. Compared with our total number of stocks, the method seems to be too strict and look arbitrary. Later in robustness tests, we try several different values of cut-off size to see if our choice affects the results significantly. It is important to note that the main purpose of these procedures is to remove the effects of illiquid stocks on the estimates of betas, since the negative bias in the OLS estimates of betas of infrequently traded stocks is far more severe than the positive bias of frequently traded stocks. 9 Controlling firm sizes is also important in our study for another purpose. The statistics proposed in Theorem 1 are for equally weighted herd measure rather than value weighted measure since value weights are hard to be included in the principal component analysis. Therefore our study is herding behaviour in large and liquid stocks rather than small and illiquid stocks. 9 See Tables 1 and 3 of Scholes and Williams (1977). 20

23 3.1.2 Herd Behaviour We use the four factor model r it = α bs it + β bs imtr mt + β bs istr smbt + β bs ihtr hmlt + β bs immtr mmt + ε it. (18) The estimated betas and t-statistics are used to calculate the herd measure as in (14). We also calculate the herd measurebasedonolsestimatesof betas for comparison purpose. Table 1 reports some basic statistical properties of the herd measures. We find considerable differences between the herd measures based on the OLS estimates of betas and t-statistics. For example, the US market shows that the rank correlations between H O mt and H mt are close to zero or negative. The results indicate that the cross-sectional average of variances of P 1 the estimation error in equation (11) (i.e., Nt N t i=1 η2 imt) couldaffect the herd measures based on OLS estimates of betas. To examine the effects of heteroskedasticity of the estimation error, we calculate rank correlations between the cross-sectional average of variances of the estimation error and herd measures. Table 1 shows that the cross-sectional average of variances of the estimation error and estimated betas-based herd measures are highly correlated, i.e., these are close to 0.9. Therefore most of the dynamics of the herd measures based on the OLS estimates of the betas are explained by the heteroskedasticity of the OLS estimation errors. On the other hand, we find negative correlation between the t-statistics-based herd measures and the variances of the estimation errors. Another result is that there are noticeable differences in herding towards the market portfolio between the market model and the four factor model, in particular with the t-statistic-based measure. We will discuss this multicollinearity issue later in detail in the robustness tests section. 21

24 3.1.3 Macroeconomic Variables We test if the proposed measure can be explained by the level of factor returns or volatility or macroeconomic activity which is known to affect stock returns. Note that the herd measure H mt is calculated using rolling windows (overlapping samples), and is highly persistent. A conventional method is to run the following linear regression; H mt = c m0 + MX c mi f it + ξ mt, (19) i=1 where f it represents factor i at time t and M is the number of such factors, andthenapplyaheteroskedasticconsistent standard deviation. In our study we estimate the Cochrane-Orcutt recursive method; H mt = c m0 + MX c mi f it + ξ mt, (20) i=1 ξ mt = ρξ mt + v mt, where v mt iid N(0,σ 2 mv ). Using an initial guess for ρ (usually ρ =0)weperform regression (20) to estimate parameters c mi, i =0,..,M. Then an updated estimate of ρ can be obtained by regressing the OLS residuals b ξ mt = Hmt bc m0 P M i=1 bc mif it on its own lagged value. The new updated ρ will be used to perform regression (20) again. This procedure is repeated until the estimates converge. 10 We use four macroeconomic variables; the dividend price ratio (DP t ), the relative treasury bill rate (RT B t ), the term spread (TS t ), and the default spread (DS t ). Thechoiceofthesefourmacroeconomicvariablesfollowsfrom previous studies such as those of Chen, Roll, Ross (1986), Fama and French (1988, 1989), Ferson and Harvey (1991), and Goyal and Santa-Clara (2003). 10 See Hamilton (1994) for detailed explanation on the Cochrane-Orcutt recursive procedure and how to calculate standard deviation of the estimates. 22

25 We also add market returns and market volatility. 11 We use log-dividend price ratio of S&P500 index for DP t,thedifference between the US 3 month treasury bill rate and its 12 month moving average for RTB t, the difference between the US 30 year treasury bond rate and the US 3 month treasury bill rate for TS t, and the difference between Moody s AAA and BAA rated corporate bonds for DS t. The results of the linear regression using the Cochrane-Orcutt recursive method are reported in Table Durbin-Watson statistics show no significant serial correlation in the error term. The herd measures either calculated with the four factor model or the market model are not explained by the macroeconomic variables. However, we find some evidence that our herd measure based on t-statistics increases when volatility increases. Overall R 2 values are not high suggesting that these factors do not explain the herd behaviour Herding and Economic Events Figure 1 shows the evolution of our herd measure towards the market portfolio. We find that H O mt has wider confidence levels than H mt (not reported). This result indicates that there are many cases of insignificantly large estimates of betas. As expected, the dynamic behaviour of H O mt is almost the same as that of the cross-sectional average of variances of the estimation error P 1 in equation (11) (i.e., Nt N t i=1 η2 imt). This confirms our preference towards herd measure based on t-statistics, which shows a narrow confidence band. In the following we focus on our t-statistics-based herd measure. 11 market volatility is calcualted by summing squared daily returns as in Schwert (1989). 12 We also used herd measure based on the OLS betas and find that the results are not different from those reported in Tables 2 in the sense that the macroeconomic variables and the additional stock market variables do not explain the herd measure. 23

26 Let us investigate the relationship between herding towards the market portfolio with economic events. There are several significant sudden changes in Hmt inthesensethatthesechangesarefaraboveorbelowthepreviousupper and lower boundaries at the 90% confidence level. Several sharp positive jumps can be found in 1970 (Recession), 1973 (Oil Shock), 1982 (Maxican Crisis), 1987 (Market Crash), 1991 (Desolution of USSR), 1998 (Russian Crisis), 2001 (September the Eleventh). The events in the parentheses show what has happened during the positive jumps. When these shocks happens, herding level decreases significantly. On the other hand there are three sharp declines in the measure during 1980, 1989, and 1993, of which the last two simply reflect the reversals from the positive increase of 1987 and 1991 after 24 months (τ). The large drop in the herd measure during 1980 comes from sudden drops in the interest rate during the recession of We could interpret that the sudden drops in the interest rate increased herding by increased expectation or confidence of economic recovery. This could be explained better with the recent market data. Herding happened before the Russian crisis in 1998 and between late 1999 and August The first herding period could be characterized by a bull market, and the second period by a bear market. When there were shocks such as Russian crisis or September the Eleventh, herding disappears. Interestingly the US market do not seem to be affected significantly by the Asian crisis despite the sudden jump in the market volatility. Two implications can be withdrawn from the results. The first is that herding happens either in bull or bear markets. When economy is in recession like early 1980 s or during 2001, we still observe high level of herding. On the other hand we also observe herding when economy is boom, for example late 1990 s or mid-1980 s. The second result is that when there are crises 24

27 or unexpected shocks, herding disappears. See for example 1987 crash, 1988 Russian crisis, and September the Eleventh. Our findings are not necessarily inconsistent with previous studies. We note that many empirical studies on herding in advanced markets find little concrete evidence of herd behavior, see Bikhchandani and Sharma (2000). However, in the South Korean case, Kim and Wei (1999) and Choe, Kho, andstulz(1999)studyherdbehavioraroundtheasiancrisisin1997and find some evidence during the Crisis. Note that both these studies use the Lakonishok, Shleifer, and Vishny (1992) measure which focuses a subset of market participants. Therefore, we cannot conclude that their results are inconsistent with ours since our measure considers herding in the whole market rather than a subset of participants. Chang, Cheng and Khorana (2000), using a variant of the method of Christie and Huang (1995), suggest the presence of herding in emerging markets such as South Korea and Taiwan, but failed to find evidence in the US, Hong Kong and Japanese markets. However, our evidence is not consistent with the view that herding happens when financial markets are in stress (or in crisis). On the contrary, the figures show that herd behavior can be clearly detected when the markets are not in stress and thus investors are confident on the outlook of the future stock market. If the direction towards which the market is heading is assured, herding begins to occur regardless of whether it is a bull or a bear market; these periods are the late 1960 s, before the Oil Shock in 1973, early 1980 s and before the 1987 crash, the 1990 s and early 2000 s. We could interpret this as suggesting that it is the investors over-confidence that induces herding. When market is in stress, however, investors lose confidence and begin to focus more on fundamentals. In this sense market stress is beneficial to the market rather harmful, although it creates stress for market participants. 25

28 From this point of view, we do not agree with the view that herding arises when financial markets are in stress. When a market is in crisis, we may observe large negative returns and the majority of the individual assets will also show negative returns, which could be interpreted as herding in whole market. However, as far as individual asset returns move following their systematic risks, we do not agree with the argument. Instead in popular linear factor models we could claim herding only when the factor loadings of individual assets are systematically biased by the crisis, and thus the longrun relationship between individual asset returns and factor returns does not work. So the fact that the majority of assets show negative returns during the market crisis is not enough for the evidence of herding. 3.2 Robustness of the Herd Measure We have proposed a measure of herding and argued that the measure is designed to capture investor sentiment and biased pricing. In this section we test how robust our herding measure is. Using the US data, we investigate various cases The Effects of Small Stocks By removing small stocks whose market values are less than 0.01 percent of the total market capitalisation, the analysis above used larger stocks. To investigate the effects of small stocks on our herd measure we calculate herd measure with stocks whose market values are 0.1 percent and percent of the total market capicalisation. We also calculate herd measure with stocks whose returns are available for the entire sample period (287 stocks), and with the constituents included in the S&P500 index definedondecember In particular for the last two cases we suspect surviviourship bias in 26

29 the data. Changes in the structure of the indices and the non-existence of some equity returns in the early part of our sample imply that the number of equities at the beginning of the sample period is necessarily less than the number of constituents for each index. Figure 2A shows that the herd measures calculated with these five different sets of stocks are close to each other. A few noticeable differences in our herd measure from including small stocks whose market values are less than 0.01 percent of the total market capitalisation can be found in the late 1990 s and early 2000 s. During this period herd measure calculated with small stocks became very large compared with those with relatively larger stocks. These results suggest that investors began to herd less (or even adverse herd) towards the end of 1990 s for small (Nasdaq) stocks. As the prices of small Nasdaq stocks increased exceptionally during the 1990 s, investors herded less (or more vigilant) for these stocks. This is consistent with our argument above that herding happens only when investors are confident. However, except the exceptional period there is little difference between the herd measures. Therefore our measure appears to be robust against survivourship bias. The herd measure calculated with the S&P500 index constituents is important in our study since we use similar methods to calculate herd measure in the UK and South Korean stock markets The Effects of Betas Since our sample is a subgroup, our results may also be exposed to selection bias. Forexamplewehaveremovecertainstockswhichwebelieveareless liquid and thus may cause bias in the estimates of betas. Since our measure is based on the estimates of betas, we need to investigate how robust herd measure for different subsets of betas. Using the estimated betas to rank 27

30 the stocks, we make four subgroups; large beta stocks (top 70%), small beta stocks (bottom 70%), middle beta stocks (middle 70%) and high-low beta stocks (except middle 30%). Then for each of these sub-samples we apply the same procedure outlined above. We plot the calculated herd measures for the four subgroups in Figure 2B. The figure shows that there is little difference between these subgroups in the sense that the dynamics of these subgroups are not different from each other. However there are some differences in the levels of herd measures between these groups. Unless we switch from one group to another over time to calculate herd measure, our selection ofstocksdoesnotcreatebias Factors in Linear Factor Models We use factor mimicking portfolios such as SMB, HML and momentum as control variables. Some correlation between the factors within the sample is inevitable given that firm specific characteristics are used to construct the factors. During our sample period positive correlations are found between the excess market return, SMB, and HML, whereas SMB is negatively correlated with momentum (not reported). Another impact from using additional factors could come from changes in the standard deviation of estimated beta. Adding factors can change the idiosyncratic errors and thus the standard deviation of the estimated beta even if there is no multicollinearity problem. To evaluate the effects of using additional factors, we calculate herd measures using the simple market model, the FF model, and the four factor model. we find that there is no significant difference in herding towards the market portfolio between the FF model and the four factor model. However, the herd measure based on the simple market model is generally larger than the two models, and is more volatile than the two. For example the large 28

31 increase of the herd measure with the market model during the late 1990 s and 2000 s is attributed to the factors SMB and HML. Therefore without considering these factors, our measure may not represent herd behaviour towards the market portfolio. Finally we note that there is little difference between the FF model and the four factor model except for 1980 and the end of For the UK and South Korean markets we use the FF model to reduce calculation burden Choice of τ As mentioned earlier, when τ is small, we could capture the time variation in betas, but small sample effects may be significant. For example in many academic studies 60 months is common, and in practice 5 to 7 years are used. To evaluate the effects of different periods of τ on herd measure, we also use 36, 48, and 60 monthly observations to estimate betas. Figure 2D shows that the herd measures move in similar directions. As expected the longer the time period to calculate betas, the smoother the herd measure becomes. However we have an unfavourable effect that herd measure lags when τ becomes larger. These results suggest that the negative effects of using 24 months are not serious, and we could enjoy more dynamics from using a short window. 3.3 International Comparison with UK and South Korean Stock Markets Data We use monthly data from January 1993 to November 2002 to compare slowly moving herding in the UK and South Korean stock markets. The 29