dependence between stock returns and investor sentiment in chinese ...

J Syst Sci Complex (2012) 25: 529–548

DEPENDENCE BETWEEN STOCK RETURNS AND INVESTOR SENTIMENT IN CHINESE MARKETS: A COPULA APPROACH∗ Xunfa LU · Kin Keung LAI · Liang LIANG

DOI: 10.1007/s11424-012-9332-0 Received: 19 November 2009 / Revised: 5 January 2011 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2012 Abstract Using data of newly opened stock trading accounts in China as a proxy of investor sentiment index, the authors employ the time-varying copula-GARCH model with Hansen’s skewed Student-t innovations to investigate the dynamic dependence between investor sentiment and stock returns. The empirical findings show that shifts in investor sentiment are asymptotically positively correlated to stock returns in extreme value situations in both A shares market and B shares market in China, that is to say, stock prices will increase (decrease) more when investors become more bullish (bearish). Also, results show that the dependence between investor sentiment and stock returns is time-varying, which means that the traditional Pearson’s correlation based on normal distribution is not enough to describe the relationship between stock market behavior and investor behavior. Key words accounts.

Behavioral finance, copula, GARCH, investor sentiment, newly opened stock trading

1 Introduction More and more attention is being paid to the investor sentiment theory by researchers working on behaviors of market participants in the financial industry. The classical finance theory supposes that investors are completely rational, resulting in researchers ignoring the existence of noise traders and their role in determining market movement. Based on this theory, many elegant financial models have been constructed, such as the capital asset pricing model (CAPM)[1−4] , the arbitrage pricing theory (APT)[5] , the efficient-market hypothesis (EMH)[6] , and so on. However, the increase in complexity of behavior of financial markets from the 1980s onwards has challenged the classical finance theory in several ways. Consequently, the behavioral finance theory has emerged, at least in part, in response to distortions in the traditional Xunfa LU Department of Management Sciences, City University of Hong Kong, Hong Kong, China; School of Business, University of Science and Technology of China, Hefei 230026, China. Email: [email protected]. Kin Keung LAI Department of Management Sciences, City University of Hong Kong, Hong Kong, China. Email: [email protected]. Liang LIANG School of Business, University of Science and Technology of China, Hefei 230026, China. Email: [email protected]. ∗ This research is supported by the National Natural Science Foundation of China under Grant No. 70821001. This paper was recommended for publication by Editor Shouyang WANG.

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XUNFA LU · KIN KEUNG LAI · LIANG LIANG

paradigm. Based on two building blocks (limits to arbitrage and cognitive psychology), it generally studies the features of financial markets more accurately, using models that are less narrow than those based on the expected utility theory (Von Neumann-Morgenstern) and arbitrage assumptions. During the 1980s, an important paper in behavioral finance field was written by Kahneman and Tversky[7]. They found that most traders were not normative and fully rational financial investors and proposed the prospect theory as an alternative to the classical finance theory (or the expected utility theory). Black[8] incorporated noise traders into the CAPM, and constructed the noise model. He found that the existence of noise traders improved liquidity of capital in stock markets, but caused markets to be somewhat inefficient. After Black, De Long, et al. (DSSW)[9] argued that changes in noise traders’ sentiments were difficult to predict for avoiding arbitrage. In other words, investor sentiment is a systematic factor that influences equilibrium prices. Different from the DSSW model, Barberis, et al. (BSV)[10] explained the influence of investor sentiment on stock prices on the basis of cognitive psychological biases. They found that in making forecasts, people pay too much attention to the strength of the evidence they are presented with, and too little attention to its statistical weight. In addition, Daniel, et al.[11] and Hong and Stein[12] investigated the influence of investor sentiment on stock prices by studying interactions between heterogeneous points of view. The increasing importance of investor sentiment in behavioral finance motivates us to investigate the relationship between investor sentiment and stock returns in the Chinese market, which is one of important emerging stock markets, but has not yet been studied seriously. Unlike previous research, we don’t first use investor sentiment as a factor of change in stock returns and then estimate its statistical significance, according to regression analysis. As demonstrated by Brown and Cliff[13] , investor sentiment does not necessarily impact stock returns directly, though there is vigorous evidence of a strong correlation between the two. Therefore, the idea of directly using investor sentiment as a regressive variable of stock returns may be questionable, per se. The traditional approach used to estimate the dependence between any two financial series is computation of their correlation, which hypothesizes that marginal distributions of financial series and the corresponding joint distributions are elliptical. However, it is commonly recognized that asset returns in financial markets are usually non-normal, especially fat-tailed, peaked and skewed. Therefore, the linear correlation used to model the dependence structure cannot correctly capture the real features of financial series. Although the conditional normality assumption in generalized autoregressive conditional heteroskedasticity (ARCH/GARCH) models generates some degree of unconditional excess kurtosis, it remains typically less than adequate to fully account for fat-tailed properties of the data. In order to overcome these problems, this paper resorts to a prevalent copula method to examine the dependence between investor sentiment and stock returns. To the best of our knowledge, to date, this relationship has not been examined, using the copula approach, in behavioral finance research. A copula is a powerful and user-friendly tool to describe dependent risks. Embrechts, et al.[14] first introduced this concept to finance literature. Since then, applications of copulas in finance have developed rapidly. Cherubini, et al.[15] gave a brief review of copula functions in finance, especially copulas for derivative pricing. Patton[16] extended the constant copula to the time-varying copula, or the unconditional copula to the conditional copula. After the methodological expansion of Patton, the conditional copula began to be used in financial models. Due to the success of GARCH models in univariate time series, many researchers use them to construct marginal distributions of asset returns, and then combine these marginal distributions with a copula to build the conditional joint distribution of asset returns[17−19] . For further applications of copula theory, readers can refer to [20–22]. Similar to [17], this paper employs a GARCH model with innovations modeled by Hansen’s

DEPENDENCE OF INVESTOR SENTIMENT: A COPULA APPROACH

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skewed Student-t distribution to examine the conditional volatility of investor sentiment and stock returns in the Chinese market, and then constructs the joint distribution of variables, based on a copula. Furthermore, we construct a time-varying copula-GARCH model to fit the data. But unlike extant literature, the paper applies the symmetrized Joe-Clayton copula, as used in [16], to investigate the dependence between investor sentiment and stock returns, because this copula can contemporarily capture the symmetric and asymmetric tails’ extreme dependence. This paper selects data of the Shanghai Stock Exchange (SSE) as the proxy of Chinese stock markets. According to the policy of Chinese stock markets, a listed company can issue two types of shares: “A” shares, traded in RMB, for domestic investors, and “B” shares, traded in USD, for both domestic and foreign investors, including overseas Chinese residing in Hong Kong, Macau or Taiwan. These two classes of shares are otherwise identical in every respect including voting rights and dividend payments[23] . In completely rational conditions, investors will have the same expectations for A and B shares of a company, and both types of shares of a company will behave in the same way. However, it is obvious that A and B shares of the same company do behave differently. This stimulates us to investigate the dependence between investor sentiment and stock returns in A and B shares market and to compare the difference. This study makes three contributions. First, we introduce the copula to the field of behavioral finance and investigate the dependence between investor sentiment and stock returns in the Chinese market. The second contribution is to show how a time-varying copula model with marginal distributions fitted by the GARCH model with skewed student-t innovations is applied to measure the time-varying dependence between investor sentiment and stock returns. It is found that the strong persistence, in the dynamics of the dependence, reflects fluctuations in the dependence parameter, over the sample period. Finally, we investigate the relationship between investor sentiment and returns in A and B shares markets, and compare the difference. It is shown that fluctuations in the dependence are more violent in B shares market than in A shares market. That is to say, investor sentiment and stock returns are more uncertain in B shares market than in A shares market. The remainder of the paper is organized as follows. Section 2 reviews the recent literature on theoretical considerations concerning investor sentiment and market volatility, and measures of sentiment. Section 3 presents marginal models, including ARMA and GARCH models, copula theory, and methods for copula estimation. In Section 4, we describe the data and the empirical results. Finally, we summarize the results and derive conclusions.

2 Measurement of Investor Sentiment Index in Chinese Market Investor sentiment is broadly defined as a belief about future cash flows and investment risks that is not justified by the facts at hand[24] , and thus reflects the optimism or pessimism of investors. However, investor sentiment is a vague and intangible concept that does not cause uniform perceptions for all. One can feel its existence, but one can’t point out what it is, how high it is at a given point of time, and what the difference is between investor sentiment at different points of time. Therefore, the important question when investigating the relationship between investor sentiment and stock market returns is how to measure investor sentiment. In previous literature, methods used to measure investor sentiment are not identical. Broadly, methods to measure investor sentiment can be put into two categories. The first is to directly survey investors to derive a direct index reflecting investor sentiment at different points of time. The other is to extract objective data (indirectly reflecting investor sentiment) from financial markets data, and use the objective data as proxies of investor sentiment, viz.

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the indirect index. Direct indices measuring investor sentiment include investors intelligence sentiment index (IIS)[25] , American association of individual investors (AAII)[26] , and bearish sentiment index (BSI). In Chinese markets, direct indices include the bullish sentiment index of China Central Television (CCTV) and the bull/bear index of Stock Market Trend Analysis weekly. Indirect indices can be further categorized into five types: 1) Indices reflecting the overall market performance, such as Advance Decline Line, Arms Index, and New High-New Low Index; 2) Trading indices, such as Short Sales to Total Sales, Odd-lot Sales to Purchases, Change in Short Interest, Change in Margin, Turnover, and Newly Opened Stock Trading Accounts; 3) Trading indices for derivative products, such as Put/Call Ratio, and Change in Net Position of S&P 500 Index; 4) Indices of special products in stock markets, such as Wall Street Analyst Sentiment Index, and the initial 1-day return on IPOs (IPORET); and 5) Consumer Confidence indices, such as University of Michigan Consumer Confidence Index, Conference Board Consumer Confidence Index, and China Consumer Confidence Index. From the review above, it can be seen that ways of measuring investor sentiment still vary widely, among hundreds of different schools of thoughts. In Chinese markets, research analysts mainly use the bullish sentiment index of CCTV and the relative survey of China Security News as proxies of investor sentiment. In view of the limited availability and continuity of reliable data, a good direct index that can be used to investigate investor sentiment in China is still not available. Therefore, we consider indirect methods of using trading data to construct the investor sentiment index (ISI) of the Chinese stock market. We select the weekly newly opened stock trading accounts data as the proxy reflecting the change of investor sentiment in the Chinese market. This variable has three advantages. First, shifts in newly opened stock trading accounts can directly reflect investment intentions and willingness of investors to invest. When investors predict a continued rise of stock market, they behave in an optimistic manner, i.e., more new investors enter the market. This leads to rapid increase of newly opened stock trading accounts. Contrarily, when investors predict a bearish tendency, they behave in a pessimistic manner. Hereupon, the market sentiment takes on a wait and see attitude; external investors either delay entering the stock market, or transfer their investments to other avenues. Therefore, not only do newly opened stock trading accounts decrease, but some of the existing investors either quit the stock market or reduce their investments quantitatively. Thus, the index of newly opened stock trading accounts can efficiently measure the change of investor sentiment in the stock market, especially by reflecting expectations of “Bullish”, “Fair”, and “Bearish” phases. The second advantage is the good continuity of the data. Although the bullish sentiment index of CCTV is widely used in analysis of behavior of investors in financial markets in China, it had once been discontinued by its publisher. The newly opened stock trading accounts data have been released un-interruptedly since 2005. The data can be directly obtained from the homepage of China Securities Depository and Clearing Corporation Limited. In addition, this data is more objective and reliable than the subjective index obtained from a sampling survey. The survey data in Chinese stock market is usually aimed at institutional investors or investment analysts. On one hand, neutrality and objectivity of survey objects are questionable. On the other hand, size of the sample often varies, from a dozen to nearly a hundred, which can result in large errors. On the basis of the above advantages, it is rational to select the newly opened stock trading accounts as the proxy of investor sentiment.

3 Models and Methodology As stated before, empirical studies in finance literature have highlighted the importance of allowing for fat-tailness, skewness, and non-normality of returns for asset allocation, portfolio


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management and pricing models. Moreover, the inter-dependence between returns of different assets often results in nonlinear structures and asymmetric extremal behavior, which the classical correlation coefficient is unable to describe adequately. Therefore, copula is a natural choice to solve the problem, due to its successful applications in finance. Briefly, a copula is a multivariate distribution of marginally uniform random variables. In contrast, the copula concept which originally dates back to Sklar[27] but was made popular in finance applications through the pioneering work of [14], provides a flexible tool to capture different patterns of dependence. For simplicity, we assume that two random variables are considered in copula functions in this work, while the higher dimensional copulas are not; high-dimensional copulas have been investigated in several publications[28−29] . 3.1 The Copula Theory The intuitional and important idea of copula modeling is that a joint distribution can be factored into its marginals, and a dependence function called a copula[20] . Nelsen[30] presented a nice introduction to copula’s theoretical and practical aspects. A bivariate copula can be defined as follows. Definition 1 A two-dimensional copula is a bivariate cumulative distribution function (c.d.f.), C, with uniform distribution margins in I = [0, 1], and has the following properties: 1) Dom I 2 = [0, 1]2 2) For every u, v in I, C(u, 0) = C(0, v) = 0, and C(u, 1) = u, and C(1, v) = v;

(1)

3) For every u1 , u2 , v1 , v2 in I such that u1 ≤ u2 and v1 ≤ v2 , C(u2 , v2 ) − C(u2 , v1 ) − C(u1 , v2 ) + C(u1 , v1 ) ≥ 0.

(2)

Following the above definition, if F1 and F2 are univariate distribution functions, C(F1 (x1 ), F2 (x2 )) is a bivariate c.d.f. with margins F1 and F2 , because u = F1 (x1 ) and v = F2 (x2 ) are uniform random variables. Sklar’s theorem[27] provides the theoretical proof interpreting why the margins and the dependence structure can be separated. Theorem 1 (Sklar’s theorem for unconditional distributions) Let F be a joint distribution ¯ function with margins F1 and F2 . Then there exists a copula C such that, for all x1 , x2 ∈ , F (x1 , x2 ) = C(F1 (x1 ), F2 (x2 )).

(3)

If F1 and F2 are continuous, then C is unique; otherwise, C is uniquely determined on Ran F1 ×Ran F2 . Conversely, if C is a copula and F1 and F2 are distribution functions, then function F , defined by Equation (3), is a joint distribution function with margins F1 and F2 . This theorem provides a way to analyze the dependence structure of multivariate distributions without studying marginal distributions and shows that it is possible to separately specify the dependence among variables, and marginal density of each variable. Because we’d like to investigate whether the dependence between investor sentiment and stock returns is time-varying, like the usual stock returns in financial markets, we need to allow for the conditional copula. Actually, Patton[16] extended the unconditional copula theorem to the conditional case. For exact depiction, we use the following conventions: X1 and X2 denote two continuous random variables, Ω is a conditioning variable or vector of variables, F12Ω is the joint distribution of (X1 , X2 , Ω ), F12|Ω is the conditional distribution of (X1 , X2 ), given Ω , and conditional distributions of X1 |Ω and X2 |Ω are denoted as F1|Ω and F2|Ω , respectively. Furthermore, we adopt the usual convention of denoting cumulative distribution functions (c.d.f)


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and random variables using upper case letters, while lower case letters are used for probability density functions (p.d.f) and realization of random variables. Throughout this paper, we assume that F12Ω is sufficiently smooth for all required derivatives to exist, and that F12|Ω , F1|Ω and F2|Ω are continuous. Theorem 2 (Sklar’s theorem for conditional distributions) Let F1|Ω and F2|Ω denote conditional distributions of X1 and X2 given the information set Ω . If F12|Ω is the joint multivariate conditional distribution and F1|Ω and F2|Ω are continuous in x1 and x2 , then there exists a unique conditional copula C such that F12|Ω (x1 , x2 |Ω ) = C(F1|Ω (x1 |Ω ), F2|Ω (x2 |Ω )|Ω ).

(4)

Conversely, if we let F1|Ω and F2|Ω be the conditional distributions of X1 |Ω and X2 |Ω , respectively, and {C(·, ·|Ω )} be a family of conditional copulas measurable in Ω , then function F12|Ω , defined by Equation (4), is a conditional bivariate distribution function with conditional marginal distributions F1|Ω and F2|Ω . These Sklar’s theorems imply that any two univariate distributions, even of different types, may be linked together via a copula to define a valid bivariate distribution, as long as the information set used is unchanged. The bivariate distribution constructed via a copula, not necessarily a bivariate normal distribution, can be used in financial research. Many copulas have been constructed by academics in accordance with Sklar’s theorem. Here we just introduce some simple and commonly used copulas, such as Normal copula (also Gaussian copula), Student-t copula, Gumbel copula, Plackett copula, and the symmetrized Joe-Clayton (SJC) copula. Copula Family Normal Student-t Gumbel

Table 1 Distribution functions of five copulas C(u, v) Φ −1 (u) Φ −1 (v) 2 −2ρst+t2 −1 −1 1 Φρ (Φ (u), Φ (v)) = −∞ exp{− s 2(1−ρ }dsdt 2) −∞ 2π(1−ρ2 )1/2 t−1 t−1 s2 −2ρst+t2 − υ+2 −1 −1 1 υ (u) υ (v) tρ,υ (tυ (u), tυ (v)) = −∞ {1 + υ(1−ρ2 ) } 2 dsdt −∞ 2π(1−ρ2 )1/2 θ exp{−[(− log(u)) + (− log(v))θ ]1/θ } √

1+(θ−1)(u+v)−

[1+(θ−1)(u+v)]2 −4uvθ(θ−1)

Plackett 2(θ−1) SJC 0.5(CJ C (u, v|τ U , τ L ) + CJ C (1 − u, 1 − v|τ L , τ U ) + u + v − 1) Note: Φ −1 : The inversion of c.d.f. of a standard normal distribution, Φρ : The c.d.f. of a standard bivariate normal distribution with Pearson correlation ρ, t−1 υ : The inversion of c.d.f. of a Student-t distribution with υ degrees of freedom, t−1 ρ,υ The c.d.f. of a bivariate Student-t distribution with υ degrees of freedom and Pearson correlation ρ, θ: Parameter of Gumbel and Plackett copulas.

In this paper, we use the SJC copula because we want to know how the market will behave when investors have a more bearish (bullish) mood. The dependence between returns of different assets in stock markets exhibits greater correlation during market downturns than during market upturns, where the SJC copula has special advantages in capturing this market feature. Following [16], the SJC copula can be expressed as follows. CSJC (u, v|τ U , τ L ) = 0.5(CJC (u, v|τ U , τ L ) + CJC (1 − u, 1 − v|τ L , τ U ) + u + v − 1),

(5)

where, CJC is the Joe-Clayton copula (usually called BB7 copula). CJC (u, v|τ U , τ L ) = 1 − (1 − {[1 − (1 − u)κ ]−γ + [1 − (1 − v)κ ]−γ − 1}−1/γ )1/κ , where, κ = 1 log2 (2 − τ U ), γ = −1 log2 (τ L ), and τ U ∈ (0, 1), τ L ∈ (0, 1).

(6)


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The SJC copula has two parameters, τ U and τ L , which are measures of dependence, known as tail dependence (This concept is illustrated in the following section). 3.2 Dependence Measures One motivation for considering copulas for financial data is that in this area, the linear correlation coefficient, Pearson’s ρ, is often the only measure of dependence used to explain the association between variables. However, analysts have the other perceptions of risk that are not captured by ρ, such as the dependence structure. In addition to linear correlation ρ, there are various other measures of dependence, among which Spearman’s ρs and Kendall’s τ , usually called rank correlations, are two scale free measures of dependence which are commonly studied with copula models. Let (X1 , Y1 ), (X2 , Y2 ), (X3 , Y3 ) be three independent vectors of continuous random variables with joint distribution functions F and Copula C. Then, Spearman’s ρs and Kendall’s τ can be defined as follows[30] : ρX,Y = ρs = 3(P [(X1 − X2 )(Y1 − Y3 ) > 0] − P [(X1 − X2 )(Y1 − Y3 ) < 0]) uvdC(u, v) − 3, = 12

(7)

τX,Y = τ = P [(X1 − X2 )(Y1 − Y2 ) > 0] − P [(X1 − X2 )(Y1 − Y2 ) < 0] C(u, v)dC(u, v) − 1, =4

(8)

I2

I2

where, P is the probability of events, ρs ∈ [−1, 1] and τ ∈ [−1, 1]. The higher the ρs or τ value, the stronger is the dependence. From Equations (7) and (8), it can be seen that these two rank statistics are very important because of their relationship with copula. It has been proved that under weak regularity conditions in the copula family, inversion of Kendall’s τ yields a consistent estimator of the dependence parameter of the copula[31] . For simplicity, we only show the relationship of Kendall’s τ and copulas’ parameters in Table 2. Copula Family Normal Student-t Gumbel Plackett SJC

Table 2 Copulas and their parameters Range of parameters Estimates of parameters [−1,1] ρ = sin(πτ /2) [−1,1] ρ = sin(πτ /2) [1, ∞) θ = 1/(1 − τ )forτ > 0 11 (0, ∞) − {1} Solution of 4 0 0 C(u, v; θ) − 1 − τ = 0 v; θ)dC(u, k ∈ (1, ∞) k=1 log2 (2 − τ U ) r ∈ (0, ∞) r = −1 log2 (τ L )

Another useful dependence measure defined by copulas is the tail dependence, which measures the probability that both variables are in their lower or upper joint tails. One of the key properties of copulas is that they are invariant under increasing and continuous transformations. For example, the copula does not change with returns or logarithm of returns, which is usually not true for the correlation. The tail dependence preserves the property of copula, and is invariant under strictly increasing transformations. Let X and Y be two random variables with joint distribution F (whose margins are F1 and F2 ) and copula C. The lower (left) and upper (right) tail dependence coefficients are defined as λL = lim+ P (Y < F2−1 (α)|X < F1−1 (α)) = lim+ α→0

α→0

C(α, α) , α

(9)


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λU = lim− P (Y > F2−1 (α)|X > F1−1 (α)) = lim− α→1

x→1

1 − 2α + C(α, α) , 1−α

(10)

where, α is the probability level and F1−1 (α) and F2−1 (α) are the 100α-th percentiles of F1 and F2 , respectively, λL ∈ [0, 1] and λU ∈ [0, 1]. It is worth noting that if λL ∈ (0, 1] (λL = 0), C has lower tail dependence (independence); if λU ∈ (0, 1] (λU = 0),C has upper tail dependence (independence). For unification of the notation, we substitute τ L (τ U ) for λL (λU ). According to the definition above, the upper (lower) tail dependence of each copula can be computed. For copulas in Table 2, upper (lower) tail dependences are listed in Table 3. Copula Family Normal Student-t Gumbel Plackett SJC

Table 3 Upper and lower tail dependence parameters τL τU 0 √ √ 0 √ √ √ √ τ = 2tυ+1 (− υ + 1 1 − ρ/ 1 + ρ) τ = 2tυ+1 (− υ + 1 1 − ρ/ 1 + ρ) 0 0 0 0 τ L = 2−1/r τ U = 2 − 21/k

As the dependence of financial returns is not always uniform, we follow [16] and allow for time-varying dependence. Based on our empirical data, we propose the following evolution equation for tail dependences of the SJC copula, on lines similar to[16] : L τtL = Λ(ωL + βL τt−1 + αL |ut−1 − vt−1 |),

(11)

U τtU = Λ(ωU + βU τt−1 + αU |ut−1 − vt−1 |),

(12)

where Λ = (1 + e−x )−1 is the logistic transformation, used to keep τ L and τ U within (0, 1) at all times. This paper only considers dependence of investor sentiment and stock returns; it doesn’t consider the impacts of different copulas on the dependence. Therefore, we mainly use the SJC copula to analyze the empirical data in the following sections. 3.3 Estimation for Parameters: The Inference for Margins Method (IFM) We now assume that the copula function depends on a set of unknown parameters through the function Θ (x1t−1 , x2t−1 ; θC ). We also denote f1 (x1t ; θ1 ) and f2 (x2t ; θ2 ) as the marginal p.d.f of random variables X1 and X2 , respectively. We set θ = (θ1 , θ2 , θC ) to be the vector of all parameters to be estimated. According to Sklar’s theory and based on Equations (3) and (4), if F1 (F1|Ω ) and F2 (F2|Ω ) are differentiable, and F (F12|Ω ) and C are twice differentiable, then the density function equivalents of (3) and (4) are easily obtained. ∂ 2 F12|Ω (x1 , x2 |Ω ) ∂x1 ∂x2 ∂F1|Ω (x1 |Ω ) ∂F2|Ω (x2 |Ω ) ∂ 2 C(F1|Ω (x1 |Ω ), F2|Ω (x2 |Ω )|Ω ) = · · ∂x1 ∂x2 ∂u∂v = f1|Ω (x1 |Ω ) · f2|Ω (x2 |Ω ) · c(u, v|Ω ),

f12|Ω (x1 , y2 |Ω ) =

(13)

where, u = F1 (x1 ) (u = F1|Ω (x1 |Ω )) and v = F2 (x2 ) (v = F2|Ω (x2 |Ω )). Consequently, the expression for the log-likelihood function of bivariate distribution function


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is given by l(θ) =

T

ln c(F1|Ω (x1t |Ω ; θ1 ), F2|Ω (x2t |Ω ; θ2 )|Ω ; Θ(x1t−1 , x2t−2 ; θC ))

t=1

+

2 T

ln fj|Ω (xjt |Ω ; θj ).

(14)

t=1 j=1

For unconditional copula, the formulas are similar. We only need to ignore information set Ω for the unconditional case. Without loss of generality, we denote the log-likelihood as l(θ) = l1 (θ1 ) + l2 (θ2 ) + lC (θ1 , θ2 , θC ),

(15)

where, l1 (θ1 ) and l2 (θ2 ) are log-likelihood of marginal distributions, and lC (θ1 , θ2 , θC ) is the log-likelihood of p.d.f. of copula. Maximum-likelihood estimation involves maximizing the log-likelihood function (15) or (16)

simultaneously with all parameters, yielding parameter estimators denoted by θ ML = ( θ 1 , θ 2 , θ 3 ) ,

such that θ ML =

arg max l(θ).

(16)

Under regularity conditions, ML estimators are consistent, efficient, and asymptotically normal. In some applications, however, the ML estimation method may be difficult to implement, because of a large number of unknown parameters or the complexity of the model. In such a case, it may be necessary to adopt a two-step ML procedure, also called inference functions for margins (IFM). This approach[23,36] can be viewed as the ML estimation of the dependence structure, given the estimated margins, and is computationally simpler than the ML method[32−33] . Joe[34] , for unconditional copula, and Patton[16], for conditional copula, prove that under regular conditions, the IFM estimator is consistent and asymptotically normal. Throughout this section, we assume that the usual regularity conditions for asymptotic maximum likelihood theory hold for the multivariate model (i.e., the copula), as well as for all of its margins (i.e., the univariate distribution). According to the IFM method, parameters of marginal distributions are estimated separately from parameters of the copula. In the first stage, we estimate marginal parameters θ1 and θ2 by implementing the ML method of univariate marginal distributions: θ j=

arg max lj (θj ), j = 1, 2.

(17)

In the second stage, given θj (j=1, 2), we estimate the copula parameter θC : θ C=

arg max lC ( θ 1 , θ 2 , θC ).

(18)

3.4 Models for Marginal Distributions It is well documented in the literature that daily asset returns show fat-tailness and heteroskedasticity. As usual, the error variance is unknown and must be estimated from the data. The generalized autoregressive conditional heteroskedasticity (GARCH) model is a common approach to model time series with heteroskedastic errors. Also, it is well known that the residuals obtained from a GARCH model are generally non-normal, with features such as fat-tailness

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and asymmetry. This observation has led to introduction of fat-tailed distributions for innovations. For instance, Nelson[35] considered the generalized error distribution, while Bollerslev and Wooldridge[36] focused on Student-t innovations. A notable work in the field of GARCH is that Hansen[37] generalized symmetric Student-t innovations to asymmetric skewed Student-t innovations. In our comparisons, GARCH models driven by Hansen’s skewed Student-t distributed innovations are employed and they often fit returns clearly better than the others with Student-t or normal innovations. Without loss of generality, let the logarithm of financial time series be {rt }Tt=1 . We fit the autoregression-moving average (ARMA) model to the mean model of variables, and the GARCH model to the conditional variance model of variables. These general models are given as follows. For mean model: ARMA(r, s) rt = μ +

r

φi rt−i + εt −

i=1

s

ϑj εt−j .

(19)

βn ht−n ,

(20)

j=1

For conditional variance model: GARCH(p, q) ht = α0 +

p

αm ε2t−m +

m=1

εt =

ht ηt

q n=1

ηt ∼ iid Skewed − t(υ,λ),

(21)

max(p,q) (αi + βi ) < 1, where, μ is a constant, and {εt } are residuals. α0 > 0, αm ≥ 0, βn ≥ 0, i=1 ht is the conditional variance of εt , {ηt } are the independent and identically distributed random variables, distributed to Hansen’s skewed Student-t distribution (with parameter υ and λ), E(η) = 0, and Var(η) = 1, and r, s, p, q ≥ 0. Hansen’s skewed Student-t distribution, Skewed − t(υ, λ), is defined by ⎧ 1 2 −(υ+1) bc(1 + υ−2 ( bz+a ⎪ 1−λ ) ) ⎪ ⎪ if z < −a/b, ⎨ 2 f (z; υ, λ) = ⎪ 1 bz+a 2 −(υ+1) ⎪ ⎪ bc(1 + υ−2 ( 1+λ ) ) ⎩ if z ≥ −a/b, 2 υ−2 2 where, a ≡ 4λc υ−1 , b ≡ 1 + 3λ2 − a2 , c ≡ √

Γ ( υ+1 2 )

π(υ−2)Γ ( υ 2)

, and 2 < υ ≤ ∞ and −1 < λ < 1 denote

the degree-of-freedom parameter and the asymmetry parameter, respectively. If a random variable Z has the density f (z; υ, λ), we write Z ∼ Skewed − t(υ, λ).

4 Empirical Results 4.1 The Data This paper targets the dependence between investor sentiment and stock returns in the Chinese market, using the time-varying copula-GARCH model with Hansen’s skewed Student-t innovations. We investigate the interactions between newly opened stock trading accounts and stock returns in A and B shares market in the Shanghai Stock Exchange. We use the SSE A and B share index to represent the performances of A and B shares markets, respectively. The data are from the SSE and the website of China Securities Depository and Clearing Corporation


539

Limited[38] , sampled at a weekly frequency, and cover the period from 3 June 2005 to 10 July 2009. Table 4 presents some summary statistics of the data. Similar to typical analysis of financial time series, this paper adopts the logarithm of newly Aopened stockBtrading accounts B and stock A B index as the variables. We denote {rxt = log(xA t xt−1 )} and {rxt = log(xt xt−1 )} as logrates of newly opened stock trading accounts B for A and B shares markets, respectively, and A A B {ryt = log(ytA yt−1 )} and {ryt = log(ytB yt−1 )} as log-returns of SSE A and B share indices, A B respectively, where xt (A shares) and xt (B shares) are values of newly opened stock trading accounts at time t; and ytA and ytB are values of the two stock indices at time t. Table 4 Descriptive statistics for ISI and share market index Mean Std.Dev. Minimum Maximum Skewness Kurtosis A Shares Market: ISI Share Index B Shares Market: ISI Share Index

Jarque-Bera

0.015716 0.311122 −0.760159 1.161789 0.681065 4.526997 35.2416 (0.0000) 0.005557 0.046342 −0.149236 0.139306 −0.253303 3.891264 8.84593 (0.0000) 0.010213 0.408247 −1.012645 2.260710 0.005636 0.063381 −0.181694 0.291941

1.901828 11.21908 690.344 (0.0000) 0.445484 5.552170 61.5039 (0.0000)

The above table shows that each mean of newly opened stock trading accounts is larger than the mean of its corresponding stock index, just as the standard deviation. Skewnesses of series are different from zero; three of them skew to the right, and only log-returns of A share skews to the left. All series exhibit excess kurtosis. The kurtosises in A shares market are less than those in B shares market. Both skewness and excess kurtosis imply that the series are not normally distributed. This is confirmed by the Jarque-Bera test. All Jarque-Bera statistics have very low p-values, meaning that normality of unconditional distribution of each series is strongly rejected. In order to observe the dynamics, we plot the series in Figures 1 and 2. For comparability of these curves, we multiply log-rates of newly opened stock trading accounts by 10, and log-returns of stock returns by 100. Sentiment Index for A Shares SSE A Share Index

15

10

Log−values

5

0

−5

−10

−15 2005−6−10

2006−4−7

2007−1−26

2007−11−23 Time

2008−9−5

2009−7−10

Figure 1 Newly opened stock trading accounts for A shares and SSE A share index


540

30 Sentiment Index for B Shares SSE B Share Index

25 20 15

log−values

10 5 0 −5 −10 −15 −20 2005−6−10

2006−4−7

2007−1−26

2007−11−23 Time

2008−9−5

2009−7−10

Figure 2 Newly opened stock trading accounts for B shares and SSE B share index

Figures 1 and 2 show that changes of investor sentiment are not consistent with those of stock market indices. It is noticeable that both have obvious volatility clustering and heteroskedasticity, that is, larger (smaller) volatility follows another larger (smaller) volatility, and the variance changes over time. Also, Figure 1 shows that investor sentiment index and share index in A shares market present larger volatility from March 2008 to May 2008, while investor sentiment index and share index in B shares market present larger volatility from February 2007 to April 2007 in Figure 2. This implies that volatilities of investor sentiment index and share index in A and B shares market are different. 4.2 Estimation of the Marginal Models In this section we apply the methodology discussed above to investigate the structure of dependence between investor sentiment and stock returns. According to the IFM approach, we first estimate marginal distributions for each series. Then, we estimate the parameters of copula. From unit tests ADF and PP (Table 5), it can be seen that each series is stationary at 0.05 level. The Ljung-Box tests in Table 6 show that each series has autocorrelation. This is confirmed by the sample autocorrelation function (ACF) and partial autocorrelation function (PACF) (Figures 3 and 4). Therefore, we resort to ARMA model to fit the conditional mean of each series in order to eliminate the autocorrelation. In light of ACF and PACF (Figures 3 and 4), and the AIC and SC criteria, the appropriate conditional mean model of each series is A B B illustrated in Table 7 which shows that {rxt }, {rxt }, and {ryt } are mainly influenced by lag 1, A while {ryt } is influenced by lag 1, lag 2, and lag 3. This implies that investor sentiment in A shares market is influenced by returns of the past three weeks, and persists for a long time.

541

DEPENDENCE OF INVESTOR SENTIMENT: A COPULA APPROACH Table 5 Unit root test of ISI and share index ADF PP A Shares Market: ISI −17.80021 −17.79487 (0.0000) (0.0000) Share Index −13.38213 −13.64826 (0.0000) (0.0000) B Shares Market: ISI −13.02135 −13.58669 (0.0000) (0.0000) Share Index −11.90971 −11.94371 (0.0000) (0.0000) Sample ACF of SSE A Share Index 0.3

0.2

0.2

Sample Autocorrelation


Sample ACF of ISI for A Share 0.3

0.1 0 −0.1 −0.2 0

5

10 Lag

15

0.1 0 −0.1 −0.2

20

0

Sample PACFof ISI for A Share

10 Lag

15

20

Sample PACF of SSE A Share Index 0.3 Sample Partial Autocorrelations

0.3 Sample Partial Autocorrelations

5

0.2 0.1 0 −0.1 −0.2 0

5

10 Lag

15

20

0.2 0.1 0 −0.1 −0.2 0

5

10 Lag

15

20

Figure 3 Sample correlogram of ISI and share index for A market Table 6 Ljung-Box test of ISI and share index Lags 1 A Shares Market: ISI 8.6913 (0.0032) Share Index 0.5537 (0.4568) B Shares Market: ISI 1.5928 (0.2069) Share Index 5.7906 (0.0161) Note: * indicates 7 lags, values in the parentheses are P-values.

5

10

10.2462 (0.0686) 7.6620 (0.1759)

10.9260 (0.3633) 17.4575 (0.0648)

8.6842 (0.1223) 6.9914 (0.2213)

13.5226* (0.0604) 12.6599 (0.2433)


542

Sample ACF of SSE B Share Index 0.3

0.2

0.2



Sample ACF of ISI for B Share 0.3

0.1 0 −0.1 −0.2 0

5

10 Lag

15

0.1 0 −0.1 −0.2

20

0

Sample PACF of ISI for B Share

10 Lag

15

20

Sample PACF of SSE B Share Index 0.3 Sample Partial Autocorrelations

0.3 Sample Partial Autocorrelations

5

0.2 0.1 0 −0.1 −0.2 0

5

10 Lag

15

20

0.2 0.1 0 −0.1 −0.2 0

5

10 Lag

15

20

Figure 4 Sample correlogram of ISI and share index for B market Table 7 Conditional mean equations of each series Conditional mean equations Log-likelihood AIC ISI in A shares market AR(3) −41.77891 0.425661 SSE A Share Index ARMA(3,3) 337.2798 −3.319395 ISI in B shares market ARMA(1,1) −101.7654 1.032491 SSE B Share Index AR(3) 272.3896 −2.700394

SC 0.442095 −3.203550 1.065360 −2.683960

Based on the ARMA model of each series, we perform Ljung-Box and ARCH tests on residuals of each model. Table 8 shows that autocorrelations of four series have been completely eliminated, and three series have ARCH effect, except the series of ISI in A shares market. This means that we can use the GARCH model to fit these series, except the series of ISI in A shares A market. According to these tests, we use AR(3) model to fit {rxt }, ARMA(3,3)-GARCH(1,1) A B model to fit {ryt }, ARMA(1,1)-GARCH(1,1) model to fit {rxt }, and AR(3)-GARCH(1,1) model B }. Distributions of innovations for all four models are distributed to Hansen’s skewed to fit {ryt Student-t distributions. We use EViews6.0 to estimate parameters of the four marginal distributions. The results are displayed in Table 9. The ARMA terms are significant autoregressive terms and reject the A B null hypothesis at significance level of 0.05. Both ARCH and GARCH terms for {ryt }, {rxt }, B and {ryt } are strongly significant at 0.05 level, indicating heteroskedasticity of data. Kurtosis parameters ν of the four marginal models range from 3.538 to 7.274, and skewness parameters λ range from −0.068 to 0.343. All of them reject the null hypothesis, meaning that the normal or Student-t distribution is not more appropriate than the skewed Student-t distribution to fit the residuals. In order to evaluate the goodness of fit of the marginal models, we obtain Quantile-Quantile (QQ) plots of standardized residuals, and compare these QQ plots with QQ plots of standard normal distribution of standardized residuals. The results are displayed in Figures 5 and 6.


Lags

Table 8 Ljung-Box test and ARCH test of residuals of conditional mean equation Ljung-Box test for autocorARCH test for heteroskedasticrelation ity

A

ISI Share Index

B

ISI

Share Index

1 3.6714e-08 (1.0000) 0.0047 (0.9451) 0.0038 (0.9508) 4.7644 (0.4453) 0.0047 (0.9456)

5 1.6777 (0.8917) 2.2205 (0.8178) 10.7758 (0.3752)

10 2.5853 (0.9896) 9.4281 (0.4920) 0.4325 (0.5108)

1 2.6188 (0.1056) 1.5571 (0.2121) 2.6125 (0.7595)

5 5.9817 (0.3080) 12.1116 (0.0333) 50.3575* (0.0000)

10 13.5749 (0.1933) 21.7905 (0.0162)

1.4167 (0.9225)

5.2223 12.8963 (0.8758) (0.0003)

13.9565 (0.0159)

14.9889 (0.1325)

* indicates 15 lags Table 9 Estimators of parameters in the four marginal distributions ARMA(r, s) GARCH(p, q) Skewed − t(ν, λ) A

ISI Share Index

B

ISI

Share Index

AR(1): −0.203538 Constant: 0.005373 AR(1): 0.319544 AR(2): 0.311449 AR(3): −0.860450 MA(1): −0.284083 MA(2): −0.276237 MA(3): 0.969085 AR(1): −0.539475 MA(1): 0.653545 AR(1): 0.174061

/

ν = 5.345

λ = 0.142

Constant: 1.584028e−05 ARCH: 4.053572e−09 GARCH: 0.9997999959

ν = 7.274

λ =-0.129

Constant: 0.0380 ARCH: 0.4712 GARCH: 0.3965 Constant: 0.0008 ARCH: 0.2418 GARCH: 0.6060

ν = 3.538

λ = 0.343

ν = 4.111

λ =-0.068

543


544

0.8 0.6 0.4 0.2 0

0 0.2 0.4 0.6 0.8 1 Quantile of Standard Normal Distribution

1 0.8 0.6 0.4 0.2 0

0

0.2 0.4 0.6 0.8 Quantile of Skewed−t Distribution

Figure 5

1

Quantile of Empirical Distribution

QQ Plot of SSE A Share Index




QQ Plot of ISI for A Share 1

1 0.8 0.6 0.4 0.2 0


1 0.8 0.6 0.4 0.2 0

0

0.8 0.6 0.4 0.2


1 0.8 0.6 0.4 0.2

0


Figure 6

1


QQ Plot of SSE B Share Index




QQ Plot of ISI for B Share

0

1

QQ plots of ISI and share index for A market

1

0


1 0.8 0.6 0.4 0.2 0


1 0.8 0.6 0.4 0.2 0

0


1

QQ plots of ISI and share index for B market

From Figures 5 and 6, it is obvious that GARCH models with skewed Student-t innovations are better than those with normal innovations. Similar results are found in Student-t innovations (not drawn in the figures to save space). This is because the skewed Student-t distribution can contemporarily capture fat-tailness and asymmetry in the financial data. It implies that the skewed Student-t distribution fits the data quite well.


545

4.3 Estimation of Copula Parameters According to the IFM method, given the estimators of marginal parameters, we can perform estimation of copula parameters using Equation (13) to (18). We estimate parameters of both constant SJC copula and time-varying SJC copula. Estimations of lower tail dependence and upper tail dependence for A shares market and B shares market are provided in Table 10. Table 10 Estimations of lower and upper tail dependence Constant parameters Time-varying parameters τL

τU

A Share Market

0.137768

0.142370

B Share Market

0.014063

0.332445

τL ωL = −0.456813 βL = 1.604082 αL = −6.163277 ωL = −11.670260 βL = 14.178350 αL = 0.041855

τU ωU βU αU ωU βU αU

= −0.572210 = −5.563675 = −3.739442 = −0.907784 = 1.772238 = −1.392455

Table 10 shows that the constant lower and upper tail dependences are significantly greater than zero. This implies that both investor sentiment and stock returns have extremely smaller (larger) values contemporarily. When the stock market is in an upturn (downturn), investors believe it is (not) time to invest in the stock market. Therefore, more (less) investors would like to enter the stock market, that is, newly opened stock trading accounts will increase (decrease). Conversely, if investor sentiment becomes more bullish (bearish), stock returns will rapidly increase (decrease). Another finding is that the upper tail dependence is larger than the lower in both A and B shares markets. This finding is not consistent with results of typical financial assets, where usually the lower tail dependence is larger than the upper. That is to say, when the market is in an upturn, the dependence between investor sentiment and stock returns is stronger. This is because the reflection of investor sentiment on the increase in a given stock price is faster than the reflection of factors on the increase in a given stock price. It is obvious that the dependence between investor sentiment and stock returns varies over time. All parameters in evolution Equations (11) and (12) are significant at 0.05 significance level. In Figures 7 and 8, we plot the time path of conditional tail dependence implied by the time-varying SJC copula model. Both figures present that the lower tail dependence is more volatile, tempestuously. The lower tail dependences in A shares market (B shares market) increase from 0.0173 (0.0001) to 0.6318 (0.7231), while the corresponding upper tail dependences increase from 0.0109 (0.1724) to 0.3344 (0.4831). Also, these two figures show that the volatility of dependence in B shares market is larger than in A shares market. Standard deviation of upper tail dependences in B shares market is 11.25%, which is larger than the 7.94% in A shares market. Standard deviation of lower tail dependences in B shares market is 40.36%, which is dramatically larger than the 13.34% in A shares market. This is easy to understand. B shares market presents less liquidity, smaller market capacity, and poor transparency of market. These characteristics of B shares market cause investors to face larger risks. Therefore, investors are more sensitive to market changes. This leads to more tempestuous volatility of dependence in extreme cases in B shares market.

5 Conclusions This paper uses the copula function, prevalent in financial research, to analyze the dynamics and asymmetry of the dependence between investor sentiment and stock returns. We select


546

newly opened stock trading accounts as the proxy of investor sentiment index. The data of stock returns are from the SSE. Upper Tail Dependence in A Share Market 0.4 0.3 0.2 0.1 0 2005−7−1

2006−4−28

2007−2−16

2007−12−14

2008−9−26

2009−7−10

2008−9−26

2009−7−10

Lower Tail Dependence in A Share Market 0.8 0.6 0.4 0.2 0 2005−7−1

2006−4−28

2007−2−16

2007−12−14

Figure 7 Conditional tail dependence in A shares market

Figure 8 Conditional tail dependence in B shares market

The copula method has been widely used in a variety of applications in financial research, but less in behavioral finance. Copulas can allow us to take into consideration important characteristics of financial data, such as non-normality, which challenges the theory of traditional finance. The results show that the data do deviate from the null hypothesis of normality at the univariate. Even when we have removed autoregression and heteroskedasticity using the GARCH models, the data series still show asymmetries. Therefore, we use Hansen’s skewed Student-t distribution to fit the marginal distributions. QQ plots show that GARCH models with skewed Student-t innovations are better than those with normal innovations. The non-normality of data also means that the traditional Pearson’s correlation, based on normal


547

distribution, is not enough to describe the relationship between stock market behavior and investor sentiment. Copulas also facilitate flexible and convenient computation because these functions can be estimated separately from marginal distributions. We find that there exists significant dependence between investor sentiment and stock returns, even in extreme cases. This implies that when the stock market is in an upturn (downturn), investors will become bullish (bearish), and vice versa. In addition, we especially find that the upper tail dependence is larger than the lower. This is not consistent with results of typical financial assets, where usually lower tail dependence of asset returns is larger than the upper [19,39] . That is to say, when the market is in an upturn, the dependence between investor sentiment and stock returns is stronger. This is because the reflection of investor sentiment on the increase in a given stock price is faster than the reflection of other factors on the increase in a given stock price. When considering the time-varying case, the dependence is found to be dynamic over time. This tells us that the relationship between investor sentiment and stock returns is not constant. Therefore, attempts to find a stationary or consistent relationship between the two may fail. Investors should recognize that the influence of investor sentiment on stock market returns is time-varying, so that they can make relatively more accurate decisions. Finally, according to the policy of Chinese stock markets, we investigate the dependences between investor sentiment and stock returns in A and B shares markets separately. It is found that the volatility of dependence in B shares market is larger than in A shares market. This is because B shares market is characterized by less liquidity, smaller market capacity, and poor transparency. These characteristics of B shares market expose investors to larger risks. Therefore, investors are more sensitive to market changes. This leads to more tempestuous volatility of dependence in extreme cases in B shares market. References [1] J. Treynor, Toward a theory of market value of risky assets. Unpublished manuscript. A final version was published in 1999, in Asset Pricing and Portfolio Performance: Models, Strategy and Performance Metrics, Robert A. Korajczyk (editor) London: Risk Books, 1962. [2] W. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 1964, 19(3): 425–442. [3] J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, The Review of Economics and Statistics, 1965, 47(1): 13–37. [4] J. Mossin, Equilibrium in a capital asset market, Econometrica, 1966, 34(4): 768–783. [5] S. A. Ross, The arbitrage theory of capital asset pricing, Journal of Economic Theory, 1976, 13(3): 341–360. [6] E. Fama, Efficient capital markets: A review of theory and empirical work, Journal of Finance, 1970, 25(2): 383–417. [7] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 1979, 47(2): 263–291. [8] F. Black, Noise, Journal of Finance, 1986, 41(3): 529–543. [9] J. De Long, A. Shleifer, L. G. Summers, and R. Waldmann, Noise trader risk in financial markets, Journal of Political Economy, 1990, 98(4): 703–738. [10] N. Barberis, A. Shleifer, and R. Vishny, A model of investor sentiment, Journal of Financial Economics, 1998, 49(3): 307–343. [11] K. Daniel, D. Hirshleifer and A. Subrahmanyam, Investor psychology and security market underand overreactions, Journal of Finance, 1998, 53(6): 1839–1885. [12] H. Hong and J. Stein, A unified theory of underreaction, momentum trading, and overreaction in asset markets, Journal of Finance, 1999, 54(6): 2143–2184.

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