Ruanmin Cao

Lajos Horváth

Zhenya Liu
email: zhenyaliu@hotmail.com

Yuqian Zhao

A Study of Data-driven Momentum and Disposition Effects in the Chinese Stock Market by Functional Data Analysis

Keywords: Momentum effect, Disposition effect, Functional principal component analysis, Portfolio selection, Chinese stock market JEL Classification: C58, G12, G15, G40

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A study of data-driven momentum and disposition effects in the Chinese stock market by functional data analysis

INTRODUCTION

Introduction

Understanding market anomalies in the Chinese stock market gives great value to both investors and market regulators. As the largest emerging market, the Chinese stock market is of enormous interest among international investors, especially after the MSCI added A shares 1 to its benchmark emerging market index in June 2017. Meanwhile, even though considerable progresses in financial liberalization and market regulation have been made since the 1990s, the Chinese stock market still shows differences from other developed markets. One typical feature of the Chinese stock market, similar to many other emerging markets, is its high volatility. Consequently, it is questionable whether the market anomalies discovered in the developed markets are still feasible in the Chinese stock market.

One controversial topic in the literature is the momentum effect in the Chinese stock market (see, [START_REF] Kang | Contrarian and momentum strategies in the china stock market[END_REF][START_REF] Wong | The cross-section of stock returns on the shanghai stock exchange[END_REF][START_REF] Wu | Momentum trading, mean reversal and overreaction in Chinese stock market[END_REF][START_REF] Cheema | Momentum returns and information uncertainty: Evidence from China[END_REF][START_REF] Cheema | Momentum returns and information uncertainty: Evidence from China[END_REF][START_REF] Cakici | Cross-sectional stock return predictability in China[END_REF]. Results suggest that the momentum anomaly in China gives a relatively reduced predictive power. However, these results contradict the knowledge that the momentum factor is the most pervasive and well-documented market anomaly [START_REF] Fama | Size, value, and momentum in international stock returns[END_REF]; [START_REF] Asness | Value and momentum everywhere[END_REF]. The standard approach to find market anomalies is to sort cross-sectional returns according to firm-specific historical characteristics, such as size, value, momentum, and dividends (see, [START_REF] Fama | The cross-section of expected stock returns[END_REF], 2012, 2015;[START_REF] Carhart | On persistence in mutual fund performance[END_REF][START_REF] Daniel | Evidence on the characteristics of cross sectional variation in stock returns[END_REF]; amongst others), and one expects to obtain abnormal returns if anomalies exist. As a result, we can use multi-factor models, such as the Fama-French (F-F) three-, five-factor and the Carhart four-factor models [START_REF] Fama | The cross-section of expected stock returns[END_REF], 1993, 2015;[START_REF] Carhart | On persistence in mutual fund performance[END_REF] to explain cross-sectional returns. Thus, due to these empirical contradictions, it is natural to ask if these traditional methodologies are still appropriate in the highly volatile Chinese stock market? Our impetus is to use updated statistical techniques to investigate market anomalies in Chinese A-shares.

INTRODUCTION

Briefly, we apply a recently developed functional data analysis technique to decompose the total variation of the cross-sectional residuals. The intuition is that in a highly volatile market, some behavioral biases are hard to explain with the well-known asset pricing factors, and the residuals might play a more critical role. [START_REF] Blitz | Residual momentum[END_REF] constructed a residual-weighted portfolio and argued that the volatilities of total returns are too high to propose a well-performing trading strategy. Thus, we explore underlying anomalies based on the F-F three-factor residuals instead of the total stock returns. In [START_REF] Gandhi | Size anomalies in us bank stock returns[END_REF], the authors focused on a relatively chaotic market of U.S. bank shares and applied principal component analysis (PCA) to decompose a size-sorted cross-section of U.S. bank residuals, and they found that the second principal component suggests a size factor in highly leveraged bank shares.

Enlightened by their methodology, we propose a two-step functional data analysis approach to decompose the residual returns. In the first step, we smooth the cross-sectional residuals into functional curves in the Hilbert L 2 [0, 1] space by using B-spline orthonormal basis functions. Secondly, we use the functional principal component analysis (FPCA) to obtain data-driven eigenfunctions. These eigenfunctions exploit the underlying risk patterns and suggest asset pricing factors. For a related work to study the cross-sectional returns under the framework of FDA we refer to [START_REF] Kokoszka | Dynamic Functional Regression with Application to the Cross-section of Returns[END_REF].

To understand the differences between our method and the method used by [START_REF] Gandhi | Size anomalies in us bank stock returns[END_REF], we can compare FPCA with PCA. In the context of FDA, since discrete high dimensional data are treated as continuous functional curves, FPCA extracts eigenfunctions rather than eigenvectors, so in some sense, FPCA is a continuous version of the PCA. In equity markets, although the cross-sectional returns are discretely observed, the risk patterns on the cross-sections are naturally continuous, which makes FPCA more appropriate in this situation. An additional benefit from using the smoothing technique is that the FPCA controls the level of roughness by filtering noise while preserving main features. For details, we refer to the monographs of [START_REF] Ramsay | Functional Data Analysis[END_REF] and [START_REF] Horváth | Inference for functional data with applications[END_REF].

In the empirical study, we apply the two-step method to explore the F-F three-factor residuals. Because

INTRODUCTION

our interest lies in the momentum effect, we first sort the cross-sectional returns based on 12-month firmspecific accumulated past returns. The cross-sections are ranked based on their historical performances, and we form ten equal-weighted portfolios from a loser group to a winner group. Next, we decompose the variability of the residual returns to estimate eigenfunctions. In our case, we estimated 13 eigenfunctions of which the first four eigenfunctions explain 87.77% of the total variation of the F-F three-factor residuals, with the first one already explaining 64.62%, and the second one 14.73%. We then construct two asset pricing factors by using the standardized weights suggested by the first two eigenfunctions, leaving the remaining eigenfunctions as noise.

These two data-driven risk factors help us to understand the Chinese stock market better. The first factor is a data-driven momentum winner-minus-loser (wml) factor. Note that the behavioral bias toward losers and winners can either be explained by the winner-minus-loser effect [START_REF] Jegadeesh | Returns to buying winners and selling losers: Implications for stock market efficiency[END_REF] or the contrarian loser-minus-winner effect [START_REF] Bondt | Does the stock market overreact?[END_REF]. Both scenarios reveal the momentum effect. We construct relative strength portfolios [START_REF] Jegadeesh | Returns to buying winners and selling losers: Implications for stock market efficiency[END_REF]) and find that the data-driven wml factor is belonging to the latter case. From the perspective of behavioral finance, the momentum effect may originate from three sources: (1) investor sentiments of underreacting short-run information and overreacting information in the long-run [START_REF] Chan | Evaluating the performance of value versus glamour stocks the impact of selection bias[END_REF][START_REF] Barberis | A model of investor sentiment[END_REF][START_REF] Jegadeesh | Profitability of momentum strategies: An evaluation of alternative explanations[END_REF]); (2) investor overconfidence [START_REF] Bondt | Does the stock market overreact?[END_REF][START_REF] Kang | Contrarian and momentum strategies in the china stock market[END_REF]; (3) the lead-lag effect [START_REF] Lo | When are contrarian profits due to stock market overreaction?[END_REF]. Based on an analysis of market data, our results show that investors in the Chinese stock market are overreact to short-run information. The lead-lag effect also exists, which is consistent with the conclusion made by [START_REF] Kang | Contrarian and momentum strategies in the china stock market[END_REF].

The second factor, indicating the distinction between the extreme-performing and mediocre-performing groups, can be explained by the asymmetric disposition effect. As an extension of the prospect theory, the disposition effect [START_REF] Shefrin | The disposition to sell winners too early and ride losers too long: Theory and evidence[END_REF] describes the investors' risk attitude toward stock performances, which indicates that investors sell winning stock too early and ride losers too long (also see Fazzini, 2006; [START_REF] Barberis | What Drives the Disposition Effect? An Analysis of a Long-Standing Preference-Based Explanation[END_REF]. Some developments in the disposition effect were discussed by Ben-David

INTRODUCTION

and Hirshleifer (2012; also see An, 2015), who argued that the selling/buying function of the disposition effect for past returns is asymmetric U-shaped and explained this with investor overconfidence. Our findings are also in agreement with [START_REF] Chen | Trading performance, disposition effect, overconfidence, representativeness bias, and experience of emerging market investors[END_REF], who documented the disposition effect by using brokerage account data in the Chinese stock market. Overall, these two new asset pricing factors increase the adjusted R 2 in the F-F three-factor model from 0.79 to 0.89 and bring less significant adjusted returns (alpha). The evidence of the robustness of these two factors is shown by using the Fama-Macbeth regression.

Our paper contributes to the literature in several ways. We suggest a functional data analysis approach to decompose cross-sectional returns and calibrate market anomalies from samples. Our findings show that the momentum effect in the Chinese stock market is not as weak as mentioned in the literature. We also complement the literature by empirically showing the disposition effect in the Chinese stock market.

Instead of using individual investor-level data, we show that the asymmetric U-shape disposition effect can be uncovered from market data. The well-known finding that disposition effect is a source of momentum effect in [START_REF] Grinblatt | Prospect theory, mental accounting, and momentum[END_REF] has been challenged recently. [START_REF] Birru | Confusion of confusions: a test of the disposition effect and momentum[END_REF] did not find disposition effect after a stock split and he argued that the disposition effect is not able to explain the momentum effect. We complement this vein of research by showing that momentum and disposition effects give two orthonormal risk premiums in the Chinese stock market. Additionally, we use behavioral finance theories to explain our results, this providing a better understanding of the impact of investors' behavior and regulation rules on the pricing equilibrium in the Chinese stock market. This provides a deeper market insight for international investment practitioners. Our results can be easily applied to portfolio selection in practice.

We incorporate the exploited risk factors into a buy-and-hold portfolio, and the Sharpe ratio of the meanvariance optimization increased to 1.27.

The outline of the paper is as follows. In Section 2, we discuss the dataset and apply conventional multi-factor models to study the market anomalies in the Chinese stock market. In Section 3 we introduce a two-step functional data analysis approach to decompose cross-sectional residuals. Section 4 exploits the residuals in the Fama-French three-factor model and introduces two data-driven common risk factors, which 2. DATA AND BENCHMARK MODELS are further elaborated in Section 5. Section 6 incorporates them into eigenfunction portfolios, and concluding remarks are made in Section 7.

Data and benchmark models

In this section we introduce the dataset and revisit the results of classic multi-factor models in Chinese A-shares. We regress past return-sorted cross-sectional returns onto the classical asset pricing factors [START_REF] Fama | The cross-section of expected stock returns[END_REF][START_REF] Carhart | On persistence in mutual fund performance[END_REF]. The F-F three-factor residuals are kept for the use of later sections.

Dataset

We collect monthly adjusted closing prices2 for all Chinese A-shares to avoid erratic behavior in daily and weekly frequency. To maintain the consistency with relevant studies, we exclude the financial sector due to its high leverages, as well as listings without 12-month past returns information. The dataset ranges between January 2007 and June 2017 in order to eliminate the influence of the structural reform in the Chinese stock market during the year of 20063 . Note that the sample period over the last decade contains two business cycles. There are 1470 shares listed in 2007, and this number increases to 3226 in June 2017. We use the log return transformations of the raw price data.

We use the Shanghai Stock Exchange Composite index and three-month Treasury bill rate as the market index r m t and the risk-free rate r f t , respectively. For constructing the size and value factors, we collect firmspecific market capitalization and book-to-market ratios. The short-term and long-term firm-specific past returns are computed by the equations

P R i,t = t-2 j=t-K r i,j , LP R i,t = t-48 j=t-60 r i,j , (1)
where the subscript i denotes stock i. Unless otherwise stated, K = 12. The long-term LP R i,t is only used to assess the performances of relative strength portfolios in section 5.1.

The market risk premium is computed as rmrf t = r m t -r f t . We first sort cross-sectional returns into deciles according to firm-specific size, value and past returns in ascending order, and then construct equal-weighted small-minus-big (smb), high-minus-low (hml) and wml common risk factors. Additionally, we collect the monthly firm-level turnovers to analyze their trading activities in section 5.2. All data was obtained from the Wind database.

Multi-factor models

We start to build portfolios according to the standard portfolio strategy of [START_REF] Fama | Common risk factors in the returns on stocks and bonds[END_REF]. Crosssectional returns are ranked by their 12-month past returns. We group stocks into ten portfolios ranging from "loser" to "winner" according to their firm-specific P R i,t . The portfolio returns are computed with equal weights. For completeness, we also consider a "winner-minus-loser" portfolio. We first assess the explanatory power of the F-F three-factor model,

r i,t -r f t = α i + β 1,i rmrf t + β 2,i smb t + β 3,i hml t + i,t , (2)
where r i,t denotes the ith decile portfolio return for i ∈ [START_REF] An | Asset pricing when traders sell extreme winners and losers[END_REF][START_REF] Cakici | Cross-sectional stock return predictability in China[END_REF]. The F-F three-factor model studies the cross-sectional market anomalies through firm-level fundamental information (also see the five-factor model, [START_REF] Fama | A five-factor asset pricing model[END_REF]. Meanwhile, in order to verify existing results about the momentum effect, we examine the Carhart (1997) four-factor model,

r i,t -r f t = α i + β 1,i rmrf t + β 2,i smb t + β 3,i hml t + β 4,i wml t + i,t . (3)

DATA AND BENCHMARK MODELS

Once all multi-factor models are estimated, the next issue is to assess their performance. Since a better factor model always delivers less significant risk-adjusted returns (intercept α i ), the GRS test [START_REF] Gibbons | A test of the efficiency of a given portfolio[END_REF] is used here to test the joint significance of intercept coefficients. We aim to test the null hypothesis H 0 : α i = 0, for all i = 1, 2, . . . , 10. When comparing two multi-factor models M 1 and M 2 , if M 1 rejects H 0 at a lower significance level, we then conclude that M 1 better predicts cross-sectional returns than M 2 .

[Insert Table 1 Here] Table 1 displays the estimators of the regression models specified in Equations ( 2) and (3). Panel A

shows the estimation results in the F-F three-factor model, and we observe three remarkable facts. First, the GRS test does not find an overall statistically significant risk-adjusted return, and the estimated intercepts in the majority of deciles are not statistically significant. The only exceptions are the two winner (9th and 10th) groups and the winner-minus-loser portfolio, where these portfolios show significant risk-adjusted returns with negative signs. Second, the model exhibits better explanatory power to 4th-6th groups with adjusted R 2 = 0.86, while it becomes less efficient for the loser and winner groups, e.g., adjusted R 2 = 0.79 in the winner group. Lastly, we notice that the coefficients of F-F three factors are statistically significant.

The market factor and size factor positively explain the excess returns, while the value factor shows negative coefficients to all of the deciles under consideration. Similar results have been discovered in the literature (see [START_REF] Kang | Contrarian and momentum strategies in the china stock market[END_REF][START_REF] Wong | The cross-section of stock returns on the shanghai stock exchange[END_REF][START_REF] Cakici | Cross-sectional stock return predictability in China[END_REF]. The size effect is significant because initial public offerings (IPO) in China are regulated by the China Securities Regulatory Commission (CRSC) rather than based on a system of registration like in the developed markets. This barrier makes newly listed small shares more likely to be over-valued in the stock market. The negative signs of the value factor are possible because retail investors pay less attention to the value factor; they mainly make their investment decisions based on market rumors.

Panel B of Table 1 shows that, not surprisingly, the Carhart four-factor model outperforms the F-F three-factor model with a higher adjusted R 2 , especially in the loser and winner groups. The coefficients

FUNCTIONAL DATA DECOMPOSITION APPROACH

of the wml factor are negative and positive at the losers and winners groups, respectively. The result of the GRS test implies that the Carhart four-factor model generates less significant risk-adjusted returns than the F-F three-factor model. Nevertheless, the variation in the cross-sectional returns is still not to be fully explained, given the fact that risk-adjusted returns are jointly significant at the 1% significance level in both of the models. Therefore, more risk patterns are hidden in the residuals and worth decomposing.

Functional data decomposition approach

In this section, we introduce a functional data analysis approach to decompose cross-sections of portfolio residuals. Converting the panel data

y i,t = [y 1,t , y 2,t , . . . , y N,t ], 1 ≤ t ≤ T , 1 ≤ i ≤ N to functional
observations, we smooth the cross-sectional entities into the [0, 1] interval without loss of generality and obtain y t (u) for 1 ≤ t ≤ T and 0 ≤ u ≤ 1. We assume that the random curve y t (u) is square integrable and that it is in the Hilbert space

L 2 [0, 1] for all 1 ≤ t ≤ T . The space L 2 [0, 1] is equipped with inner product x(u), y(u) = 1 0 x(u)y(u)du, and the corresponding norm is x(u) = ( 1 0 x 2 (u)du) 1/2
. The functional data y t (u) can be expanded by the Karhunen-Loéve Theorem (Horváth and Kokoszka, 2012),

y t (u) = ∞ m=1 η t,m φ m (u) ≈ M m=1 η t,m φ m (u), (4)
where {φ m (u), 0 ≤ u ≤ 1} are orthonormal functions on [0, 1], i.e., φ i , φ j = 0 if i = j and φ i , φ i =

||φ i || 2 = 1.
The most commonly used basis functions φ m (u) are Fourier basis and B-splines basis. The former is mainly used for cases in which data contains periodical or nearly-periodical patterns, and the latter usually adapts to non-periodic data. The number of bases M can be crucial, and it is determined by optimizing the trade-off between fitting features and noises in the smoothing process. The coefficients η t,m 1 ≤ m ≤ M , 1 ≤ t ≤ T are given by the inner product of φ m (u) and y t (u),

η t,m = φ m (u), y t (u) = 1 0 φ m (u)y t (u)du. ( 5
c(u, v) = E[(y t (u) -µ(u))(y t (v) -µ(v))],
where c(u, v) is a symmetric, non-negative definite function. We aim to decompose c(u, v). There are eigenvalues and corresponding eigenfunctions as

λ 1 ≥ λ 2 ≥ • • • ≥ 0 and ψ 1 (u), ψ 2 (u), . . . satisfying λ i ψ i (u) = 1 0 c(u, v)ψ i (v)dv, i = 1, 2, . . . . (6)
In principal component analysis as well as in functional data analysis, φ m = ψ m is a suitable choice in (4) because M m=1 y t (u), ψ m ψ m (u) is the best approximation for y t (u) in L 2 [0, 1], i.e., when y t (u) is approximated with 1, 2, . . . , M orthogonal functions in H, we get the smallest E y t (u) -

M m=1 y t (u), , ψ m ψ m 2 .
However, c(u, v) is unknown and it should be estimated with

ĉT (u, v) = 1 T T t=1 (y t (u) -ȳT (u))(y t (v) -ȳT (v)), where ȳT (u) = 1 T T t=1 y t (u).
Now the eigenvalues and eigenfunctions of the theoretical c(u, v) are estimated by λ1

≥ λ2 ≥ • • • ≥ 0 and ψ1 (u), ψ2 (u), . . . satisfying λi ψi (u) = 1 0 ĉT (u, v) ψi (v)dv, i = 1, 2, . . . , (7)
so due to the optimality properties of the eigenfunctions of ĉT (u, v), a popular way to decompose functional

3. FUNCTIONAL DATA DECOMPOSITION APPROACH observations y t (u) is y t (u) ≈ ŷt (u) = M m=1 ηt,m ψm (u), (8)
with ηt,m = ψm (u), y t (u) =

1 0 ψm (u)y t (u)du.
This means that y t (u) is projected into the finite dimensional space spanned by { ψ1 , ψ2 , . . . , ψM }. This projection works well if y t (u) is a weakly dependent sequence. Then, Theorem 3.1 If Assumptions 3.1 holds, then we have that for each From the computational point of view, the integral is approximated by Riemann sums; for instance, ηt,m in ( 8) is approximated by

1 ≤ i ≤ M | λi -λ i | p → 0, ||ĉ i ψi -ψ i || p → 0,
ηt,m = 1 N N i=1 y t (u i ) ψ(u i ),
where y t (u i ) is y t (u) on the regular grid u i , 1 ≤ i ≤ N .

EMPIRICAL RESULTS: DECOMPOSITION OF CHINESE A-SHARE CROSS-SECTIONAL RETURNS

Empirical results: decomposition of Chinese A-share cross-sectional returns

This section uses the method proposed above to decompose the F-F three-factor residuals, i.e., ˆ i,t in Equation [START_REF] Asness | Value and momentum everywhere[END_REF]. In order to accommodate the notation in the last section, we rewrite ˆ i,t into ˆ t (u i ). The reason for omitting the Carhart four-factor residuals is because our dependent variables are past return-sorted crosssectional returns, the wml factor is suspected to contain overlaps and should not be controlled here.

Decomposing variations in residuals

We explore the estimated residuals

ˆ t (u i ), i ∈ [1, N ],
where N is the total number of sorted groups. As a pre-analysis, we again sort cross-sectional returns into 100 groups (N = 100) using the algorithm mentioned in section 2.2, and this is because the previous ten groups are not enough for implementing the functional smoothing technique. In each of the groups, we regress excess portfolio returns on the F-F three factors and save the residuals ˆ t (u i ), 1 ≤ i ≤ 100. Before adopting the two-step decomposition method, the estimated residual vectors are demeaned and standardized for emphasizing variations and reducing noise as suggested by [START_REF] Blitz | Residual momentum[END_REF]. In the first step, we smooth these residuals into functional curves. At each time t, we set 11 knots on ˆ t (u i ). Considering that there is hardly any periodical pattern in the cross-sectional returns, we choose 13 B-spline functions as the smoothing bases as shown in Figure 1. The smoothing technique leads to N=126 functional curves ˆ t (u) spanning over 2007-2017, appearing in the upper panel of Figure 2.

By smoothing residual observations to curves, Assumption 3.1 is satisfied.

[Insert Figure 1, Figure 2 and Figure 3 Here] Next, we apply the FPCA method to decompose the functional residual curves ˆ t (u) and find 13 empirical

eigenfunctions (ef), ψ(u) = { ψj (u)| λj > 0, 1 ≤ j ≤ 13}
, where ψj (u) is the jth eigenfunction and λj is the corresponding eigenvalue. The lower panel of Figure 2 illustrates that the first four eigenfunctions explain 87.77% of the total variation. The first eigenfunction ψ1 (u) explains 64.62% and displays an upward trend.

EMPIRICAL RESULTS: DECOMPOSITION OF CHINESE A-SHARE CROSS-SECTIONAL RETURNS

This pattern implies a risk factor that loads negatively on the loser groups and positively on the winner groups. It is similar but not identical to the momentum wml factor. The second eigenfunction ψ2 (u), plotted as an asymmetric U/V shaped curve, accounts for 14.73% of the total variation. This risk pattern suggests that investors behave differently toward extreme and mediocre stocks, thereby suggesting a risk factor that loads positively on the extreme-performing groups and negatively on the mediocre-performing groups. We compute the 95% confidence envelopes of the first four eigenfunctions via a bootstrapping approach (5,000 iterations) [START_REF] Hall | On properties of functional princi-pal components analysis[END_REF], as shown in Figure 3. Table 2 displays the percentages of variations explained by the first four eigenfunctions, where the results exploited from the real data are consistent with the bootstrapped ones. The eigenfunctions from the third and below are hard to align with cross-sectional portfolio returns. Also because they account for minor variations, we treat them as noise.

[Insert Table 2 Here]

Another method to select useful eigenfunctions is by using of the information coefficient (IC). Figure 4 plots ICs between the past returns-sorted cross-sectional returns and eigenfunctions. Over the entire period, ψ1 (u)

shows the highest average IC, followed by ψ2 (u), while the average IC of ψ3 (u) and ψ4 (u) are flatter and close to zero. Therefore, we use ψ1 (u) and ψ2 (u) to construct asset pricing factors.

[Insert Figure 4 Here]

Data-driven asset pricing factors: F P C1 t and F P C2 t

In order to check whether the uncovered asset pricing factors actually explain the cross-sectional returns, we build two data-driven variables:

F P C1 t (r t (u i ) × ψ1 (u i ) ), and
F P C2 t (r t (u i ) × ψ2 (u i ) ), where r t (u i ) = [r t (u 1 ), r t (u 2 ), . . . , r t (u N )
] is a 126 × 100 matrix, and ψj (u i ) is a 100 × 1 weights vector with the grid values of ψj (u). The first variable is a portfolio that shorts loser shares and longs winner shares, while the second variable is a portfolio that longs extreme-performing shares and shorts mediocre-performing shares. Note

EMPIRICAL RESULTS: DECOMPOSITION OF CHINESE A-SHARE CROSS-SECTIONAL RETURNS

that the weights are standardized and their sum is equal to one.

We apply the Fama-MacBeth regression [START_REF] Fama | Risk, return, and equilibrium: Empirical tests[END_REF] on F P C1 t and F P C2 t , which is the standard method to estimate the coefficients of asset pricing factors. Table 3 shows the results: F P C1 t has significant risk premium on past returns-sorted portfolio returns with a strong January effect; F P C2 t is statistically significant at the 10% level, and surprisingly it is less significant over the course of January.

The final column shows that, due to orthogonality, the coefficients of risk premiums for F P C1 t and F P C2 t remain unchanged and the adjusted R 2 equals the sum of single-factor regressions. This finding reveals that both F P C1 t and F P C2 t are strong asset pricing factors in the Chinese stock market.

[Insert Table 3 Here]

In order to have a contest with the benchmark factor models, we again compute the multi-factor regression models while adding the F P C1 t and F P C2 t factors. Table 4 reports the results of the regression models specified in Equations ( 9) and ( 10). Compared with Table 1, adding F P C1 t and F P C2 t improves the explanatory power of the models, which we expected given the higher adjusted R 2 s and the statistically less significant risk-adjusted returns are reported.

r i,t -r f t = α i + β 1,i rmrf t + β 2,i smb t + β 3,i hml t + β 4,i F P CX t + i,t , X = 1, 2., (9)
r i,t -r f t = α i + β 1,i rmrf t + β 2,i smb t + β 3,i hml t + β 4,i F P C1 t + β 5,i F P C2 t + i,t . (10)
Remarkably, panel A of Table 4 shows that the estimated coefficients of the F-F three factors remain almost the same as in Table 1. The estimated coefficients of F P C1 t are similar to the wml factor in panel B of Table 1. Also, we note that the sign of the coefficients of F P C1 t are also identical to the sign of the wml factor, and the adjusted R 2 is equal to 0.95 when we regress F P C1 t on wml. Hence, it is not hard to see that F P C1 t is a data-driven form of the wml factor.

EXPLANATIONS OF DATA-DRIVEN ASSET PRICING FACTORS

Panel B of Table 4 shows that F P C2 t significantly explains excess returns in the mediocre and loser groups. These findings are reasonable because F P C2 t affirms the behavioral biases toward mediocre and extreme-performing shares. The coefficients of F P C2 t in the mediocre groups are around -0.14, and this coefficient changes to 0.08 in the loser group. Another remarkable fact is that F P C2 t cannot explain the wml portfolio return (adjusted R 2 = 0.15), which suggests that F P C2 t explains phenomena which can not be explained by the wml or F P C1 t factor.

Due to orthogonality, panel C of Table 4 shows the enhanced explanatory power of the multi-factor models when adding F P C1 t and F P C2 t , with adjusted R 2 over 0.87 for all groups. This model takes advantage of the observation that the risk-adjusted returns in all of the groups under study become insignificant. The relatively lower P -values of the GRS tests indicate that model [START_REF] Cakici | Cross-sectional stock return predictability in China[END_REF] outperforms the other factor models discussed above. Finally, in order to assess the influence of market state on our asset pricing factors, we follow [START_REF] Cooper | Market states and momentum[END_REF] to inspect the role of market states on the FPC factors. By defining an upward market (downward market) that accumulated historical 6-month or 12-month market returns as positive (negative), our results rarely show a notable impact on F P C1 t and a slightly significant impact on F P C2 t .

We omit these results here.

[Insert Table 4 Here]

Explanations of data-driven asset pricing factors

We now explain the new asset pricing factors from the perspective of behavioral finance. Since the F P C factors are constructed based on exploited cross-sectional risk patterns, together with the fact that crosssectional returns are sorted by past return, these factors give insight to investors' behavioral biases regarding persistence of equities' performances.

EXPLANATIONS OF DATA-DRIVEN ASSET PRICING FACTORS

How to explain F P C1 t ?

The previous section showed that F P C1 t is a data-driven wml factor describing the investors' behavioral bias toward the winner and loser shares. Here we investigate the underlying psychological structures of the risk factor F P C1 t . We first follow Jegadeesh and Titman (1993) in constructing relative strength portfolios. We sort cross-sectional returns in ascending order by firm-specific past returns with different look-back periods K, where K varies from short-term (3, 6, 9 and 12 months) to long-term (60 months, cf. Equation ( 1)).

The holding periods are 3, 6, 9, 12 and 24 months. In total, there are 50 portfolios (25 momentum winnerminus-loser and 25 contrarian loser-minus-winner). The average monthly returns are reported in Table 5.

The momentum strategy generates no profit while the portfolio with the contrarian strategy earns positive profits under any formation and holding periods. Our results are consistent with the findings of Kang et al.

(2002), [START_REF] Griffin | Momentum investing and business cycle risk: Evidence from pole to pole[END_REF] and [START_REF] Wu | Momentum trading, mean reversal and overreaction in Chinese stock market[END_REF].

[Insert Table 5 Here]

One reason for observing profitable contrarian portfolios, as suggested by [START_REF] Bondt | Does the stock market overreact?[END_REF], is investors' overreaction, as evidenced by calculating autocorrelations of portfolio returns with negative signs.

Lo and Mackinlay (1990) observed that the cross lead-lag effect is another reason for a profitable contrarian strategy. The lead-lag effect is observed when a higher return on stock i at period t -1 leads to a higher return on stock j at period t, i.e., calculating positive cross-serial correlations in portfolio returns. We thus expect to get negative autocorrelations and positive cross-serial correlations when overreaction and lead-lag effects exist.

To study the dependence structure of past returns-sorted cross-sectional portfolio returns, we calculate serial correlations among these portfolios using lags ranging from 1 to 4. Panel A of Table 6 6 gives cross-autocorrelations with lags 1. Even though the cross-serial correlations vary between positive numbers and negative numbers, note that the historical returns of the loser group is positively correlated with the contemporary returns of the winner group (with correlation value 0.06). A similar situation occurs between the historical returns of the winner group and the contemporary returns of the loser group (with correlation value 0.02). This result implies that there is a lead-lag effect between winner and loser portfolios. We thus confirm that investors' overreaction and the lead-lag structure are the driving forces of the data-driven momentum factor.

[Insert Table 6 Here]

How to explain F P C2 t ?

F P C2 t captures investors' asymmetric investment behavior regarding extreme and mediocre-performing shares. This pattern is in accordance with the asymmetric U-shaped disposition effect observed by Ben-David and Hirshleifer (2012), see Panel B of Figure 2 in their paper. The authors claimed this effect reflects investors' overconfidence in the U.S. market, as investors always over-value their knowledge and are inclined to trade stocks with big news; these stocks are more likely to be extreme-performing stocks.

Moreover, another potential source of F P C2 t is the selling pressure driven by the disposition effect on both outperforming and underperforming stocks. Deviations from the current price level increases liquidations in extreme-performing stocks. Hence, investors have to pay the risk premium to 'mediocre-performing stocks' for their safety. This argument parallels with the low-volatility anomaly. We note that our work, from this perspective, demonstrates another measure of "safety", in addition to idiosyncratic risk [START_REF] Blitz | The volatility effect[END_REF] and beta (Frazzini and Pederson, 2014). Regulation rules in the Chinese stock market restrict the daily price ceiling and floor, i.e., maximum 10% price changes, which have been widely adopted in Asian exchanges [START_REF] Chan | Price limit performance: evidence from transactions data and the limit order book[END_REF]. A significant price change in extreme-performing shares thus leads to drainage of liquidity and severely exacerbates order imbalance, resulting in a higher return compensating

THE EIGENFUNCTION PORTFOLIO

for the illiquidity.

Another notable pattern is that the F P C2 t factor shows a right skewness toward losers. This asymmetric pattern can be interpreted by prospect theory. Prospect theory states that investors are more inclined to trade winning stocks and to hold losers. In another word, investors are more likely to re-examine positions or to update their beliefs about profitable stocks but are relatively inattentive to losers. Also see An (2015).

[Insert Figure 5 Here]

In order to empirically assess the disposition effect, we use firm-specific turnover to measure their trading activities. Table 7 shows that the turnover ratio between the extreme and mediocre groups is as high as 1.4124. Table 8 further reports turnover ratios between each of individual groups. The ratios between the loser groups and mediocre groups are more than 1.0 but less than 1.1, while these ratios rise to 1.8245 in cases of the winner and group 5. Figure 5 plots the turnover ratio for each group during the sample period. Although trading turnover in the loser group is not always higher than in the mediocre groups, the asymmetric U-shaped pattern is observable, particularly during the periods of 2009-2010, 2011-2014 and 2017.

[Insert Table 7 and Table 8 Here]

Meanwhile, since the eigenfunctions ψ1 (u) and ψ2 (u) are orthonormal, the behavioral biases explained by F P C1 t and F P C2 t cannot account for each other. In other words, this result indicates that the disposition effect can be separated from the momentum effect so that it deliveries a different risk premium4 .

The eigenfunction portfolio

In this section, we apply the F P C1 and F P C2 factors to develop a portfolio construction strategy. The relative strength portfolios have shown that the contrarian strategy earns positive returns while the momen-

THE EIGENFUNCTION PORTFOLIO

tum strategy does not. Thus, together with the market index, we treat the contrarian loser-minus-winner portfolio as the benchmark trading strategy. Considering a buy-and-hold strategy without any transaction cost, we conducted an in-sample simulation. Recall that cross-sectional returns were sorted into 100 groups from the loser to the winner. We constructed extreme spread portfolios, which portfolio weights are assigned by using the standardized values of ψ1 (u) and ψ2 (u). We name these portfolios as eigenfunction portfolios and denote them with EP 1 and EP 2, respectively. We also apply a mean-variance optimization on the EP 1 and EP 2, where the optimal weights are calculated by using historical three-month rolling windows, in order to avoid any future information.

Table 9 reports some popular statistics. The first eigenfunction strategy performs similarly to the contrarian-lmw strategy with a Sharpe ratio of around 0.48. The second eigenfunction portfolio is also profitable with Sharpe ratio of 0.33, which is lower than EP1 as fewer variations are explained. Lastly, benefiting from the orthogonality, the mean-variance optimized portfolio records a boosting of the Sharpe ratio to 1.27 and a much lower maximum draw-down of -0.02. The monthly performance shows that the mean-variance optimization obtains positive returns in all of the months under study. Therefore, the use of common risk patterns exploited by F P C1 t and F P C2 t leads to better portfolio selection. Note that as long as EP 1 and EP 2 persist out-of-sample, the momentum and disposition payoff structures should always be profitable in the active portfolio selection. Next, we discuss the profitability of eigenfunction portfolios under the framework of the FDA.

[Insert Table 9 and Figure 6 Here]

In order to discuss the profitability of eigenfunctions in the L 2 [0, 1] space, we consider the recalling functional objective r t (u). Recalling Equation [START_REF] Blitz | The volatility effect[END_REF], where the empirical eigenfunctions ψj (u) and eigenvalues λj are estimated, we incorporate these data-driven cross-sectional patterns into curves r t (u). The curves 6. THE EIGENFUNCTION PORTFOLIO r t (u) thus can be approximated well with finite K eigenfunctions,

r t (u) ≈ µ(u) + K j=1 r t (u), ψj (u) ψj (u) + ε t (u), (11)
where µ(u) is the functional mean, and the projection r t (u), ψj (u) is the time-varying functional loading for ψj (u).

We take the jth eigenfunction portfolio return r pj ,t as an example. It is not hard to see that r pj ,t is the inner production between the jth eigenfunction ψj (u) and the return r t (u),

r pj ,t = r t (u), ψj (u) . ( 12
)
Substituting Equation ( 11) into above equation, we then get

r pj ,t = (µ(u) + K j=1 r t (u), ψj (u) ), ψj (u) + ε t (u), ψj (u)
= µ(u), ψj (u) + r t (u), ψj (u) , ψj (u) + ε t (u), ψj (u) .

(

) 13
According to Theorem 3.1, the third term t (u), ψj (u) is noise. It is clear that two sources generate the profits of jth eigenfunction portfolio, namely, a deterministic part and a stochastic part. The former is an inner product between ψj (u) and the functional mean µ(u), and the latter is a time-varying functional loading that projects r t (u) on the jth eigenfunction ψj (u). Since cross-sectional behavioral patterns in one market are likely to be formed in the long run, we can easily obtain profits by taking the deterministic part.

But these behavioral patterns might vary in the short run, so that r pj ,t has to take a risk (variance) from the stochastic part. One potential way to improve the profitability of the eigenfunction portfolio is to predict the functional loadings; this will be difficult and more studies will be required.

CONCLUSION

Conclusion

This paper employed a functional principal component analysis approach to decompose cross-sectional Fama-

French three-factor model residuals in the Chinese stock market. Based on our empirical results, two asset pricing factors were derived from the market data, which we interpreted as the momentum effect and the asymmetric disposition effect, respectively. According to behavioral finance theories, we explain these two data-driven factors through investors' overreaction, overconfidence, and the lead-lag effect. These two factors, combined with the Fama-French three factors explain almost all variation of cross-sectional returns in the Chinese stock market. Our findings empirically verified the existence of the momentum effect and the disposition effect in the Chinese stock market, and the orthogonality of empirical eigenfunctions suggests that the disposition effect could not explain the momentum effect. By incorporating these cross-sectional patterns into the portfolio selection, we found that the Sharpe ratio of the mean-variance optimized portfolio could be boosted to 1.27. Our functional data analysis approach can be generally applied to any market of interest, particularly to markets suffering from high volatilities. Future works can focus on the following issues:

understanding the role of higher order eigenfunctions; investigating the relationship between data-driven factors and macroeconomic variables; and modeling market anomalies by using a functional multi-factor model.

Table 1: Results for F-F three-factor and Carhart four-factor models

Table 1 shows the OLS regression estimates on the past returns-sorted equal-weighted excess returns and asset pricing factors. Each column represents excess portfolio returns from the first decile (loser) to the last decile (winner). The returns on the winner-minus-loser portfolio are in column "wml". Panel A shows the estimation results for the three-factor Fama French model [START_REF] Fama | The cross-section of expected stock returns[END_REF]. Panel B gives the estimation results of Carhart's four-factor model with an additional wml factor [START_REF] Carhart | On persistence in mutual fund performance[END_REF]

→

  where p denotes convergence in probability and ĉ1 , ĉ2 , . . . are random signs defined by ĉi = sign( ψi , ψ i ). Theorem 3.1 is an immediate consequence of the ergodic theorem in Hilbert spaces, i.e., ||ĉ T -c|| p → 0 (see Hörmann and Kokoszka, 2010 for the proof). Another consequence of the theorem is that ŷt (u) in (8) is close to y t (u) if M and T are both large. Therefore, the cross-sectional patterns in the variation of y t (u) are decomposed and expressed by a finite number of estimated eigenfunctions ψi , and their variance contributions are captured by their corresponding λi .

  documents serial correlation coefficients, which shows that 33 out of 40 coefficients are negative. This result indicates a short-run overreaction effect in the Chinese stock market. Similar results were found by Tan et al. (2008) 5. EXPLANATIONS OF DATA-DRIVEN ASSET PRICING FACTORS and Ni et al. (2015). Panel B of Table

Figure 6

 6 Figure 6 plots the cumulative returns for contrarian, EP 1, EP 2, and mean-variance optimized portfolios.

Figure 1 :

 1 Figure 1: The 13 B-spline basis functions The figure plots 13 basis functions used on 100 grouped cross-sectional returns. Nine dashed lines are interior breakpoints or knots, splitting 100 groups into ten intervals and requiring 13 basis functions for smoothing.

  Assumption 3.1 is very mild as it is fairly acceptable that financial data follows autoregressive and GARCHtype processes in L 2 [0, 1] (Hörmann and Kokoszka, 2010). Due to stationarity, the functional mean Ey t (u) = µ(u) does not depend on time. The covariance function of {y t (u), 0 ≤ u ≤ 1} is defined by

)

3. FUNCTIONAL DATA DECOMPOSITION APPROACH

We assume that Assumption 3.1 {y t (u), 0 ≤ u ≤ 1} is a stationary sequence with sample paths in L 2 [0, 1], and also the moment condition E||y t (u)|| 4 < ∞ is satisfied.

  . The final two columns show the F statistics and P values of the GRS test (Gibbons et al., 1989). *, ** and *** denote statistical significance at 10%, 5% and 1%, respectively. * * -1.31 * * * -1.44 * * * -1.57 * * * -1.54 * * * -1.69 * * * -1.78 * * * -1.86 * * * -1.75 * * *

		Loser	2	3	4	5	6	7	8	9	Winner	wml	F(GRS) p(GRS)
							Panel A: Fama-French model
	α	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-0.01 *	-0.02 * * * -0.01 * *
	rmrf	1.10 * * *	1.09 * * *	1.10 * * *	1.09 * * *	1.08 * * *	1.07 * * *	1.08 * * *	1.07 * * *	1.04 * * *	1.06 * * *	-0.02
	smb	0.97 * * *	0.92 * * *	0.88 * * *	0.91 * * *	0.89 * * *	0.84 * * *	0.90 * * *	0.80 * * *	0.85 * * *	1.71 * * *	0.37 * * *	4.82	0.0000
	hml	-0.92 -0.79	0.45 *
	AdjR 2	0.83	0.85	0.85	0.86	0.86	0.86	0.85	0.86	0.82	0.79	0.15
							Panel B: Carhart model
	α	-0.01 * *	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-0.01 * *	0.00
	rmrf	1.08 * * *	1.08 * * *	1.09 * * *	1.09 * * *	1.08 * * *	1.07 * * *	1.08 * * *	1.07 * * *	1.04 * * *	1.08 * * *	0.00
	smb	1.24 * * *	1.13 * * *	1.05 * * *	1.05 * * *	0.99 * * *	0.89 * * *	0.90 * * *	0.73 * * *	0.64 * * *	1.24 * * *	0.00 * *	3.65	0.0003
	hml	-0.87 87 * * *	0.00
	wml	-0.73 * * * -0.58 * * * -0.45 * * * -0.39 * * * -0.28 * *	-0.15	-0.02	0.20	0.56 * * *	1.27 * * *	1.00 * * *
	AdjR 2	0.87	0.88	0.86	0.87	0.86	0.87	0.86	0.86	0.84	0.89	1.00

* * * * -1.28 * * * -1.41 * * * -1.54 * * * -1.53 * * * -1.68 * * * -1.78 * * * -1.87 * * * -1.78 * * * -0.

Table 2 :

 2 The percentage of variations explained by the first four eigenfunctionsThe columns show the percentage of variability accounted for by the first four eigenfunctions of cross-sectional returns. The last column documents the total variations explained by the first four eigenfunctions. The first row shows the percentages computed from the real data, and the second row shows the percentage computed from the bootstrap with 5,000 replications.

		1st eigenfunction 2nd eigenfunction 3rd eigenfunction 4th eigenfunction	Total
	Real data	64.62%	14.73%	5.16%	3.26%	87.77%
	Bootstrap	74.72%	14.78%	5.22%	3.39%	88.11%

Table 3 :

 3 Fama-MacBeth regression on F P C1 t and F P C2 tThe estimated coefficients of the Fama-MacBeth regression on F P C1t and F P C2t with dependent variable past returnssorted equally weighted portfolio returns are documented. First, we computed the factor risk exposures for each portfolio by regressing the portfolio returns on the risk factors. Second, we obtained coefficients for risk premiums by regressing the portfolio returns on the risk exposures month to month. To measure the strength of risk premiums instead of their directions, we used the absolute value of their coefficients. Bracketed values are Newey-West robust t-statistics.

			Jan.2007-Jun.2017			Jan.2007-Jun.2017			Jan.2007-Jun.2017
		Entire	Jan	Feb-Nov	Dec	Entire	Jan	Feb-Nov	Dec	Entire	Jan	Feb-Nov	Dec
		0.13	0.18	0.12	0.15					0.13	0.17	0.12	0.14
	F P C1 t
		(3.43)	(4.56)	(3.23)	(4.45)					(3.99)	(5.53)	(3.75)	(5.09)
						0.07	0.08	0.07	0.07	0.07	0.08	0.07	0.07
	F P C2 t
						(1.92)	(1.87)	(1.93)	(1.92)	(2.47)	(2.93)	(2.41)	(2.62)
	AdjR 2	0.17	0.28	0.16	0.20	0.08	0.09	0.08	0.09	0.25	0.37	0.24	0.26

Table 4 :

 4 Results for multi-factor models with FPCsThe OLS estimates of the coefficients in regressing past returns-sorted excess portfolio returns on Fama-French risk factors, and functional momentum factors F P C1t and F P C2t are presented. Columns represent excess portfolio returns from the first decile (loser) to last decile (winner), and returns on the winner-minus-loser portfolio (wml). Panel A displays the three Fama-French risk factors and F P C1t. Panel B replaces F P C1t with F P C2t. Panel C considers both F P C1t and F P C2t, forming a five-factor model. The final two columns present F statistics and P values of the GRS tests (Gibbons et al. 1989). * * -1.36 * * * -1.48 * * * -1.60 * * * -1.56 * * * -1.69 * * * -1.78 * * * -1.84 * * * -1.70 * * * * * -1.17 * * * -1.29 * * * -1.24 * * * -1.17 * * * -1.22 * * * -1.29 * * * -1.36 * * * * * -0.12 * * * -0.14 * * * -0.16 * * * -0.16 * * * -0.18 * * * * * -1.21 * * * -1.31 * * * -1.26 * * * -1.18 * * * -1.23 * * * -1.28 * * * -1.34 * * * -1.13 * * * -1.01 * * * * * -0.12 * * * -0.14 * * * -0.16 * * * -0.16 * * * -0.18 * * * 0.09 * 0.01

	*, ** and *** denote statistical significance at 10%, 5% and 1% level.
		Loser	2	3	4	5	6	7	8	9	Winner	wml	F(GRS) p(GRS)
							Panel A: F-F with F P C1t
	α	-0.01 * *	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-0.01 *	0.00
	rmrf	1.07 * * *	1.07 * * *	1.09 * * *	1.08 * * *	1.07 * * *	1.07 * * *	1.08 * * *	1.08 * * *	1.06 * * *	1.10 * * *	0.02 *
	smb	1.15 * * *	1.07 * * *	0.99 * * *	1.01 * * *	0.96 * * *	0.87 * * *	0.88 * * *	0.74 * * *	0.68 * * *	1.39 * * *	0.12 * * *	2.99	0.0022
	hml	-0.97 -0.70 *	0.14 * *
	F P C1t -0.12 * * * -0.10 * * * -0.08 * * * -0.07 * * *	-0.05 *	-0.02	0.01	0.04 *	0.11 * * *	0.22 * * *	0.17 * * *
	AdjR 2	0.87	0.87	0.86	0.86	0.85	0.85	0.85	0.86	0.85	0.89	0.95
							Panel B: F-F with F P C2t
	α	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-0.01 * * * -0.02 * * * -0.01 * * *
	rmrf	1.09 * * *	1.09 * * *	1.10 * * *	1.09 * * *	1.08 * * *	1.07 * * *	1.08 * * *	1.07 * * *	1.04 * * *	1.06 * * *	-0.02
	smb	0.82 * * *	0.99 * * *	0.97 * * *	1.09 * * *	1.10 * * *	1.09 * * *	1.17 * * *	1.09 * * *	1.18 * * *	1.55 * * *	0.36 * * *	3.44	0.0006
	hml	-1.18 -1.16 *	-1.09 *	0.05
	F P C2t	0.08 * *	-0.04	-0.05	-0.10 0.09	0.00
	AdjR 2	0.84	0.85	0.85	0.87	0.87	0.88	0.87	0.88	0.84	0.79	0.15
							Panel C: F-F with FPC1 and FPC2
	α	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
	rmrf	1.07 * * *	1.07 * * *	1.08 * * *	1.08 * * *	1.07 * * *	1.07 * * *	1.08 * * *	1.08 * * *	1.06 * * *	1.10 * * *	0.02 *
	smb	1.01 * * *	1.15 * * *	1.09 * * *	1.19 * * *	1.17 * * *	1.13 * * *	1.17 * * *	1.02 * * *	1.00 * * *	1.21 * * *	0.10 * * *
	hml	-1.22 0.10 *	2.16	0.0243
	F P C1t -0.12 * * * -0.10 * * * -0.08 * * * -0.07 * *	-0.05 *	-0.02	0.00	0.04	0.11 * * *	0.22 * * *	0.17 * * *
	F P C2t -0.10 AdjR 2 0.08 * * -0.05 -0.05 0.87 0.88 0.87 0.88	0.87	0.88	0.87	0.88	0.88	0.89	0.95

* * * * *

Table 5 :

 5 Relative strength of momentum and contrarian portfolios

Table 5

 5 shows the relative strength of the average monthly returns from momentum and contrarian portfolio strategies.Formation periods are denoted by K, which equals to 3, 6, 9 and 12 months for short-run past returns and 60 months for long-run past returns. Holding periods vary from 3 to 24 months in each column. We apply the method proposed in[START_REF] Jegadeesh | Returns to buying winners and selling losers: Implications for stock market efficiency[END_REF] to construct equally weighted portfolios with overlapping holding periods. Bracketed values are Newey-West robust t-statistics.

				Momentum					Contrarian
	K	3	6	9	12	24	3	6	9	12	24
		-0.0030	-0.0028	-0.0025	-0.0025	-0.0024	0.0028	0.0025	0.0023	0.0023	0.0023
	3 months
		(-1.52)	(-1.90)	(-2.36)	(-2.85)	(-4.48)	(1.42)	(1.78)	(2.19)	(2.63)	(4.24)
		-0.0042	-0.0039	-0.0036	-0.0034	-0.0032	0.0042	0.0038	0.0035	0.0034	0.0031
	6 months
		(-2.30)	(-2.85)	(-3.14)	(-3.43)	(-4.23)	(2.25)	(2.78)	(3.05)	(3.34)	(4.15)
		-0.0036	-0.0032	-0.0030	-0.0028	-0.0023	0.0034	0.0030	0.0028	0.0026	0.0021
	9 months
		(-2.00)	(-2.55)	(-2.74)	(-2.85)	(-2.77)	(1.92)	(2.39)	(2.55)	(2.66)	(2.55)
		-0.0042	-0.0039	-0.0036	-0.0034	-0.0027	0.0038	0.0034	0.0032	0.0029	0.0022
	12 months
		(-2.19)	(-2.78)	(-3.09)	(-3.25)	(-3.09)	(2.00)	(2.45)	(2.68)	(2.81)	(2.58)
		-0.0025	-0.0023	-0.0022	-0.0021	-0.0018	0.0018	0.0013	0.0011	0.0008	0.0002
	60 months
		(-1.64)	(-2.21)	(-2.48)	(-2.63)	(-2.89)	(1.14)	(1.21)	(1.15)	(0.99)	(0.47)

Table 9 :

 9 Statistics of portfoliosAverage monthly returns, Sharpe ratios and the maximum draw-down of five portfolios are displayed. The first two columns are the benchmarks market index and contrarian-lmw portfolios. Considering that the momentum-wml portfolio is unprofitable and the contrarian-lmw is profitable, the portfolio weight for EP 1 is -ψ1 instead of ψ1 , and the weight for EP2 is ψ2 . The last three columns represent portfolios EP 1, EP 2 and their mean-variance optimization. Assuming no transaction costs, we hold all portfolios from January 2007 to June 2017.

		Benchmark Portfolios		Eigenfunction Portfolios
		Market index Contrarian	EP 1	EP 2	Optimized portfolio
	average return	0.0168	0.0182	0.0236	0.0248	0.0264
	Sharpe ratio	0.0550	0.4757	0.4936	0.3272	1.2736
	maximum draw-down	-1.2368	-0.0773	-0.1245 -0.2141	-0.0208

In this paper, we term the China A-shares market as the Chinese stock market, which is dominated in Chinese Yuan/CNY, and is mainly accessible for local investors. Compared with the developed markets, individual investors in this market outweigh institutions regarding the stock capitalization held.

The closing price is adjusted by its stock splits and dividend payments.

There are non-tradable shares in the Chinese stock market before 2005 because of the liquidity shortage. In order to reconstruct the ownership structure of listed companies, the authority launched a structural reform to eliminate the nontradable shares during2005-2006.

We also applied the same approach to the U.S. market including more than 7,000 shares in the NYSE, AMEX and NASDAQ from 2,000 to 2017. The result showed that the F P C2t factor still exists but is not as strong as in the Chinese stock market.

Acknowledgements. We are grateful to Mr. Curtis Miller for his carefully reading our paper. We also thank the editor, Cheng-Few Lee, one anonymous referee, whose detailed and constructive comments helped to improve the quality of the paper.

Table 6: Serial and cross-Serial correlations

Serial and cross-serial correlations in the past returns-sorted portfolio returns are displayed. Panel A describes the serial correlations up to lag 4. Panel B describes the cross-serial correlation with lag 1. The correlation between lagged loser returns and contemporary winner returns is 0.06, and the correlation between lagged winner returns and contemporary loser returns is 0.02. The sample includes stocks without missing data from January 2007 to June 2017 (1307 stocks).


Oriane Morel

Corentin Spriet

Cédric Lion

Fabien Baldacci-Cresp

Garance Pontier

Marie Baucher

Christophe Biot

Simon Hawkins

Godfrey Neutelings
email: godfrey.neutelings@univ-lille.fr

Ratiometric fluorescent safranin-O staining allows the quantification of lignin contents in muro

Keywords: Lignin quantification, safranin-O, cell wall, segmentation, confocal microscopy

In some specific vascular plant tissues, lignin can impregnate the entire cell wall to make it more rigid and hydrophobic. Different techniques have been developed in the past years to make possible the quantification of this polyphenolic polymer at the organ or tissue level but difficulties of access to the cellular level remain. Here we describe an approach based on ratiometric emission measurements using safranin-O and the development of a macro adapted for the FIJI software which makes possible the quantification of lignin in three different layers of the cell wall on images captured on a fluorescent confocal microscope.

Introduction

Lignin is a major component of plant cell walls present in some specialized tissues of vascular plants. This polyphenolic polymer is mainly derived from the polymerization of p-coumaryl, coniferyl and sinapyl alcohols differing in their degree of methoxylation. After their incorporation, these three phydroxycinnamyl alcohol precursors (monolignols) give rise to p-hydroxyphenyl (H), guaiacyl (G), and syringyl (S) units respectively. Lignin is responsible for water transport in the xylem vessel elements and confers strength to cell walls contributing to the general mechanical support necessary for plant upright growth [START_REF] Boerjan | Lignin biosynthesis[END_REF]. Studies on lignin biosynthesis are motivated by the need to understand its role in plant physiology but also by the increasing use of biomass in agro-industry [START_REF] Li | Improvement of biomass through lignin modification[END_REF].

A good understanding of lignin metabolism necessarily requires the use of techniques allowing their quantification. Two main "wet chemistry" approaches are commonly used for quantifying lignin, depending upon the plant species and more importantly, whether the tissues contain more or less quantities of lignins. Thus, the method using the solubilization of lignins by acetyl bromide [START_REF] Morrison | Improvements in the acetyl bromide technique to determine lignin and digestibility and its application to legumes[END_REF][START_REF] Morrison | A semi-micro method for the determination of lignin and its use in predicting the digestibility of forage crops[END_REF] is well adapted to herbaceous species whereas the gravimetric Klason method [START_REF] Johnson | The spectrophotometric determination of lignin in small wood samples[END_REF] is more suitable for woody species. There are also different methods to determine the H, G, and S subunit composition and the S/G ratio of lignins. Historically, thioacidolysis [START_REF] Lapierre | Thioacidolyse von Pappelholzligninen: Nachweis von monomeren Syringylverbindungen und Charakterisierung von Guajacyl-Syringyl-Ligninfraktionen[END_REF] and nitrobenzene oxidation [START_REF] Monties | Lignins[END_REF] methods were used, but they are now gradually replaced by pyrolysis coupled with GC/MS [START_REF] Wagner | Exploring lignification in conifers by silencing hydroxycinnamoyl-CoA: shikimate hydroxycinnamoyltransferase in Pinus radiata[END_REF] or NMR [START_REF] Mansfield | Whole plant cell wall characterization using solution-state 2D NMR[END_REF].

These approaches provide relevant information on the different tissues/organs studied but their use leads to irreversible tissue destruction. It is therefore not possible to obtain information concerning lignin quantities and compositions at the cellular level. Therefore, histochemical approaches were also developed such as phloroglucinol staining (Wiesner reaction), which specifically reveals the cinnamaldehyde functions of S and G unit-derivatives. In this case, the intensity of the coloration will depend on the quantity of the red chromogen produced by the reaction [START_REF] Clifford | Specificity of acidic phloroglucinol reagents[END_REF]. The Maüle test, on the other hand, uses potassium permanganate and hydrochloric acid to transform guaiacyl and syringyl residues into catechols, which then give rise to orange-brown quinones in the case of G lignins and purple-red quinones for G-S lignins [START_REF] Day | Lignification in the flax stem: Evidence for an unusual lignin in bast fibers[END_REF]. Other approaches exploit the autofluorescence of lignins [START_REF] Donaldson | Fluorescence lifetime imaging of lignin autofluorescence in normal and compression wood[END_REF] or the possibility to visualize them by using fluorescent compounds such as auramine [START_REF] Pesquet | Novel markers of xylogenesis in zinnia are differentially regulated by auxin and cytokinin[END_REF] and acriflavine [START_REF] Donaldson | Lignification and lignin topochemistry -An ultrastructural view[END_REF].

All these techniques can therefore provide relatively detailed spatial information, but without offering the possibility to produce quantitative data.

Safranin-O has an interesting potential as a fluorescent probe [START_REF] Bond | Safranine fluorescent staining of wood cell walls[END_REF]. Indeed, the emission spectrum of safranin-O depends on the amount of lignin present in the cell wall (Fig. 1). When present in high amounts, the emission will be more intense in the red part. The emission spectrum (at an excitation of 488 nm) always shows a maximum at 568 nm. Before this value, the aspect of the graph is invariable whatever the amount of lignin. On the other hand, in the orange/red area between 568 and 600 nm, it is dependent on the cell wall environment. Indeed, emission values are higher in the lignified tissues (xylem) compared to the non-lignified tissues. Based on these observations, our group developed an approach including a ratiometric method allowing the relative quantification of lignins by comparing the emissions between 540-560 nm (stable portion) and 570-600 nm (variable portion) [START_REF] Baldacci-Cresp | A rapid and quantitative safranin-based fluorescent microscopy method to evaluate cell wall lignification[END_REF]. In this chapter, we first describe the methodology allowing the quantification of lignin at the cellular level following confocal microscopy acquisition. In a second step, we demonstrate the possibility to apply a segmentation approach [START_REF] Morel | REPRISAL: mapping lignification dynamics using chemistry, data segmentation, and ratiometric analysis[END_REF] to subdivise the cell wall on these images in an automatic way in order to obtain information on lignification in the cell corners, the compound middle lamella and the secondary cell 3. Growth chamber in which temperature and artificial light variations are possible.

Sample preparation

1. 50% and 70% (v/v) ethanol solutions.

2. Agarose solution at 4% (w/v) in ultrapure water.

3. 0.2% safranin-O in a 50% (v/v) ethanol solution kept in the dark. The solution can be stored for several months at -20°C. [START_REF] Morrison | A semi-micro method for the determination of lignin and its use in predicting the digestibility of forage crops[END_REF]. Vibroslicer (e.g. VT-1000S, Leica, Wetzlar, Germany) for cutting sections to a thickness of 80 μm.

Image capture and analysis

1. A confocal laser scanning microscope. We used a Nikon A1R confocal (Nikon, Tokyo, Japan) equipped with a 60x/1.4 aperture oil immersion objective (Plan APO VC) and the NIS Element AR3.0 software.

2. FIJI v2.0 [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] . This can be downloaded at https://fiji.sc 3. The associated FIJI macro developed for the quantification of fluorescence. This can be downloaded at https://zenodo.org/record/5775647#. YbiaNi17RhE.

Methods

Plant growth

Arabidopsis thaliana seeds were stratified during 72 h at 4 °C in the dark. Plants were grown in a GroBank chamber (CLF Plant Climatics GmbH, Wertingen, Germany) under 12 h light (100 PAR light intensity) at 22 °C and 12 h dark at 20 °C for 6 weeks. The photoperiod was then changed to 16 h light, 8 h dark and the floral stems were cut when they reached a size of 30 cm.

Sample preparation

1. Collect floral stem samples of 0.5 cm length 2. Melt the 4% agarose solution and maintain it at 55°C. Pour 1.5 mL of this solution in enough wells of a 48-well cell culture plate. Let the agarose cool down until it becomes thicker but still stays liquid and place the plant fragment in the center of each well with tweezers. Maintain in a vertical position until the agarose hardens.

3. Leave the plate on the bench for another 10 minutes and then unmold with a thin spatula. Cut the basis of the agarose perfectly perpendicular to the fragment with a razor blade. Glue the agarose-embedded fragment on the specimen holder with cyanoacrylate glue. On the Leica VT-1000S vibroslicer, it is possible to place up to three samples on one holder. 5. Place the fragments in a 48-well cell culture plate containing 1 mL 50% (v/v) ethanol solution.

Sample staining

1. When all the sections are done, replace the ethanol by a fresh or daily thawed 0.2% safranin-O (w/v) solution and incubate for 10 min on a microplate shaker in the dark at room temperature. The shaking intensity should be high enough to allow good mobility of the fragments in the liquid.

2. Remove all safranin-O solution and replace by 50% ethanol. Incubate with shaking for 10 min in the dark. Remove the solution and rinse twice with the same parameters during 15 min in ultrapure water.

3. Mount sections in water on a glass slide and place a coverslip.

Image capture

1. Perform the acquisition on the tissues by using 3 independent channels. The first is on autofluorescence excitation at 405 nm, and emission at 450/500 nm), the second is on the green channel with excitation at 488 nm and emission at 530-560 nm. The third is on the red channel with excitation at 561 nm and emission at 570-600 nm. All tracks must be acquired sequentially and parameters must be kept constant for all acquisitions.

2. Select a flat region and adjust your focus to perform images between 6 and 9 µm deep from the surface.

Image analysis

1. Crop the sharp area of the images acquired in 3.4 2. Treat images using FIJI v2.0 and the associated homemade macro developed for this method. First open the Fiji app, open the macro and finally the image cropped in 1. Run the macro and follow the instructions. The macro includes 2 steps: 2.1 First, a ratio image is generated as previously described [START_REF] Baldacci-Cresp | A rapid and quantitative safranin-based fluorescent microscopy method to evaluate cell wall lignification[END_REF] (Fig. 2). The result appears as a ratiometric image with a color range between 0 (purple) and 2 (red). The intensity ratio values also appear in the results window and can be used for quantification.

2.2 Combining safranin fluorescence and lignin autofluorescence information, the ratio image obtained in 3.5.1.1 is then segmented, as previously described [START_REF] Morel | REPRISAL: mapping lignification dynamics using chemistry, data segmentation, and ratiometric analysis[END_REF], into 3 cell walls regions: the cell corner (CC), the compound middle lamella (CML) consisting of the middle lamella and the primary cell wall, and the secondary cell wall (SCW). (Fig. 3).

Notes

1. The use of a vibratome ensures regular homogenous sectioning of the plant tissues. However, hand sectioning with a sharp razor blade can also be performed. The sections should then be checked under the microscope and only the most fine and regular ones should be selected for the safranin-O staining.

2. If using a vibratome, the agarose solution must be used within the week. In our protocol, we suggest to use multi-well cell culture plates but it is also possible to use 1.5 mL microtubes. Their bottom conical part should first be cut-off and placed inside the rest of the tube. Both parts are then sealed with a piece of Parafilm Ⓡ . The melded agarose can then be poured in the tube and the sample placed in the center. When the gel has cooled down, it can be unmolded by pressing on the bottom conical portion of the tube.

3. The 0.2% safranin-O in a 50% (v/v) ethanol solution can be stored for several months at -20°C.

When the solution is thawed, it should be stored in the dark and used within 24 hours. 4. The samples must be stained and processed in the same day in order to be compared. Each characterization of mutants for example must be done with the corresponding WT. 5. When using the confocal microscope, the focus is more accurate when using the red channel.

6. Accurate quantification should be performed by analyzing at least 3 different plants on which 5 sections were performed and at least 2 images captured per section (total = 30 images) 7. The segmentation macro also allows access to the intensity ratio values in the three layers of the cell wall. It is therefore possible to perform accurate quantification and statistics on lignin amounts.

1 .

 1 Seeds of Arabidopsis thaliana2. Pots filled with potting soil.

4 .

 4 Cut 80 µm thick fragments with the vibratome. The speed and the frequency depend on the tissue. For a fragment taken at the base of the floral stem, use a speed of 0.4 mm/s (position 6 on the Leica model) and a frequency of 70 Hz (position 7).

Figure 1 :

 1 Figure 1: Average safranin-O spectra (with excitation at 488 nm) of cell walls from different flax (Linum usitatissimum) tissues containing low amounts (bast fiber and cambial cell walls) or high amounts (xylem cell walls) of lignins.

Figure 2 :

 2 Figure 2: Local lignin content determination on Arabidopsis floral stems (wild type and mutants) using safranin O fluorescence-based method. A: Green and red channels correspond to safranin fluorescence.The ratiometric image was obtained by applying the specific FIJI macro on both images. B: The normalized safranin intensity ratios (± SD) are represented (WT=100%). The quantification was performed on interfascicular fibers. Means (± SD) with an asterisk are significantly different according to Student t-test (p-value < 0.05, n = 3). WT: wild type; prx72: Salk_136893 mutant; gaut9: SALK_040287 mutant.

Figure 3 :

 3 Figure 3: Automatic segmentation of different cell wall zones. Three masks were applied to images of Arabidopsis flowering stem cross-sections stained with safranin-O and observed by confocal laser scanning microscopy. Green and red channels correspond to safranin-O fluorescence. WCW: whole cell walls; SCW: secondary cell walls; CML: compound middle lamella; CC: cell corner.


Walter H Aschbacher
email: aschbacher@ma.tum.de

Jean-Marie Barbaroux
email: barbarou@univ-tln.fr

J.-M Barbaroux

Out of equilibrium correlations in the XY chain

Keywords: Mathematics Subject Classifications (2000). 46L60, 47B35, 82C10, 82C23 Non-equilibrium steady state, XY chain, correlations, Toeplitz operators

We study the transversal spin-spin correlations in the non-equilibrium steady state of the XY chain constructed by coupling a finite cutout of the chain to the two infinite parts to its left and right acting as thermal reservoirs at different temperatures. We prove that the spatial decay of these correlations is at least exponentially fast.

Introduction

The XY chain is the one-dimensional integrable spin system, introduced in [START_REF] Lieb | Two soluble models of an antiferromagnetic chain[END_REF] (see also [START_REF] Bratteli | Operator algebras and quantum statistical mechanics 2[END_REF]), whose formal Hamiltonian is specified by

H = - 1 4 ∑ x∈Z (1 + γ) σ (x) 1 σ (x+1) 1 + (1 -γ) σ (x) 2 σ (x+1) 2 + 2λ σ (x) 3 , (1.1)
where σ (x) j denotes the Pauli matrix at site x ∈ Z in the transversal directions, j = 1, 2, and in the longitudinal direction, j = 3. The parameter γ ∈ (-1, 1) describes the anisotropy of the spin-spin coupling whereas λ ∈ R stands for an external magnetic field.

Since the discovery of their ideal thermal conductivity, quasi-one-dimensional Heisenberg spin-1/2 systems, made from different materials, have been intensively studied experimentally (see [START_REF] Sologubenko | Thermal conductivity and specific heat of the linear chain cuprate Sr 2 CuO 3 : Evidence for the thermal transport via spinons[END_REF][START_REF] Sologubenko | Heat transport by lattice and spin excitations in the spin-chain compounds SrCuO 2 and Sr 2 CuO 3[END_REF]; SrCuO 2 and Sr 2 CuO 3 are often considered to be the best physical realizations of onedimensional XYZ Heisenberg models).

Not only this highly unusual transport property motivates the theoretical study of such nonequilibrium models, but the XY chain also represents one of the simplest non-trivial testing grounds for the development of general ideas in rigorous non-equilibrium theory.

Already in [START_REF] Lieb | Two soluble models of an antiferromagnetic chain[END_REF], for vanishing external magnetic field λ = 0, the transversal spin-spin correlations in the ground state and at nonzero temperature could be expressed by means of determinants of large truncated Toeplitz matrices. Moreover, bounds on these correlations implied vanishing long-range order, at least in the isotropic case γ = 0. Later, in [START_REF] Mccoy | Spin correlation functions of the X -Y model[END_REF], this study was continued yielding an asymptotic evaluation of the transversal correlations with the help of Szegö's theorem. Afterwards, in [START_REF] Barouch | Statistical mechanics of the XY model. II. Spin-correlation functions[END_REF], almost the complete phase diagram in γ and λ for the behavior of the correlation functions both in the longitudinal and in the transversal directions were obtained, and so for zero and nonzero temperature: The result of this study for the case of nonzero temperature is that all the correlation functions are asymptotically exponentially decaying with a decay rate which depends on the magnetic field λ and the anisotropy γ.

In this note, we study the large n behavior of the transversal spin-spin correlations,

C j (n) = ω(σ (0) j σ (n) j ), j = 1, 2, n ∈ N, (1.2)
in the non-equilibrium steady state (NESS) ω constructed in [START_REF] Aschbacher | Non-equilibrium steady states of the XY chain[END_REF] (and already in [START_REF] Araki | Asymptotic time evolution of a partitioned infinite two-sided isotropic XY-chain[END_REF] for γ = 0) in a setting which has become to serve as paradigm in non-equilibrium statistical mechanics: a "small" system which is coupled to two infinite reservoirs which are in thermal equilibrium at different inverse temperatures β L and β R . We prove that the spatial decay of C j (n) is at least exponentially fast. This behavior contrasts with the one generally expected by the folklore which predicts quasi-long-range order out of equilibrium.

In the following section, we give a brief informal description of our non-equilibrium setting for the XY chain. We refer to [START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF][START_REF] Aschbacher | Non-equilibrium steady states of the XY chain[END_REF][START_REF] Jakšić | Mathematical theory of non-equilibrium quantum statistical mechanics[END_REF] for a precise formulation within the framework of C * algebraic quantum statistical mechanics.

The non-equilibrium setting for the XY chain

Consider the XY chain described by the Hamiltonian (1.1) and remove the two bonds at the sites ±M, M > 0. Doing so, the initial chain divides into a compound of three noninteracting subsystems. This configuration is what we call the free system whose Hamiltonian

H 0 = H L + H S + H R is built as in (1.1) according to the union Z L ∪ Z S ∪ Z R , where Z L = {x ∈ Z | x ≤ -M -2}, Z R = {x ∈ Z | x ≥ M + 1}, and Z S = {x ∈ Z | -M ≤ x ≤ M -1}.
The infinite pieces Z L and Z R will play the role of thermal reservoirs to which the finite system on Z S is coupled by means of V = H -H 0 . In contrast, the configuration described by H, i.e. the original XY chain on the whole of Z, is considered to be the perturbed system.

In order to construct a NESS in the sense of [START_REF] Ruelle | Entropy production in quantum spin systems[END_REF], we choose the initial state ω 0 to be composed of (τ j , β j )-KMS states ω j on Z j , j = L, R, and of the normalized trace state ω S on Z S (τ j denotes the time evolution generated by H j , and β j is the inverse temperature), i.e. we set

ω 0 = ω L ⊗ ω S ⊗ ω R .
It is well-known that the Jordan-Wigner transformation maps the XY model on a model of free fermions with annihilation and creation operators c x , c * x . Using scattering theory on the 1particle Hilbert space of these fermions, the NESS ω w.r.t. the initial state ω 0 and the perturbed time evolution τ t have been constructed in [5],

ω(c x ) = lim t→ +∞ ω 0 (τ t (c x )).
It has been shown in [START_REF] Aschbacher | Non-equilibrium steady states of the XY chain[END_REF] that ω is a quasi-free state with 2-point operator S,

ω(B * ( f )B(g)) = ( f , Sg), (2.3) where f , g ∈ l 2 (Z)⊗C 2 ≃ l 2 (Z)⊕l 2 (Z), and
B( f ) = ∑ x∈Z ( f + (x)c * x + f -(x)c x ) for f = ( f + , f -).
Moreover, S has been explicitly computed in [START_REF] Aschbacher | Non-equilibrium steady states of the XY chain[END_REF]. In the Fourier picture,

l 2 (Z) ≃ L 2 (T) (with T = {z ∈ C | |z| = 1}), it reads S(ξ ) = 1 + e β h(ξ )+δ k(ξ ) -1 , (2.4)
with the parameters β and δ given by

β = 1 2 (β R + β L ) , δ = 1 2 (β R -β L ) .
(2.5)

The 1-particle operators h and k have the form

h(ξ ) = (cos ξ -λ ) ⊗ σ 3 -γ sin ξ ⊗ σ 2 , k(ξ ) = sign(κ(ξ )) µ(ξ ) ⊗ σ 0 ,
where the functions κ(ξ ) and µ(ξ ) are defined by

κ(ξ ) = 2λ sin ξ -(1 -γ 2 ) sin 2ξ , µ(ξ ) = (cos ξ -λ ) 2 + γ 2 sin 2 ξ 1/2 . (2.6)
Furthermore, σ 0 is the identity on C 2 , and σ 1 , σ 2 , σ 3 denote the Pauli matrices,

σ 0 = 1 0 0 1 , σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 . (2.7)

Exponential decay of the correlation function

Let us now state our result.

Theorem Let ω be the quasi-free NESS characterized by (2.3), and C j (n), j = 1, 2, the correlation functions (1.2). For all values of the parameters 0 ≤ δ < β < ∞, γ ∈ (-1, 1), and λ ∈ R, the transversal spin-spin correlations C j (n) decay at least exponentially fast, and

lim sup n→∞ log C j (n) n ≤ 1 2 2π 0 dξ 2π log [th(β L µ(ξ )/2) th(β R µ(ξ )/2)] < 0.
Remarks 1) If the temperature difference of the reservoirs vanishes, i.e. δ = 0, the total system is in thermal equilibrium at inverse temperature β L = β R , and our bound estimates from above the asymptotically exponentially decaying behavior from [START_REF] Barouch | Statistical mechanics of the XY model. II. Spin-correlation functions[END_REF].

2) For δ = 0, the correlation in the 3-direction decays like 1/n 2 at infinity for all γ ∈ (-1, 1), λ ∈ R, see [START_REF] Aschbacher | Non-equilibrium steady states of the XY chain[END_REF]. In contrast to this result, our theorem does not confirm the folklore about the change in the type of decay -from short range to long range -when passing from equilibrium to non-equilibrium.

Before we begin the proof of the theorem, we rewrite the correlation function C j (n) as the determinant of a 2n × 2n block Toeplitz matrix, in the same way as it has been done at zero and non-zero temperature in [START_REF] Barouch | Statistical mechanics of the XY model. II. Spin-correlation functions[END_REF][START_REF] Lieb | Two soluble models of an antiferromagnetic chain[END_REF]. The reader not familiar with Toeplitz theory may consult Appendix A before entering the proof. We restrict our attention to the transversal correlation function C 1 (n) ≡ C(n) in the 1-direction (the 2-direction being analogous, see the Jordan-Wigner transformation for σ 2 in (3.8) below),

C(n) = ω(σ (0) 1 σ (n) 1 ), n ∈ N.
The well-known Jordan-Wigner transformation expresses the spins σ

(x) 1 , σ (x) 2 , σ (x)
3 , x ∈ Z, by means of fermionic creation and annihilation operators c

♯ x (= c * x , c x ), σ (x) 1 = T S (x) (c x + c * x ), σ (x) 2 = iT S (x) (c x -c * x ), σ (x) 3 = 2c * x c x -1, (3.8)
where

S (x) = σ (1)
3 . . . σ (x-1) 3

for x > 1, S (1) = 1, and

S (x) = σ (x) 3 . . . σ (0)
3 for x < 1, see for example [START_REF] Aschbacher | Non-equilibrium steady states of the XY chain[END_REF]. The element T stems from Araki's C * crossed product extension of the CAR algebra for the two-sided chain, see [START_REF] Araki | On the XY-model on two-sided infinite chain[END_REF]. It has the properties T 2 = 1, T c ♯ 0 = -c ♯ 0 T , and T c ♯ x = c ♯ x T for x > 0. Plugging the Jordan-Wigner transformation (3.8) into the product σ (0)

1 σ (n) 1 , we find σ (0) 1 σ (n) 1 = (-1) n a (0) -a (1) + a (1) -a (2) + a (2) -• ... • a (n-1) + a (n-1) - a (n) + ,
where a

(x) ± = c *
x ± c x . Moreover, we define η ± = (1, ±1) ∈ C 2 and α (x) ± = δ x ⊗ η ± (with δ x ∈ l 2 (Z), δ x (y) = 1 for y = x and zero otherwise). With the linear map

B introduced in [1], l 2 (Z) ⊕ l 2 (Z) ∋ f = ( f + , f -) → B( f ) = ∑ x∈Z ( f + (x)c * x + f -(x)c x ), we have a (n) ± = B(α (n) ± )
. Hence, we can express the correlation function C(n) as

C(n) = (-1) n ω(B(α (0) -) B(α (1) + )B(α (1) -) • ... • B(α (n-1) + )B(α (n-1) - ) B(α (n) + )).
Since ω is quasi-free [START_REF] Aschbacher | Non-equilibrium steady states of the XY chain[END_REF], the correlation function C(n) is a Pfaffian,

C(n) = Pf Ω(n),
where the matrix (Ω(n) jk ) 2n j,k=1 ∈ C 2n×2n is defined to be skew-symmetric, and, for j < k,

Ω(n) jk = ω(B( f j )B( f k )) with f 2i-1 = α (i-1) - , f 2i = α (i) + , i = 1, 2, ..., n.
The Pfaffian Pf A of a matrix A ∈ C 2n×2n is defined by Pf A = ∑ π sign(π) ∏ n k=1 A π 2k-1 ,π 2k with the sum running over all π in the permutation group S 2n which satisfy π 2k , π 2k+1 > π 2k-1 . If A is skew-symmetric, A T = -A (A T denotes the transpose of A), the Pfaffian of A is related to the determinant of A through (Pf A) 2 = det A. Thus, we are led to study the large n asymptotics of the determinant of Ω(n),

C(n) 2 = det Ω(n).
(3.9)

In order to cast Ω(n) into the form of a truncated block Toeplitz matrix, we compute the matrix entries

A jk ±± = ω(B(α ( j) ± )B(α (k) ± )), 1 ≤ j < k ≤ n. (3.10)
For this purpose, we rewrite the 2-point operator S(ξ ) on L 2 (T) ⊗ C 2 from (2.4) as

S(ξ ) = s 0 (ξ ) ⊗ σ 0 + 3 ∑ k=1 s k (ξ ) ⊗ σ k , (3.11)
using the Pauli matrices from (2.7). The first component s 0 (ξ ) looks like

s 0 (ξ ) = 1 2 + 1 2 sign(κ(ξ )) ϕ δ (ξ ), (3.12)
and s(ξ ) = (s 1 (ξ ), s 2 (ξ ), s 3 (ξ )) has the form

s(ξ ) = 1 2 ϕ β (ξ ) r(ξ ), r(ξ ) = 1 µ(ξ ) (0, -γ sin ξ , cos ξ -λ ), (3.13)
with the functions κ(ξ ) and µ(ξ ) from (2.6). Moreover, we used the definition

ϕ α (ξ ) = sh(α µ(ξ )) ch(β µ(ξ )) + ch(δ µ(ξ )) , α ∈ R. (3.14)
Let us start with the computation of A jk ++ in (3.10). Using the property [START_REF] Aschbacher | Non-equilibrium steady states of the XY chain[END_REF], and the expressions in (3.11), (3.12), and (3.13), we find, for 1 ≤ j < k ≤ n, and with the definition e j (ξ ) = e -i jξ , that

B * ( f ) = B(J f ) for J : f = ( f + , f -) → ( f-, f+ ) from
A jk ++ = (e j ⊗ η + , S e k ⊗ η + ) L 2 ⊗C 2 = 2 2π 0 dξ 2π s 0 (ξ )e -i(k-j)ξ = 2π 0 dξ 2π sign(κ(ξ )) ϕ δ (ξ ) e -i(k-j)ξ . The computation of A jk --yields A jk --= -A jk ++ . Next, the entry A jk +-looks like A jk +-= (e j ⊗ η + , S e k ⊗ η -) L 2 ⊗C 2 = 2 2π 0 dξ 2π (s 3 (ξ ) + is 2 (ξ ))e -i(k-j)ξ = 2π 0 dξ 2π cos ξ -λ -iγ sin ξ µ(ξ ) ϕ β (ξ ) e -i(k-j)ξ . Finally, A jk -+ = -A k j +-. Due to translation invariance, we can define A x ±± = A j j+x ±± with x ∈ Z. Using A x --= -A x ++ , A x -+ = -A -x
+-, and the symmetry property A -x ++ = -A x ++ , we can write Ω(n) in (3.9) in the form of a 2n × 2n truncated block Toeplitz matrix with 2 × 2 blocks a x ,

Ω(n) =      a 0 a -1 . . . a -(n-1) a 1 a 0 . . . a -(n-2) . . . . . . . . . . . . a n-1 a n-2 . . . a 0      , (3.15)
where the blocks are given by

a x = A x ++ -A x-1 +- A -x-1 +- -A x ++ , x ∈ Z. (3.16)
Using the relations a x = 2π 0 dξ /(2π) a(ξ )e -ixξ from (A.20), we can extract from (3.16) the symbol a(ξ ) a(ξ ) = sign(κ(ξ )) ϕ δ (ξ ) -q(ξ ) ϕ β (ξ ) q(ξ ) ϕ β (ξ ) -sign(κ(ξ )) ϕ δ (ξ ) .

(3.17)

Here, ϕ α (ξ ) is defined in (3.14), κ(ξ ), µ(ξ ) in (2.6), and q(ξ ) is the complex conjugate of

q(ξ ) = cos ξ -λ -iγ sin ξ µ(ξ ) e iξ .
Therefore, denoting by T [a] the block Toeplitz matrix associated with the symbol a given by (3.17), we can write Ω(n) as a truncated block Toeplitz matrix (see (A.23)),

Ω(n) = T n [a] = P n T [a]P n .
Proof of the Theorem For n ∈ N, let s

(n) 1 ≤ s (n) 2 ≤ . . . ≤ s (n)
2n be the singular values of T n [a], and denote by N ε,n the number of singular values in [0, ε], 0 < ε < 1, including multiplicity. From the above computations, we have

|C(n)| 2 = | det Ω(n)| = | det T n [a]| = 2n ∏ j=1 s (n) j ≤ ε N ε,n 2n ∏ j=N ε,n +1 s (n) j Thus, log | det T n [a]| ≤ 2n ∑ j=N ε,n +1 log s (n) j ≤ 2n ∑ j=1 χ ε (s (n) j ) log s (n) j ,
where 0 ≤ χ ε ≤ 1 is a smooth characteristic function with support in (ε, T [a] + 1) and value 1 on [ε + ε 2 , T [a] ]. Since the symbol a of T n [a] has the two singular values ϕ β -ϕ δ and ϕ β + ϕ δ , the Avram-Parter theorem [8, p.203] (see also (A.26)) yields lim sup

n→∞ 1 n log | det T n [a]| ≤ 2π 0 dξ 2π tr [(χ ε log)(|a(ξ )|)] = 2π 0 dξ 2π (χ ε log)(ϕ β (ξ ) -ϕ δ (ξ )) + (χ ε log)(ϕ β (ξ ) + ϕ δ (ξ )).
Since the inequality holds for all ε > 0, and since for all ξ ∈ [0, 2π] we have 0

< ϕ β (ξ ) - ϕ δ (ξ ) ≤ ϕ β (ξ ) + ϕ δ (ξ ) < 1, we obtain lim sup n→∞ 1 n log | det T n [a]| ≤ 2π 0 dξ 2π log [th(β L µ(ξ )/2) th(β R µ(ξ )/2)] (3.18)
which concludes the proof.

Remarks 3) We do not provide a lower bound in our proof (but see remark 1 for δ = 0). Due to the lack of results on the invertibility of block Toeplitz matrices T [a], the behavior of the singular values of T n [a] close to zero is, in general, rather difficult to control. (see [START_REF] Böttcher | Introduction to large truncated Toeplitz matrices[END_REF]

A Toeplitz operators

Toeplitz operators [8, p.185] Let N ∈ N. We define the space l 2 N of all C N -valued sequences

f = { f i } ∞ i=1 , f i ∈ C N , by l 2 N = { f : N → C N | f < ∞}, f = ∞ ∑ i=1 f i 2 C N 1/2 , (A.19)
where • C N denotes the l 2 norm on C N . We write l 2 ≡ l 2 1 . Let {a x } x∈Z be a sequence of N × N matrices, a x ∈ C N×N . The Toeplitz operator defined through its action on elements of l 2 N by f → {∑ ∞ j=1 a i-j f j } ∞ i=1 is a bounded operator on l 2 N , if and only if

a x =

4 )

 4 In the critical regime of the XY chain, γ = 0, λ ∈ [-1, 1], and γ = 0, λ = ±1, where the nonnegative function µ(ξ ) from (2.6) can have zeroes on [0, 2π], the bound (3.18) remains finite due to the integrability of log x at the origin.5) A straightforward proof using the inequalities| det T n [a]| ≤ T n [a] 2n and T n [a] ≤ T [a] = ess sup ξ ∈T a(ξ ) L (C N ) = ess sup ξ ∈T s 2 (a(ξ ))

  )e -ixξ (A.20) for some a ∈ L ∞ N×N (see[8, p.186]), where we define (withT = {z ∈ C | |z| = 1}) L ∞ N×N = {φ : T → C N×N | φ i j ∈ L ∞ (T), i, j = 1, ..., N}. (A.21)In this case, we write the Toeplitz operator as -1 a -2 ... a 1 a 0 a -1 ... a 2 a 1 a 0 ... ... ... ... ...

  and (A.24)-(A.25)) yields the weaker bound| det T n [a]| ≤ th 2n (β R µ ∞ /2) which, in contrast to(3.18), is valid for all n. On the other hand, in the limit β R → ∞, this bound is void whereas (3.18) still implies exponential decay as long as β L < ∞.

Acknowledgements

We gratefully acknowledge the financial support by ACI "Modélisation stochastique des systèmes hors équilibre", Ministère délégué à la Recherche, France. Moreover, we would like to thank the referee for his constructive remarks.

The function a ∈ L ∞ N×N is called the symbol of T [a]. If N = 1 the symbol a ∈ L ∞ ≡ L ∞ 1×1 and the Toeplitz operator T [a] are called scalar, whereas for N > 1 they are called block.

For n ∈ N, let P n be the projections on l 2 N , P n ({x 1 , ..., x n , x n+1 , ...}) = {x 1 , ..., x n , 0, 0, ...}.

With the help of these P n , we define the truncated Nn × Nn Toeplitz matrices as

Norm of Toeplitz operators [8, p.186] The norm of a Toeplitz operator is related to its symbol,

where a ∞ is defined to be the operator norm of the multiplication operator acting on ⊕ N 1 L 2 (T) by multiplication with the matrix function a ∈ L ∞ N×N . From [7, p. 95], we have

where

is the operator norm induced by the l 2 norm on C N .

Avram-Parter theorem [8, p.203