Analysis of the efficiency of Hong Kong REITs market based on Hurst exponent

Accepted Manuscript Analysis of the efficiency of Hong Kong REITs market based on Hurst exponent Jian Liu, Cheng Cheng, Xianglin Yang, Lizhao Yan, Yongzeng Lai

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S0378-4371(19)31172-0 https://doi.org/10.1016/j.physa.2019.122035 122035 PHYSA 122035

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Physica A

Received date : 16 January 2019 Revised date : 12 June 2019 Please cite this article as: J. Liu, C. Cheng, X. Yang et al., Analysis of the efficiency of Hong Kong REITs market based on Hurst exponent, Physica A (2019), https://doi.org/10.1016/j.physa.2019.122035 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Highlights: 1. Based on fractal market theory, Hurst exponent is used to examine the market efficiency. 2. The result shows that the Hong Kong REITs market has long memory, and we further measure the length of its long memory. 3. We use the time-varying Hurst exponent to analyze the dynamic changes of Hong Kong REITs market efficiency. 4. We compared the efficiency of the REITs market, the stock market and the real estate market in Hong Kong.

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Analysis of the Efficiency of Hong Kong REITs Market Based on Hurst Exponent Jian Liu1 Cheng Cheng1 Xianglin Yang2 Lizhao Yan3 Yongzeng Lai4* 1. School of Economics and Management, Changsha University of Science and Technology, Changsha, Hunan, China 410114, [email protected], [email protected] 2. School of Economics and Management, Wuhan University, Wuhan, Hubei, China, 430072, [email protected] 3. School of Business, Hunan Normal University, Changsha, Hunan, China 410081, [email protected] 4. Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5, [email protected] Abstract: At present, as China promotes the virtuous circle of housing leasing market, Real Estate Investment Trusts (REITs) has become an important investment and financing tool. In this paper, Hurst exponent is used to examine the efficiency of Hong Kong REITs market in China, and time-varying Hurst exponent is used to explore the dynamic changes of its efficiency. The empirical results of the Hong Kong REITs market show that the Hong Kong REITs market has not yet reached weak efficiency, and it is basically in a state of inefficiency from November 25, 2005 to October 10, 2018, with only three times approximate to weak-form efficiency, but the duration is very short. Furthermore, compared with the Hong Kong stock market and the real estate market, the degree of efficiency of the Hong Kong REITs market is the lowest. Finally, on this basis, some countermeasures and suggestions for the development of China's REITs market are put forward. This paper not only enriches the study of REITs market, but also provides the relevant basis for investors to make investment decisions. Keywords: REITs; Market efficiency; Hurst exponent; Long memory 1. Introduction REITs (Real Estate Investment Trusts) is a real estate trust investment fund, which is an important means of real estate securitization. Specifically, REITs is a trust fund that pools the fund of many investors by issuing stocks or income certificates, real estate investment management by specialized investment institutions, and proportionally distributes investment income to investors. As of October 2018, 37 countries have implemented legislation and issued REITs. In 2005, China's first REITs product was listed in Hong Kong on November 25 of that year. In recent years, with the development of the country's housing rental market, REITs has become an important investment and financing tool. The scale and number of releases of REITs-like products have doubled and their development potential is huge. The data from Wind Information Database shows that the cumulative issuance of REITs-like products in China has reached USD *

Corresponding author: [email protected]

1

$64.9 billion, with a total of 30 products including mortgage and hybrid products by the end of 2017. At present, China's REITs products mainly exist in the Hong Kong market. According to the Asian REITs Research Report, as of the second quarter of 2017, the total market value of REITs in Hong Kong reached USD $30.77 billion, accounting for 15.47% of the total market value of major Asian exchanges, making it one of the major REITs markets in Asia. The development of REITs can not only solve the problem that the real estate industry of our country relies excessively on the indirect financing of banks with rigid payment for a long time, but also bring new investment channels for investors. For financial markets, market information efficiency, that is, market efficiency, often directly affects market mechanisms. In an efficient market, investors cannot earn abnormal investment returns by relying on the information set they can get (Jensen, 1978). Studying the efficiency of REITs market can help investors make investment decisions. At the same time, the efficiency of the market reflects the efficiency of market information. The higher the degree of efficiency is, the higher the efficiency of information transmission is, and the faster the government's implementation of regulatory policies is. Therefore, the study of the efficiency of the REITs market can provide reliable information for managers to effectively regulate the REITs market, which can be used as the basis for the government to implement real estate policy. In addition, studying the efficiency of the REITs market can also enrich the research system of the REITs market and provide an important theoretical basis for the asset pricing of REITs. REITs originated in the United States, and early studies on the market efficiency of REITs focused on this market (Nelling & Gyourko, 1998; Kuhle & Alvayay, 2000). With the continuous development of REITs in various countries, many researchers have examined the efficiency of multiple international REITs markets. On the one hand, scholars mainly discuss whether the REITs market is an efficient market. However, as the selected markets and the time periods studied by researchers are different, the conclusions are also inconsistent. Some empirical results show that REITs market is efficient (Kleiman, Payne & Sahu, 2002; Hao & Yenhsien, 2013; Zhang, 2017). However, there are also empirical results that most REITs market does not obey the random walk and has not yet reached an efficient market (Belaire-Franch et al., 2007; Schindler, Rottke & Füss, 2009; Su, Cheung & Roca, 2012; Zheng, 2016). Schindler (2010) further divided the REITs market into four developed REITs markets and 12 emerging REITs markets, and found that the developed REITs market was not an efficient market, while the emerging REITs market including Hong Kong and other six REITs markets had reached a weak efficiency. On the other hand, some scholars focus on the changes in the efficiency of REITs market and its influencing factors. The main research of these scholars finds that (1) the efficiency of the REITs market has increased over time (Jirasakuldech & Knight, 2005), and (2) changes in efficiency are influenced by market conditions and the real estate market. In terms of market conditions, factors such as tax reform, market development level, regulatory level, and inflation level will have impacts on the efficiency of REITs market (Jirasakuldech & Knight, 2005; Zhou & Lee, 2013). In terms of the real estate market, as the investment object of REITs is 2

real estate, the real estate market is also an important factor affecting the market efficiency of REITs (Huang et al., 2009), where real estate climate change can also cause changes in the efficiency of REITs (Jamal, 2013). In terms of research methods for REITs market efficiency, there are the early use of simple autocorrelation analysis method, random walk hypothesis based test method and traditional variance ratio test method (Kleiman, Payne & Sahu, 2002; Jirasakuldech & Knight, 2005). Later, various improved variance ratio tests are gradually applied (Huang et al., 2009; Schindler, 2010; Zhang and Sun, 2015). In recent years, scholars have also begun to use new methods to study the market efficiency of REITs. For example, the binomial option pricing tree method is used to study the efficiency of the REITs market (Ho & Tay, 2016), or the present value model is used for analysis (Zhang, 2017), such methods have certain complexity. These research methods are mostly based on the theory of efficient market. Many hypotheses of efficient market theory use a linear paradigm to describe the market. However, the financial market is essentially non-linear (Fan and Zhang, 2002), and it often has the characteristics of sharp peaks and thick tails (Wen et al., 2019). Based on the efficient market theory, the price fluctuation behavior of financial markets cannot be well explained and judged. Compared with the Efficient Market Hypothesis with strict premise hypothesis, the fractal market theory has a simpler premise, and it has strong applicability and interpretability when studying financial time series with nonlinear, sharp peak and thick tail characteristics. If the market has long memory, it indicates that the market has fractal characteristics, implying a rejection of the Efficient Market Hypothesis (Sánchez-Granero et al., 2008), and then the market is not an efficient market at this time. In various long-term memory tests, evidence of the persistence of market returns and volatility of REITs are found (Cotter & Stevenson, 2008; Assaf, 2015). These results provide support for the fractal characteristics of the REITs market. The Hurst exponent plays an important role in the theory of fractal market, and it is a measure exponent reflecting fractal characteristics. The Hurst exponent method is a nonparametric method. It has fewer assumptions about the system requirements studied and the time requirements measured, and it can be widely used in all time series (Hurst, 1951). The Hurst exponent is originally used to study the relationship between water flow and storage capacity, and is gradually used in financial markets. As early as the 1970s, scholars used the Hurst exponent to study the long-term dependence in common stock returns (Greene & Fielitz, 1977). After that, the Hurst exponent is widely used in the stock market for studying (Lo, 1991; Ye & Cao, 2001; Chen et al., 2005; Rejichi and Aloui, 2012; Wang et al., 2015; Jin, 2016). At the same time, some scholars introduced the Hurst exponent when studying the term structure of interest rates and inflation rates (Shea, 1991; Hassler & Wolters, 1995). Some scholars also used the Hurst exponent to analyze the Chaos and Order characteristics of capital markets (Peters, 1996). In recent years, the Hurst exponent has also been applied to other financial markets. In the foreign exchange market, Liashenko and Kravets (2016) explored the long memory of the foreign exchange market using the Hurst exponent. In derivative market, Serletis and Rosenberg (2012) used the Hurst exponent to analyze the longterm memory of American energy futures market. In emerging markets, Zhang et al. 3

(2016) used the single-scale and multi-scale Hurst exponent to examine the overall efficiency of China's pilot carbon trading market. With the wide application of Hurst exponent in financial market, the market efficiency index (EI) based on Hurst exponents was proposed. Sukpitak & Hengpunya (2016) used it to measure the efficiency of capital market. The Hurst exponent does not depend on any distribution, and it can distinguish between random and non-random, find long memory and periodic cycle, etc. So the Hurst exponent can meet the requirements of this paper. At the same time, compared with other efficiency test methods, the Hurst exponent method can further judge whether it is a persistent or anti-persistent sequence based on whether the time series obeys the random walk, so it can also analyze the market trend. Based on the fractal market theory, this paper examines the long memory of the Hong Kong REITs market through the Hurst exponent, and then explores the efficiency of the Hong Kong REITs market. The research on the market efficiency of REITs has been rich, but the research on the market of REITs in Hong Kong mainly focuses on the feasibility of development of REITs, the problems in development and the research on risk and return (Chiang & Joinkey, 2006; Wing, 2010; Ju and Chen, 2015; Zhang, Shu and Sun, 2016). There are few discussions on efficiency, and few papers in the literature analyze the dynamic changes in the efficiency of Hong Kong REITs. Due to the different research methods and data samples adopted by scholars, different conclusions have been drawn: Hao and Yenhsien (2013) believed that the Hong Kong REITs market was an efficient market, while Zhang and Sun (2015) discussed the efficiency of Hong Kong REITs market through Wild Bootstrap variance ratio test, and found that the Hong Kong REITs market was in an inefficient state. In recent years, due to the influence of many important factors, such as complex economic environment, financial crisis, geopolitical tensions, the uncertainty of market has increased (Xiao et al., 2019). In different periods, the relative importance of each affecting factor to the change in a market is different (Wen et al., 2018; Liu et al., 2019). Therefore, with the constant changes in the economic situation and changes in the market structure, the efficiency of the market cannot be static in the long process of time, but should be characterized by time (Lo, 2004). Lim & Brooks (2011) analyzed the changes in the efficiency of the U.S. stock market and found that the market efficiency in the U.S. stock market had a cyclical fluctuation with very long periodicity, from 30 to 40 years. Zhou & Lee (2013) found that the market efficiency of the REITs in the United States had been increasing over time. Is there any such change in the efficiency of the Hong Kong REITs market? What is the specific situation of the change in its efficiency? At present, very few articles in the literature explore these related issues. Therefore, this study makes two main contributions to the literature with respect to the efficiency of the Hong Kong REITs market. The first one is that to use new methods to study whether the REITs market is an efficient market. At the same time, we not only investigate the efficiency but also explore its dynamic changes of the Hong Kong REITs market through the Hurst exponent method. Different from the previous literature, when studying the dynamic change of efficiency, the efficiency level can be specifically described by the Hurst exponent. The second one is that this paper makes 4

a comparative analysis of the efficiency of the Hong Kong REITs market, stock market and real estate market and examines the relationship among these three markets, taking into account the impact of the real estate market and the stock market on the REITs market. Through this analysis, we can know whether the efficiency of the REITs market can be judged by changes in the efficiency of the stock market or the real estate market. The remainder of the paper is organized as follows: Section 2 introduces the judgment principle of market efficiency and expounds the Hurst exponent and its calculation method. Section 3 analyzes the efficiency of the Hong Kong REITs market. Section 4 presents the comparative analyses of REITs market, stock market and real estate market. Section 5 gives the conclusion and recommendations. 2. Theoretical hypothesis and selection of model 2. 1 Theoretical basis and judgment principle of market efficiency The study of market efficiency mainly relies on two important theories: The first is the Efficient Markets Hypothesis (EMH), which was first proposed by Cootner (1964) and Samuelson (1965) but the development of this theory was done by Eugene Fama (1965). EMH means that if the price in a securities market fully reflects all available information, then such a market is called an efficient market. According to the degree of information reflected, the market efficiency is divided into three different forms: weak-form efficiency, semi-strong-form efficiency and strong-form efficiency. The second is the Fractal Market Hypothesis (FMH) proposed by Edgar E. Peters in 1991. From the viewpoint of non-linearity of financial market, EMH means that not all financial markets satisfy the assumptions of independence, normality or limited variance. It emphasizes that the behavior of market participants is influenced by the degree of acceptance of information and the investment time. Stable financial markets have fractal structure. That is to say, if there are a large number of investors in the financial markets, with different investment periods and different responses to market information, the market will become an organic combination of global certainty and local randomness. Fractal market theory extends the linear market hypothesis of efficient market theory into a non-linear market that can reveal the authenticity of the market. It means that the change of the transaction price is not always a random walk, but a biased random walk called fractional Brownian movement. The transaction behavior includes the relativity and memory of price information. In nonlinear system theory, linear system is a special case of nonlinear system. Therefore, the case of efficient market is also included in the framework of FMH, and the efficient market is a special fractal market under linear conditions. With the deepening of research, the efficient market theory has many controversies in the analysis of market efficiency because of its strict assumptions. The emergence of fractal market theory effectively compensates for the applicability of the Efficient Market Hypothesis, which is closer to the financial market. Therefore, this paper starts from the Fractal Market Hypothesis and tests the market efficiency. The Hong Kong REITs market is an emerging market, so we discuss whether the Hong Kong REITs market is efficient in the weak-form. In the case of weak-form efficiency, the market price has fully reflected all the information of the security price in the past history. Therefore, the time series of market price changes should follow a random walk. To 5

explore the weak-form efficiency of REITs market is to analyze whether the price changes of REITs conform to the random walk, in other words, to examine whether the REITs market has long memory. Long memory refers to the significant autocorrelation between time series and observations that are far apart from each other. Simply put, long memory means that historical information affects the future, and we can use the historical yield values to predict the future yield values. If the REITs market has long memory, it indicates that the market has fractal characteristics, implying the rejection of the Efficient Market Hypothesis, and it means that the REITs market has not achieved weak efficiency. 2. 2 Hurst exponent and market efficiency The main purpose of examining the efficiency of REITs market is to see whether it has long memory, that is, whether two observations with a long time interval are related. The analysis of the long memory process of the capital market has always been the subject of finance, because the existence of market memory may imply a rejection of the Efficient Market Hypothesis. When verifying the efficiency of the market by testing fractal characteristics, the commonly used method is the rescaled range analysis method proposed by Hurst, that is, the R/S analysis method (Sánchez-Granero et al., 2008). The R/S analysis is a method of analyzing long-term memory and non-periodic cyclic time series, which is proposed by English hydrologist H.E. Hurst (1951) on the basis of a large number of empirical studies. Based on experience, he proposed a new statistic: Hurst exponent. This paper determines whether the market is efficient by the Hurst exponent value, that is, the value of H. According to the definition of H value, as long as the H value is not equal to 0.5, it means that the time series is self-correlated, and the stock price can be predicted to a certain extent, which contradicts to the Efficient market Hypothesis, thus judging that the market is inefficient. There are several cases of H values: (1) If H = 0.5, the process will be a random walk. It shows that the time series to be studied is uncorrelated and its movement is unpredictable. At this time, the market has reached the weak efficiency described by the Efficient Market Hypothesis. (2) If 0.5 < H < 1, the time series will be persistent. It indicates that the time series to be studied is positively correlated and has long-term memory. (3) If 0 < H < 0.5, the time series will be anti-persistent. It indicates that the time series to be studied is negatively correlated, and there is pink noise in the time series, which is the mean recovery process. Therefore, the less the value of the Hurst exponent deviates from 0.5, the closer the time series is to a random walk, and the more efficient the market is. It is generally believed that if the H value falls within the range of [0.45, 0.55], then the market has approximated the state of random walk and achieves weak-form efficiency (Xia, 2018). 2. 3 Calculation of Hurst exponent There is a wide range of approaches in scientific literature to deal with Hurst exponent estimation. Originally there was a classic R/S analysis, but due to some shortcomings of the classic R/S, such as the large sample size required, it is sensitive to short-term memory (Couillard & Davison, 2005). After that, various methods such as modified R/S analysis, detrended fluctuation analysis (DFA), multifractal detrended 6

fluctuation analysis (MF-DFA) and generalized Hurst exponent (GHE) analysis appeared. This paper introduces the classical R/S analysis method and the GHE analysis method, and uses these two methods to calculate the Hurst exponent values. 2.3.1 Classical R/S analysis method The calculation steps of the Hurst exponent by this method are as follows: Step 1. Determine a time series and divide the sub-intervals. We define a time series {Xi, i=1, 2...N}, then divide the time series into M sub intervals called Aa with length n, a=1, 2...M, where M = int( N / n) ,

Aa = { Xa , j}nj =1 , a = 1, 2,..., M .

（2.1）

Step 2. Calculate the sample mean Ea, sample standard deviation Sa, accumulated dispersion Da,j and extreme difference Ra for each sub interval Aa as follows:

1 n Ea =   Xa , j , n j Sa =

1 n   ( Xa , j − Ea ) 2 , n j

（2.2）

（2.3）

j

Da , j =  ( X a , k − E a ) ,

（2.4）

Ra = max( Da , j ) − min( Da , j ) .

（2.5）

k =1

Step 3. Calculate the rescaled range Ra Sa of each sub interval Aa, so that the average rescaled range of the entire time series with length n defined ( R S ) n is calculated as follows:

( R / S )n =

1 M   ( Ra Sa ) . M a =1

（2.6）

Step 4. Increase the value of n, repeat the above steps until n= N/2, and obtain the average remark range sequence. Step 5. Calculate the Hurst exponent H as follows:

( R S )n = c  n H ,

（2.7）

log ( R S ) n = log (c) + H  log (n) .

（2.8）

The least squares linear regression is performed on the above equation, and the slope of the fitted straight line is the estimated value of the Hurst exponent H. 2.3.2 Generalized Hurst Exponent analysis method The generalized Hurst exponent is a tool to study the scaling properties of the data via the qth-order moments of the distribution of the increments. The exponent is a 7

common sign of long dependence, and it is necessarily related to fundamental statistical quantities that is a good amount to describe the statistical evolution of the random variable X(t) with t = (1, 2,…, k,…, T), defined as follows:

Fq ( ) =

X (t +  ) − X (t ) X (t )

q

q

,

（2.9）

where  can vary between 1 and  max ，and ... represents the sample average over the time window. X(t) is taken to be log-prices in financial applications (Sensoy, 2013). If X (t ) is scaled, then there is a power law change as follows:

Fq( )   qH ( q ) ,

（2.10）

where H(q) is a generalized Hurst exponent of qth-order . Furthermore, in order to study the change of H(q) with q, a linear relationship can be obtained by taking the natural logarithm on both sides of the above formula: 1 ( ) ln Fq ( )  H (q) ln  , q

（2.11）

where H(q) is estimated as an average of several linear fits of Eq. (2.11) with   1, max  and  max varying between 5 and 19 days. For different values of q, H(q) describes different characteristics of time series. For q=1, Eq. (2.9) describes the scaling behavior of the absolute increments, and it is expected to be closely related to the original Hurst exponent. H(1) is used to describe the persistence characteristics of trend changes. If the trend change is persistent, H(1)>0.5; if it is anti-persistent, H(1)<0.5; for a random walk, H(1) = 0.5. Therefore, this paper calculates the H value for q=1 to judge the efficiency of the REITs market. 2. 4 V-Statistic The V-Statistic is an indicator related to the Hurst exponent, and its expression is given by: . （2.12） Vn = ( R / S ) n n The V-Statistic is originally used to test the stability of R/S analysis, and then gradually it is used to estimate the average cycle of time series, that is, to measure the length of long memory of time series. By plotting the graph with log(n) as the X axis and V-Statistic ( Vn ) as the Y axis, and observing the curve trend in the graph, we can not only judge the market efficiency, but also find the long memory length of the time series. There are several situations: (1) If the R/S value changes in a scale based on the square root of time, a horizontal line will appear in this ratio. In other words, if the V-Statistic graph is in a horizontal state, the process is independent and random. Then this market is an efficient market. (2) If the R/S value changes in a scale that is faster than the square root of time, then 8

this ratio will show an upwardly sloping curve. It shows that the time series is not random, but a long memory process. If the R/S value changes scale on a ratio below the square root of time, the process is anti-persistent. In the graph with the log (n) as the X axis and the V-Statistic as the Y axis, we can observe the trend of the curve in the graph. The breakpoint from the rise to the constant or the decrease in the curve is the critical point of the long-term memory disappearance of the time series. The time corresponding to the critical point is the average cycle period of the sequence, i.e., the long memory length of the sequence. Since Peters (1994) did not specify that how much the trend of the V-Statistic change meant the end of the long memory of the system when judging the acyclic period according to the V-Statistic, and there is no calculation standard for judging the breakpoint of V-Statistic graph at present, we usually observe the curve turning points by subjective observation, and use a turning point as a breakpoint to judge the average cycle period (Chen et al., 2005; Xie et al., 2010). Therefore, this paper mainly judges the turning points through visual observation. At the same time, because of the frequent fluctuation of V-Statistic graph, it is not obvious to find the larger turning points of the curve. So, a new smooth curve is obtained by linear interpolating the V-Statistic in order to find the larger turning points of the curve more clearly (Liu, 2009). Then an average cycle test is performed on all turning points on the curve after linear interpolation. Finally, the long memory length of the sequence is judged by the results of the test. 3. Analysis of the efficiency of the Hong Kong REITs market 3.1 Data source This paper selects the data from the Hong Kong REITs market and uses the Hang Seng REITs Index (HSRI) as the research object to explore the efficiency of the Hong Kong REITs market. HSRI contains the major REITs products in Hong Kong, which can better reflect the development of REITs in Hong Kong. As the first REITs product was listed on the market in Hong Kong on November 25, 2005, the daily closing price of the HSRI is selected for analysis from November 25, 2005 to October 10, 2018 in this paper. There are 3,040 data in total, and 3,039 valid data after taking the logarithmic rate of return, and the data is obtained from the Wind Information Database. 3.2 Analysis of Hurst exponent We first calculate the logarithmic rate of return for HSRI, and then perform a market efficiency analysis of the logarithmic rate of return series. If Pt is the trading price of the t-day HSRI, then the logarithmic rate of return can be expressed in Equation 3.1: Rt = log( Pt ) − log( Pt − 1) .

9

（3.1）

Fig. 3.1 Histogram of logarithmic rate of return of HSRI Table 3.1 Descriptive statistics of HSRI logarithmic rate of return Max

Min

Mean

Std

Skew

Kurt

JarqueBera

p-Value

Rt 0.04044 -0.04391 0.00012 0.00445 -0.53011 18.8390 31909.2 0.00000 Fig. 3.1 presents that the logarithmic return of HSRI is characterized by a sharp peak and a thick tail. Table 3.1 shows that the skewness of the logarithmic rate of return of HSRI is -0.53011, and the kurtosis is 18.8390. The negative skewness implies that the logarithmic rate of return of HSRI is flatter to the left compared to the normal distribution. The kurtosis indicates that there are phenomena of sharp peak and fat tail in the yield series. At the same time, the Jarque-Bera statistic that can test whether the sample data has a goodness of fit for the skewness and kurtosis of a normal distribution, is 31909.2, and the corresponding p-Value is p=0.0000<0.01. So it indicates that the logarithmic yield time series does not obey the normal distribution. Statistical methods based on normal distribution, such as the run test, are not applicable to test the efficiency of REITs market. Therefore, this paper applies the Hurst exponent to the REITs market to explore the long memory of the REITs market.

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Fig. 3.2 Graph of HSRI log (n) and log (R/S)n Table 3.2 Analysis results of Hurst exponent of HSRI HSRI

Hurst exponent

R2

F-Value

Significance

0.68084

0.99050

158129.4

7.12**

Note: ** indicates significant at 1% level, the critical value is Z0.05=1.96，Z0.01=2.5758. Fig. 3.2 is a rescaled range analysis of the daily logarithmic yield series of the Hong Kong HSRI. Table 3.2 displays the results of the analysis of the fitted line in Figure 3.2. In the Table 3.2, the significance is obtained by using Equation 3.2, this is the calculation formula given by Peters (1994) after doing relevant research:

Z=

H − E(H ) . Var ( H )

（3.2）

From the results in Table 3.2, we can see that the Hurst exponent of the Hong Kong REITs market is 0.68084, and its value passes the significance test, indicating that the price changes of Hong Kong REITs market is not random, but it has long-term memory, and there is a positive correlation between variables. The time series shows a strong trend continuity, that is, the future trend is likely to continue the current trend. This shows that the Hong Kong REITs market has not yet achieved weak efficiency. The shortcoming of the classical R/S analysis method is the sensitivity to shortterm memory. When a sequence contains short-term memory and has heterogeneity or non-stationary, classical R/S analysis may give erroneous and biased arguments, which tend to overestimate the long-term memory of time series (Ye, 2010; Sánchez-Granero et al., 2008). The GHE method does not require a large sample, and it combines sensitivity to any type of dependence in the data and simplicity (Sensoy, 2013). Therefore, this paper combines the Generalized Hurst Exponent method with high robustness to calculate the Hurst exponent and judge the long memory of the Hong Kong REITs market to ensure the accuracy of the results.

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Table 3.3 Analysis results of Hurst exponent of HSRI Estimated H E(H) H(Shuffle) deviation R/S

0.68084

0.55175

0.05175

0.59783

GHE

0.59535

0.49670

0.00330

0.50550

Note: E(H) is the Hurst exponent estimate of the random walk sequence, and H (Shuffle) is the Hurst exponent recalculated after the original data is randomly scrambled. Table 3.3 presents the results of the classical R/S analysis and the GHE analysis for comparative analysis. As can be seen from Table 3.3, the Hurst exponent value calculated by the GHE method is 0.59535, and this value is also greater than 0.5, indicating the Hong Kong REITs market has not yet achieved weak efficiency. This is consistent with the results obtained by the classical R/S analysis. However, the underlying distribution of data affects the interpretation of the Hurst exponent (Sánchez-Granero, 2015). In order to further explore if the memory is in the distribution or in the series. We randomly shuffle the data so that the order of the observations is completely different from the original order. Because the actual observation data are still there, the distribution of the observation data remains unchanged. This is consistent with Sánchez-Granero 's approach to explore long memory in the series of stock market (Sánchez-Granero, 2015). For the calculation of the Hurst exponent for disrupted data, if the sequence is a truly independent sequence or its long memory is derived from the data distribution, there will be no long memory effect or correlation between the observations, then the Hurst exponent should remain essentially unchanged. The disruption of data has no effect on the qualitative nature of the data. Conversely, if the sequence really has long memory, then the order between the observations is important. Disrupting the data will destroy the structure of the system, and the calculated Hurst exponent should be much smaller than or close to 0.5. Table 3.3 shows that after disrupting the data, the H values calculated by the two methods have changed greatly, and the H values have decreased significantly, which indicates that there is indeed a long memory in the return series of the Hong Kong REITs market. At the same time, we use the two methods to estimate the Hurst exponent value of the random walk sequence. As can be seen from Table 3.3, The R/S method overestimates the long memory of the time series, and the estimated deviation of the GHE method is smaller. Therefore, the GHE method is used to estimate Hurst exponent latter part of the paper to improve the accuracy of the results. In order to further test the validity of the Hurst exponent results, this paper uses the traditional methods - unit root test and the Wild Bootstrap variance ratio test to analyze the Hong Kong REITs market efficiency. As Table 3.4 reports that the Augmented Dickey-Fuller (ADF) test statistic value of the unit root test is -34.01188, and the corresponding p-value is p=0.0000<0.01. The test statistic cannot reject the unit root of the sequence for 1% significant level. It shows that there is no unit root in the REITs market return series in Hong Kong, and the assumption of random walk is rejected. That is to say, the REITs return is a stationary time series, and the REITs market has not achieved weak-form efficiency, which is consistent with the conclusion 12

of Hurst exponent analysis. In addition, the Wild Bootstrap variance ratio test is used to analyze the efficiency of the Hong Kong REITs market during the entire sample period. The variance ratio test is widely used to test the hypothesis of the random walk, and its principle is that if the sequence follows a random walk, the ratio of variance of k-period return to variance of 1/k of one-period return will be 1. The k is equal to the days in the observation interval in this paper. Variance ratio test statistic VR is defined as the ratio of 1/k of the variance of the k-period return to the variance of the 1-period return，then the MV represents the joint test statistic of the variance ratio when taking multiple kvalues. Table 3.5 displays the results of Wild Bootstrap variance ratio test. The test is conducted for the repeated time of 1000 times and various lags of k (i.e., 2, 4, 8, 16 days). It can be seen from the results that the joint test statistic MV is 8.202343, and the corresponding p-Value is p=0.0000<0.01. The test results indicate that the hypothesis that the sequence follows the martingale process is rejected at significance level of 1%. That is to say, its logarithmic rate of return sequence does not follow the random walk. Therefore, the Hong Kong REITs market has not reached the weak-form efficiency. The weak-form efficiency found in REITs market is also consistent with the result based on the Hurst exponent. Table 3.4 Unit root test of logarithmic rate of return t-Statistic

p-Value

-34.01188

0.0000

1%

-3.432313

-

5%

-2.862293

-

10%

-2.567215

-

ADF test statistic Test critical values

Table 3.5 Wild Bootstrap variance ratio test Joint Tests MV

Value

Prob

8.202343

0.0000

Individual Tests k

VR

z-Statistic

Prob

2

0.512672

-8.202343

0.0000

4

0.264523

-7.521129

0.0000

8

0.130305

-0.647531

0.0000

16

0.066102

-5.170650

0.0000

If the time series follows a random walk, i.e., the value of the Hurst exponent is 0.5, then the graph of the V-Statistic should be a horizontal line. In other words, if a time series is random and independent, then the V-Statistics drawn for log (n) should be a straight line. In addition, the V-Statistic can also better reflect the average cycle by estimating the average cycle length and finding breakpoints. So we further make the V13

Statistic chart of the HSRI. As shown in Fig. 3.3, it presents an upward sloping curve, displaying an obvious upward trend. According to the definition of the V-statistic, R/S changes in a scale that is faster than the square root of time, indicating that the time series is not a random walk, but is a long memory process. In short, the price of Hong Kong REITs is persistent, that is, when the price of the REITs is upward, the next moment is likely to continue upward. Thus, the Hong Kong REITs market is not an efficient market and has long-term memory.

Fig. 3.3 V-Statistics of Hang Seng REITs Index

Fig 3.4 Linear interpolation graph of V-Statistics Fig. 3.4 shows the curve after linear interpolation of the V-Statistic. Further analysis of Fig. 3.4 shows that there are many breakpoints where the curve changes from rising to constant or to decreasing trend, and the curve has the large turning at log(n)=5.3, 5.6, 6.3, 7.1, where the corresponding times are 200, 270, 545 and 1212 days. The average cycle time test is performed on these breakpoints, whose results are shown in Table 3.6. From the test of the average cycle time, the H values after the breakpoints are all greater than 0.5, and so the time series still has long memory. Although the long memory is weakened to some extent, it has not disappeared. That is 14

to say, there is no obvious long memory length in REITs market.

Time (days) 200 270 545 1212

Table 3.6 Test of average cycle time H value before breakpoint H value after breakpoint 0.5553 0.5980 0.5248 0.5895 0.5073 0.5982 0.6051 0.5883

3.3 Analysis of time-varying Hurst exponent The classical R/S method and the GHE method have shown that the REITs market in Hong Kong is inefficient, which describes the efficiency of the REITs market in an average meaning. However, it can be seen from Fig. 3.2 that the local Hurst exponent is also changing with the change of coordinate points. We further use the time-varying Hurst exponent for efficiency analysis in order to reflect the dynamic change of Hurst exponent more clearly. The time-varying Hurst exponent is obtained by using the cyclic program rolling calculation for the GHE method. It can not only reflect changes in market efficiency, but also serve as a quantitative timing model for judging market trends and strong and weak emotions. The larger the H value deviates from 0.5, the more tense the market sentiment is or the higher the sentiment is, and the stronger the reversal or continuation of the trend is. The greater the overall volatility of the timevarying Hurst exponent is, the more unclear the market sentiment is. The time-varying Hurst exponent is a method of calculating the corresponding Hurst exponential sequence using a GHE analysis method by rolling forward with a fixed time window length in a large period of time. Fig. 3.5 is a flow chart for calculating the time-varying Hurst exponent by performing a cycle degree rolling calculation with a window length of N and a step size of 1. Generally speaking, the choice of time window should be moderate. If it is too large, it will easily cover part of the information, and it will not maintain its sensitivity with time. If the window is too small, then there will not be enough points to calculate the Hurst exponent, which is not statistically significant. N

2005.11.25

m m+1

m+N m+N+1

2018.10.10

Calculate the H exponent by rolling forward with a window length of N and a step length of 1. Fig. 3.5 Flow chart of time-varying Hurst exponent calculation method In this paper, the window length of N=500 is selected for cyclic rolling calculation, and the result is shown in Fig. 3.6. This is consistent with the window size chosen by 15

Hiremath et al. when using time-varying Hurst exponent to analyze the efficiency of Indian stock market (Hiremath et al., 2016). At the same time, this paper also uses the window lengths of 300, 400 and 600 to calculate time-varying Hurst exponent, and finds that the results are not much different. From the time-varying Hurst exponent of Hong Kong's REITs market in Fig. 3.6, we can see that the Hurst exponent values are mostly greater than 0.5, and they are basically in the high position. It shows that the price of Hong Kong REITs market does not follow the situation of the random walk, so the market is inefficient, and the inefficiency of the market has basically not changed. The only three times that approximate to weak-form of efficiency state are very short. However, the Hurst exponent is constantly changing with time, its value changes significantly, and market efficiency is constantly changing. Fig. 3.6 shows that there are two fluctuations in the value of the Hurst exponent in the REITs market between 2008 and 2012. The market efficiency has improved to some extent in late 2008 and late 2010, especially during the period of 2010, the efficiency of the REITs market has increased significantly, reaching a state of weak efficiency. This may be related to the policy incidents in which the State Council of China issued a document encouraging REITs in 2008 and the Chinese Ministry of Housing and Urban-Rural Development in 2010 encouraged trust funds to expand financing channels. Because the incentive policy improves the liquidity of REITs market to a certain extent, and the improvement of market liquidity is propitious to the transmission of market information, which is propitious to the improvement of market efficiency. From 2012 to 2014, the market efficiency has stabilized compared with the previous period, but the value of the Hurst exponent is basically at a high level. The market efficiency of REITs is low and market sentiment is high. There were also no major policy events during this time period. After 2015, the efficiency of the REITs market rise again, and the REITs market reaches weak efficiency. This may be caused by the intensive introduction of REITs development policies in China since 2015. In the later period, the Hurst exponent value shows a downward trend, indicating that the market efficiency of the Hong Kong REITs market has increased in recent years. This may be related to the policy of “Improving the Implementing Scheme of Promoting Consumption System and Mechanism (2018-2020)” issued by the State Council of China, which proposes to vigorously develop the market for housing leasing.

16

Fig. 3.6 Time-varying Hurst exponent plot of the HSRI 4. Comparative analysis of the efficiency of Hong Kong REITs market, stock market and real estate market 4. 1 Comparative analysis of Hurst exponent The Hurst exponent is also applicable to the Hong Kong stock market and the real estate market. We use the GHE method to analyze the efficiency of the Hong Kong stock market and the real estate market. The Hang Seng Composite Index (HSCI) and the Hang Seng Property Index (HSPI) in the same period are used as the comparative analysis. These two indices represent the Hong Kong stock market and the Hong Kong real estate market, respectively. Table 4.1 Analysis results of Hurst exponent Index

Hurst exponent

Deviation from 0.5

HSRI

0.5951

0.0951

HSCI

0.5184

0.0184

HSPI

0.5152

0.0152

Table 4.1 shows the result of corresponding Hurst exponent analysis. As can be seen from Table 4.1, the Hurst exponent of the Hong Kong stock market is 0.5184, and the Hurst exponent of the Hong Kong real estate market is 0.5152. Among the three markets, the Hurst exponent value of Hong Kong REITs market deviates from 0.5 to the greatest extent, with the least efficiency and the slowest information transmission speed. This may be because China's REITs market started late and the development is not mature enough yet. We further conduct the V-Statistic analysis. As can be seen from the V-Statistics of Figures 4.1 and 4.2, both the HSCI and the HSPI have similar horizontal stages of presentation. This phenomenon also indicates that Hong Kong stock market and real estate market are more efficient than REITs market. We use the same linear interpolation method to get the turning point. Table 4.2 shows the average cycle test. In the Hong 17

Kong real estate market, the V-Statistics are an oscillation rising process while the H value of this sequence is 0.5959 when n<60, indicating that there is an obvious persistence characteristic. The V-Statistics graph has a breakpoint when n=60，and the H value after the breakpoint is 0.5161，indicating that the sequence is approximately a random walk and its persistence, that is, the long memory disappears. Therefore, we can see that the Hong Kong real estate market has an average cycle of 60 days. In the Hong Kong stock market, the H values before and after the breakpoints are all close to 0.5. It indicates that the time series basically follows a random walk. Therefore, there is no obvious average cycle in the Hong Kong stock market.

Fig. 4.1 V-Statistics of Hang Seng Composite Index

Fig. 4.2 V-Statistics of Hang Seng Property Index

18

Index

HSCI HSPI

Table 4.2 Test of average cycle time H value before Time (days) breakpoint 181 0.4808 365 0.5038 446 0.4983 1212 0.5221 60 0.5959

H value after breakpoint 0.5170 0.5084 0.5044 0.5233 0.5161

4. 2 Comparative analysis of time-varying Hurst exponent Since the window used for the time-varying Hurst exponent analysis of the REITs market is 500, the same window size is used to analyze the time-varying Hurst exponent of the Hong Kong stock market and the real estate market for comparison with the REITs market.

Fig. 4.3 Time-varying Hurst exponent plot for each index

19

Table 4.3 Descriptive statistics of time-varying EI exponent Index

Min

Max

Mean

Median

Std

Range

HSRI

0.00024

0.24828

0.09889

0.09403

0.05811

0.24804

HSCI

0.05533

0.09471

0.02979

0.02385

0.02343

0.03938

HSPI

0.04448

0.14868

0.04675

0.04299

0.03086

0.10420

Fig. 4.3 shows that the Hurst exponents of the three markets are not all greater than 0.5 or less than 0.5. In order to more clearly measure the efficiency of the market, we introduce a market efficiency exponent EI to measure market efficiency. For a market, the farther the value of H deviates from 0.5, the lower the efficiency of this market is. Therefore, the expression of the EI exponent is defined as follows:

EI = H − 0.5

（4.1）

Among them, the greater the value of the EI exponent is, the lower the efficiency of the market is, and the more nervous the market sentiment is. Fig. 4.3 presents the variation of the Hurst exponents over time in three markets. From the trends of the time-varying Hurst exponents in Fig. 4.3, it can be seen that the time-varying Hurst exponents of the Hong Kong stock market and the real estate market basically fluctuate around the Hurst exponent value 0.5 of the random walk sequence. However, the time-varying Hurst exponent of REITs in Hong Kong is basically at a high level, which is significantly higher than the other two markets. Compared with the stock market and the real estate market, it is found that the H value of the REITs market fluctuates the most. Table 4.3 reports that the standard deviation of the EI exponent of the Hong Kong REITs market is the largest, reaching 0.05811, indicating that the market sentiment of REITs is the most uncertain. Meanwhile, its average value is 0.09889, which is higher than those of the other two markets. This shows that compared with the other two markets, the market efficiency of REITs market is the lowest, and the market sentiment is tenser than those in the other two markets. This is a comparison of the EI exponents between Hong Kong REITs market and the other two markets in an average meaning. In order to make the result more reliable, this paper carries out a test on the magnitude of the efficiency in different time periods of the three markets. Since the EI value does not obey the normal distribution, the Wilcoxon signed rank test is used to analyze the difference. In the Wilcoxon signed rank test, it adds the ranks of the absolute values of the difference between the observed value and the center position of the null hypothesis as their test statistics according to different symbols. It is suitable for pairwise comparison in the test, and it does not require that the difference of the paired data obeys a normal distribution, and only requires a symmetric distribution. It checks whether the difference between paired observations comes from the population with a mean of 0 (whether the population producing the data has the same mean).

20

Table 4.4 Wilcoxon signed rank test Rank ZN Rank sum mean Statistic

Group

HSCI-HSRI

HSPI-HSRI

HSPI-HSCI

Negative rank Positive rank

2301

1351.19

3109080.00

238

485.08

115450.00

1941

1462.48

2838675.00

598

645.24

385855.00

734

884.10

648932.00

1805

1426.92

2575598.00

Knot

0

Total

2539

Negative rank Positive rank Knot

0

Total

2539

Negative rank Positive rank Knot

0

Total

2539

p-Value

-40.517a

.000

-33.197a

.000

-26.076b

.000

Note: a. based on positive rank; b. based on negative rank. The Wilcoxon signed rank test results are shown in Table 4.4. The Z-Statistics for the three groups in the table are -40.517, -33.197, -26.076, respectively. The corresponding p-values are all p=0.000<0.01, indicating that the difference is statistically significant. The value of the Hurst exponent in the REITs market is significantly larger than that in the stock market and the real estate market, thus indicating that the REITs market has the lowest degree of efficiency. Fig. 4.3 shows that the Hurst exponents of the stock market and the real estate market change roughly the same, while the change of the Hurst exponent of the REITs market is quite different from the other two markets. In the period when the stock market and the real estate market become more efficient, the efficiency of the REITs market is lower. It shows that the efficiency of the Hong Kong REITs market has no obvious correlation with the stock market and the real estate market, or the degree of correlation is low. In order to verify this result, the correlation analysis of the three markets efficiency is further carried out. The analysis results are shown in Table 4.5.

21

Table 4.5 Correlation coefficients of market efficiency in the three markets Correlation HSRI-HSCI HSRI-HSPI HSCI-HSPI coefficient Pearson Correlation -0.097** -0.049** 0.445** coefficient Spearman Correlation 0.016 -0.050* 0.407** coefficient ** Note: and * indicate that the correlation is significant when the confidence is 0.01 and 0.05, respectively. Table 4.5 shows the analysis of the pearson correlation coefficient and the spearman correlation coefficient. From the results of the pearson correlation coefficient analysis, we know that the correlation coefficient between the Hong Kong stock market and the real estate market EI exponent is as high as 0.445, the correlation between the two markets is highly correlated. The correlation coefficient between the Hurst exponents of the Hong Kong stock market and real estate market are -0.097 and -0.049, respectively. It shows that although the efficiency of Hong Kong REITs market has certain correlation with the efficiency of the other two markets, the degree of correlation is relatively low, and the change in its efficiency is quite different from the other two markets. The spearman correlation coefficient analysis results are consistent with this. At the same time, Fig. 4.3 shows that the differences mainly concentrate on the intervals of 2008-2011 and 2014-2016. From 2008 to early 2010, the market efficiency of Hong Kong stock market and real estate market fluctuate little, and basically near the baseline, while Hong Kong REITs market efficiency drops significantly. During this period, there are no new REITs products listed in the Hong Kong REITs market, which has affected the enthusiasm of REITs market development to a certain extent. And this may affect the market efficiency. From mid-late 2010 to 2011, the market efficiency of the stock market and the real estate market shows a certain degree of decline. But the market efficiency of the REITs market fluctuates greatly, and its efficiency is greatly increased to weak efficiency and then greatly reduced. This may be related to the liquidation and delisting of RuiFu REITs on April 1, 2010. RuiFu REITs completed its listing in June 2007 in the form of counterfeiting, and its delisting has brought some impact on the REITs market. At the same time, half of the REITs listed in Hong Kong are based on real estate in the Mainland China. At the end of 2010, the Mainland China adopted strict control policies on the property market. The prospects of the property market were unknown at the moment, which might also cause fluctuations in the efficiency of the REITs market. During the period from 2014 to 2016, the market efficiencies of the stock market and the real estate market fluctuate moderately, while that of the REITs market showed a valley value in 2015, and the efficiency change is more obvious. This may be caused by the intensive introduction of REITs development policies in China since 2015. The intensive introduction of policies makes the investor sentiment in REITs market unstable, which results in the fluctuation of market efficiency. 5. Conclusions and recommendations 22

In this paper, the classic R/S analysis method is first used to describe the efficiency of Hong Kong REITs market in an average meaning, and then the time-varying Hurst exponent is further used to describe the dynamic changes of the market efficiency of Hong Kong REITs, and the Hong Kong stock market and real estate market are used as the comparative analysis. The results show that the Hong Kong REITs market has not yet reached the state of weak efficiency, there is long-term memory, and there is a positive correlation between returns of the REITs market. The time series of its return has the characteristics of persistence or trend enhancement, but it does not have an obvious long memory length. At the same time, through time-varying Hurst exponent, it is found that the Hong Kong REITs market has been in a state of inefficiency from November 25, 2005 to October 10, 2018, with weak efficiency only appears approximately three times, and their duration is very short. In addition, compared with the Hong Kong stock market and the real estate market, the REITs market has the lowest level of efficiency, its H value is the most volatile, and the market sentiment is the most uncertain. Through the time-varying Hurst exponent trend chart, we can preliminarily judge that the correlation between the efficiency of Hong Kong REITs market and the efficiency of the stock market and the real estate market is low. Based on the above research conclusions, this paper provides some suggestions for investment decisions and the development of REITs market in China. According to our research results, the Hong Kong REITs market is inefficient and persistent. If the price of the REITs is up at a certain moment, then the next moment is likely to continue upward. As an investor, technical analysis can be carried out in the Hong Kong REITs market, and trading decision-making methods based on technical analysis can play a guiding role. Investors can use the relevant information in the market to obtain abnormal returns. From the results of the time-varying Hurst exponent, the efficiency of the Hong Kong REITs market has no obvious correlation with the stock market and the real estate market, and the market sentiment is unclear, and investors cannot conduct market sentiment analysis. Therefore, investors should make rational analysis when investing. The inefficiency of the market also reflects the immaturity of the market, the imperfection of the trading system, the imperfection of the market system and the inadaptability of the regulatory system. At the same time, China's REITs products mainly exist in Hong Kong. The low efficiency of the Hong Kong REITs market compared with the Hong Kong stock market and the real estate market also increases the necessity of the efficiency construction of the REITs market in China. Therefore, combined with the conclusions of this study, this paper puts forward relevant suggestions for the government to implement efficient regulation and control. According to the time-varying Hurst exponent analysis, the improvement of market efficiency of REITs in Hong Kong of China is accompanied by the emergence of corresponding incentive policies, and the government should increase the incentives and release more policies to encourage the development of REITs. At the same time, the efficiency of REITs market in Hong Kong has a low correlation with the stock market and the real estate market. When implementing policies to improve the efficiency of the REITs market, the government should not blindly copy the stock 23

market and real estate market, but should formulate a targeted development policy according to the relevant characteristics of REITs market. Acknowledgments The authors are very grateful to the journal editors and anonymous reviewers for their useful suggestions and would like to express gratitude for the support given by the National Natural Science Foundation of China (No. 71871030, No.71501069), the Provincial Natural Science Foundation of Hunan (No. 2017JJ3330) and the Hu-Xiang Youth Talents Program of Hunan, the Scientific Research Project of Hunan Provincial Education Department in China (No.18A037, No.18B128).

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