Time-varying return predictability in the Chinese stock market

China's stock market is the largest emerging market all over the world. It is widely accepted that the Chinese stock market is far from efficiency and it possesses possible linear and nonlinear dependence. We study the predictability of returns in the Chinese stock market by employing the wild bootstrap automatic variance ratio test and the generalized spectral test. We find that the return predictability vary over time and significant return predictability is observed around market turmoils. Our findings are consistent with the Adaptive Markets Hypothesis and have practical implications for market participants.


Introduction
The Efficient Markets Hypothesis (EMH) is one of the cornerstones of modern finance. 18,19 There are three forms of market efficiency, including weak-form, semistrong form and strong form. The weak-form market efficiency hypothesis suggests the failure of detecting mispriced assets and furthermore the futility of return prediction in terms of related past information. The EMH stimulated numerous studies reporting "abnormal" phenomena that are inconsistent with the EMH. One abnormal phenomenon is related to return predictability in terms of historical firmspecific information, such as market capitalization or the size effect, 6 price-earnings ratio, 7 book-to-market ratio or the value effect, 47,5 past prices or the momen- Time-varying return predictability in the Chinese stock market 3 return series, are employed to check the return predictability.
The rest of this paper is organized as follows. Section 2 describes the methods employed in this work. Section 3 presents the data in the study. Section 4 reports the empirical results. Section 5 concludes.

Methodology
The EMH states that asset prices fully reflect all available information, which implies the failure of technical analyses. Various methods for testing Martingale difference Hypothesis (MDH) are frequently used by studying return predictability. Specifically, it is a classic assumption in the EMH that where {Y t } +∞ −∞ is a stationary time series, I(t) is the set of all available information before time t, and I t = {Y t , Y t−1 , ...}. According to the Eq. (1), Y t is a martingale difference sequence (MDS). The MDH generalizes the notion of MDS such that the unconditional mean of Y t could be nonzero: According to equation (2), the conditional expectation of Y t is a constant. The MDH implies that conditional mean is independent, which is consistent with the EMH. It means that historical information is useless in forecasting future values. Eq. (2) can be rewritten as follows, where ω(I t−1 ) represents the transformation of past information. Different function forms of ω lead to different methods for testing linear and nonlinear dependence in the return time series. Charles et al. compared different MDH testing methods through conducting Monte Carlo experiments. 10 They concluded that the wild bootstrap automatic variance ratio test (hereafter, AVR test) 33 and the generalized spectral test (hereafter, GS test) 17 are more favorable to test the linear and nonlinear dependence in return time series. Hence, we use both the AVR test and the GS test to investigate the return predictability in the Chinese stock market. We review briefly these two methods.

Wild bootstrap automatic variance ratio test
The variance ratio test was developed by Lo and MacKinlay, 42 which has been widely employed to test if a market is efficient in the weak form. Let Y t be the asset return at time t (t = 1, ..., T ), the AVR test statistic can be written as follows: 11 When Y t is i.i.d. and has finite fourth moment as well, under the null hypothesis that Y t is serially uncorrelated, we have 11 as k → ∞, T → ∞, T /k → ∞. The optimal value of lag truncation point (or holding period) k can be determined by the fully data-dependent method. 2 One thus obtains the AVR test statistic AV R(k) with the optimal choicek for k. It is argued that the small sample properties of the AVR test 11 can be substantially improved after employing the wild bootstrap. Extensive Monte Carlo experiments have been conducted to show that the wild bootstrap provides accurate statistical inference in small samples under conditional heteroskedasticity. 32,33 Specifically, wild bootstrap of AV R(k) could be performed as the following three steps: 32,33 (1) Generate a bootstrap sample of size T , Y * t = η t Y t (t = 1, ..., T ), where η t is random variable with zero mean and unit variance; (2) Obtain the AV R * (k * ) through calculating AVR statistic from {Y * t } T t=1 ; and (3) Repeat the first two steps many times and construct a bootstrap distribution {AV R * (k * ; j)} B j=1 .

Generalized spectral test
On the other hand, ω can be nonlinear functions. Exponential function and indicator function are popularly adopted. The former is to detect the general nonlinear conditional mean dependence, and the latter is to test for no directional predictability. Escanciano and Velasco proposed the null of the MDH in a form of pairwise regression function. 17 The null hypothesis is that , and the alternative hypothesis is that H 1 : P {m j (y) = 0} > 0 for some j. In fact, the above null hypothesis is consistent with the exponential weighting function as follows, where γ j (x) plays a role of an autocovariance measure in a nonlinear framework with x being any real number. They also proposed the use of the generalized spectral distribution function, 17 Time-varying return predictability in the Chinese stock market 5 whose sample estimate is written aŝ , and the test statistic for H 0 is written as To evaluate the value of S T for all possible values of λ and x, the Cramer-von Mises norm is used to obtain the statistic 17 To improve small sample properties, Escanciano and Velasco recommended the use of the wild bootstrap, 17 whose process is similar to that in the AVR test mentioned above.

Data sets
The data used to study the return predictability through the AVR and GS tests are retrieved from RESSET (http://www.resset.cn), which contains the daily and weekly returns for all A-share individual stocks listed on the Shanghai Stock Exchange (SHSE) and the Shenzhen Stock Exchange (SZSE), covering the period from December 1990 to September 2015. The equally-weighted average daily and weekly returns of all individual stocks are calculated for both exchanges. Table 1 presents some related descriptive statistics. It is found that SHSE stocks have higher daily and weekly average returns than SZSE stocks during the sample period, which is consistent with the fact that the average returns of SHSE stocks have larger skewness. In addition, the average returns of SHSE stocks have higher kurtosis than SZSE stocks. The Jarque-Bera test also shows that the averages are not normally distributed.

Empirical results
We perform the AVR test and the GS test for the return predictability of the daily and weekly data in the SHSE and the SZSE. Because the observations in the corresponding time window are sufficient to guarantee the precise estimation, the size of time window for weekly data and daily data is set as 2 years and 5 years, respectively. 34 Each time window includes nearly 250 observations for weekly data and almost 500 observations for daily data. The sample is moved one year forward for the re-estimation of the AVR and GS statistics. The time-varying AVR statistic and its corresponding 5% confidence intervals (CIs) are illustrated in Fig. 1. When the AVR statistic is greater than the upper CI value, the returns exhibit statistically significant positive serial correlation. When the AVR statistic is less than the lower CI value, the returns exhibit statistically significant negative serial correlation. It is evident that the AVR statistic rises and Kim et al. found that stock market bubbles and crashes generally correspond to higher AVR statistics and sometimes wider confidence bands, which can serve as the measure of market uncertainty. 34 It implies that the market efficiency is dependent on market conditions. 40,34 Specifically, when the Chinese stock exchanges (including the SHSE and SZSE) were established in the early 1990s, the markets were more volatile and had higher uncertainty mainly due to the market trading mechanism of T + 0 and no implementation of mature pricing limits. Accordingly, for daily data in the early 1990s, the AVR statistics are positive with statistical significance and the confidence band is wider. Similarly, during the time period from 1996 to 1997, the results for daily data show that the AVR statistics have a sudden and sharp increase with the CI band getting wider, which corresponds to the market bubble in Chinese stock market from 1996 to 1997. Significant nonlinear dependence appears in several time periods, similar to the results of the AVR test. The most evident is again around the 2007 crash. However, it seems that the AVR statistic has stronger predictive power of large crashes. Rigorous evaluation of the predictive power can be carried out based on the pattern recognition framework. 50 Apparently, the AVR and GS statistics, as measures of market efficiency, indicate that the efficiency of the Chinese stock market is time-varying and dependent on market conditions. Specifically, there would be higher degree of return predictability during market bubbles and crashes, which is different from the findings in US market. 34

Conclusion
We have adopted the wild bootstrap automatic variance ratio test and the generalized spectral test to check the return predictability in the Chinese market. The results show taht the return predictability is time-varying and dependent on market conditions. More specifically, the high levels of return predictability coincide with historical market turmoils around crashes. In some cases, there are clear evidence showing that cumulating return predictability is able to predict market crashes.
Our findings indicate that the Chinese stock market is inefficient and the level of market inefficiency varies over time. These results provide evidence supporting the Adaptive Markets Hypothesis.