This article discusses the autocorrelation in daily returns of the China Stock Index 500 (CSI500) from the perspective of price latency due to price limit mechanism. We propose limit-up/limit-down (LULD) indices to quantify the price latency in CSI500 as an aggregated number of component stocks closing with LULD in a given trading day. We found that the positive autocorrelation in the CSI500 market index during the data period disappeared after the price latency was controlled. This implies that the autocorrelation we observed may be attributable to the price latency measured by LULD indices. Our findings provide new insight into the dynamic features of market indices and may serve as a workable reference for practical usage of the market index.

Keywords: Autocorrelation; price limit; dynamic feature

I. Introduction

This article investigates the contribution of price stabilization mechanism to the observed autocorrelation in the daily returns on stock market indices from the perspective of price latency. The source of stock return autocorrelation is crucial for a better understanding of the price-formation process. Some researchers have considered autocorrelation in stock returns as an indicator of market inefficiency (Worthington and Higgs [10]; Griffin, Kelly, and Nardari [5]; Wei [9]; Shynkevich [7]), while some others have taken the observed autocorrelation as a statistical artefact caused by price latency (Atchison, Butler, and Simonds [1]). Identifying whether the observed autocorrelation is a statistical artefact is of significant meaning, it will shape our understanding of market microstructures and the design of trading strategies (Harris [6]).

Price latency refers to delays in transaction price adjustments due to market friction, mostly insufficient liquidity. When the market is lack of liquidity, a divergence between the reported transaction-based returns and true returns will be observed and thus introduce autocorrelation into asset prices. This phenomenon is also referred to as nonsynchronous trading and was believed responsible for the observed autocorrelation in asset prices (Dimson [4]). However, nonsynchronous trading may not be the only reason for the observed autocorrelation, since empirically lack of liquidity cannot explain all the autocorrelation in stock returns (Cohen et al. [2], [3]; Theobald and Yallup [8]). Harris ([6]) indicated that other market frictions, such as price stabilizations, may also contribute to price latency, but currently, there is little direct evidence for that claim in literature.

Price stabilization refers to the mechanisms designed to control volatility by halting trades for a short period (e.g. circuit breakers) or by introducing price limits. In this study, we explored the possible contribution of price stabilization measures, specifically price limit-up and limit-down (LULD), to the observed stock return autocorrelation. LULD literally limits stock prices from rising above or falling below predetermined price levels. Apparently, such mechanism can cause price latency: when the price hits price limit at the current trading session, any additional price changes would be postponed to the next trading session and therefore causes the autocorrelation in stock returns. In other words, LULD as one source of price latency could be responsible for part of the observed autocorrelation in the market.

The role of LULD as a source of price latency has rarely been explored. This is possibly because the price limits need to be set appropriately to observe their effects on price latency. We find that China's stock market is an ideal test field for addressing these concerns. Its price limit, ±10%, is set neither too wide to dilute the effect of price latency nor too narrow to arouse amplified magnet effect and confound the true effect of price latency due to price limits.

II. Data and analysis

Data

We gathered market performance data from 16 April 2015 to 16 May 2018, a total of 753 trading days over which Chinese stocks experienced multiple historical ups and downs. Since an index composed of small capital stocks is more likely to be autocorrelated (Atchison, Butler, and Simonds [1]), we used China Stock Index 500 (CSI500) for this study, an index composed by 500 mid-sized and small-sized listed firm in A-share of China's stock market. Figure 1 shows the market performance of the Shanghai Composite Index (SCI) and CSI500 during that period, where CSI500 shows greater volatility than SCI.

PHOTO (COLOR): Figure 1. Time-series plot of returns of CSI500 and SCI from 16 April 2015 to 16 May 2018.

We collected all the stocks that were (or once were) listed as CSI500 composites during that period, which amounts to 762 (the index changed its composites 8 times in that period). Among them, 748 stocks experienced at least one LULD; all those CSI500 composite stocks had 3066 limit-ups and 8145 limit-downs during the data period. Such frequent LULD are expected to make the autocorrelation more significant and easier to detect from the stock indices. All data were obtained from the CSMAR database.

Daily returns

We calculated the daily index return,

Graph

$R_t$ , as the difference between the logged closing market index of the current trading day and the previous. That is,

(2-1)

Graph

$R_t=\mathrm{l}\mathrm{g}{P}_t-\mathrm{l}\mathrm{g}{P}_{t-1},$

where

Graph

$P_t$ is the closing index of CSI500 at day

Graph

$t$ . The daily index return

Graph

$R_t$ has a negative mean (−0.0004) but positive median (0.0013), with heavy tails (kurtosis of 7.2448) and left skewed (skewness of −1.1724), indicating more positive return days than negative but greater volatility in downturns than upturns (maximum of 0.0639 and minimum of 0.0893). This observation gives us a hint that we may see a stronger effect in limit-downs than limit-ups across our data.

Indices for LULDs

We defined two indices to measure the price latency in the CSI500 market index due to price stabilization,

Graph

$LU{I}_t$ and

Graph

$LD{I}_t$ , as the counts of stocks that closed with LULDs, respectively, in a given trading day

Graph

$t$ . Specifically,

(2-2)

Graph

$LU{I}_t=\sum _{i=1}^{500}{\chi }_{i,t}^{\mathrm{U}},{\chi }_{i,t}^{\mathrm{U}}=\{\begin{array}{cc}1,& \mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}i\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}-\mathrm{u}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{y}t\\ 0,& \mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\end{array};$

(2-3)

Graph

$LD{I}_t=\sum _{i=1}^{500}{\chi }_{i,t}^{\mathrm{D}},{\chi }_{i,t}^{\mathrm{D}}=\{\begin{array}{cc}1,& \mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{k}i\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}-\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{y}t\\ 0,& \mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\end{array}.$

These two indices represent an aggregated effect of price latency over the CSI500 composite stocks, and thus can also be referred as a proxy for the price latency on index performance of CSI500. Table 1 shows the summaries of

Graph

$LU{I}_t$ and

Graph

$LD{I}_t$ . Within the data period, on average, there are four stocks closed with a limit-up in a day and 10 stocks closed with a limit-down. Some extreme days are remarkable. On 16 July 2015, 18 CSI500 composite stocks closed with limit-ups (356 composite stocks was going up), bringing the entire CSI500 index up by 173.47 (2.34%). On 24 August 2015, as many as 425 composite stocks closed with limit-downs, bringing the entire CSI500 index down to 606.71 (7.97%). Generally, a typical day will see one stock closing with limit-up and one with limit-down.

Table 1. Summary statistics of

Graph

$LU{I}_t$ and

Graph

$LD{I}_t$ during the data period

	Mean	Median	Std	Max	Min
$LUt$	4.0717	1	11.9523	18	0
$LDIt$	10.8167	1	49.3806	425	0

Autocorrelation and the contribution of price latency

We first investigated whether there is first-order autocorrelation in the daily return of CSI500 index within the data period by fitting model 4. The estimated

Graph

${\varphi }_1$ is 0.0920 (with t-statistics of 2.5279 and p-value of 0.012), indicating a significant first-order positive autocorrelation in the daily return of CSI500.

(2-4)

Graph

$R_t={\varphi }_1{R}_{t-1}+{\epsilon }_{1t}.$

In order to further investigate the role of price stabilization mechanism in such autocorrelation, we try to take out the influence of LULD from the daily return of CSI500 index and re-examine the first-order autocorrelation in the residual. Specifically, we examine the following two models:

(2-5)

Graph

$R_t=c_1+a_{1}LU{I}_t+a_{2}LD{I}_t+u_t,$

(2-6)

Graph

$u_t={\varphi }_2{u}_{t-1}+{\epsilon }_{2t},$

where

Graph

$u_t$ is the random error of model 5. Both

Graph

$LU{I}_t$ and

Graph

$LD{I}_t$ are standardized here. Specifically, the

Graph

$LU{I}_t$ is standardized to be −1 to 0 so that the more price limit-ups the larger the standardized

Graph

$LU{I}_t$ . The

Graph

$LD{I}_t$ is standardized to be 0–1 so that the more limit-ups the smaller the standardized

Graph

$LD{I}_t$ .

Model fitting results of Equations 5 and 6 are shown in Table 2. It indicates that both

Graph

$LU{I}_t$ and

Graph

$LD{I}_t$ have a significant influence on the same day market return

Graph

$R_t$ . More importantly, after taking out the influence of price stabilization mechanism, the autocorrelation in model 6 becomes insignificant. This suggests that the observed autocorrelation in market index return of CSI500 is a statistical artefact that can be attributable to the price latency due to price stabilization mechanism.

Table 2. Results of model fitting for models 5 and 6

	Variable	Coefficient	SE	t-Stat	p-Vue	$R2$
Model 5	$LUIt$	0.0569	0.0037	15.15	0.0000	0.5677
$LDIt$	0.0564	0.0021	26.87	0.0000
$c$	0.0004	0.0042	0.11	0.9090
Model 6	$ut−1$	−0.0211	0.0365	−0.57	0.5630	0.0004

To further consolidate our findings above, another way to take out the impact of

Graph

$LU{I}_t$ and

Graph

$LD{I}_t$ is tried. We fit the following model

(2-7)

Graph

$R_t=c_2+a_{3}LU{I}_t+{\epsilon }_{3t},$

to obtain fitted value of

Graph

${\hat{R}}_{t,LUI}$ , and

(2-8)

Graph

$R_t=c_3+a_{4}LD{I}_t+{\epsilon }_{4t},$

to obtain fitted value of

Graph

${\hat{R}}_{t,LDI}$ . We define

Graph

${\eta }_t=R_t-{\hat{R}}_{t,LUI}-{\hat{R}}_{t,LDI}$ to measure the part of market return that is not associated with

Graph

$LU{I}_t$ and

Graph

$LD{I}_t$ and fit the following model to re-examine the autocorrelation:

(2-9)

Graph

${\eta }_t={\varphi }_3{\eta }_{t-1}+{\epsilon }_{5t}.$

We compare the coefficient estimate of

Graph

${\varphi }_3$ in Equation 9 with that of

Graph

${\varphi }_2$ in Equation 4. If the autocorrelation is introduced by price limits, then we should see

Graph

${\varphi }_3$ in Equation 9 becomes insignificant.

Results in Table 3 confirm that after taking out the influences of

Graph

$LU{I}_t$ and

Graph

$LD{I}_t$ from the

Graph

$R_t,$ the autocorrelation in model 9 diminishes. This provides additional evidence to our conclusion.

Table 3. Results of model fitting for models 7–9

	Variable	Coefficient	SE	t-Stat	p-Value	$R2$
Model 7	$LUIt$	0.0606	0.0052	11.52	0.0000	0.1494
$c$	0.0576	0.0051	11.35	0.0000
Model 8	$LDIt$	0.0576	0.0024	24.02	0.0000	0.4344
$c$	−0.0550	0.0023	−23.48	0.0000
Model 9	$ηt−1$	−0.0104	0.0365	−0.28	0.7760	0.0012

III. Conclusion

This article examined how price latency resulting from stabilization mechanism could possibly lead to the autocorrelation in market index. By using daily return of CSI500 and its composite stocks from year 2015 to 2018, we found that the price latency due to price stabilization, when measured by LULD indices, appears to be responsible for the observed autocorrelation in the market index. This provides an alternative explanation for market index autocorrelation and may provide new insight into market dynamic features.

Disclosure statement

No potential conflict of interest was reported by the authors.

1 Atchison, M. D., K. C. Butler, and R. R. Simonds. 1987. " Nonsynchronous Security Trading and Market Index Autocorrelation." Journal of Finance 42 (1): 111 – 118. doi: 10.1111/j.1540-6261.1987.tb02553.x. 2 Cohen, K., S. Maier, R. Schwartz, and D. Whitcomb. 1979. " On the Existence of Serial Correlation in an Efficient Securities Market." Journal of Political Economy 11 : 151 – 168. 3 Cohen, K. J., G. Hawawini, S. F. Maier, R. A. Schwartz, and D. K. Whitcomb. 1983. " Friction in the Trading Process and the Estimation of Systematic Risk." Journal of Financial Economics 12 (2): 263 – 278. doi: 10.1016/0304-405X(83)90038-7. 4 Dimson, E. 1979. " Risk Measurement When Shares are Subject to Infrequent Trading." Journal of Financial Economics 7 (2): 197 – 226. doi: 10.1016/0304-405X(79)90013-8. 5 Griffin, J. M., P. J. Kelly, and F. Nardari. 2010. " Do Market Efficiency Measures Yield Correct Inferences? A Comparison of Developed and Emerging Markets." Review of Financial Studies 23 (8): 3225 – 3277. doi: 10.1093/rfs/hhq044. 6 Harris, L. 1989. " The October 1987 S&P 500 Stock‐Futures Basis." Journal of Finance 44 (1): 77 – 99. doi: 10.1111/j.1540-6261.1989.tb02405.x. 7 Shynkevich, A. 2019. " Pricing Efficiency and Market Efficiency of Two Bitcoin Funds." Applied Economics Letters. doi: 10.1080/13504851.2019.1707760. 8 Theobald, M., and P. Yallup. 2001. " Mean Reversion and Basis Dynamics." Journal of Futures Markets 21 (9): 797 – 818. doi: 10.1002/fut.1901. 9 Wei, W. C. 2018. " Liquidity and Market Efficiency in Cryptocurrencies." Economics Letters 168 : 21 – 24. doi: 10.1016/j.econlet.2018.04.003. Worthington, A. C., and H. Higgs. 2009. " Efficiency in the Australian Stock Market, 1875-2006: A Note on Extreme Long-Run Random Walk Behaviour." Applied Economics Letters 16 (3): 301 – 306. doi: 10.1080/13504850601018379.

By Meng Li; Lixin Qiao and Fangfang Sun

Reported by Author; Author; Author