University of Amsterdam
Faculty of Economics and Business, Amsterdam School of Economics
Forecasting intraday returns in the
United Kingdom
including an analysis of the impact on predictability by volume, volatility and the ability of investors to process information
Name: Tim Everaert Student Number: 10548971
Course: Bachelor Thesis Econometrics
Subject: High-frequency financial time series modeling and forecasting:
investigating the relations between the overnight returns and the subsequent intraday returns.
Thesis Supervisor: mw. H. (Hao) Li MSc Date: 4th of December, 2017
Statement of originality
This document is written by Student Tim Everaert who declares to take full responsi-bility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
To contribute to prior research, the predictive power(s) of half hourly returns during the day on the
predictability of the last half hour return will be tested in this thesis including an analysis of the impact
of volatility and volume on predictability in the U.K. stock market. Also, the ability of investors to
process overnight information will be tested. According to advocates of the efficient capital market
theory, it is not possible to significantly predict the last half hour return and to profit from this
phenomenon. This thesis has contributed to the scientific community since it has found that the first
half hour return (r1) and the twelfth half hour return (r12) contain significant predictive powers both
in-sample and out-of-in-sample. In line with the research of Gao et al. (2017), this thesis has found evidence
that predictability rises with volatility. However, regarding the impact of volume on predictability this
thesis has found no evidence that a higher volume positively impacts the predictability of the last half
hour return. In contradiction with the research methods applied by Liu and Tse (2017) and Gao et al.
(2017), this thesis has found evidence that investors process information in the first half hour. The
Contents
1 Introduction 1
2 Literature Review 3
2.1 Efficient capital market theory . . . 3
2.2 Momentum market anomaly . . . 5
2.3 Intraday momentum market anomaly . . . 6
2.4 Forecasting based on intraday momentum . . . 9
2.5 Model and hypotheses . . . 10
3 Data and research methodology 11 3.1 Data . . . 11
3.2 Research methodology . . . 12
4 Results and analysis 14
1
Introduction
Current financial market conditions featured by low volatility and rising operational costs are taking a toll on High-Frequency Trading (HFT) firms who are forced to shift their attention to alternative businesses, according to Chapparo (2017) from Business Insider. In his article, he states that the total revenues obtained by HFT firms have dropped from 7.2 Billion dollars in 2009 to 1.1 billion dollars in 2016, which indicates a massive drop of 85 percent. Consequently, Chaparro states that high frequency trading firms are changing their main strategy of being "the fastest" into being "the smartest" by exploring new strategies to cope with this challenging environment.
Being "the smartest" entails developing algorithmic trading strategies to profit from certain market anomalies, which ought not to exist according to the famous effi-cient market hypothesis of Fama and Malkiel (1970). The effieffi-cient market hypothesis states that asset prices fully reflect all public information and consequently an investor could not profit from exploiting market anomalies. However, a lot of research has been done regarding market anomalies contradicting the efficient market hypothesis such as the January effect (Thaler, 1987), day-of-the-week effect (Kamara, 1997), size-effect (Reinganum, 1981), low-price-to-earnings effect (Banz, 1981) and several other anoma-lies like value versus growth stock performance (Porta et al., 1997).
Recently, the academic community has focused on stock market return volatility and intraday momentum as a new market anomaly which can be exploited by HFT firms. For instance, Liu and Tse (2017) have found that returns during trading hours are insignificant or negative, while stock returns overnight are significantly positive. Also, they claim that overnight returns have less tail risk (crash risk) indicating lower risk during non-trading hours. According to Liu and Tse, this phenomenon is a clear market anomaly since this trading strategy provides a higher return and lower variance
than regular trading which contradicts the efficient market hypothesis. Furthermore, Gao, Han, Li, and Zhou (2017) find that the first half hour return significantly predicts the last half hour return and predictability rises with volatility in the United States. They have also examined whether volume has an impact on their findings.
The research of both Liu and Tse (2017) and Gao et al. (2017) has been applied to the U.S. stock market and it is interesting to examine whether their results on intraday predictability are also applicable to the U.K. stock market. Hence, this thesis will examine whether a half hour during the day can significantly predict the last half hour return in the U.K. stock market. Also, in line with the research of Gao et al. (2017) this thesis will examine to what extent volume and volatility play a role in the predictive power of this half hour. Furthermore, the research methods applied by Liu and Tse (2017) and Gao et al. (2017) differ in their assumptions on the time investors need to process information. Therefore, this thesis will also examine both the method of Gao et al. and Liu and Tse with regard to incorporating overnight information in comparison with the information processed only in the first half hour.
The research methodology of this thesis is based on the work of Gao et al. (2017) where the predictive powers of several half hours are investigated and the impact of volume and volatility on predictability is determined based upon the R2, the Aikaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) values of each regression. Moreover, Gao et al. claim that almost all momentum studies are restricted to return patterns at the weekly or monthly frequency and are confined to the U.S. stock market. Therefore, this thesis will contribute to the scientific community regarding the intraday momentum by applying the model of Gao et al. (2017) to daily returns and to the U.K. stock market.
In line with the results of Liu and Tse (2017) and Gao et al. (2017), it is expected that the significant predictability of the last half hour return is possible based on another
half hour during the day. Also, it is expected that volatility positively impacts the predictability of the last half hour return. Whether trading volume or the different assumptions on information processing of investors have an impact on the predictability of the last half hour return remains to be seen.
The setup of this thesis is as follows. In section two a literature review is provided containing a review of the efficient capital market theory and the intraday momentum market anomaly including the model used and hypotheses tested in this thesis. Then, a data description and the research method are presented. The following section presents the results including an analysis. The last section concludes.
2
Literature Review
This section contains a literature review on the efficient capital market theory and the (intraday) momentum market anomaly. Firstly, the theoretical framework is examined to explain the efficient capital market theory which claims that market anomalies do not exist. Secondly, the general momentum market anomaly is discussed. Furthermore, the intraday momentum and its implications are reviewed. Moreover, forecasting based on the intraday momentum is discussed. At last, the model is presented and the hypotheses investigated in this thesis are formulated.
2.1
Efficient capital market theory
Malkiel (2003) claims that new public available information is reflected in the stock price which leads to a pattern of stock prices following a random walk, which is a metaphor for the pattern of stock returns where consecutive price changes incorporate random shifts from previous prices. The efficient capital market theory embraces the random walk theory in an informationally efficient market. Mandelbrot (1966) and
Samuelson (1965) were the first to discuss the random walk theory and find evidence that price changes must be unpredictable in an efficient market.
In addition, Malkiel (2003) states that the Efficient Market Hypothesis (EMH) is related to the random walk theory of Samuelson (1965). According to Malkiel, the EMH entails that a higher efficiency of the market leads to a higher level of randomness in price changes and that the most efficient market is characterized by certainly random and unpredictable price fluctuations. The EMH relies on profit driven investors who move on any informational advantage available leading to a consolidation of public information into the stock prices, resulting in an elimination of the profit opportunities that initiated their actions.
By revising the efficient capital market theory of Samuelson (1965), Fama and Malkiel (1970) formulate three forms of the EMH. Firstly, the weak-form EMH assumes that all price information is fully represented in stock prices. Secondly, the semi-strong form of the EMH which integrates the weak-form EMH and adds the assumption that stock price changes represent all publicly available information including information on historical stock prices. Thirdly, the strong-form of the EMH entails that both public and private information is reflected in the stock prices. These three forms of the EMH imply that no investor can outperform the average investor.
In their research, Fama and Malkiel (1970) find evidence for the weak-form and semi-form of the EMH. Furthermore, Malkiel (2005) claims that an efficient market ex-ists since active equity management consistently under performs the market. However, Hamid et al. (2017) find evidence contradicting the weak-form EMH in Asian markets. In summary, the efficient capital market theory and the three forms of the EMH are theoretical frameworks which lead to great discussions in practice. The claim of these theoretical frameworks that an investor cannot profit from market anomalies is very interesting since high frequency trading firms are exploiting these anomalies in
practice, like for instance the market anomaly (intraday) momentum.
2.2
Momentum market anomaly
The work of Jegadeesh and Titman (1993) marks the birth of the momentum market anomaly implying the phenomenon where winners (losers) over the last half year to a year are likely to remain winners (losers) over the next half year to a year. Jegadeesh and Titman find evidence for the momentum market anomaly by achieving significant abnormal returns through employing several trading strategies that sell past losers and buy past winners over a three to twelve month holding period. In line with the results of Jegadeesh and Titman, Carhart (1997) also finds evidence for the momentum market anomaly by investigating mutual fund performance. He states that a fund which performed strong last year produces a higher than average expected return the following year, although not from that time forward.
Jegadeesh and Titman (1993) derive their thought process from Bondt and Thaler (1985) who apply the common view that people overreact to information to the stock market. After observing the market momentum anomaly, Jegadeesh and Titman pro-vide two possible explanations for this phenomenon. One possible explanation is that investors who sell past losers and buy past winners shift prices temporarily away from their fundamentals and therefore from their long-run intrinsic value, which leads to an overreaction of prices. This explanation is in line with the findings of De Long et al. (1990) who examine the consequences of "positive feedback traders" on stock prices. Alternately, Jegadeesh and Titman argue that it is possible that investors overreact to information on short-term prospects but overreact to long-term prospects of companies.
2.3
Intraday momentum market anomaly
Recently, the academic world has shifted its attention from the time-series momentum to the intraday momentum as an interesting market anomaly to examine with the focus on the relations between the overnight returns and the subsequent intraday returns.
For instance, Hong and Wang (2000) model a competitive stock market and ex-amine how market closures influence investors’ trading strategies and the following return-generating process. The purpose of their research is to be able to increase com-prehension of the time variation in security trading and the associated returns with market closures. In their research, Hong and Wang find evidence for and refer to com-prehensive literature on the empirical patterns of stock returns and trading activities. Among these patterns are predominantly: (1) Intraday mean return and volatility are U-shaped. (2) Intraday trading volume is U-shaped. (3) close-to-close returns are less volatile than open-to-open returns. (4) Returns over trading periods are more volatile than returns over non-trading periods.
Hong and Wang (2000) state that the logic behind these phenomena can be ex-plained as follows. During an open market, investors adjust their portfolios to hedge the risk of illiquid assets or to speculate on future stock payoffs. During a closed mar-ket, investors hold onto their positions of the last trading period, regardless whether they would like to trade on new information. This allows investors to optimally adjust their trading strategies which causes time variations in equilibrium returns. However, the market closure causes a lack of market prices which gives rise to time variation in the information asymmetry among investors. The interaction between the effect of time varying hedging trade and the effect of time varying information asymmetry can generate patterns in stock returns. When the effect of time-varying hedging trade dom-inates, both the volatility and mean of stock returns decrease. By contrast, when the effect of time-varying information asymmetry dominates, both the mean and volatility
of stock returns increase over time. This phenomenon leads to a mean and volatility return which is U-shaped during trading periods, higher during the opening and closing and lower during mid-period(Hong and Wang, 2000).
In line with the research of Hong and Wang (2000), Stoll and Whaley (1990) also find that close-to-close returns are less volatile than open-to-open returns. According to Stoll and Whaley, this greater volatility implies that the correlation between the daytime and the next overnight returns are greater than the correlation between the overnight and the next daytime returns. Stoll and Whaley claim this result indicates that overnight returns are likely to be reversed by the next daytime return. They view this as a manifestation of a temporary price anomaly at the opening.
Moreover, Liu and Tse (2017) find that while returns during trading hours are in-significant or negative, stock returns overnight are in-significantly positive. Paradoxically, the variance of daytime returns is higher than the variance of overnight returns. Also, they claim that overnight returns have less tail risk (crash risk) indicating lower risk during non-trading hours.
Branch and Ma (2012) reach to the same conclusions and find a substantial degree of negative autocorrelation between adjacent overnight and intraday returns. Their first explanation for negative autocorrelation is the behavior of market makers. In the case of an actively traded security, both buy and sell orders tend to have entered the market before opening. Consequently, the market maker could set the opening price approaching the order imbalance. So, if the imbalance consists of an excess of buy (sell) orders, the asset could be opened above (below) the prior close price to set off enough limit orders above (below) the previous close to offset the imbalance. This approach has several advantages for the market maker. Firstly, more limit orders are set off and the market maker earns a fee for each transaction. Secondly, the market maker restraints fluctuations in his inventory by employing limit orders to cover part of the shortfall
which reduces exposure to unexpected developments. Thirdly, shifting the price away from its previous close boosts trading which increases the ability of the market maker to gain more trading profits. Concluding, Branch and Ma claim that a market maker has motivations to set the opening price below or above the prior close and usually overshoots the equilibrium price. They argue that throughout the day the market price reverts to its intrinsic value, which causes the negative autocorrelation. However, Liu and Tse (2017) do not find evidence for the hypothesis that market maker behavior causes negative autocorrelation.
The second explanation of Branch and Ma (2012) is the bid-ask bounce where the observed negative autocorrelation is produced by the bouncing of the stock price back and forth between the bid and ask, although the underlying price has not changed at all. This occurs for example when a stock closes on the bid price and opens at the ask which causes a positive overnight return despite an unchanged underlying quote. The phenomenon observed can be adjusted by employing a midpoint between the bid and ask price according to Berkman et al. (2012). However, employing this midpoint quote did not have a meaningful impact on the results of Branch and Ma (2012) indicating not much explanation power of the bid-ask bounce.
In the context of micro-foundations, there are two economic reasons for the day momentum according to Gao et al. (2017). They claim that theoretically the intra-day momentum market anomaly can be driven by the infrequent re-balancing of port-folios of investors. Due to slow-moving capital and institutional factors, re-balancing of portfolios may happen in the first half-hour or in the last half-hour. In addition, late-informed investors trade in the last half hour in order to take advantage of the high liquidity and to avoid overnight risk.
In summary, in contradiction with the efficient market hypothesis several intraday patterns are observed regarding stock returns and volatility of which the causes lead to
debate among academics. However, some academics have shifted from explaining these anomalies to finding ways in how to profit from these market irregularities.
2.4
Forecasting based on intraday momentum
Profitable opportunities appear in the stock market due to the intraday momentum market anomaly and a few researchers investigate models to predict stock price returns and generate a trading strategy. For example, Liu and Tse (2017) examine the fore-casting possibilities of overnight returns for different periods of trading hours. They find that overnight returns significantly forecast the intraday first-30-min-returns nega-tively, and the last-30-min-returns positively. However, Liu and Tse find no significant prediction power between these intraday returns and there is no significant relation between other trading hours and overnight returns.
Alternatively, Gao et al. (2017) find that the first half hour return since the previous market close significantly predicts the last half hour return and predictability rises with volatility. Gao et al. focus on the first and last half hour since they argue that news often takes 30 minutes to sink in which is evident based on the findings explained in the previous section. On top of that, they examine the role of volume on the predictability of the last half hour return.
A substantial difference between the above stated papers is the method of calcu-lating overnight returns. Liu and Tse (2017) use the closing price of the previous day and the opening price of today at 08:00 am. However, Gao et al. (2017) use the closing price of the previous day and the price at 08:30 am, when the market is already open for 30 minutes. This difference arises due to different assumptions on the ability of investors to process information.
2.5
Model and hypotheses
In order to test the predictability of intraday returns on the basis of overnight returns in the United Kingdom, this thesis builds upon the model of Gao et al. (2017). Gao et al. examine the predictive powers of all half hours during the day by using formula (1) with the Newey-West correction (Newey and Kenneth, 1987) for heteroskedasticity and autocorrelation:
r17,t = α + βiri,t + t (1)
for i = 1,..., 16, t = 1,..., T.
Where i stands for half hour i and T is the total number of trading days.
The hypotheses stated below test whether the findings of Gao et al. (2017) and Liu and Tse (2017) in the U.S. stock market are also applicable to the U.K. stock market. Also, the different assumptions on how long it takes for investors to process information are tested. The focus of this thesis is set on whether overnight returns (in combination with the first half hour) or a certain half hour during the day can significantly predict the last half hour. Also, to what extent volume and volatility play a role in the predictive power of this certain half hour is examined.
Hypothesis 1 H0: Investors process information overnight.
Hypothesis 2 H0: Half hour i is not able to significantly predict the last half hour using formula (1).
Hypothesis 3 H0: Volatility does not positively impact the significant predictive powers of half hour i.
Hypothesis 4 H0: Volume does not positively impact the significant predictive powers of half hour i.
Hypothesis (1) is tested by comparing the R2, AIC and BIC values of three regressions based on formula (1). The first regression has the first half hour return as independent
variable. The second regression has the overnight return of Liu and Tse (2017) as inde-pendent variable. The third regression has the overnight including first half hour return of Gao et al. (2017) as independent variable. Hypothesis (2) is tested by performing t-tests on the coefficients of the examined half hours and hypotheses (3) and (4) are tested based upon the level of the R2, AIC and BIC values of each regression after splitting the sample into groups based on volatility and volume.
3
Data and research methodology
This section contains a data description and a review of the research methodology applied in this thesis. The aim of this section is to empower the reader to reproduce the results of this thesis. In addition to the explanation presented in this section, the appendix is used as a referencing point by providing tables and figures to provide a better understanding of the research method.
3.1
Data
To conduct the research performed in this thesis, data has been retrieved from
www.Dukascopy.com which is a reliable source for historical tick data on the UK 100 index. The sample period examined is 01-02-2012 until 27-10-2017 due to lack of avail-able data on the UK 100 index. However, this period is sufficient for the research of this thesis. After retrieving the five-minute tick data, the data has been cleaned in excel by filtering out the weekends, adjusting for summertime, excluding trading days with less than 500 trades and calculating the half hourly returns. Regressions, and residual and stability diagnostics are performed in Eviews. See table 13 in the appendix for descriptive statistics, and figure 1 and 2 for the U-shape patterns observed in average daily volume and volatility levels.
3.2
Research methodology
In pursuance of enabling the reader to replicate the results, a clear description of the research methodology is provided. Each subsection contains an explanation of the research method used to test one of the four hypotheses stated in section 2.5.
Before it is possible to test hypothesis (1) by investigating the different assump-tions of Liu and Tse (2017) and Gao et al. (2017) on the ability of investors to process information, it is important to explain how each dependent variable is calculated. First, consider the general case of calculating intraday returns. Every half hour return is cal-culated using the opening price and the closing price 30 minutes later, which results in 17 half hours from 08:00 am to 4:30 pm Eastern time. On a given trading day t this provides a total of 17 observations per day by using formula (2):
rj,t = log(
pj,t
pj−1,t
) (2)
for j = 1,..., 17. Here p(j, t) is the price of the index at the jth half-hour, and pj−1,t is
the price of the index at the previous half-hour, for j = 1,..., 17. Note that the formula used by Gao et al. (2017) is enhanced by using logarithmic returns (Shang, 2017).
Now consider the case of Liu and Tse (2017) who incorporate the processing of overnight information in the independent variable OR. By exercising formula (2), this independent variable is calculated by using the prior trading day’s closing price at 04:30 pm Eastern time and the opening price of today at 08:00 am Eastern time.
However, Gao et al. (2017) formulate the independent variable ORH1 to capture the effect of information released overnight. Gao et al. argue that overnight information is still processed in the first half hour (r1), so ORH1 is calculated with formula (2) based
on the prior trading day’s closing price at 04:30 pm Eastern time and the opening price at 08:30 am Eastern time, which is the end of the first half hour.
To test hypothesis (1) stating that investors process information overnight, three separate regressions are performed with (r1), OR and ORH1 as independent variables,
respectively. The output of these regression are compared and evaluated on the basis of coefficient significance, R2, AIC and BIC values. It is expected that one of the independent variables which incorporate overnight return will outperform the regression with only the first half hour return as independent variable.
In order to test hypothesis (2), formula (1) (see section 2.5) is performed with the Newey-West correction (Newey and Kenneth, 1987) for heteroskedasticity and autocor-relation. In this regression the half hour returns are the independent variables and the last half hour return is the dependent variable. Consequently, t-tests are performed on the coefficients of each half hour return i (except the last half hour return) to determine which half hour i has significant predictive powers. Based on the findings of Gao et al. (2017) it is expected that the return of the first half hour (08:00-08:30) and the return of the half hour before the last one (15:30-16:00) contain significant predictive powers. For brevity, only half hours for which evidence of significant predictive powers has been found are used as independent variables to test hypothesis (3) and (4).
To test hypothesis (3) the effect of return volatility on return predictability is determined by splitting the returns of the significantly predicting half hour i into three groups (low, medium, high) based on the level of volatility. Then for each group, formula (1) is executed and an analysis of the effect of return volatility is performed by discussing theR2 value of each regression.
Like the calculation for volatility, hypothesis (4) is tested by making three groups (low, medium, high) of trading days based on the level of volume. For each group formula (1) is performed and consequently an analysis of the effect of volume on the return predictability of half hour i is provided based on R2 value of each regression.
4
Results and analysis
In this section the results of this thesis are presented and analyzed. First, the abil-ity of investors to process information overnight (H1) is examined by evaluating two
independent variables which incorporate overnight information and one independent variable excluding overnight information. Then, all half hour returns are examined for significant predictive powers (H2). Moreover, several tests on residual and stability
diagnostics are performed on these significantly predictive half hours. Also, the impact of volume and volatility on predictability is examined (H3 & H4).
To test whether investors process information overnight (H1), three separate
re-gressions are performed with r1, OR and ORH1 as independent variables. An F-test on
a single regression testing that every independent variable equals zero is not possible due to the presence of multicollinearity since the independent variables r1, OR and
ORH1 overlap in data. Gao et al. (2017) use HAC Newey-West (Newey and Kenneth, 1987) standard errors to correct for heteroskedasticity and serial correlation. To justify the use of HAC Newey-West errors the presence of heteroskedasticity and serial corre-lation is investigated. Therefore, on the basis of each regression a White test without cross-terms for heteroskedasticity is performed. The White test without cross-terms is used since this form of the White test only tests heteroskedasticity. When cross-terms are included, the correct specification of the model is also tested but that is examined in a later stadium of this section. The results of the White tests on the three regressions with r1, OR and ORH1 as independent variables are stated in the table 1.
For these three variables, the LM value of OR is the lowest with 14.70623 and the LM value of ORH1 is the highest with 49.23327. For all three regressions evidence is obtained for heteroskedasticity and the null hypothesis of homoskedasticity is rejected at the 1% significance level. Table 1 also provides evidence for heteroskedasticity at the
1% significance level for r12 and the combination of r1 and r12. However, this result is
used later in this section.
Table 1: White test for heteroskedasticity
Table 1 provides a summary of the results obtained by performing White tests for heteroskedasticity without crossterms. The number of observations in the sample is 1409 per (in)dependent variable. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Predictor r1 OR ORH1 r12 r1 and r12.
LM = n ∗ R2 34.78946*** 14.70623*** 49.23327*** 3.101075*** 39.44888***
* p<0.10, ** p<0.05, *** p<0.01
Furthermore, on the same regressions performed for the white test, several Breusch-Godfrey LM tests are executed with multiple lags and there is no evidence found for serial correlation. However, since it is appropriate in time series to correct for serial correlation the HAC Newey-West (Newey and Kenneth, 1987) standard errors are used in line with the research of Gao et al. (2017). The essentials of the regression output of each regression are stated below in table 2.
Table 2: Regression r1, OR, and ORH1.
Table 2 provides a summary of the results obtained by performing predictive regressions on r1, OR,
and ORH1. The number of observations in the sample is 1409 per (in)dependent variable. HAC Newey-West standard errors are stated in parentheses. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Predictor r1 OR ORH1
Intercept 2.34E-05 1.86E-05 2.17E-05
(2.48E-05) (2.50E-05) (2.50E-05)
β -0.043607** 0.026275 0.005618 (0.021519) (0.018518) (0.014956) R2 0.004606 0.003782 -0.000435 AIC -11.30321 -11.30239 -11.29816 BIC -11.30043 -11.29493 -11.29071 * p<0.10, ** p<0.05, *** p<0.01
To determine whether investors process information overnight it is necessary to look at the significance of the estimated coefficients of the independent variables. The estimated coefficient of r1 is −0.043607 and is significant at the 5% level. TheR
2
value of the regression with r1as independent variable is 0.004606 which is higher than the R
2
value of the other regressions. Furthermore, to determine which model is the best model the AIC and BIC value of each regression is compared since both measures include a penalty for the use of extra independent variables. The model with r1 as independent
variable has the lowest AIC value of −11.30321 and the lowest BIC value of −11.30043, therefore this is the best model. Also, out of sample analysis is performed based on comparing the Root Mean Squared Error (RMSE) and the Mean Absolute Error (MAE) of each regression. The result is presented in table 4 and it shows that the first half hour seems to contain a better forecasting performance than OR and ORH1, since r1
has a lower RMSE of 0.00868 and a lower MAE of 0.000631 than obtained for OR and ORH1. Therefore, in contradiction with the results of Gao et al. (2017) and Liu and Tse (2017), this thesis seems to provide evidence that investors process information in the first half hour. Hence, the null hypothesis of H1 is rejected. Moreover, table 2 shows that on the basis of the first half hour return it is possible to predict the last half hour return at the 5% significance level in the U.K. stock market, so the null hypothesis of H2 is also rejected.
Consequently, the results presented above also provide evidence for the rejection of the null hypothesis of H2. However, in line with the research of Gao et al. (2017) this
thesis also examines the predictive powers of other half hourly returns. To determine whether other half hours contain significant predictive powers a regression is performed with all half hours as independent variables. The regression output is presented in the appendix in table 12. After performing multiple regressions and eliminating indepen-dent variables step by step, evidence is found for significant predictive powers of r12
which is the half hour return between 14:00 and 14:30. It seems that investors adjust their portfolio earlier in the day than expected. This result is in contradiction with the findings of Gao et al. (2017) who find evidence for significant predictive powers of r15 which is the half hour before the last half hour(15:30-16:00). Although, theoretical
support lacks for the inclusion of r12 (14:00-14:30), it is a significant variable which
cannot be ignored. Perhaps further research can investigate the significant predictive powers of r12.
To determine which model has the best fit on the data, three separate regressions are performed with r1 combined with r12, r1 and r12 as independent variable(s). These
regressions are performed with the HAC Newey-West (Newey and Kenneth, 1987) stan-dard errors.
Table 3: Regression r1, r12, r1 and r12.
Table 3 provides a summary of the results obtained by performing predictive regressions on r1, r12, r1
and r12. The number of observations in the sample is 1409 per (in)dependent variable. HAC
Newey-West standard errors are stated in parentheses. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Predictor r1 r12 r1 and r12
Intercept 2.34E-05 2.48E-05 2.56E-05
(2.48E-05) (2.46E-05) (2.46E-05)
βr1 -0.043607** - -0.044663** (0.021519) - (0.021693) βr12 - 0.086026** 0.088459** - (0.037709) (0.037844) R2 0.004606 3 0.003822 0.008691 AIC -11.30321 -11.30243 -11.30662 BIC -11.30043 -11.29497 -11.29544 * p<0.10, ** p<0.05, *** p<0.01
Table 3 provides evidence for the significant prediction power of r1, r12 and the
combination of r1 with r12 at the 5% significance level. It is clear that the combination
of r1 with r12 has the highest R 2
−11.30043 and the BIC value of the combination of r1 with r12 is −11.29544. The AIC
value of the combination of r1 with r12 is −11.30662 which is lower than the AIC value
of the model with r1 which is −11.30321. The comparison of the AIC and BIC value
of the regressions provide contradicting results to chose the best model. Consequently, out of sample analysis is performed to chose between the model with r1 as independent
variable and the model with the combination of r1 with r12 as independent variables.
So, in line with the research of Gao et al. (2017) out-of-sample analysis is per-formed to examine forecasting powers of the independent variables stated in table 2 and table 3 based on the RMSE and MAE values. To execute this out-of-sample analysis, the sample is split in half. The in-sample period spans from from 1 February, 2012 until 12 January, 2015. The out-of-sample period spans from 13 January, 2015 until 27 October, 2017. The in-sample period contains 705 and the out-of-sample period con-tains 704 observations. Table 4 shows the results of the out-of-sample analysis which provides evidence for the best forecasting performance of r1 and r12 since they obtain
the lowest RMSE and MAE which are respectively 0.000868 and 0.000631. Therefore, the model with the combination of r1 with r12 is the best and is used to test H3 and
H4. Also, the first half hour seems to contain a better forecasting performance than
OR and ORH1, since r1 has a lower RMSE of 0.00868 and a lower MAE of 0.000631
than obtained for OR and ORH1.
Table 4: Out of sample analysis of r1 and r12
Table 4 provides a summary of the results obtained by performing out of sample analysis for r1 and
r12. The in-sample period spans from 1 February, 2012 until 12 January, 2015. The out-of-sample
period spans from 13 January, 2015 until 27 October, 2017. The number of observations in the sample is 1409 per (in)dependent variable, 705 in the in-sample period and 704 in the out-of-sample period. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Statistic r1 r1 and r12 OR ORH1
RM SE 0.000868 0.000865 0.000884 0.000880
Before the impact of volume and volatility on predictability can be tested, it is necessary to investigate the chosen model with the combination of r1 with r12 on
residual and stability diagnostics. Firstly, the stationarity of the included variables in the regression is investigated by performing an Augmented Dickey Fuller test to determine whether each variable has a unit root. If the sample of a variable has no unit root then this sample is stationary. It is important to test for stationarity to determine whether the unconditional joint probability of the stochastic process does not shift over time, which implies constant parameters such as the mean and variance of the sample. Table 5 shows the results of the Augmented Dickey Fuller test on the variables r1, r12
and r17. The t-statistic of r1 is the most negative with -37.94225 so for this variable
the most evidence is found for no unit root. The variable r17 has the highest t-statistic
with -35.801411. However, for all three the variables the null hypothesis, stating that a unit root is present, is rejected at the 1% significance level. Hence, there is enough evidence for stationarity of r1, r12 and r17.
Table 5: Augmented Dickey Fuller test for stationarity on r1, r12 and r17.
Table 5 provides a summary of the results obtained by thee separate Augmented Dickey Fuller test on stationarity of r1, r12, and r17. The number of observations in the sample is 1409 per (in)dependent
variable. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Predictor r1 r12 r17
t − statistic -37.94225*** -36.16789*** -35.801411***
* p<0.10, ** p<0.05, *** p<0.01
Secondly, it is important to test with the Ramsey Regression Equation Specifica-tion Error Test (RESET) test whether the regression has a correct linear specificaSpecifica-tion. The Ramsey RESET test investigates whether the dependent variable can be explained by non-linear combinations of the fitted values. If these non-linear explanatory vari-ables have any power in explaining the dependent variable, the model is misspecified and the linear model might not be appropriate in this case. However, the null
hypoth-esis, stating linear specification, is not rejected since the pvalue is 0.7134 as stated in
table 6. Hence, in this case enough evidence is found for the linear specification of the model used for predictive regression analysis with r1 and r12.
Table 6: Ramsey Reset test on r1 and r12
Table 6 provides a summary of the results obtained by a Ramsey Reset test on correct linear speci-fication. The number of observations in the sample is 1409 per (in)dependent variable. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Predictor r1 and r12
F − statistic 0.134941
P rob F (1, 1405) 0.7134
* p<0.10, ** p<0.05, *** p<0.01
Although the residual and stability diagnostics of the chosen model are examined, it is first necessary to investigate the data on potential structural breaks regarding volume and volatility. Therefore, a CHOW break test is performed separately on the sorted data from low to high for both volatility and volume. Table 7 shows the results of the CHOW break test on sorted volatility. There is no evidence for structural breaks at the 10% level, however the pvalue for the F-statistic at the break-point from medium
to high is 0.1396 which might indicate some effect of high volatility.
Table 7: Chow break test on sorted volatility
Table 7 provides a summary of the results obtained by a Chow Break test on sorted volatility. The trading days are sorted on volatility, from low to high. The number of observations in the sample is 1409 per (in)dependent variable. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Period Low to Medium Medium to High
F − statistic 0.257568 1.823986
P rob F (3, 1403) 0.8552 0.1396
* p<0.10, ** p<0.05, *** p<0.01
and high. Based on Gao et al. (2017), it is clear that high volatility has an impact since the R2 value is 0.019971 which is substantially higher than the other R2 values which are −0.001668 and 0.002657, for respectively low and medium volatility levels. Hence, the null hypothesis of H3 is rejected. This result was expected, as stated in the research
methodology, since Gao et al. (2017) also find evidence for impact of higher volatility on predictability.
Table 8: Regression on sorted volatility with r1 and r12.
Table 8 provides a summary of the results obtained by performing predictive regression on r17 based
on three tercentiles sorted on volatility: Low, medium and High. The number of observations in the sample is 1409 per (in)dependent variable. HAC Newey-West standard errors are stated in parentheses. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Volatility Low Medium High
Intercept 1.05E-07*** 2.83E-05 4.98E-05
(3.34E-05) (3.82E-05) (4.48E-05)
βr1 -0.024360 0.041323 -0.061608** (0.066712) (0.040856) (0.023987) βr12 0.063858 0.081907 0.112567* (0.072664) (0.057003) (0.066020) R2 -0.001668 0.002657 0.019971 * p<0.10, ** p<0.05, *** p<0.01
Also, with regard to volume a CHOW break test is performed of which the out-come is stated in Table 9. Just as with volatility, the data is sorted on the basis of volume level. Then the data is divided into three tercentiles: low, medium and high. Table 9 indicates that the break-point at the volume level from medium to high con-tains a structural break since the F-statistic 2.907059 is significant at the 5% level. The break-point at the volume level from low to medium does not seem to have a structural break since the pvalue of the F-statistic is 0.8015.
In their research, Gao et al. find evidence for impact of higher volume on pre-dictability. However, table 10 does not provide evidence for impact by a higher volume
Table 9: Chow break test on sorted volume
Table 9 provides a summary of the results obtained by a Chow Break test on sorted volume. The trading days are sorted on volume, from low to high. The number of observations in the sample is 1409 per (in)dependent variable. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Period Low to Medium Medium to High
F − statistic 0.332983 2.907059**
P rob F (3, 1403) 0.8015 0.0336
* p<0.10, ** p<0.05, *** p<0.01
level since the R2 value of −0.001553 is substantially lower than for the other volume levels which explains the structural break between medium and high. The volume lev-els low and medium have a similar impact based on their R2 value of 0.013411 and 0.015472, respectively.
Table 10: Regression on sorted volume with r1 and r12.
Table 10 provides a summary of the results obtained by performing predictive regression on r17 based
on three tercentiles sorted on volume: low, medium and high. The number of observations in the sample is 1409 per (in)dependent variable. HAC Newey-West standard errors are stated in parentheses. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Volume Low Medium High
Intercept 1.65E-05 -1.98E-05 7.73E-05*
(3.39E-05) (4.94E-05) (3.95E-05)
βr1 -0.072010* -0.081358** -0.001244 (0.038948) (0.039238) (0.023480) βr12 0.102441 0.102249 0.062428 (0.081351) (0.064279) (0.070638) R2 0.013411 0.015472 -0.001553 * p<0.10, ** p<0.05, *** p<0.01
Since a higher volume level does not impact the predictability, the null hypothesis of H4 is not rejected. This result is not in line with the research of Gao et al. (2017),
has different effects in different markets.
Although, all four the hypotheses are examined, the sample period of this thesis is prone to a historical economic and political event since it incorporates the referendum about the U.K. leaving the European Union, also known as the Brexit. It is appropri-ate to look at structural breaks in the data regarding the Brexit just like Gao et al. (2017) examined the financial crisis. Hence, two CHOW break tests are performed with two structural break-points. The first break-point is the announcement of the Brexit referendum on 20 February, 2016 and the second break-point is the outcome of the referendum on 23 June, 2016. Table 11 contains the outcome of these two CHOW break tests. For both break-points there is no evidence for a structural break since the pvalues are 0.2949 and 0.7663. Since there is no evidence for structural breaks, it is not
appropriate to investigate the impact of the Brexit on predictability.
Table 11: Chow break test on Brexit period
Table 11 provides a summary of the results obtained by a Chow Break test on Brexit. The potential break points investigated in the data are the announcement of the referendum on 20 February, 2016 and the outcome of the referendum on 23 June, 2016. The number of observations in the sample is 1409 per (in)dependent variable. The sample period spans from 1 February, 2012 until the 27 October, 2017.
Period Announcement of Referendum Referendum Outcome
F − statistic 1.236929 0.381533
P rob F (3, 1403) 0.2949 0.7663
* p<0.10, ** p<0.05, *** p<0.01
In summary, in this section the hypotheses of section 2.5 have been tested. With regard to the ability of investors to process information overnight (H1), the null
hy-pothesis is rejected since the estimated coefficient of r1 is −0.043607 and is significant
at the 5% level. Furthermore, the model with r1 as independent variable has the lowest
AIC and BIC value. On top of that, the first half hour seems to contain a better fore-casting performance than OR and ORH1, since r1 has a lower RMSE of 0.00868 and
a lower MAE of 0.000631 than obtained for OR and ORH1. Therefore, it seems that investors process overnight information in the first half hour. However, this result is in contradiction with the research of Gao et al. (2017) and Liu and Tse (2017), therefore further research is needed on the digesting of overnight information by investors.
The null hypothesis of H2is also rejected since r1and r12have significant predictive
powers at the 5% significance level. The predictive power of r12contradicts the findings
of Gao et al. (2017) who find evidence for significant predictive powers of r15
(15:30-16:00). Hence, further research is needed to explain the significant predictive powers of r12 (14:00-14:30). Furthermore, the model including r1 and r12 is the best model since
it obtained the best forecasting performance with the lowest RMSE and MAE which are respectively 0.000868 and 0.000631. Therefore, the model including r1 and r12 is
used to test H3 and H4.
The null hypothesis of H3 is rejected since a higher volatility positively impacts
the predictability of the last half hour return. However, by contrast to the results of Gao et al. (2017) the null hypothesis of H4 is not rejected since there is no evidence
for a positive impact of higher volume. Hence, future research could examine whether volume has different effects in different markets.
5
Conclusion
Lately, the academic community has focused on stock market return volatility and intraday momentum as a new market anomaly which can be exploited by HFT firms. For instance, Liu and Tse (2017) claim that overnight return significantly predicts the last half hour return positively. Furthermore, Gao et al. (2017) argue that the first half hour return significantly predicts the last half hour return and predictability rises with volatility and volume in the United States.
This thesis has contributed to the scientific community by applying the model of Gao et al. (2017) to the U.K. stock market where it has examined the predictability of the last half hour returns with other half hour returns as explanatory variables. Furthermore, the research methods applied by Liu and Tse (2017) and Gao et al. (2017) differ in their assumptions on the time investors need to process information. Therefore, this thesis has examined both the method of Gao et al. and Liu and Tse with regard to incorporating overnight information in comparison with the information processed in the first half hour. Furthermore, this thesis has examined whether volume and volatility increase predictability of the last half hour return based upon half hour return(s) with significant predictive powers.
In contradiction with the research methods applied by Liu and Tse (2017) and Gao et al. (2017), this thesis has found evidence that investors process information in the first half hour. The estimated coefficient of r1 is −0.043607 which significantly predicts
the last half hour returns at the 5% level. Also, the model with r1 as independent
variable has the lowest AIC and BIC values of -11.30321 and −11.30043 in comparison with the AIC and BIC values of the models with OR and ORH1 as independent variables, respectively. Also, the first half hour seems to contain a better forecasting performance than OR and ORH1, since r1 has a lower RMSE of 0.00868 and a lower
MAE of 0.000631 than obtained for OR and ORH1. Since this thesis seems to have found contradicting evidence, further research is needed on the digesting of overnight information by investors.
On top of that, this thesis has found that in the U.K. stock market the estimated coefficient of r12 which is 0.086026 also has significant predictive powers at the 5%
level. The predictive power of r12 contradicts the findings of Gao et al. (2017) who
find evidence for significant predictive powers of r15 (15:30-16:00). It appears that
is needed to explain the significant predictive power of r12 (14:00-14:30). Furthermore,
the model including r1 and r12 is the best model since it obtained the best forecasting
performance with the lowest RMSE and MAE which are respectively 0.000868 and 0.000631. Therefore, the model including r1 and r12 is used to test the impact of
volatility and volume on predictability.
In line with the research of Gao et al. (2017), this thesis has found evidence that predictability rises with volatility since a higher level of volatility provides a higher value of R2. However, regarding the impact of volume on predictability this thesis has found no evidence that a higher volume positively impacts the predictability of the last half hour return. Consequently, it is interesting for future research to examine whether volume has different effects in different markets.
Thus, on the basis of the results of this research it can seem that the efficient capital market theory is rejected and that the results of the research of Gao et al. (2017) and Liu and Tse (2017) are not correct. Admittedly, it is essential to contemplate the limitations regarding the results of this thesis. For example, the current period examined in this thesis spans from 1 February, 2012 until 27 October, 2017. In future research, this period could be expanded which could lead to different results. Also, since prior research is only done in the U.S. and this thesis has focused on the U.K., it might be interesting for future research to examine whether the results of Gao et al. (2017) and Liu and Tse (2017) are applicable to other stock markets than the U.K. or the U.S. stock market.
As stated above, future research is needed to examine and clarify the digesting of overnight information by investors, the significant predictive power of r12, and whether
volume has different effects in different markets. Furthermore, this thesis can be im-proved by investigating individual stocks, several indexes or a different period. However, this advanced research stretches well beyond the scope of this thesis.
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Appendix
Regression Table and Descriptive Statistics
Table 12: Regression on all half hour returns
Table 12 provides a summary of the results obtained by performing predictive regression on all half hour returns 1 until 16 with r17 as dependent variable.The number of observations in the sample is
1409 per (in)dependent variable. HAC Newey-West standard errors are stated under std. error. The sample period spans from 1 February, 2012 until the 27 October, 2017. Days with lower than 500 trades are excluded from the sample.
Variable Coefficient std. error t-statistic Prob
Intercept 2.79E-05 2.47E-05 1.127015 0.2599
r1 -0.039812** 0.019782 -2.012527 0.0444 r2 -0.019089 0.035721 -0.534409 0.5931 r3 0.03942 0.038077 1.035265 0.3007 r4 0.059712 0.049104 1.216038 0.2242 r5 -0.004399 0.036765 -0.119644 0.9048 r6 -0.014026 0.056964 -0.246233 0.8055 r7 0.043791 0.0616 0.710884 0.4773 r8 -0.094846** 0.047727 -1.987271 0.0471 r9 -0.005476 0.054487 -0.100495 0.92 r10 -0.03904 0.063633 -0.613525 0.5396 r11 -0.063299 0.068617 -0.922496 0.3564 r12 0.098589*** 0.038059 2.590452 0.0097 r13 0.061331 0.065547 0.93567 0.3496 r14 -0.061633* 0.035972 -1.713361 0.0869 r15 0.075696* 0.043945 1.722514 0.0852 r16 0.038403 0.046316 0.82915 0.4072 * p<0.10, ** p<0.05, *** p<0.01
Figure 1: Volume U-shape pattern
Figure 1 shows a U-shape pattern of the average daily volume per half hour. The 12th and the 14th half hour seem to have a high level of volume in comparison with a perfect
Figure 2: Volatility U-shape pattern
Figure 2 shows a U-shape pattern of the average daily volatility per half hour. The 9th, 12th and the 14th half hour seem to have a high level of volatility in comparison with
a perfect U-shape pattern of volatility. Furthermore, it appears that at the end of the day, in the 15th and 16th half hour, volume is lower than expected.
T able 13: Descriptiv e statistics T able 13 pro vides a summary of the descriptiv e statistics of all (in)dep enden t v ariables. The n um b er of observ ations in the sample is 1409 p er (in)dep enden t v ariable. The sample p erio d spans from 1 F ebruary , 2012 un til the 27 Octob er, 2017. Da y s with lo w er than 500 trades are excluded from the sample. V ariable Mean Median Maxim um Min im um Std. Dev . S k ewness Kurtosis Jarque-Bera n O R 0.000153 7 .9 5E-05 0.010892 -0.03 777 0.00217 -3.624098 72.45066 2 86257.7 1 409 O R H 1 0.000169 0 .0 0013 0 .009958 -0.030047 0.0 02514 -1.273484 19.0 8091 15562.56 1409 r1 1.65E-05 0 0.007 724 -0.006552 0.0014 23 0.03161 5.49541 365.8156 1409 r2 -3.09E-05 -8.74E-06 0.008266 -0.0041 8 0.000818 0.582 407 12.4045 8 5272.182 1409 r3 -5.34E-05 -3.33E-05 0.00383 -0.003957 0.000745 -0.376 74 5.868784 516.49 56 1409 r4 6.68E-06 6.60E-08 0.004419 -0.005927 0.000697 -0.690743 11.78981 4647.897 1409 r5 -2.04E-05 0 0.003012 -0.004126 0.000633 -0.083 957 5.88125 489.02 83 1409 r6 -2.06E-05 0 0.00239 -0.0040 88 0.000583 -0.298886 6.218725 629.20 77 1409 r7 2.07E-05 2.92E-05 0.003529 -0.002745 0.000549 0.206289 7.257689 10 74.253 1 409 r8 -1.30E-05 6.00E-06 0.002703 -0.003012 0.000583 -0.221287 5.543232 391.2264 1409 r9 3.06E-05 5.18E-05 0.005297 -0.003829 0.000608 0.356081 13.11573 60 37.285 1 409 r10 -2.53E-06 3.69E-06 0.003649 -0.004165 0.000499 -0.219714 10.36244 3193.651 1409 r11 1.55E-05 0 0.003 604 -0.002258 0.0005 11 0.265788 6.435455 709.4858 1409 r12 -2.52E-05 -1.92E-05 0.003529 -0.0060 76 0.000666 -0.661233 13.31917 6354 .2 49 1409 r13 2.50E-05 2.83E-05 0.002463 -0.006058 0.00056 -1.051678 14.38619 7870.989 1409 r14 2.38E-05 0 0.008 165 -0.00357 0.000828 0.599317 10.77619 3634.386 1409 r15 -3.30E-05 -1.39E-05 0.007292 -0.0050 47 0.000816 0.181688 10.91033 3681.324 1409 r16 -2.80E-07 0 0.003333 -0.003919 0.000744 -0.226 89 5.108732 273.15 02 1409 r17 2.26E-05 4.33E-05 0.004705 -0.003826 0.000851 0.148946 5.768692 45 5.2476 1 409 * p<0.10, ** p<0.05, *** p<0.01
