High-frequency Financial Time Series Modelling
Using momentum variables to forecast intraday stock returns
Radmir M. Leushuis (10988270) June 26, 2018
Supervisor MSc Hao Li
Abstract
This thesis analyses the contribution of modelling intraday momentum to the forecast accu-racy of stock price returns. Models for both the sign and magnitude of the returns are con-structed utilizing economic literature and econometric tests. Utilizing the EGARCH(1,5) model the last half hour returns could be predicted out-of-sample with an adjusted R2 of 4.32%. Using the boosted model out-of-sample to create timing signals leads to a Sharpe ratio of 0.51.
This document is written by Radmir M. Leushuis who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are 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
1 Introduction 1 2 Literature Review 3 2.1 Return patterns . . . 3 2.2 Economic forces . . . 3 2.3 Intraday decomposition . . . 5 2.4 Conceptual models . . . 6 2.5 Hypotheses . . . 8 3 Methodology 10 3.1 Data . . . 10 3.1.1 Characteristics . . . 10
3.1.2 Additional explanatory variables . . . 11
3.2 Analysis . . . 12 3.2.1 Analysis . . . 12 3.2.2 Transformations . . . 14 3.3 Models . . . 15 3.3.1 Sign . . . 15 3.3.2 Magnitude . . . 16
3.3.3 Boosting and forecast accuracy . . . 18
4 Results 21 4.1 Models . . . 21 4.1.1 Coefficients . . . 21 4.1.2 Forecast accuracy . . . 23 4.2 Robustness . . . 24 4.3 Trading Strategy . . . 25 5 Conclusion 28 6 Appendix 33
A.C. Clarke, an influential futurist and inventor, stated in 1964 that predicting the future is a discouraging and hazardous occupation. Still, this has not discouraged investors in the stock market from desperately trying to make predictions about the future prices of financial securities in order to increase their profits. Using every possible tool available, investors used a variety of economic and financial variables in the hope of increasing their forecast accuracy. As a result, these variables started playing a large role in the discussion of the predictability of stock price movements.
As the computational power of computers improved exponentially in the last decade more complex variables emerged. It is still uncertain, however, if these variables, used in the framework of statistical models, actually lead to higher forecast accuracies. Goyal and Welch (2008), for instance, argue that most models do not only fail to beat the simple average of the historical excess stock return, but outright underperform it. This results in a debate about whether these models are functional at all and predicting stock returns is simply impossible.
A critical argument against stock return predictability is given by the efficient market hypothesis (EMH), which states that current stock prices reflect all available information. As a result of the competition between investors, the full effects of new information are immediately processed in current prices of financial securities (Fama, 1970). The usage of financial predictors to make investment decisions is thus illogical, since the EMH implies that stock price movements are random and can not be predicted.
However, one of the most important assumptions of the EMH, the law of one price, often does not hold. Dodd and Graham (1934), for example, noticed that the closing prices of mutual funds are often different from the sum of the prices of the individual stocks. Likewise, other examples of violations of the law of one price are the recent housing bubble in the United States and the internet bubble of 2001 (Thaler, 2009). This would imply that the movement of stock prices is not completely random and follows some hidden pattern. This is further confirmed by Ferson (2007), who found compelling evidence for the predictability of stock prices using financial variables.
Although there are many stock price prediction methods, the one, which shows the most promise, is momentum forecasting. Momentum forecasting is a simple strategy, which has at its core the principle that consecutive returns are correlated with each other (Jegadeesh
& Titman, 1993). Albeit there is plenty of research on momentum, most of it is restricted to returns at the monthly frequency. Despite evidence that models predicting intraday returns repeatedly yield a higher forecast accuracy than the models predicting interday returns (Angelidis & Degiannakis, 2008).
One of the few studies on this matter found statistically significant intraday predictabil-ity for the SP 500 ETF both in-sample and out-of-sample (Gao, Han, Li, & Zhou, 2017). The researchers in this study used first half hour returns of the market to forecast the last half hour returns. When expanding the model with the returns of the half hour prior to the last half hour, the forecast accuracy increased even further. This raises the question what the contribution of modelling intraday momentum is to the forecast accuracy of stock price returns.
The main objective of this thesis is to use advanced time series models to capture the intraday momentum and compare their forecast accuracies. Selecting the models is done by consulting existing literature and performing diagnostic tests on the data. After the selection, the specific configurations and settings of these models are determined by doing econometric tests. Subsequently, the forecast accuracies of the different models are evaluated and compared. This thesis contributes to available literature on intraday momentum by the strong focus on modelling instead of economic inference.
The remainder of this thesis is organized as follows. Section 2 provides a literature re-view on existing articles available on intraday momentum. Section 3 describes the method-ology and data. Section 4 presents the main results of the different models and discusses possible market timing signals. Section 5 concludes.
This literature review is made up of five sections. The first section analyzes return patterns of momentum trading strategies. Afterwards, the second section discusses the economic forces behind intraday momentum. Thereafter, in section three, intraday de-composition based on volume and volatility is analyzed. Subsequently, the advantages and disadvantages of different models are analyzed in section four. Lastly, in section five, different expectations for this thesis are presented in the form of hypotheses.
2.1
Return patterns
As stated in the previous chapter the influence of momentum in stock price movements is found to be persistent. This is further confirmed by Moskowitz, Ooi and Pedersen (2013), who find a significant momentum effect present in indices, currencies, commodities, bonds and future prices. Using 58 liquid financial instruments, they show that momentum strategies across all asset classes deliver considerable abnormal returns with a relatively smaller variance compared to other strategies. Likewise, Asness, Moskowitz and Pedersen (2013) find similar results and argue that momentum profits are positive in periods of both negative and positive economic growth. The best performance was achieved during extreme market conditions, when the volatility was relatively high.
On intraday level, these patterns of significant positive returns are also supported by the research of Gao et al. (2017). They found that the profit, generated by predicting the last hour returns using the first hour returns, is 6.02 percent per annum. This corresponds to a Sharpe ratio of 1.06, which is considerably higher than the 0.29 of a daily Buy-and-Hold strategy. This is further supported by a similar study done by Lou, Pol and Skouras (2017), who decomposed the cross section of expected returns into overnight and intraday components. After analyzing both components, they found a similar outperformance of momentum trading strategies with respect to other trading strategies. In the next section the economic forces behind intraday momentum are discussed.
2.2
Economic forces
Given the consistent outperformance of intraday momentum strategies, it is relevant to consider the economic forces driving it. First of which is the violation of the EMH. As
stated in the introduction, the assumption of instant processing of market information by investors is a key element of the hypothesis. This would imply that prices are always in equilibrium, as they are determined by fully informed investors.
However, even with the rapid advancements of computers and algorithms, the assump-tion that every investor in the world is able to instantaneously process informaassump-tion is unlikely. Bouchas, Farmer and Lillo (2008), for instance, state that stock prices can not be instantaneously in equilibrium and reflect all available market information at the same time. They argue that there is almost always a considerable counterbalance between la-tent demand and supply, which leads to investors slowly processing new information. The implication being that not every investor is as well informed at a certain time.
This would suggest that investors have different response times to new information. According to Bogousslavsky (2016), these investors can be divided into two separate groups by an arbitrary chosen threshold for the response time. The first group consists of investors who are informed real-time and process information in a time which is below the chosen threshold. The second group is comprised of investors, who are either informed late or have a response time above the chosen threshold.
When the first group is informed on new market information, it will immediately react to this information by trading in a certain direction. The second group, however, only has the possibility to trade this new information at a later time. After having processed the new market information, the second group proceeds to trade in the same direction, as the first group traded earlier, to minimize the risk of being wrong. Bogousslavsky (2016) continues to state that this is one of the main reasons for intraday momentum.
The second reason for intraday momentum given by Bogousslavsky (2016) is the infre-quent rebalancing of portfolios by investors. He states that due to capital constraints and various institutional factors big investors, such as hedge funds, rebalance at different times. Similar to the second group of investor, an investor which rebalances at a later time will always trade in the same direction as the investor, which came before him. This leads to a robust development of intraday momentum. In the next section intraday decomposition based on volume and volatility is analyzed.
2.3
Intraday decomposition
To better understand intraday momentum, the trading day is divided into four parts by means of volume and intraday volatility. A trading day, split into these groups, consists of the overnight period, the first half hour, the hours between the first half hour and last half hour and the last half hour. If sorted, the volume and intraday volatility of these groups form on most days an U-shape distribution (Cizeau, Liu, Meyer, Peng, & Stanley, 1997). In other words, both the volume and intraday volatility exhibit a peak at opening and closing of the market. As the market is closed during the overnight period, the accompanied volume and intraday volatility during this period are naturally non-existent. Each of these time frames plays for a different reason a crucial role in determining the intraday momentum.
The overnight period, for example, is mainly important due to company specific data and financial news typically being released before the market opens. Hence, the price difference between close and opening represents the initial reaction of investors on this new information. Having analyzed this price difference for various exchange traded funds, Clark and Kelly (2011) found that the overnight Sharpe ratio constantly surpasses the intraday Sharpe ratio. This suggests that the excess return an investor earns by accepting a higher risk is typically higher during the overnight period than during the intraday period.
Another critical discovery made by Clark and Kelly (2011) is that the overnight returns consistently move in the opposite direction of the intraday returns. This negative corre-lation between the two returns is also found by Branch and Ma (2012), who fitted basic models on the returns of the S&P 500 to investigate these patterns. Their models estimate that a 1.00 percent overnight price movement leads to a 0.37 percent subsequent intraday price movement in the opposite direction.
A possible explanation for this could be the trading behaviour of day traders. Consid-ering that day traders do not want to hold their positions during the overnight period, they accrue a large portion of their positions in the morning and liquidate the biggest chunk of it before close to avoid exposure to overnight risk (Fjeldheim & Bryhn, 2015). As prices increase when the buy side has more volume and decrease when the sell side has more volume, this leads to the negative correlation between the overnight and intraday returns as mentioned in the research of Clark and Kelly (2011). The explanation for this type of intraday momentum is also supported by the U-shape distribution of volume as found by Cizeau et al. (1997).
The influence of trading behaviour of day traders is not only limited to overnight returns. It also has a significant effect on both the first half hour and the last half hour returns. The accruing of positions in the morning and liquidation of positions in the afternoon, as described above, leads to the inverse of the momentum described in the last paragraph. This result is also confirmed by Gao et al. (2017), who found a positive relationship between the first half hour returns and the last half hour returns.
Other explanations are provided by placing the theory of Bogousslavsky (2016) in the framework of the four intraday groups. Since investors in the first group react to overnight news as soon as possible, they will trade in the first half hour of a trading day. Investors in the second group, however, have a latency in either processing or receiving information. Thus, they are limited to reacting at the time slots after the first half hour.
However, due to the known U-shaped distribution of the market volume, Gao et al. (2017) state that this group of investors will almost always choose to trade in the last half hour. Since a higher market volume leads to more liquidity on the market, the costs of trading are lowest when the volume is highest (Chordia, Roll, & Subrahmanyam, 2001). As the nearest possible time to trade on an high volume market is the last half hour, most traders will trade in the last half hour. Trading in the same direction, as the first group of traders traded in the first half hour, generates a positive correlation between first half hour returns and last half hour returns.
Likewise, the infrequent rebalancing of portfolios of large hedge funds leads in a similar way to the same intraday momentum. Minimizing trading costs, these hedge funds will choose to rebalance their portfolios either in the first half hour or the last half hour. This implies that hedge funds, which did not rebalance in the first half hour, will rebalance in the last half hour. As the hedge funds rebalance in the same manner, they trade in the same direction and contribute to the positive correlation between the first half hour returns and the last half hour returns (Gao et al., 2017). In the next section the advantages and disadvantages of different models are analyzed.
2.4
Conceptual models
A major issue of the models used by Gao et al. (2017) to capture the intraday mo-mentum is the models’ highly restrictive assumptions. Assumptions, such as constant conditional variance and linearity, are embedded in the structure of these models.
Al-though holding for regular time series, Ding, Granger and Engle (1993) state that these assumptions do not hold for financial time series data. Ignoring these violations and mod-elling financial time series data using these basic models will lead to biased estimators and incorrect conclusions.
Many time series in business and economics exhibit changes in volatility over time. According to Heij, de Boer, Franses, Kloek and Dijk (2004) this property, called clustered volatility, especially holds true for many financial time series. A powerful model, which is able to account for the changes in volatility, is the generalized autoregressive conditional heteroskedasticity model (GARCH). By including volatility parameters for the conditional variance, this model manages to correctly specify the relationships found in financial time series (Heij et al., 2004).
Another model, which is able to effectively model the conditional variance, is the expo-nential generalized autoregressive conditional heteroscedastic (EGARCH) model (Boller-slev, Engle, & Nelson, 1994). Similar to the GARCH model the EGARCH model describes the variance of the current error terms as a function of the sizes of the previous time peri-ods’ error terms (Heij et al., 2004). The EGARCH model, however, models the logarithm of the variance instead of the variance itself.
This leads to two major advantages. Firstly, since the logarithm of the variance may take on negative values, no additional sign restrictions on the coefficients are needed. Sec-ondly, as the logarithm is a symmetric transformation, the asymmetries found in the data have little to no influence on the parameter estimation. This implies that the EGARCH model, unlike the GARCH model, is able to react asymmetrically to the good and bad news on the markets. According to Kat and Heynen (1994) these advantages combined lead to a significant outperformance in financial time series prediction accuracy of the EGARCH model compared to both standard GARCH models and stochastic volatility models.
Although great for accounting for clustered asymmetrical volatility, an EGARCH model still utilizes a linear framework. As financial time series are generally erratic, relationships in these time series often can not be summarized adequately by a linear model (Ding et al., 1993). According to Belloni, Chen, Chernozhukov and Hansen (2012) the solution to this problem is the boosting of linear models. By combining linear models, which use the residuals or classifications of the other models, non-linear relationships can be modelled by the combination of coefficients in the different models. This also provides flexibility in estimating the coefficients, since each of the different models explains a different part of
the data. Belloni et al. (2012) also found that boosting leads to higher forecast accuracies compared to using traditional linear models, despite the risk of overfitting on the in-sample training set. As can be noticed, each of the specified models has its advantages and disadvantages. In order to find the model that best fits the data, econometric tests need to be performed on the data and models. In the next section, different expectations for this thesis are presented in the form of hypotheses.
2.5
Hypotheses
The question what the contribution of modelling intraday momentum is to the forecast accuracy of stock prices is analyzed in four parts. Firstly, the overnight returns and the intraday returns are analyzed. As stated in section 2.3, Clark and Kelly (2011) found the overnight returns being negatively correlated with the last half hour returns. The first half hour returns, however, are given the results of Gao et al. (2017) likely to be positively correlated with the last half hour returns. The expectation is that similar results are found in this thesis. Thus, the following hypothesis are formulated:
• Hypothesis 1: The overnight returns are negatively correlated with the last half hour returns
• Hypothesis 2: The first half hour returns are positively correlated with the last half hour returns
Afterwards, the forecast accuracies of the different models are evaluated and compared. As stated in subsection 2.4, Belloni et al. (2012) suggest that boosted models outperform linear models in modelling financial time series data. The expectation is that a similar outperformance will be found in this thesis. Hence, the following hypothesis is formed:
• Hypothesis 3: The out-of-sample forecast accuracy of the boosted model is higher than that of the EGARCH model
Lastly, the models are used to create timing signals. The timing signals are in turn used to create a momentum trading strategy. The performance of this strategy is compared to a traditional Buy-and-Hold strategy. As in subsection 2.1 Gao et al. (2017) provide adequate proof for the outperformance of the momentum trading strategy on American markets, the expectation is that similar results will be found on the European market analyzed in this thesis. This leads to the following hypothesis:
• Hypothesis 4: The constructed momentum trading strategy outperforms the Buy-and-Hold strategy
In the next chapter, first, basic characteristics of the data are described. Subsequently, basic statistical test are performed on the data. Finally, the models which describe the data best are analyzed.
This methodology is made up of three sections. Firstly, the first section describes the basic characteristics of the data and evaluates additional explanatory variables. Secondly, in section two, the data is analyzed in depth and possible transformations are considered. Lastly, section three analyzes the models mentioned in the literature review and finds suitable configurations for them.
3.1
Data
This section is made up of two subsections. The first subsections discusses basic char-acteristics of the data and defines the variables used in the models. Thereafter, the second subsection explores the option of adding additional explanatory variables to the models.
3.1.1 Characteristics
The data, used in this thesis, is retrieved from the official site of Dukascopy: a Swiss bank and brokerage firm, which provides historical data to private investors. The data consist of financial time series of the Europe 50 STOXX (EUROXX 50). The EUROXX 50 is Europe’s leading blue-chip index and provides a representation of the super-sector leaders in Europe. The index covers 50 stocks from 17 European countries. By analyzing the largest European index, a deeper look can be taken at the momentum effects in Eu-ropean markets. This is a contribution to current research, as prior research analyzes the momentum effects on American markets.
A key characteristic of the data is its frequency. The data set is comprised of high-frequency data of bid and ask prices at an one minute interval. This leads to a total sum of 922,339 observations ranging from January 1, 2016 till June 15, 2018. The first step, which is performed to clean up the raw data, is the deletion of non-trading days, such as weekend and holidays (A.1). Subsequently, the data is grouped by half hours using a custom Python script (A.2). This leads to a total of 637 trading days each consisting of 17 half hours and 1 overnight period. Afterwards, the data is split into an in-sample part, consisting of the first 437 observations, and an out-of-sample part, consisting of the 200 other observations. Thereafter, the percentage returns of both parts are calculated using
the following formula: rt,h = Pt,hbid+ Pt,hask Pbid t,h−1+ Pt,h−1ask (1) where: t = tth trading day; t ∈ Z+; t ∈ [1, 637]
h = hth half hour in a day; h ∈ Z+; h ∈ [0, 17] Pbid
t,h = bid price in the hth half hour on the tth trading day; Pbidt,h ∈ R+
Pask
t,h = ask price in the hth half hour on the tth trading day; Paskt,h ∈ R+
In order to stay as close to reality as possible, the midpoint price between the bid and ask side will be used as both the buy and sell price. As limit orders placed at the bid and ask prices are often not filled, using a hypothetical midpoint price leads to the highest chance of hypothetical returns being equal to realized returns (Aber, Li, & Can, 2009). Additionally, using these prices also leads to more reliable results, as it prevents returns, resulting from a possible bid-ask bounce, from playing a role in the analysis of momentum returns. In the next subsection each of the variables defined above is analyzed in order to determine whether additional variables should be added to the original model as defined by Gao et al. (2017).
3.1.2 Additional explanatory variables
As mentioned in subsection 2.3, Cizeau et al. (1997) state that the intraday volume and volatility form an U-shape distribution. By indexing both the average intraday volume and volatility of the EUROXX 50 (A.3), one can clearly see that the mentioned distribution is nowhere to be found. A shape which comes close to representing the distribution of the intraday volume is an upside down V. This is due to a steadily increasing intraday volume, which attains its maximum value between the ninth and tenth half hour and then proceeds to decrease again. The intraday volatility, however, starts with being a decreasing function, which attains its minimum value between the sixth and seventh half hour. After attaining its minimum value the intraday volatility morphs into a strictly increasing function. As the intraday volatility does not decrease in the seven latest half hours, like the intraday volume does, its shape looks more like a horizontal J.
Gao et al. (2017) argue that the U-shape of the intraday volume and volatility found in their data is the main reason for the predictive power of the first half hour returns when predicting the last half hour returns. This would imply that, given the upside down V-shape of the intraday volume and J-V-shape of the intraday volatility found in the EUROXX
50, it is important to include the returns of either the ninth half hour or the tenth half hour. A possible criteria which can quantify the explanatory power of these returns is the average of the squared correlations:
Φh = 1 17 − h 18 X i=h+1 ρ2h,i (2)
where: h = hth half hour in a day; h ∈ Z+; h ∈ [0, 17]
ρh,i= correlation of the hth half hour return with the ith one; ρ ∈ R+; ρ ∈ [−1, 1]
By comparing the average of the squared correlations of the returns, specified earlier, the clear dominance of the tenth half hour surfaces (A.4). As the average of the squared correlations of the tenth half hour (0.0185) is significantly larger than the average of the squared correlations of the ninth half hour return (0.0109), it is crucial to include the tenth half hour as explanatory variable. This is further confirmed by the -8.98% correlation found between the tenth half hour and the last half hour. In the next section the descriptive statistics of the data and possible transformations for the data are considered.
3.2
Analysis
This section is made up of two subsections. The first subsections discusses both the descriptive and the distribution statistics of the entire data set. Thereafter, subsection two evaluates possible transformations for the data.
3.2.1 Analysis
Table 1 summarizes all statistical key characteristics of the relevant intraday returns of the EUROXX 50 by displaying their respective descriptive statistics. The results show that the average return of most half hours is positive. As higher returns are commonly accompanied by a higher risk in the form of an higher standard deviation, it is odd to find that the relative standard deviation (compared to the average) is smallest for the overnight returns. These findings, however, are in line with the results found by Clark and Kelly (2011), who also found a higher average return combined with a lower standard deviation for the overnight period.
Variable name Average Std. Dev. Median Minimum Maximum Overnight return 0.02379% 0.00056% 0.03669% -4.60898% 2.25763% First half hour return -0.00488% 0.00138% -0.00767% -0.66509% 0.67238% Tenth half hour return 0.00373% 0.00073% 0.00000% -0.25413% 1.14708% Last half hour return 0.00844% 0.00090% 0.00608% -0.46947% 0.51810%
Table 1: Basic descriptive statistics of the explanatory and dependent variables
Another crucial finding is that the maxima of the first, tenth and last half hour returns are larger than the absolute value of their minima. This implies a skewness to the right. This is also confirmed by the positive skewness as found in Table 2.
Variable name Kurtosis Skewness J–B statistic Overnight return 12.31975 -1.14568 2440.86348 First half hour return 3.05237 0.30618 10.00998 Tenth half hour return 96.24042 6.08514 2343.10133
Last half hour return 5.54339 0.21271 176.22067
Table 2: Distribution statistics of the explanatory and dependent variables
Further analyzing the distributions of the explanatory and dependent variables, the Jarque-Bera statistics are examined. As a perfectly normal distribution has a kurtosis of three and a skewness of zero, the Jarque-Bera statistic, given by the formula below, takes on large values for abnormal distributions and small values for normal distributions:
J B = n − 1 6 S2+ 1 4(K − 3) 2 (3) where: S = skewness; S ∈ R K = kurtosis; K ∈ R n = number of observations; n ∈ Z+
As all variables have an Jarque-Bera which is larger than the critical value χ20.95(2) = 5.99 of a normal distribution at a 5% significance level, there is enough statistical evidence to conclude that all of the variables do not follow a normal distribution.
Variable name µˆ σˆ vˆ Overnight return 1.00073 0.00744 2.70499 First half hour return 0.99985 0.00230 3.88196 Tenth half hour return 1.00004 0.00090 3.12809 Last half hour return 1.00018 0.00134 3.17325
Table 3: Distribution statistics of the approximation of the distribution of the variables
Given that the data does not follow a normal distribution, a fit which describes the characteristics of the data better would be the t-distribution. By running maximum likeli-hood optimizations (A.5) on the data, estimations for the parameters of the t-distributions are given by the results presented in Table 3. The estimations found for the parameters are mostly in line with prior analysis of the data. Knowing the characteristics and distributions of the data, the next subsection discusses possible transformations for the data.
3.2.2 Transformations
An important component of time series prediction is the preprocessing of data. By changing the structure of certain characteristics of the data, assumptions of an used model, which were first violated, could hold. Although there are many possible transformations, one, which is particularly good at modelling financial time series, is the Box-Cox transfor-mation (Proietti & L¨utkepohl, 2013). Hence, in this thesis the two parameters Box-Cox transformation is used, as shown below:
rt,h∗ = log(rt,h+ λ2) if λ1 = 0 (rt,h + λ2)λ1 − 1 λ1 else (4)
where: rt,h = return in half hour h on day t
λ1 = parameter for the curvature; λ1 ∈ R+; λ1 ∈ [0, 1]
λ2 = parameter for the location; λ2 ∈ R+; λ2 ∈ [0, ∞); λ2 > −rt,h
The biggest benefit of the Box-Cox transformation is the ability to transform asymmetrical distributions into normal symmetrical distributions. Using two parameters in this trans-formation instead of one also gives the added benefit of combining power transtrans-formations (λ1) with linear transformations (λ2). This gives the transformation extra flexibility when
In the optimal scenario the parameters are chosen in such a way that the Jarque-Bera statistic is minimized. The simplest way to find these parameters is trying every possible combination of parameters and determining which combination leads to the lowest Jarque-Bera. However, this is both computationally intensive and is limited by the maximum accuracy of Python.
Another method is analyzing the structure of the Jarque-Bera with respect to the data. This leads to the discovery that the Jarque-Bera is decreasing in both λ1 and λ2 (A.6
and A.7). Thus, the optimal transformation given the structure of the data in the sample would be λ1 = 0 and λ2 = 0, which corresponds to the natural logarithm of the data.
This finding is logical as the percentage returns are concentrated around the value of 1. Taking the logarithm leads to the concentration moving to the value of 0, transforming the distribution into a normal one and thus minimizing the Jarque-Bera statistic. Another added benefit of using the logarithm of the percentage returns as a transformation is the fact that it often leads to stationarity of financial time series (Wang, Bao and Chen, 2017). Having analyzed the data in depth, the next section builds on these findings and covers time series models, which are able to describe the data accurately.
3.3
Models
This section is made up of three subsections. The first subsections discusses a model for predicting the sign of the last half hour returns. Thereafter, different versions of the autoregressive conditional heteroscedastic (ARCH) model are analyzed in subsection two. Lastly, subsection three defines the boosted model.
3.3.1 Sign
First, a model to predict the sign of the last half hour returns is constructed. The model, which will be used for this will be a probit model. For a binary dependent variable, which can take the value 0 if the return is negative and 1 if the return is either positive or zero, the model would be given by:
where: rt,h = return in half hour h on day t
βi = coefficient i; βi ∈ R;
yt,17 = binary variable for the return in the last half hour; yt,17 ∈ Z+; yt,17 ∈ [0, 1]
t = residual t; t ∈ R; t∼ N (0, σ2)
The estimates for the parameters are found by maximizing the log-likelihood using the Newton–Raphson optimization method. As the model for the sign of the returns only uses observations that occurred the same day (first half hour and tenth half hour returns), the goal of the model is to perform a cross-sectional classification. In order to guarantee a correctly specified model for this, two crucial underlying assumptions need to hold. Both heteroskedasticity and auto-correlation need to be absent from the residuals of the estimated model.
To test whether heteroskedasticity is present in the residuals a Huber-White test is performed (A.8). As the test rejects the null hypothesis of homoskedasticity, using normal residuals leads to a mispecified model. A solution for this problem is using Hubert-White residuals. These residuals are not constant over the observations, but vary observation from observation, making them ideal for data with a large number of outliers.
Subsequently, a test for auto-correlation is performed by doing a Breusch-Godfrey serial correlation LM test on the residuals (A.9). Given that the null hypothesis of no auto-correlation is not rejected at a significance level of 5%, the assumptions for the model hold and no further action needs to be taken. Lastly, the explanatory variable are tested on their significance by performing a joint F-test and a LR-test (A.10). Both reject the null hypothesis at a significance level of 5%, which leads to the conclusion that the variables are jointly significant. This implies that the model has explanatory power. In the next subsection a deeper look is taken into not only predicting the sign of the returns but also the magnitude.
3.3.2 Magnitude
As stated in the literature review, financial time series tend to exhibit the clustered volatility property. This is the case when one period has significantly larger price move-ments than other periods. To test whether this is also the case in the data used in this thesis, an Engle’s ARCH test is performed on the residuals of the ordinary least squares model (A.11). Since the null hypothesis is rejected at a significance level of 5%, the as-sumption of homoskedastic residuals is violated. As predicting the magnitude is no longer
a cross-sectional analysis, as was the case in subsection 3.2.1, simply using Huber-White residuals does not solve this problem.
Using a version of the ARCH model, however, provides the possibility to not only model the returns but also the heteroskedastic variance. To decide whether to use the GARCH version or the EGARCH version of the model, as specified in the literature review, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) are analyzed. As estimators of the the relative quality of statistical models, they contain information on the mismatching of observations with estimated models. Thus, a lower AIC and BIC score translates into a better fit of the model on the data. Hence, the EGARCH model is the preferable pick, since both the AIC and BIC of the EGARCH model are lower than the AIC and BIC of the GARCH model.
Just like any other time series model, the EGARCH model has at its core the criti-cal assumption of stationarity of the dependent variable. If the statisticriti-cal properties of the dependent variable do not remain constant over time, the model is often found to be misspecified or even spurious (Heij et al., 2004). In order to still be able to investigate cer-tain time series relationship, a transformation, like taking the difference of the dependent variable, can guarantee the stationarity of the time series. To test whether the transforma-tion specified in 3.2.2 leads to a statransforma-tionary time series, an Augmented Dickey-Fuller Unit Root is performed on the dependent variable (A.12). As the null hypothesis of an unit root is rejected at a 5% significance level, the assumptions of the EGARCH model are not violated.
As time series data is strongly correlated over time, past observations often have in-fluence on present observations (Heij et al., 2004). By taking a look at the correlation table and the accompanied autocorrelations (AC), partial autocorrelations (PAC) and Q-statistic, the potential addition of autoregressive–moving-average (ARMA) terms to the EGARCH model can be evaluated (A.13). However, since none of the terms up to lagg 36 fail to be significant at a 5% significance level, no additional ARMA terms should be included in in the EGARCH model. This implies that all of the information of past trading days is decently summarized by the overnight, first half hour and tenth half hour returns of the same day.
Lastly, the local configurations of the EGARCH model itself are analyzed. As stated in the prior paragraph, time series data is strongly correlated over time. This is not only the case for variables used in the regression, but also for the variances of these variables. To
select the optimal number of lagg terms for the variance and residuals, both the AIC and BIC terms are compared for all possible combinations up to lagg term 10. The configuration which results in the lowest AIC and BIC terms is EGARCH(1,5). This results in the following model:
log(rt,17) = β0+ β1log(rt,1) + β2log(rt,10) + t
log(σt) = ω + α1t−1+
P5
i=1γilog(σt−i) + ηt
(6)
where: rt,h = return in half hour h on day t
βt = coefficient for logarithm of the return t; β ∈ R;
αt = coefficient for error lagg term ; α ∈ R;
γt = coefficient for variance lagg term ; γ ∈ R;
t = white noise residual t main regression; ∈ R; ∼ LOGN (0, σ2)
ηt = white noise residual t secondary regression; η ∈ R; η ∼ N(0, ω2)
An added benefit of using the EGARCH model is that no further sign restrictions for the parameters are needed, since log(σt) may take on negative values. In the next subsection,
a model is constructed which utilizes both the model for sign prediction as the model for magnitude prediction. Lastly, criteria to evaluate the performance of different models is discussed.
3.3.3 Boosting and forecast accuracy
As stated in the literature review, the combination of models, provides flexibility in estimating the coefficients. This leads to different situations having an individual model, which in most cases leads to an higher model forecast accuracy. In the case of returns, research suggests that the forecast accuracy for positive returns is higher than for negative returns (Chen, 2009). Hence, making a model for expected positive returns and a model for expected negative returns makes sense. By using the probit model, as specified in 3.3.1, the signs of the returns can be predicted. Subsequently, a linear model is made for predicting the magnitude of both the positive returns and negative returns. This leads to the following model structure:
Figure 1: Visualization of the structure of the boosted model
Another more formal way of presenting the structure of the model is displayed below: log(rt,17) = β0,++ β1,+log(rt,1) + β2,+log(rt,10) + t,+ if yt ≥ 0.5 β0,−+ β1,−log(rt,1) + β2,−log(rt,10) + t,− if yt < 0.5 yt= α0+ α1log(rt,1) + α2log(rt,10) + ηt (7)
where: rt,h = return in half hour h on day t
yt = binary variable for the return in the last half hour; yt ∈ Z+; yt ∈ [0, 1]
βi,+ = coefficient i for logarithm of the expected positive return; βi,+ ∈ R;
βi,− = coefficient i for logarithm of the expected negative return; βi,− ∈ R;
αi = coefficient i for binary return of the last half hour; αi ∈ R;
t = white noise residual t main regression; t ∈ R; t∼ N (0, ω2)
ηt = white noise residual t secondary regression; ηt ∈ R; ηt∼ N (0, ω2)
Finally, the main criteria to evaluate the forecast accuracy of the different models out-of-sample is discussed. The first criterion which will be used to evaluate the relative forecast accuracy of the models is the out of sample relative R2 as specified by Gao et al. (2017):
R2OS = 1 − PT t=1(rt,17− ˆrt,17)2 PT t=1(rt,17− ¯r17)2 (8)
where: rt,17 = return in half hour 17 on day t
ˆ
rt,17 = predicted return in half hour 17 on day t
¯
r17 = average return in half hour 17
T = number of out-of-sample observations
As this criterion only compares the relative performance of the model to using the simple average, it contains no information on the overall absolute performance of the model. Hence, the mean absolute error (MAE), as specified below, also is used:
M AEOS =
PT
t=1|rt,17− ˆrt,17|
T (9)
where: rt,17 = return in half hour 17 on day t
ˆ
rt,17 = predicted return in half hour 17 on day t
¯
r17 = average return in half hour 17
T = number of out of sample observations
Lastly, given the fact that the prior criteria do not account for the number of explanatory variables used, the adjusted R2 also is used. This criterion corrects for the phenomena
of R2 automatically increasing when extra explanatory variables are added to the model. The criterion is given by:
¯ R2OS = 1 − T − 1 T − k − 1 1 − PT t=1(rt,17− ¯r17)2 PT t=1(rt,17− ˆrt,17)2 ! (10)
where: rt,17 = return in half hour 17 on day t
ˆ
rt,17 = predicted return in half hour 17 on day t
¯
r17 = average return in half hour 17
T = number of out of sample observations k = number of explanatory variables
All of the prior criteria are used in order to evaluate the forecast accuracy of the models in order to get an unbiased evaluation. In the next chapter, the results and forecast accuracies of the models, specified in the methodology, are discussed.
The results are made up of three sections. The first section analyzes the different models introduced in the methodology and compares them. The second section analyzes the parameter stability of the model with the highest forecast accuracy. Lastly, the third section creates a momentum trading strategy and evaluates its profitability.
4.1
Models
This section is made up of two subsections. The first subsection discusses different estimations of the coefficient found in each model. The second subsection analyzes and compares the model forecast accuracies.
4.1.1 Coefficients
Table 4: Table of the estimations of the coefficients of the in-sample models
Variable EGARCH(1,5) Probit Boosted+ Boosted− OLS
constant 7.78E-05** 0.1883*** -1.73E-06 0.000334* 4.32E-05 (3.31E-05) (0.06198) (4.48E-05) (0.000193) (4.08E-05) log(rnight) 0.018828*** 45.3271*** 0.05016*** 0.074462*** 0.048157*** (0.005714) (13.1563) (0.010166) (0.022201) (0.00784) log(r1) -0.039373** 15.2648 0.13179 -0.048336 -0.015411 (0.016027) (47.3785) (0.34606) (0.70097) (0.030904) log(r10) -0.04938** -241.1059** -0.196132*** -0.10457 -0.09349* (0.020324) (99.5231) (0.075461) (0.96165) (0.049273) * Significant at a 10% level ** Significant at a 5% level ***Significant at a 1% level
Firstly, the coefficients of the overnight returns are analyzed. An important feature of these coefficients is their significance. As can be seen in Table 4, the coefficients for the overnight returns are the only coefficients that are significant at a 1% level in every model. This implies a persistent explanatory effect of the overnight returns with respect to the last half hour returns. Analyzing the signs of these coefficient, one can clearly see
that each models suggests that the overnight returns have a positive effect on the last half hour returns. A 1% increase in the overnight returns, leads to an increase ranging from 0.0189%, in the EGARCH(1,5) model, to 0.0745%, in the expected positive part of the boosted model. These positive estimated effects combined with the 18.04% correlation found between the overnight and first half hour returns (A.4) provide enough statistical evidence to conclude that a positive relationship exists between the two returns. This is in contradiction with the results of Clark and Kelly (2011), who estimated this relationship to be negative. Thus, the first hypothesis, formulated by applying the findings of Clark and Kelly (2011), is rejected.
Secondly, the coefficients for the first half hour returns are analyzed. As the estimated coefficient for the first half hour returns is only significant for the EGARCH(1,5) model, the estimated effects of the first half hour in other models are inconclusive. The EGARCH(1,5) model, however, suggests that a 1% increase in the first half hour returns leads to a 0.0394% decrease in the last half hour returns. This is not in line with the estimated effects as found by Gao et al. (2017), who found a positive relationship between the first half hour and last half hour returns on the US stock markets. Thus, the second hypothesis, formulated by applying the findings of Gao et al. (2017), is rejected.
Lastly, the coefficients for the tenth half hour returns are analyzed. As the estimated coefficients for the tenth half hour return are significant in all models except the expected negative part of the boosted model, the explanatory power of the tenth half hour returns with respect to the last half hour returns is found to be persistent. The difference in significance between the expected positive part and expected negative part of the boosted model implies that the increase in explanatory power of the model when adding the tenth half hour as variable is most apparent for positive returns. Analyzing the signs of the coefficients, one can see that the estimated effect of the tenth half hour returns is negative in every model. The EGARCH(1,5) model, for instance, estimates that 1% increase in the tenth half hour returns leads to a 0.0494% decrease in the last half hour returns. The boosted model and the ordinary least squares estimate this coefficient being considerably larger with the expected positive part of the boosted model and the ordinary least squares model placing the estimated coefficient at around -0.1961% and the ordinary least squares model at -0.0935%.
4.1.2 Forecast accuracy
Table 5: Table of the estimations of the coefficients of the in-sample models Variable EGARCH(1,5) Boosted model OLS
R2 IS 7.0427938% 7.1393825% 6.9824295% M AEIS 0.000589921 0.00058313 0.000594415 ¯ R2IS 5.5124426% 5.6427345% 5.35137315% R2 OS 5.7837191% 5.7829636% 5.7792292% M AEOS 0.000663118 0.000680556 0.000775375 ¯ R2OS 4.3249515% 4.10352734% 3.9853732%
Firstly, the in-sample accuracy is analyzed by consulting the first three rows in Table 5. An important observation is that both the EGARCH(1,5) and boosted model have an higher accuracy than the OLS model, as the ¯R2
IS of these models is higher and the M AE is
lower. Comparing the EGARCH(1,5) model to the boosted model, leads to the conclusion that the boosted model outperforms the EGARCH(1,5) model in every criteria. The differences, however, are extremely small. This is especially visible in the mean absolute error, which is only 6.791E-06 (1%) larger for the EGARCH(1,5) model.
Secondly, the out-of-sample forecast accuracy is analyzed by consulting the last three rows in Table 5. As was the case in the in-sample results, the out-of-sample results also imply that both the EGARCH(1,5) model and boosted model outperform the OLS model in forecast accuracy. A crucial difference with the in-sample results, however, is the com-parison of the EGARCH(1,5) model to the boosted model. The out-of-sample forecast accuracy of the EGARCH(1,5) model is slightly higher than the forecast accuracy of the boosted model. A possible explanation for the in-sample outperformance and out-of-sample underperformance of the boosted model is the tendency of layered models to overfit on the in-sample set (Belloni et al., 2012). Hence, the third hypothesis, stating the expectation that the boosted model outperforms the EGARCH model out-of-sample, is rejected. In the next section a deeper look is taken at the robustness of the EGARCH(1,5) model and the boosted model.
4.2
Robustness
Firstly, a look is taken at the differences between the in-sample and out-of-sample set. This is done by performing Chow-Break tests on the data with September 9, 2017 as a break point. As the test rejects the null-hypothesis of no structural breaks for both the EGARCH(1,5) model (A.15) and the boosted model (A.16), the conclusion is that the coefficients estimated in the in-sample set are significantly different from the out-of-sample set. This implies a certain parameter instability in the estimation of the coefficients.
Secondly, in order to investigate this parameter instability further, 437 rolling regres-sions with a window of 200 are performed on an approximation of the boosted model. This provides an estimation of the balance of the estimated coefficients. The most important component of this analysis is the number of times the sign of a coefficient changes. Ideally, the estimations of the coefficients stay the same sign and have minimal changes in mag-nitude during every rolling regression. The estimations of the coefficients are displayed in the two line graphs below:
Figure 2: Line graphs of estimations of the coefficients
Analyzing the parameter stability as displayed in the line graphs, one can notice that the most volatile coefficient is that of the tenth half hour. A possible explanation for this parameter instability could be the fact that a large number of returns in the tenth half hour are positive between July 13, 2016 and November 10, 2016. Estimating the coefficients with a window consisting largely of these positive observations, leads to significantly different coefficients for the tenth half hour. The most stable coefficients are those of the overnight return and the constant, which do not only stay the same sign but also remain roughly the same value. This is to be expected, as those are the most significant coefficients in
the boosted model. Likewise, the coefficient of the first half hour also remains stable with only two major downfalls. Hence, the conclusion is that overall the significant coefficients are stable. In the next section momentum trading strategies using the different models are constructed and evaluated.
4.3
Trading Strategy
In order to evaluate the usefulness of the models, their performance in market timing is evaluated by constructing a trading strategy. This is done by using the overnight return, first half hour and tenth half hour as timing signals to take on positions in the last half hour. In order to utilize the model predictions to the fullest extent, both long and short positions are considered. A natural strategy would be to take on a long position in the EUROXX 50 if a prediction is positive and a short position if a prediction is negative. However, as stated in Gao et al. (2017), the addition of a boundary value to this approach could lead to a strategy, which yields a higher profit. Thus the following strategy is created:
πt(rt,0, rt,1, rt,10) = rt,17 if ˆrt,17(rt,0, rt,1, rt,10) > ηup −rt,17 if ˆrt,17(rt,0, rt,1, rt,10) < ηdown 0 else (11)
where: πt = profit generated by the trading strategy on trading day t; π ∈ R;
rt,17 = return in half hour 17 on day t
ˆ
rt,17 = predicted return in half hour 17 on day t
ηup = upper boundary value; ηup∈ R; ηup ∈ (min(ˆrt,17), max(ˆrt,17))
ηdown = lower boundary value; ηdown ∈ R; ηdown ∈ (min(ˆrt,17), max(ˆrt,17))
The function of these boundary values is only taking on long positions if the predicted value is above ηupand short positions if the predicted value is below ηdown. The advantage
of using this structure is that indecisive predictions, which usually are accompanied with a high variance and a lower or even negative profit, have less of an impact on the profit of the trading strategy. By using a custom Matlab script (A.17), an exhaustive search is performed in order to find the combination of ηup and ηdown, which maximizes the
in-sample profit for each model. The estimations of ηup and ηdown, which results in the highest
Table 6: Table of the boundary values
Variable EGARCH(1,5) Boosted model OLS Lower bound 0.000089% 0.0000095% 0.00000095%
Upper bound 0.00009% 0.00001% 0.00025%
Additionally, two benchmark trading strategies are constructed. Firstly, a random choice strategy is constructed, which consists of randomly selecting whether to take on long or short positions in the last half hour each day. Secondly, a classic Buy-and-Hold strategy is constructed in which a long position is taken in each last half hour. In order to analyze the performance of all trading strategies, the profits, standard deviations and Sharpe ratio’s of the different strategies are displayed below in Table 7:
Table 7: Table of the performance of the trading strategies in-sample and out-of-sample Variable EGARCH(1,5) Boosted OLS Random Buy-and-Hold Total profit IS 6.616% 8.4927% 5.3895% -2.9820% 5.5046% Std. Dev. IS 0.0872 0.0864 0.0903 0.0884 0.0819 Sharpe Ratio IS 0.5487 0.5538 0.5298 -0.1317 0.0673 Total profit OS 0.1619% 1.7804% 0.4166% -0.6985% 3.0482% Std. Dev. OS 0.0936 0.0930 0.09405 0.09398 0.09697 Sharpe Ratio OS 0.5111 0.5145 0.5087 -0.3659 0.0569
Firstly, the in-sample results are analyzed. The strategy that yields the highest profit in-sample is the market timing strategy using the boosted model. The accompanied profit corresponds to 6.616% over a period of 437 trading days (615 calender days). Taking a look at the relative profit with respect to the risk taken, one can see that the market timing strategy using the boosted model also leads to the highest Sharpe ratio. This is to be expected as the forecast accuracy of the boosted model is highest of the models in-sample. The Sharpe ratio of the strategy utilizing the boosted model is considerably higher than the Sharpe ratio’s of the two benchmark strategies, which are -0.1317 for the random choice strategy and 0.0673 for the Buy-and-Hold strategy. This is also the case for the market timing trading strategy using the EGARCH(1,5) model, which leads to a Sharpe ratio of 0.5487. This is only 0.92% smaller than the boosted model Sharpe ratio. Both the boosted
model and EGARCH(1,5) model, however, outperform the OLS model with respect to the Sharpe ratio.
Secondly, the out-of-sample results are discussed. As was the case in-sample the market timing strategy using the boosted model leads to both the highest total profit and the highest Sharpe ratio out-of-sample. This is odd, as the out-of-sample forecast accuracy is highest for the EGARCH(1,5) model. A possible explanation for this could be the fact that the boosted model has a larger upward bias than the EGARCH(1,5) model, which is also visible in the larger Theil inequality coefficient. As the percentage of positive out-of-sample last half hour returns is higher than 50%, an upward bias leads to errors upwards being smaller than downwards. This in turn leads to a higher profit and Sharpe ratio. As was the case in-sample, trading strategies utilizing either the EGARCH(1,5) or the boosted model outperform both benchmark strategies with respect to the Sharpe ratio. Hence, the last hypothesis, which formulates the expectation that a momentum trading strategy outperforms a Buy-and-Hold strategy is not rejected. In the next section a short summary, discussion and conclusion are formulated.
This thesis has analyzed the contribution of modelling intraday momentum to the forecast accuracy of stock price returns. By extending the explanatory variables introduced by Gao et al. (2017) with the tenth half hour, an attempt is made to effectively capture the intraday momentum on the EUROXX 50 for the period of January 1, 2016 till June 15, 2018. Models for both the sign and magnitude of the returns are constructed utilizing economic literature and econometric tests. The models, which lead to the most accurate specification of the data, are the EGARCH(1,5) model and the boosted model.
Firstly, the estimated coefficient of the different models have been analyzed. This has lead to the conclusion that on the EUROXX 50 the overnight returns are positively correlated with the last half hour returns for the period of January 1, 2016 till June 15, 2018. This is in contradiction with both the finding of Clark and Kelly (2011) and Brench and Ma (2012), who found that the overnight returns consistently move in the opposite direction of the intraday returns on the US markets. The estimated coefficients for the first half hour returns, however, are found to be negative, which is in contradiction with the results of Gao et al. (2017), who estimated this relationship being positive.
Secondly, the forecast accuracies of the different models have been analyzed. The model, which has the highest forecast accuracy in-sample, is the boosted model. Out-of-sample, however, the EGARCH(1,5) model slightly outperforms the boosted model. A possible explanation for this is the tendency of layered models to overfit on the in-sample set (Belloni et al., 2012). This leads to a higher forecast accuracy in-sample and a lower forecast accuracy out-of-sample.
Lastly, different trading strategies have been constructed. The trading strategy, which yields the highest out-of-sample Sharpe ratio, is the market timing strategy utilizing the boosted model. Although not having the highest forecast accuracy, the boosted model has an upward bias, which given the large number of positive returns in the out-of-sample set leads to a higher profit and a higher Sharpe ratio. Another important result is the finding that the momentum trading strategy utilizing either the EGARCH(1,5) or the boosted model outperforms the two benchmark strategies both in-sample and out-of-sample.
A major obstacle in this thesis, was the relatively small number of available observa-tions. Although enough to perform basic statistical tests and estimate the coefficients, increasing the number of observations could lead to a greater parameter stability and
ro-bustness. This in turn can entail an higher significance of the estimations of the coefficient, which adds explanatory power to the both the model and economic inference of the model. For further research, a boosted model consisting of more models and layers could be considered for modelling high-frequency financial time series. The results in this thesis found a higher out-of-sample forecast accuracy for the EGARCH(1,5) model than for the boosted model. Increasing the number of layers in the boosting process, by using the residuals of the expected negative and positive as dependent variables in the next layer, could lead to the boosted model outperforming the EGARCH models.
Despite the relatively meager number of observations, this thesis has found significant results in predicting the last half hour return. Utilizing the EGARCH(1,5) model the last half hour returns could be predicted out-of-sample with an adjusted R2 of 4.32%. Using the boosted model out-of-sample to create timing signals lead to a Sharpe ratio of 0.51.
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A.1 Holidays
The following table contains a list of all the holidays from 1 December, 2017 till 1 January, 2018 corresponding with their dates.
Date Name
December 25, 2016 Christmas Day 1 December 26, 2016 Christmas Day 2 January 1, 2017 New Years Day
April 14, 2017 Good Friday April 17, 2017 Easter Monday
May 1, 2017 Labor Day June 5, 2017 Whit Monday December 25, 2017 Christmas Day 1 December 26, 2017 Christmas Day 2 January 1, 2018 New Years Day
A.2 Code for cleaning and grouping data
This custom Python script cleans the data by removing holidays found in A.1 and groups variables by half hours.
””” T h i s c o d e c l e a n s t h e d a t a by removing h o l i d a y s and g r o u p i n g v a r i a b l e s by h a l f h o u r s . ””” # i m p o r t n e c e s s a r y modules import c s v import numpy a s np # a r r a y w i t h a l l t h e s t o c k names
s t o c k s = [ ” Euroxx ” ]
# l o o p t h r o u g h a l l s t o c k s f o r s t o c k in s t o c k s :
# s e t t h e f i l e name and l o a d t h e f i l e s t o c k n a m e = s t o c k + ” . c s v ”
f = open ( stock name , ” r ” ) r e a d e r = c s v . r e a d e r ( f ) # s k i p h e a d e r row next ( r e a d e r , None ) # c r e a t e 5 empty l i s t s t o f i l l w i t h v a r i a b l e s l a t e r o p e n p r i c e = [ ] c l o s e p r i c e = [ ] volume = [ ] d a t e = [ ] l o c a l t i m e = [ ] # d e f i n e t o c o u n t t h e m i n u t e s i n t r a d a y m i n u t e s = 0 # l o o p t h r o u g h e v e r y i n t r a d a y minute on e v e r y t r a d i n g day f o r row in r e a d e r : # i n c r e a s e p a s s e d m i n u t e s by 1 e v e r y row m i n u t e s += 1 # minute c o u n t e r s t a r t s t r a d i n g day a t 0 9 : 0 0 : 0 0 # L o c a l ( Amsterdam ) Time i f row [ 0 ] . s p l i t ( ” ” ) [ 1 ] == ’ 0 9 : 0 0 : 0 0 . 0 0 0 ’ : m i n u t e s = 1
i f row [ 0 ] . s p l i t ( ” ” ) [ 1 ] == ’ 1 7 : 3 0 : 0 0 . 0 0 0 ’ : m i n u t e s = 0 i f m i n u t e s > 0 : # g e t t h e o p e n i n g p r i c e o p e n p r i c e . append ( f l o a t ( row [ 1 ] ) ) # g e t t h e c l o s i n g p r i c e c l o s e p r i c e . append ( f l o a t ( row [ 4 ] ) ) # g e t t h e volume
volume . append ( f l o a t ( row [ 5 ] ) ) # g e t t h e d a t e d a t e . append ( row [ 0 ] . s p l i t ( ” ” ) [ 0 ] ) # g e t t h e l o c a l t i m e l o c a l t i m e . append ( row [ 0 ] . s p l i t ( ” ” ) [ 1 ] ) # c r e a t e an empty l i s t t o s t o r e a l l t h e t i m e f r a m e s n e e d e d t i m e s l o t s = [ ] # f i l l t h e l i s t w i t h 0 9 : 0 0 , 0 9 : 2 9 , . . . , 1 7 : 2 9 f o r i in range ( 9 ) : i f i == 0 : t i m e s l o t s . append ( ” 0 ” + s t r ( 9 + i ) + ” : 0 0 : 0 0 . 0 0 0 ” ) t i m e s l o t s . append ( ” 0 ” + s t r ( 9 + i ) + ” : 2 9 : 0 0 . 0 0 0 ” ) t i m e s l o t s . append ( ” 0 ” + s t r ( 9 + i ) + ” : 3 0 : 0 0 . 0 0 0 ” ) t i m e s l o t s . append ( ” 0 ” + s t r ( 9 + i ) + ” : 5 9 : 0 0 . 0 0 0 ” ) e l i f i == 8 : t i m e s l o t s . append ( s t r ( 9 + i ) + ” : 0 0 : 0 0 . 0 0 0 ” ) t i m e s l o t s . append ( s t r ( 9 + i ) + ” : 2 9 : 0 0 . 0 0 0 ” ) e l s e : t i m e s l o t s . append ( s t r ( 9 + i ) + ” : 0 0 : 0 0 . 0 0 0 ” ) t i m e s l o t s . append ( s t r ( 9 + i ) + ” : 2 9 : 0 0 . 0 0 0 ” ) t i m e s l o t s . append ( s t r ( 9 + i ) + ” : 3 0 : 0 0 . 0 0 0 ” ) t i m e s l o t s . append ( s t r ( 9 + i ) + ” : 5 9 : 0 0 . 0 0 0 ” )
# c r e a t e 3 empty m a t r i c e s t o s t o r e t h e minute d a t a g r o u p e d # by t h e h a l f hour p r i c e s t e m p = np . z e r o s ( s h a p e= ( 2 8 4 , 3 4 ) ) v o l u m e g r o u p e d = np . z e r o s ( s h a p e= ( 2 8 4 , 1 7 ) ) v o l g r o u p e d = np . z e r o s ( s h a p e= ( 2 8 4 , 1 7 ) ) d a t e t e m p = [ ] # c r e a t e 5 c o u n t e r v a r i a b l e s u t i l i z e d i n g r o u p i n g t h e d a t a c o u n t e r 1 = 0 c o u n t e r 2 = 0 c o u n t e r 3 = 0 c o u n t e r 4 = 0 c o u n t e r 5 = 0 # c r e a t e v a r i a b l e t o s t a r t t r a d i n g day on = 0 # c r e a t e r o l l i n g sum t o g r o u p t h e volume v o l u = 0 # c r e a t e r o l l i n g l i s t t o g r o u p t h e v o l a t i l i t y v o l a = [ ] # g r o u p d a t a by h a l f h o u r s and s t o r e p r i c e s by t h e h a l f hour f o r i in range ( len ( o p e n p r i c e ) ) : f o r j in range ( len ( t i m e s l o t s ) ) : # s t o r e p r i c e s by t h e h a l f h o u r s and t h e c o r r e s p o n d i n g # d a t e s i f l o c a l t i m e [ i ] == t i m e s l o t s [ j ] : p r i c e s t e m p [ c o u n t e r 2 , j ] = o p e n p r i c e [ i ] d a t e t e m p . append ( d a t e [ i ] )
c o u n t e r 1 += 1 c o u n t e r 2 = round ( c o u n t e r 1 / 3 4 ) # d e t e r m i n e w h e t h e r t h e t r a d i n g day i s s t i l l open i f l o c a l t i m e [ i ] == ” 0 9 : 0 0 : 0 0 . 0 0 0 ” : on = 1 i f l o c a l t i m e [ i ] == ” 1 7 : 2 9 : 0 0 . 0 0 0 ” : on = 0
# g r o u p volume and v o l a t i l i t y by t h e h a l f hour i f on == 1 : c o u n t e r 3 += 1 v o l u += v o l u + volume [ i ] v o l a . append ( c l o s e p r i c e [ i ] / o p e n p r i c e [ i ] ) i f c o u n t e r 3 == 3 0 : t e s t= np . a r r a y ( v o l a ) dev = np . s t d ( t e s t ) v o l u m e g r o u p e d [ c o u n t e r 5 , c o u n t e r 4 ] = v o l u v o l g r o u p e d [ c o u n t e r 5 , c o u n t e r 4 ] = dev c o u n t e r 3 = 0 v o l u = 0 c o u n t e r 4 += 1 i f c o u n t e r 4 == 1 7 : c o u n t e r 5 += 1 c o u n t e r 4 = 0 v o l a = [ ] # c r e a t e empty m a t r i c f o r a l l t h e v a r i a b l e s f i n a l d a t a = np . z e r o s ( s h a p e= ( 2 8 3 , 5 2 ) ) # s t o r e t h e d a t a i n t h e m a t r i x f o r i in range ( 1 , p r i c e s t e m p . s h a p e [ 0 ] − 1 ) : f o r j in range ( 1 7 ) :
