Machine learning trading using S&P 500 stocks
This Thesis is submitted in accordance with the requirements of
the Xi'an Jiaotong-Liverpool University for the degree of
Masters in Financial Mathematics
By Matthew Brice Delikatny
Department of Financial Mathematics
Xi'an Jiaotong-Liverpool University
December 2020
1
Acknowledgments
I would like to thank Ahmet Goncu, who guided me through my Thesis and encouraged me to pick a
topic that I was extremely passionate about. I am extremely grateful for all the time that Ahmet gave me
to support me through my Thesis as well as providing me with knowledge that will benefit me as a
financial professional.
I have to thank my parents, who have been incredibly loving and unbelievably supportive of me and all
my dreams. Words can't express the amount of love they have given me and the sacrifices they have
had to make to help me achieve my dream of becoming a Quant. I would like to thank my loving wife for
supporting me throughout my studies, encouraging me to keep going with my studies even during
difficult times.
I finally would like to thank the other professors in the department who were incredibly supportive and
helpful in my academic journey, as well as XJTLU for accepting me as a master's student in Financial
Mathematics.
2
Abstract
There are two main sections to the paper: the weak market efficiency hypothesis of the stocks in the
S&P 500, testing machine learning algorithms on stocks in the S&P 500 to test the predictability of the
stock returns.
The weak market hypothesis testing includes many types of tests conducted over the entire period of
January 3rd, 2010, to June 3rd, 2020. Weak form market efficiency has many types of tests such as the
Runs test and Kolmogorov–Smirnov goodness of fit test, autocorrelation (LBQ test), and variance ratio.
Over the entire period, these tests can have results that contradict each other due to long periods of
time, having brief periods of gross market inefficiency, making the test declare that the market is
inefficient.
A new market efficiency concept proposed by Andrew Lo in 2004, called the Adaptive Market
Hypothesis (AHM), was also applied to the stock data. AHM asserts that the market's efficiency cyclically
varies with time. When comparing the stocks in the S&P500 using first-order autocorrelations, it can be
seen that the market efficiency of these stocks fluctuates cyclically over time.
Using machine learning algorithms on S&P 500 stocks, some algorithms are more effective in predicting
the stock market's movements. Support Vector Machine (SVM) and Gradient Boosting Classifier (GB)
both have shown to be the most profitable. The machine learning algorithms were not more profitable
than the Buy and Hold strategy. However, it was seen that the machine learning algorithms had lower
drawdowns and higher Sharpe ratios.
3
Contents
1.0 Introduction ............................................................................................................................................ 6
2.0 Literature review..................................................................................................................................... 7
2.1 Market Efficiency ................................................................................................................................ 7
2.1.1 Efficient Market Hypothesis (EMH) ............................................................................................. 7
2.1.2 Adaptive Market Hypothesis ....................................................................................................... 7
2.2 Machine Learning Algorithms ........................................................................................................... 10
2.3 Backtesting Machine Learning Algorithms ....................................................................................... 11
3.0 Methodology......................................................................................................................................... 12
3.1 Efficient Market and Adaptive Market Hypothesis........................................................................... 12
3.1.1 Vratio.......................................................................................................................................... 12
3.1.5 First Order Autocorrelation........................................................................................................ 13
3.2 Machine Learning Methodology ....................................................................................................... 13
3.2.1 Logistic Regression ..................................................................................................................... 14
3.2.2 Support Vector Machine ............................................................................................................ 14
3.2.3 Decision Trees ............................................................................................................................ 14
3.2.4 Random Forest ........................................................................................................................... 15
3.3 Features ............................................................................................................................................ 15
3.3.1 Technical indicators ................................................................................................................... 15
3.3.2 Economic features ..................................................................................................................... 18
4.0 Data ....................................................................................................................................................... 20
5.0 Results ................................................................................................................................................... 23
5.1 Efficient Market Hypothesis.............................................................................................................. 23
5.2 Adaptive Market Hypothesis ............................................................................................................ 24
5.3 Machine Learning.............................................................................................................................. 27
5.4 Backtest ............................................................................................................................................. 28
6.0 Conclusion ............................................................................................................................................. 31
7.0 References ............................................................................................................................................ 32
8.0 Appendix ............................................................................................................................................... 36
8.1 Market Efficiency Tables ................................................................................................................... 36
8.2 Machine learning results................................................................................................................... 53
4
List of Figures
Figure 1-Serial Correlation Percent of the Standard &Poor's (S&P) Composite Index ................................. 8
Figure 2- Piecewise analysis of Machine Learning Algorithms (Ryll & Seidens, 2020) ............................... 12
Figure 3-Extra Tree Classifiers ..................................................................................................................... 16
Figure 4- Feature Importance Technical Indicators .................................................................................... 17
Figure 5-MACD example ............................................................................................................................. 18
Figure 6-A summary of ML papers using macroeconomic predictors. ....................................................... 19
Figure 7-Important factors for U.S. major indices & sectors ...................................................................... 19
Figure 8-Box Plot of Mean Daily Log Returns Of All Stocks ........................................................................ 20
Figure 9-Heat Map of Economic Indicators ................................................................................................ 22
Figure 10-Serial Correlations and Adjusted Close Price of S&P500 Over Time .......................................... 25
Figure 11-BAC Vratio pValues over time..................................................................................................... 26
Figure 12- Run test for BAC over time ........................................................................................................ 27
Figure 14- Portfolio Value Over Time ............................................................ Error! Bookmark not defined.
List of Tables
Table 1-Adaptive Market Hypothesis Studies ............................................................................................... 9
Table 2-Backtesting Literature Review ....................................................................................................... 11
Table 3-Machine Learning Algorithms used ............................................................................................... 13
Table 4- Technical Indicators used by Academics ....................................................................................... 15
Table 5-T-Test of Machine Learning Algorithms to Random Walk ............................................................. 28
Table 6-Backtest Results of Machine Learning Algorithms......................................................................... 30
Table 7-Vratio Test over 10 years ............................................................................................................... 36
Table 8- LQB test ......................................................................................................................................... 38
Table 9-KStest ............................................................................................................................................. 40
Table 10-TS-ADF test ................................................................................................................................... 42
Table 11-Vratio 60 day rolling window of stocks ........................................................................................ 45
Table 12- Ljung-Box Q-test for all stocks .................................................................................................... 47
Table 13-Run test for all stocks ................................................................................................................... 49
Table 14-Machine Learning Algorithms using Technical Analysis .............................................................. 53
Table 15-Machine Learning Algorithms using Technical Analysis and Economic Indicators ...................... 55
5
1.0 Introduction
In the financial world, everyone is chasing the next big strategy that will generate consistent profits. The
new trend is for people to use machine learning algorithms on the stock market.
This thesis first looks at the market efficiency of the 90 stocks selected. Looking at the efficiency of a
stock is essential to see if the market moves randomly or not. If the stock moves like a random walk, the
machine learning algorithm will not accurately predict the next movement. Using different market
efficiency tests, the stocks were not efficient at all times, allowing for profits to be made. The tests
implemented were the Runs test and Kolmogorov–Smirnov goodness of fit test, autocorrelation (LBQ
test), and variance ratio. Further market efficiency analysis was done using the Adaptive Market
Hypothesis (ADH); it could be seen that market efficiency varies with time. The markets were very
inefficient during times of uncertainty, with the recent COVID-19 being an excellent example of the
market's inefficiency and where machine learning algorithms could be profitable.
Multiple machine learning algorithms used are a wide range of classification, regression, and clustering
algorithms. The algorithms try to predict one period forward whether the stock will move up or down. If
the algorithms guess correctly, it makes money; if it fails to determine the stock's movement, it loses
money. The algorithms' average accuracy was around 50 percent, with the precision, recall, and F-score
averages were 48,48 and 41 percent, respectively. When introducing economic data alongside technical
analysis data, there were no significant improvements to the accuracy, precisions, recalls, or F-scores.
This can be due to this publicly available data is already priced into the price of the stock.
It is common practice to backtest any strategy being considered before actually start using them to
trade on the stock market. Conducting backtests on all the machine learning algorithms, most were not
more profitable than a buy and hold strategy; however, the algorithms had lower drawdowns and
higher Sharpe ratios than the buy and hold strategy.
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2.0 Literature review
2.1 Market Efficiency
An efficient market is a market in which the price fully reflects all available information. Fama defines
three levels of efficiency: weak form, semi-strong form, and strong form. Fama (1995)
The weak form efficient market, the price movements, volumes, and earnings data do not affect the
stock's price and cannot predict the stock's future direction. A semi-strong form efficient market is
where all publicly-available data are included in the price of all stocks in a market, and the price changes
to new equilibrium levels are reflections of that information. Finally, the strong form of an efficient
market, all information in a market, whether public or private, has been accounted for in the stock's
price.
2.1.1 Efficient Market Hypothesis (EMH)
The concept of market efficiency is a cornerstone of finance. The primary empirical investigation of
market efficiency involves measuring the statistical dependence between price changes. If no statistical
dependence is found in the prices, that means that the price changes are random. The lack of statistical
dependence provides evidence favoring the weak-form of EMH, which implies that no profitable
investment trading strategies can be derived based on past prices. If statistical dependence is
uncovered, that means prices do not change randomly; a trading strategy can be created around that
fact. If statistical dependence is discovered, this means that the weak form of the EMH does not exist.
Empirical evidence on the weak-form efficiency indicates mixed results. Cooper (1982) studied world
stock markets using monthly, weekly and daily data for 36 financial markets. Cooper examined the
random walk hypothesis's validity by employing correlation analysis, run tests, and spectral analysis.
When it came to the U.S. and U.K. stock markets, the evidence supported the random walk hypothesis.
Seiler and Rom (1997) examined the behavior of the U.S. market's daily stock returns from February
1885 to July 1962, partitioned annually. Using the Box-Jenkins analysis for each of the 77 years, they
indicated that changes in historical stock prices were completely random.
In 1965, Fama used 30 U.S. companies of the Dow Jones, found evidence that dependence on the price
change. The first-order autocorrelation of daily returns was positive for 23 of the 30 stocks, and they
were significant for 11 of the 30 stocks examined. In Lanne and Saikkonen (2004), monthly excess U.S.
stock returns from January 1946 to December 2002 are analyzed. The results indicated that the
presence of conditional skewness in stock returns. The evidence suggests informational inefficiency,
which means that stock prices could be predicted with a fair degree of reliability.
2.1.2 Adaptive Market Hypothesis
The EMH asserts that market prices incorporate all publicly available information, and various
researches that are familiar with behavioral economics and finance have challenged this hypothesis.
They argue that markets are not rational but are driven by fear and greed instead. Andrew Lo, in 2004,
proposed a new framework that reconciles market efficiency with behavioral alternatives by applying
the principles of evolution to approach economic interactions. The proposed idea by Andrew Lo is called
7|Page
the "Adaptive Markets Hypothesis" (AMH). Lo, and other researchers have stated in their publications
that the AMH implies complex market dynamics, such as cycles, herding, bubbles, crashes, trends, and
other phenomena in the financial markets. In their papers, emphasized that market efficiency varies
throughout time using first-order autocorrelations. They showed that the market was more efficient in
1950 than in the early 1990s. Andrew Lo used an example by computing the rolling first-order
autocorrelation for monthly returns of the Standard & Poor's (S&P) Composite Index from January 1871
to April 2003.
Figure 1-Serial Correlation Percent of the Standard &Poor's (S&P) Composite Index
The possibility that market efficiency evolves and changes with time is also stated in the paper by Self
and Mathur (2006): "The true underlying market structure of asset prices is still unknown. However, we
do know that, for some time, it behaves according to the classical definition of an efficient market; then,
for a period, it behaves in such a way that researchers can systematically find anomalies to the behavior
expected of an efficient market."
Researchers have looked at the intertemporal stability over time of the excess returns generated by
several technical trading rules in the foreign exchange market (Neely et al., 2009). Their findings were
consistent with the AMH, as they showed that both institutional and behavioral factors influence
investment strategies and opportunities in the foreign exchange market. Research from other
researchers like Brock et al. (1992) and Sullivan et al. (1997) have found evidence that, in particular,
classes of technical trading rules enabled the possibility of earning abnormal profits from the stock
market.
The effect of changing market conditions on DJIA return predictability is consistent with the adaptive
market hypothesis found (Kim et al., 2011). They concluded, return predictability is associated with
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market volatility and economic fundamentals. There are studies which point out the importance of
behavioral finance and explain investor's behavior such as overreactions and overconfidence in
particular assets which are not rational ( Barber and Ordean, 2001)
Apart from the main finding of time-varying weak-form efficiency in stock markets, the rolling window
approach also allows previous studies to gain additional insights. AMH also has several concrete
implications for the practice of investment management.
Table 1-Adaptive Market Hypothesis Studies
Study
Data and Methodology
Results
Andrew.W.Lo 2004
Monthly returns of the S&P
Composite Index analyzed over
a five-year rolling window
(1871-2003)
The degree of market efficiency
varies through time cyclically.
Markets adapt to evolutionary
selection pressures.
Self, J.K and Machur,I 2006
Daily returns of G7 national
indices (1992-2003) analyzed
using the MTAR model and E-G
stationary tests
Revealed existence of
asymmetric stationary periods,
anomalous market behavior ad
the degree of market efficiency
varies over time.
The market adapts to
evolutionary selection pressures
Neely C.L et al. 2009
Todea.A et al. 2009
Kim, Jae H, et al. 2011
Stability of excess returns
(1973-2005) to technical trading
rules in the foreign exchange
market using ARIMA models
The degree of market efficiency
varies though time in a cyclical
fashion
Daily returns of 6 indices-All
ordinary index, Hang Seng
Index, BSE National Index, Kuala
Lumpur Composite Index,
Straits Times Index, and Nikkei
225 Index( 1997-2008) analyzed
using Moving Average Strategy.
The degree of market efficiency
varies through time cyclically.
Daily returns of the DJIA(19002009) analyzed using automatic
variance ratio, automatic
portmanteau tests, and
generalized spectral tests
The degree of market efficiency
varies through time cyclically.
Markets adapt to evolutionary
selection pressures.
Markets adapt to evolutionary
selection pressures.
Markets adapt to evolutionary
selection pressures.
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2.2 Machine Learning Algorithms
Previous studies succeeded in establishing machine learning models that generate signals. One method
of signal generation is to apply piecewise linear representation to discover turning points. The buy and
sell signals generated from turning points are then denoted as the target variable for machine learning
models (Luo et al., 2017).
Research models focus on the transaction probabilities of uptrend/downtrend to downtrend/uptrend to
generate profitable trading strategies. Chavarnakul and Enke (2008) determine that neural networks can
provide more appropriate trading signals based on two technical indicators: volume adjusted moving
average and ease of movement. Meanwhile, Chang et al. (2009) identify that when training artificial
neural networks, buy and sell signals are generated according to experts' experience, but not the stock
prices for each input. A recent study generates trading signals by news based data and trains the model
through a random forest (Feuerriegel and Prendinger, 2016). Creating a consistently profitable
autonomous trading system have been attempted multiple times by single machine learning models,
such as SVM (Chen and Hao, 2018), and neural network (Chenoweth et al., 2017). Multi-layer
perceptron (MLP) is perceived to be capable of approximating arbitrary functions since the artificial
neural network is a highly elaborate analytical technique for modeling complex non-linear functions
(Principe et al., 2000). The application of neural networks on stock market trading has been studied for
years and has many research papers. For instance, Dunis and Williams (2002) concluded that neural
network regression models could forecast EUR/USD returns. According to empirical evidence from the
Madrid Stock Market, the technical trading rule is always superior to the buy-and-hold strategy for' bear'
and' stable' market episodes (Fernandez-Rodrıguez et al., 2000). Chen et al. (2003) reveal that
probabilistic neural network-based investment strategies obtain higher returns than other investment
strategies on the Taiwan Stock Index.
A new form of machine learning algorithms is called ensemble learning models. These models are
becoming very popular across many fields, including credit risk forecasts and price predictions on the oil
market, the stock futures market, the gold market, and the electricity market. Previous literature
indicates that the simple combination of neural networks outperforms individual networks when trading
S&P 500 futures contracts (Trippi and DeSieno, 1992). Paleologo et al. (2010) compared the bagging
Decision Tree with SVM when contending with credit scoring problems.
The local weighted polynomial Regression formulates a sophisticated ensemble model for exposing the
stock index (Chen et al., 2007). multiple classifiers have been shown in studies to be superior to the
single classifier in terms of prediction accuracy for stock returns (Tsai et al., 2011). The 3-stage nonlinear ensemble model proposed by Xiao et al. (2014) includes training through three neural networkbased models, optimization by improved particle swarm optimization, and SVM prediction. The 3-layer
expert trading system comprises random forest prediction, ensembles, and signal filtering alongside risk
management (Booth et al., 2014). To predict stock prices, Ballings et al. (2015) establish an innovative
idea to generate imbalanced data based on certain thresholds of annual stock price returns.
Consequently, the results were unresponsive to the selected threshold.
Ballings et al. (2015) indicate that exponential smoothing is fundamental for predicting stock price
movements through random forest.
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2.3 Backtesting Machine Learning Algorithms
Backtesting is a common practice in algorithmic trading. Backtesting is when an algorithm is developed
and tested against unseen past data or a random walk. The algorithm uses the data to decide whether it
should buy or sell a stock or index. At the end of the backtest, the user can see if they implemented this
algorithm could potentially make money or not. The table below shows researchers who have done
backtests on machine learning algorithms on the NYSE and the S&P500.
Table 2-Backtesting Literature Review
Author
Market
Period
Andrada-Felix and
FernandezRodrigues(2008)
New York
Stock
Exchange
Composite
Index
S&P 500
Index
S&P500
Index
New York
Stock
Exchange
Composite
Index
Chavarnakul and
Enke(2008)
Chenoweth et al.
(2017)
Leigh et al. (2002)
Transaction
costs
0.2 percent
per stock
Result
1993-2002
Machine learning
algorithm used
Boosting
1998-2003
Neural Network
Profitable
Returns
1982-1993
Neural Network
1981-1996
Neural Network
No
Transaction
costs
Break-even
cost at 0.35%
None
Profitable
returns
Profitable
returns
Profitable
returns
Gauch models have been tests on the NYSE index; they found that the Gauch models were profitable
even with conditional skewness in the market(Leigh et al.,2002). In 2008, Andrada-Felix and FernandezRodrigues used evolutionary algorithms to try and predict the NYSE. They found that these algorithms
can be more profitable than a buy and hold strategy with a lower standard deviation. Neural networks
have been applied to the S&P500 using well known technical indicators. It has been found that they can
beat a buy and hold strategy( Chavarnakul and Enke, 2008)
In an extensive literature review study of 150 papers, it was found that machine learning algorithms
often outperform most traditional stochastic calculus methods in market forecasting. It was also found
that recurrent neural networks outperform feed-forward neural networks along with support vector
machines on average. These findings imply that the existence of exploitable temporal dependencies in
financial time series across multiple asset classes and geographies. (Ryll & Seidens, 2020)
In Figure 2, piecewise ranking analysis has shown what percentage of time one algorithm has beaten
another. Looking at the first column is an Artificial network (ANN), and we can see when looking at the
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last row in column 1 that the ANN beats the Buy and Hold (BH) strategy 91 percent of the time. Some of
the algorithms below say no data. Values of 50 percent mean that the two strategies have similar
performance. The authors have stated that this is due to the lack of studies comparing the two. (Ryll &
Seidens, 2020)
Figure 2- Piecewise analysis of Machine Learning Algorithms (Ryll & Seidens, 2020)
3.0 Methodology
3.1 Efficient Market and Adaptive Market Hypothesis
3.1.1 Vratio
The Vratio test is available in MATLAB as a built-in function. It is based on a ratio of variance estimates
of returns r(t) = y(t)–y(t–1) and period q return horizons r(t) + ... + r(t–q+1). The overlapping horizons
increase the efficiency of the estimator and add credence to the test. Under either null, uncorrelated
innovations e(t) imply that the period q variance is asymptotically equal to q times the period one
variance. The variance of the ratio depends on the degree of heteroscedasticity; therefore, on the null.
("Variance ratio test for a random walk - MATLAB vratiotest", 2020)
Rejection of the null due to dependence of the innovations does not imply that the e(t) are correlated.
Dependence allows that nonlinear functions of the e(t) are correlated, even when the e(t) are not. For
example, it can hold that Cov[e(t),e(t–k)] = 0 for all k ≠ 0, while Cov[e(t)2,e(t–k)2] ≠ 0 for some k ≠ 0.
("Variance ratio test for random walk - MATLAB vratiotest", 2020)
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Cecchetti and Lam show that sequential testing using multiple values of q results in small-sample size
distortions beyond those that result from the asymptotic approximation of critical values. ("Variance
ratio test for a random walk - MATLAB vratiotest", 2020)
3.1.5 First Order Autocorrelation
First Order Autocorrelation is a common metric used to test for the AMH. When performing multiple
linear regression using the data in a sample of size n, there are n error terms, called residuals, defined by
ei = yi – ŷi. One of the linear regression assumptions is that there is no autocorrelation between the
residuals, i.e. for all i ≠ j, cov(ei, ej) = 0. ("Sample autocorrelation - MATLAB autocorr", 2020)
The autocorrelation (aka serial correlation) between the data is cov(ei, ej) says the data is
autocorrelated or there exists autocorrelation. If cov(ei, ej) ≠ 0 for some i ≠ j. ("Sample autocorrelation MATLAB autocorr", 2020)
In this paper, correlations were checked between today’s price and yesterday’s price using a rolling
window of 60 days.
3.2 Machine Learning Methodology
All machine learning algorithms used in this thesis were created using a python package known as Scikitlearn. Scikit-learn is a free software machine learning library that Python can read. It features various
classification, regression and clustering algorithms, including support vector machines, random forests,
gradient boosting, k-means, and DBSCAN, and is designed to interoperate with the Python numerical
and scientific libraries NumPy and SciPy. ("Scikit-learn", 2020). The machine learning algorithms used in
this thesis are listed in the table below.
Table 3-Machine Learning Algorithms used
Machine Learning Algorithms
Logistical Regression
Logistical Regression (Ridge Regression)
Linear Discrimination Analysis
K Neighbors Classifier
Gradient Boosting Classifier
Ada Boost Classifier
Random Forest Classifier
Support Machine Vectors
Support Machine Vectors Perceptron
Support Machine Vectors Hinge
Multilayer Perceptron
Abbreviation
LR
LR-L2
LDA
KNN
GB
ABC
RF
SVM
SVMP
SVMH
MLP
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3.2.1 Logistic Regression
Logistic Regression is the first machine learning model, is the simplest, and is a handy classifier when
dealing with classification problems. The advantage of Logistic Regression is that it can give a binary
class probability by producing a simple probabilistic formula for the classification. Logistical Regression
does not require strict assumptions like normally distributed input variables. A linear relationship
between the input and output variables is not required as well. Assumptions when using Linear
Regression are:
1) It requires that the dependent variable be binary, with the groups being discrete, non-overlapping,
and identifiable.
2) It considers the cost of false positives and false negatives in selecting the optimal cut-off probability
(Ohlson, 1980).
3.2.2 Support Vector Machine
Support Vector Machine (SVM) is the second classifier to be utilized. It can produce a binary classifier optimal separating hyperplanes - by an extremely non-linear mapping of the input vectors into a highdimensional feature space (Pai et al., 2011). The idea of SVM is to transform the data that is non-linear
separable in its original space to a higher dimensional space in which a simple hyperplane can separate
it. SVM, proposed by Vapnik in 1998, generates a classification hyperplane that separates two data
classes with a maximum margin. In SVM, a linear model estimates a decision function using non-linear
class boundaries based on support vectors. SVM trains to create an optimal hyperplane that separates
the data into the maximum distance between the hyperplane and the closest training points. The
training points closest to the optimal separating hyperplane are called support vectors (Kim and Sohn,
2010).
There are some advantages of SVM:
1) there are purely two free parameters, which are the upper bound and the kernel parameter. (Shin et
al., 2005).
2) SVM is a unique, optimal, and global process since the training of an SVM is conducted by solving a
linearly constrained quadratic problem; (Shin et al., 2005).
3) SVM is based on the structural risk minimization principle; therefore, this type of classifier minimizes
the upper bound of the actual risk, whereas other classifiers minimize the empirical risk (Shin et al.,
2005).
3.2.3 Decision Trees
A Decision Tree is to recursively divide data into subsets until the state of the target predictable
attribute in each subset is homogeneous. The Decision Tree classification technique consists of two
phases, namely tree building and tree pruning. Tree building is a top-down process that recursively
partitions the tree until all data items are assembled into the same class label. Meanwhile, tree pruning
is a bottom-up process. Thus, reducing over-fitting, prediction, and classification accuracy will be
improved (Imandoust and Bolandraftar, 2014).
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The Decision Tree commonly imitates the human mindset, deeming it straightforward to comprehend
data and provide accurate interpretations. Decision trees even provide the logic for interpreting data,
unlike black-box algorithms such as SVM and MLP.
3.2.4 Random Forest
A random forest's overall objective is to combine the predictions of multiple binary decision trees to
establish more accurate predictions than individual models. One advantage of the random forest
algorithm is its relative robustness when confronted with noisy data. According to Patel et al. (2015), the
utilization of sub-sampling and random decision trees often generates sounder predictive results than
single decision trees. Moreover, the random forest contains methods for balancing errors in unbalanced
data sets of class population. The random forest algorithm minimizes overall prediction errors by
maintaining low error rates on larger classes while permitting high error rates (Breiman, 1999).
3.3 Features
3.3.1 Technical indicators
Features for each observation are generated from the stock price information (open price, close price,
low price, high price, and volume), which means that technical indicators are utilized as features.
Some scholars support that technical indicator can be employed to generate positive trading strategies
(Metghalchi et al., 2008;)
Table 4- Technical Indicators used by Academics
Technical Indicator
Description
Calculation
100
100 −
(1 + 𝑆𝑀𝐴(𝑝𝑢𝑝, 𝑛1)
RSI
Relative Strength Index
STCK
Fast Slow Stochastic Oscillator
%K=(pc-min(pl)-max(pl)-min(pl))*100
STCD
Slow Stochastic Oscillator
WILLS
Williams %R
ROCP
Rate of Change Percentage
%D=SMA(%K,N)
𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝐻𝑖𝑔ℎ − 𝐶𝑙𝑜𝑠𝑒
𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝐻𝑖𝑔ℎ − 𝐿𝑜𝑤𝑒𝑠𝑡 𝐿𝑜𝑤
𝐶𝑙𝑜𝑠𝑒 𝑃𝑟𝑖𝑐𝑒 − 𝐶𝑙𝑜𝑠𝑒 𝑃𝑟𝑖𝑐𝑒𝑛 𝑝𝑒𝑟𝑜𝑖𝑑𝑠 𝑎𝑔𝑜
𝐶𝐿𝑜𝑠𝑒 𝑃𝑟𝑖𝑐𝑒𝑛 𝑝𝑒𝑟𝑜𝑖𝑑𝑠 𝑎𝑔𝑜
Moving Average Convergence
Divergence (MACD)
OBV
Moving Average Convergence
Divergence
On Balance Volume
𝐸𝑀𝐴𝑡 (𝑝𝑐 , 𝑠) − 𝐸𝑀𝐴𝑡 (𝑝𝑐 , 𝑓)
𝑂𝐵𝑉𝑡 = 𝑂𝐵𝑉𝑡 − 1
Some papers report the mixed results of technical trading strategies over different trading periods
(Schulmeister, 2009). Other scholars argue that technical trading rules are unprofitable after data
snooping bias or transaction costs are taken into account (Marshall et al., 2008).
In this thesis, a python package known as TA, which allows users to input OHLC data as well as Adjusted
Closes and volume data, was used to generate all the technical analysis data. The package takes this
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data and outputs 70 well known technical indicators. In doing feature analysis to determine which
features were the most important in predicting the stock’s movement, found the best was to use a
machine learning algorithm called Extra Trees Classifiers (ETC), which is similar to a Random Forest
classifier. With the use of Extra Trees Classifiers to introduce more variation into the ensemble, we will
change how we build trees. The features and splits are selected at random. Since splits are chosen
randomly for each feature in the Extra Trees Classifier, it is less computationally expensive than a
Random Forest.
Figure 3-Extra Tree Classifiers
Decision Trees show high variance, where Random Forests show medium variance, and Extra Trees
show low variance. After putting all 70 features through the ETC, the figure below can be created. It can
be seen in Figure 4 that the most valuable indicators are the daily returns as well as the standard
technical indicators listed in Table 4 are more important in the prediction of the stock movement.
16 | P a g e
Figure 4- Feature Importance Technical Indicators
Sullivan et al. (1999) divide trading rules into five groups: filter rules, moving averages, support and
resistance rules, channel breakouts, and on-balance volume averages. Some commonly used technical
indicators used in academia and industry are relative strength index, stochastic line, Williams, moving
average convergence divergence, rate of change percentage, On Balance, and Volume is selected from
the five categories of indicators.
Relative strength index (RSI) is a strength indicator computed as a rising rate over fall rate in a period. It
is usually used to identify an overbought or oversold state in the trading of an asset. RSI ranges from 0
to 100. If the RSI is below level 30, the asset is oversold; when RSI is above 70, it is overbought. Surges
and drops in an asset's price will influence the RSI by creating false buy or sell signals. Therefore, the RSI
is best used with other tools together to determine the trading signals (Wilder, 1978).
The stochastic lines (STCK, STCD) are composed of two smooth moving average lines (%K and %D) to
judge the buy/sell points. The fast-stochastic K represents a percent measure of the last close price
related to the highest of the highest high value and the lowest of the lowest low value of the last n
periods. The theory behind the stochastic line is that the price is likely to close near its high (or low) in
an upward-trending (or downward- trending) market. Transaction signals occur when the STCK crosses
the STCD (Basak et al., 2019).
William (WILLS) indicates stock overbuy or oversell. It can be used to discover the peak or trough of the
stock price. If the difference between the lowest stock prices and close prices is massive (or small), and
17 | P a g e
the stock is overvalued (or undervalued), traders need to wait for a short (or long) trading signal to sell
(or buy) the stock (Chang et al., 2009). Therefore, WILLS is adopted to assist the turning point decision
through examining stock is overvalued or undervalued.
Moving average convergence divergence (MACD) is to find the fluctuation of stock price via an
exponential moving average of the last n observations. When trading, traders look for a signal-line
crossover to occur. This occurs when the MACD and average lines cross each other. When the MACD
diverges from the average lines, this is an indicator that the stock should be bought. If the MACD
converges on to the average line, this is a strong indicator that the stock should be sold off.
Figure 5-MACD example
The rate of change percentage (ROCP) of a time series pc evaluates the breadth of the range between
high and low prices. ROCP can be used to confirm price moves or detect divergences (Larson, 2012).
On balance volume (OBV) uses positive and negative volume flow to predict stock price change. It adds
the volume when the close has increased and subtracts it when the close has decreased (Granville,
1976).
3.3.2 Economic features
Along with the technical indicators to help predict the stock market's movement, economic indicators
can also help the machine learning algorithms make accurate predictions. A fair amount of work has
been done in this field. The figure below shows the macroeconomic factors applied to indexes such as
the S&P500, Taiwan Stock Index, and the NIKKEI 225.
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Figure 6-A summary of ML papers using macroeconomic predictors.
In these studies, the macroeconomic indicators' correlation is strongly correlated to the target variable,
which could be the stock movement, excess stock returns, or volatility. The accuracies of the machine
learning algorithms range from 66 to 79 percent. Bin Weng and his colleagues analyzed stock exchanges
and financial sectors using the monthly closing prices in the United States. Weng and his team did
feature analysis where each feature is ranked between 0 to 1. From there, Weng and his team chose
variables that have feature importance greater than 0.6. (Weng et al., 2018)
Figure 7-Important factors for U.S. major indices & sectors
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4.0 Data
All analysis is done on the S&P500 stocks. The stocks were selected by looking at the average volume
traded daily from January 3rd, 2010, to June 3rd, 2020. Instead of analyzing all 505 stocks on the S&P500,
it was easier to select stocks with 30 of high volume traded, 30 stocks around the average, and 30 low
traded stocks. A box plot of the daily average of all 90 stocks can be seen below.
Figure 8-Box Plot of Mean Daily Log Returns Of All Stocks
Further analysis of the log-returns can be seen by the histogram below, which shows that the
distribution is relatively normal with very heavy tails. This is not abnormal for stock data; stock data is
usually a Cauchy distribution. When the log-returns is taken, it is an attempt to try and normalize the
data. It can be seen that most of the data does follow a normal distribution, but heavy tails still exist.
20 | P a g e
All pricing data was acquired using a python package called YFinance. YFinance allows the user to insert
the desired timeframe and stock symbol. From there, the package uses Yahoo Finance to collect the
Open, High, Low, Close, Adjusted Close, and Volume.
Macroeconomic data was collected by using the Federal Reserve Economic Data(FRED) and Quandl. The
FRED had M1, M2, M3, Effective Federal Fund Rate (EFFR), ten year Treasury Constant Maturity
Rate(10YT), 3 Month Treasury Bills (3MTB), Interest Rate Discount Rate for the United States (IRDR),
Unemployment rate (unemp), Consumer Price Index (CPI), Manufacturing Composite Index (MCI), Nonmanufacturing Composite Index ( NMI), as well as investment sentifimate of how many investors are
Bullish, bearish and neutral to the market. The FRED records this data on a monthly basis.
Looking at the correlations of each of the variables and plotting it onto a heatmap seen below. It can be
seen looking at the graph that many of the variables are correlated with one another.
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Figure 9-Heat Map of Economic Indicators
Further analysis can be done with a box plot. First, all the Economic Indicators need to be normalized
due to vast differences in the order of magnitude. Once this has been done, a box plot can be applied to
the Economic Indicators, and the figure below can be seen.
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It can be seen that the normalized variations in the values can be extreme, with values ranging from 0 to
1. The MCI, Bull, Bear,neu, CS, and NMI have smaller ranges due to how those values are calculated.
They often move around an average value making their movements less noticeable.
Quandl is a data management company that has access to Economic as well as Alternative data. They
sell to Hedge funds, Discretionary Funds, Fintech Companies, Corporations, Sell-Side Firms, and
Academics. The data used from Quandl was the University of Michigan Consumer Survey Index of
Consumer Sentiment, AAII investor Sentiment Data, Manufacturing Composite Index, and the NonManufacturing Index.
5.0 Results
5.1 Efficient Market Hypothesis
To test the Efficient Market Hypothesis(EMH), a commonly used test is called the Vratio test. This test
takes the data and has an output value of h, p, stat, cvalue, and ratio. The essential values are h, which is
just a comparison of the pvalue calculated vs. the default value of alpha, which is 0.05. If the value is
more significant than 0.05, the dataset is similar to that of a random walk; however, it is not random if
the p-value is less than 0.05. Looking at Table 7 in the Appendix, it can be seen that most stocks are not
random. The randomness means that most of the stocks are not market efficient, which is not shocking.
When looking at papers, it is possible to find papers that have results that are market efficient. Over ten
years, it is reasonable for this calculation to report back that the market is inefficient. This is because if
there is any herding, investor overconfidence, and loss aversion during the ten years, the dataset will no
23 | P a g e
longer be random. Researchers had argued that this is the weakness of the efficient market hypothesis
over long time periods that it did not accurately detect the times when the market was efficient and not
efficient.
The Ljung–Box test is a test that tests the overall randomness based on several lags. Using Matlab, the
number of lags by default is 20. A lag of 20 would mean for the data used that it will check 20 days of
adjusted close prices. Looking at Table 8 in the Appendix, it can be seen in the data that the data has
significant ARCH effects as well as serial correlations to each other. These results would be expected
because of the vratio test that was conducted; it is known that most of the stocks do not move
randomly, which means they can be predicted. The serial correlations are also expected because if a
stock is doing well today, it is highly likely that the stock will do well tomorrow. The same can also be
said if a stock is not doing well today, it will not do well tomorrow. In the literature, this phenomenon is
described as stocks having memory.
The One-sample Kolmogorov-Smirnov test (KStest), as well as Anderson–Darling test (ADtest), can be
used to check for normality of the data. Both tests take the data and project it onto a normal
distribution and output a p-value and an h value. The h value is 0 or 1, which is either the acceptance or
the rejection of the null hypothesis of normal distribution. The h value is determined by comparing the
p-value to the default alpha value of 0.05. Looking at Table 9 and Table 10 in the Appendix, it can be
seen that all the stocks do not have a normal distribution. This result is not shocking compared to the
literature; it is known that stock prices are a Cauchy distribution. Industry uses log-returns of stocks to
make stock returns more normal, so statistical methods that can be used for risk analysis can be done.
To give more of a perspective of how far from the normal distribution prices can move in the stock
market, during the 1987 stock market crash, the S&P 500 dropped by over 23 percent. Professor Henry
T.C. Hu of the University of Texas calculated that this event would be a "25 standard deviation event,"
which meant "if the stock market never took a holiday from the day the earth was formed, such a
decline was still unlikely." (Burch, 2020)
The Adjusted-Dicky Fuller Test (ADF test) is to test the stationarity of a dataset. If a dataset is
unstationary, it is tough to model, and the dataset needs to be made stationary. When the ADF test is
done on stock prices, it can be seen in Table 9 in the Appendix that the prices are not stable. To make
any predictions such as Arima modeling, the dataset would need to be made stationary. If the log
returns are taken of each stock, it can be seen in Table 10 in the Appendix that the log-returns are
stable.
Looking at the data, it can be seen that the stock market is inefficient and could be modeled through
Arima models. Looking at very long periods is that it gives false confidence to a trader that the market is
predictable at all times. Looking at smaller periods at different parts of the data gives different results of
how efficient the market is. Most times, the markets are efficient, but at times the market is not
efficient. The Adaptive Market Hypothesis helps put these two contradictory ideas together.
5.2 Adaptive Market Hypothesis
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The Adaptive Market Hypothesis(AMH) has been used to explain inefficiencies in the market, such as
cycles, herding, bubbles, crashes, trends, and other phenomena in the financial markets. Andrew Lo’s
first paper looked at the serial correlations between periods using a 60-month rolling window using the
S&P500. The figure below shows his graph's recreation to include more current dates and a 60-day
rolling window. It can be seen that the market cycles with the market being more inefficient when the
autocorrelations are the farthest from 0 and more efficient the closer the autocorrelations are to 0.
Figure 10-Serial Correlations and Adjusted Close Price of S&P500 Over Time
A paper done by Amini in 2010 described how positive autocorrelation often means that the next day’s
price will change similarly to yesterday’s price. This leads to a tendency of random disturbances to spill
over from one time period to the next. Negative autocorrelation means that today’s price will move in
the opposite direction of yesterday’s price. This makes a situation where there are tendencies for
successive disturbances to alter sign over time, a contrarian effect. If there is no autocorrelation, then
the prices have no relationship, hence suggesting an efficient market.
Another method that researchers have tried to use is the Vratio test. This was done on all 90 stocks, and
the figure below is a representative sample of the other stocks. The stock below is of BAC with a 60-day
rolling window. When the pValue drops to the 0.05 and lower level. This means that the returns are not
random, and they are predictable. However, it can be seen that the vast majority of the time that the
stock prices move with a degree of randomness, making the returns harder to predict.
25 | P a g e
Figure 11-BAC Vratio pValues over time
The stocks have varying efficiencies over time, with when the p-value is 1 means that the market or
stock is perfectly efficient at that time. From the figure above, it can be seen that the market is rarely
ever perfectly efficient, which supports the Grossman-Stiglitz Paradox, which argues perfectly
informationally efficient markets are an impossibility since if prices perfectly reflected available
information, there is no profit to gathering information, in which case there would be little reason to
trade. Markets would eventually collapse (Lo,2007). The stocks appear to not be predictable around 94
percent of the time. The other stock efficiency percent can be seen in Table 11 in the Appendix.
The runs test is a test that tests the randomness of a dataset. The figure below is of the BAC stock over
time. It can be seen that most of the time, the p values are larger than 0.05. This means that the returns
of the stock are random.
26 | P a g e
Figure 12- Run test for BAC over time
Looking at all the other stocks, it appears that their returns are random most of the time. Looking at
Table 13 in the Appendix, it can be seen that returns of a stock move randomly, on average, 96 percent.
The inefficient periods violate the EMH suggesting that markets are not weak market efficient when
they fluctuate between efficient and inefficient. It can be argued that the high efficiencies still prove the
EHM still holds most of the time. These high efficiencies are seen in yearly, monthly, and sometimes
daily frequency data. However, during most daily frequency data and intraday frequency, research has
shown that the market is less efficient, allowing people to arbitrage trade due to these inefficiencies.
A drawback to the Adaptive market hypothesis is that few tests are recognized to measure it accurately.
According to the literature, there are two main ways of measuring the ADH: the correlations between
periods and the Vratio test. From the tests, it can be seen that returns from a one-time frame to the
next are autocorrelated, but looking at the returns over time, the returns appear to be relatively random
most of the time. The autocorrelations and the lack of perfect randomness allow investors to create
alpha trading strategies (Smith,2017).
A significant difference between the statistical tests covered above in comparison to machine learning is
that the machine learning algorithms can use non-linear equations. This allows for more possibilities to
try and find trading strategies that could be more profitable than B&H strategies.
5.3 Machine Learning
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Since it can be seen that the market is not efficient, it may be possible to try and make money using
machine learning algorithms. In this thesis, various classification and regression machine learning
algorithms were applied to publicly available data. Machine learning algorithms used are listed below:
The machine learning algorithms had were tested for Accuracy, Precision, Recall, and F-Score on the
technical analysis data only. It can be seen in Error! Reference source not found. in the Appendix that
the Accuracies of the machine learning algorithms are around 50 percent, with the precision, recall, and
F-score averages were 48,48 and 41 percent, respectively. When adding the economic indicators to the
technical analysis data, accuracy, precisions, recalls, and F-scores had no significant changes. This can be
seen in Error! Reference source not found. of the Appendix. This outcome is expected because the
economic data is usually collected monthly, and their values are already included in the prices unless
something happens in the unforeseen market.
The figure below results from conducting a two-sided T-test on the accuracy of the machine learning
algorithms to a random walk. If the values below are greater than 1.96, then the accuracies would be
statistically significant over a random walk. Looking below, it can be seen that none of the machine
learning algorithms are able to predict the movements of the market better than a random walk.
Table 5-T-Test of Machine Learning Algorithms to Random Walk
T testing
LR
LR_L2
LDA
KNN
GB
ABC
RF
XGB
0.475167 0.474114 0.241046 -0.03247 0.036973 -0.00101 0.038182 0.127863
SVM
SVMP
0.36744 0.135689
SVMH
mlp
0.14557 0.098083
This shows that the publicly available data used does not help to predict the market. Due to the
algorithms only trying to predict one time period ahead vs multiple days ahead caused the accuracy to
be low. It is also possible that alternative data would help increase the accuracies of machine learning
algorithms. In the literature, there are machine learning algorithms that have higher accuracy. This is
often due to the lagging of specific indicators that researchers may think are leading indicators. This
increases the accuracy to a point where it would be statistically significant and would make an academic
or trader go to the next step of backtesting.
5.4 Backtest
Once the highest accuracy of machine learning is chosen from above, they are usually backtested
against previous data to see how they would perform in real life. Similar to machine learning techniques,
they have a training set and a test set. In this thesis, all the machine learning algorithms will be
compared to see which ones perform better. The backtest starts from December 2017 to June 3rd, 2020.
There are various types of metrics that are used in the backtest. This paper's metrics are: Cumulative
Return, Sharpe Ratio, Max Drawdown, and Sortino Ratio.
The figure below shows BAC with different machine learning algorithms applied to the stock trying to
make money. It can be seen that the cumulative value is very volatile over time within this instance,
KNN being the most profitable.
28 | P a g e
Figure 13-Portfolio Value Over Time
The highly volatile returns are not shocking when seen from the previous section, where the accuracies are not statistically more significant than
a random walk.
29 | P a g e
Table 6-Backtest Results of Machine Learning Algorithms
ABC
GB
KNN
LDA
LR
LR_L2
MLP
RF
SVM
Algorithm
Average of
Cumulative
Value
$9,466.69
$10,555.56
$10,370.09
$10,288.16
$9,846.25
$9,637.24
$9,805.74
$10,335.71
$10,563.31
SVMH
$10,214.58
SVMP
$9,434.89
Machine
Learning
Algorithm
Buy and Hold
Average
Cumulative
Value
$12,610.22
$12,610.22
$12,610.22
$12,610.22
$12,610.22
$12,610.22
$12,610.22
$12,610.22
$12,610.22
$12,610.22
$12,610.22
Average of
Algorithm
Sharpe
Average of
B&H Sharpe
Average of
algo
drawdown
Average of
B&H
drawdown
Average of
Sortino ratio
1.06
1.12
1.08
1.11
1.08
1.07
1.04
1.11
1.09
0.59
0.59
0.59
0.59
0.59
0.59
0.59
0.59
0.59
$3,024.74
$3,252.91
$3,250.24
$3,289.85
$3,150.28
$3,201.79
$3,275.83
$3,088.38
$3,489.61
$5,147.05
$5,147.05
$5,147.05
$5,147.05
$5,147.05
$5,147.05
$5,147.05
$5,147.05
$5,147.05
-0.15
-0.04
-0.09
-0.03
-0.1
-0.12
-0.02
-0.03
-0.07
1.11
0.59
$3,113.84
$5,147.05
0.03
1.07
0.59
$2,996.40
$5,147.05
0
30 | P a g e
It was seen in the Error! Reference source not found. that the max drawdown was lower than that of a
B&H strategy, and that the Sharpe ratios are over 1. It can also be seen that the Sortino ratios were
poor. However, this observation has been seen in the literature where the max drawdown is lower than
a B&H strategy. However, because the backtest does not include transaction costs and the algorithms
closing their positions at the end of each day. The transaction costs in a real-world situation would
destroy most, if not all, the algorithms' profits.
6.0 Conclusion
It can be seen that the EMH does not hold over long periods. If the time period selected has an extended
period of time, the market saw behavioral bias such as herding, confirmation, and confidence that the
EHM hypothesis testing will claim that the market is inefficient. Using the AMH, it was seen that the
stocks have periods where the market is efficient and inefficient and is seen cyclically. This is similar to
other observations in the literature review. Applying machine learning algorithms to the stock market, it
was seen that GB and SVM were very successful on average returning profitable portfolios. However,
when doing a t-test against a random walk, none of the algorithms were statistically better than a
random walk. It was seen that the algorithms on average had better Sharpe ratios, and lower
drawdowns, suggesting that the algorithms though not more profitable, were better at managing risk.
However, due to the algorithms not being statistically better than the random walk, it may be that they
were better with managing risk during these backtesting periods, but in other backtests, they would not.
Also, due to the algorithms only predicting one time period into the future and closing their positions
every day, there is a high likelihood that none of the algorithms would be profitable in the real world.
The Sortino ratios on average were low, suggesting that the downward movements standard deviations
were high.
31 | P a g e
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Performance, and the Bootstrap. Journal of Finance. 54, (1997): 1647-1691.
34 | P a g e
Trippi, R. R., DeSieno, D., 1992. Trading equity index futures with a neural network. Journal of
Portfolio Management 19, 27–27.
Tsai, C.-F., Lin, Y.-C., Yen, D. C., Chen, Y.-M., 2011. Predicting stock returns by classifier
ensembles. Applied Soft Computing 11 (2), 2452–2459.
Vapnik, V., 1998. Statistical learning theory. John wiley&sons. Inc., New York.
Variance ratio test for random walk - MATLAB vratiotest. (2020). Retrieved October 22nd 2020, from
https://www.mathworks.com/help/econ/vratiotest.html
Weng, B., Martinez, W., Tsai, Y., Li, C., Lu, L., Barth, J. R., & Megahed, F. M. (2018).
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Systems 7 (2), 272–290.
35 | P a g e
8.0 Appendix
8.1 Market Efficiency Tables
Table 7-Vratio Test over 10 years
Ticker
AAPL
ABMD
ADP
ADSK
AMAT
AMD
AMT
ANSS
ANTM
APTV
ARE
ATO
AVGO
AZO
BAC
BLL
BWA
C
CBRE
CERN
CMCSA
CMS
COO
COST
CSCO
CTXS
DISH
DRE
EBAY
ESS
EXPE
F
FB
FCX
FRT
h pValue
stat
cValue ratio
h analysis
1 0.037332
-2.0821
1.96 0.81541 Not a Random walk
0
0.17464
1.3575
1.96 1.0523 Random walk
1 0.012729
-2.4912
1.96 0.77985 Not a Random walk
1 0.036898
-2.0869
1.96 0.87603 Not a Random walk
1 0.013206
-2.4782
1.96 0.82069 Not a Random walk
0
0.05086
-1.9527
1.96 0.87196 Random walk
0
0.10447
-1.6236
1.96 0.80431 Random walk
1
0.02372
-2.2616
1.96 0.7844 Not a Random walk
1 0.035662
-2.1008
1.96 0.86871 Not a Random walk
0
0.77561 -0.28505
1.96 0.98418 Random walk
0
0.10072
-1.6414
1.96 0.83874 Random walk
0
0.18151
-1.3361
1.96 0.88114 Random walk
1
0.02554
-2.2331
1.96 0.87784 Not a Random walk
1 0.047147
-1.985
1.96 0.88308 Not a Random walk
0
0.16637
-1.384
1.96 0.91375 Random walk
0
0.54783 -0.60102
1.96 0.9616 Random walk
0
0.2401
1.1747
1.96 1.0271 Random walk
0
0.31549
-1.0038
1.96 0.94561 Random walk
0
0.85091 0.18796
1.96
1.01 Random walk
0
0.20173
-1.2766
1.96 0.96872 Random walk
1 0.004867
-2.8157
1.96 0.84156 Not a Random walk
0
0.3708 -0.89497
1.96 0.90546 Random walk
1 0.0073237
-2.6818
1.96 0.84076 Not a Random walk
1 0.043134
-2.0224
1.96 0.84531 Not a Random walk
1 0.020089
-2.3247
1.96 0.87111 Not a Random walk
0
0.11424
-1.5794
1.96 0.93922 Random walk
0
0.86393 0.17137
1.96
1.005 Random walk
0
0.1175
-1.5654
1.96 0.81585 Random walk
0
0.5184 -0.64582
1.96 0.9797 Random walk
0
0.1033
-1.629
1.96 0.89992 Random walk
0
0.51345 0.65348
1.96 1.0181 Random walk
0
0.22852
1.2042
1.96 1.0295 Random walk
0
0.10388
-1.6263
1.96 0.92501 Random walk
0
0.145
1.4574
1.96 1.0377 Random walk
0 0.070567
-1.8083
1.96 0.91018 Random walk
36 | P a g e
GE
GM
GWW
HII
HOG
HOLX
HPE
HPQ
IEX
INTC
IPGP
IT
JKHY
JPM
KO
LNC
LRCX
MAA
MCHP
MKTX
MMC
MS
MSFT
MSI
MTD
MU
MXIM
NFLX
NLSN
NVR
ORCL
PAYX
PFE
RE
RF
ROP
SIVB
SNA
STE
T
TDG
TFX
TRV
0
0.37771 0.88213
0
0.68702
0.4029
0
0.16242
-1.397
0
0.80137 0.25157
0
0.57677 -0.55811
0
0.69531 -0.39167
0
0.74417 -0.32633
0
0.11695
-1.5677
0 0.081724
-1.7408
1 0.010162
-2.5703
0
0.77367 -0.28757
0
0.6938 -0.39371
0
0.43071 -0.78797
0 0.070608
-1.808
0
0.41168 -0.82094
0
0.85778
-0.1792
1
0.01812
-2.3632
0
0.18723
-1.3188
0
0.17262
-1.3638
0
0.45701 -0.74378
1 0.027366
-2.2063
0 0.085189
-1.7213
1 0.0083615
-2.6371
0
0.14833
-1.4455
0
0.27186
-1.0988
0
0.26099
-1.1241
1 0.027482
-2.2046
0
0.15715
-1.4147
0
0.1971
-1.2899
0
0.84636 0.19376
1 0.031558
-2.15
1 0.032845
-2.134
0
0.19554
-1.2944
1 0.024886
-2.2432
0
0.52636 -0.63357
1 0.0077731
-2.6618
0
0.48079 -0.70503
0
0.31557
-1.0036
0 0.062547
-1.8624
0
0.18405
-1.3284
0
0.92587 0.093048
0
0.14803
-1.4465
1 0.026462
-2.2194
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.0257
1.0128
0.93914
1.0111
0.98032
0.98717
0.98593
0.95214
0.87414
0.74173
0.98868
0.98611
0.94799
0.83601
0.95347
0.99213
0.82914
0.88585
0.92217
0.94252
0.8461
0.90007
0.66597
0.87955
0.93291
0.95324
0.86677
0.93757
0.97053
1.013
0.84456
0.82462
0.93719
0.85601
0.97154
0.77467
0.96989
0.96083
0.87397
0.92132
1.0087
0.89451
0.81587
Random walk
Random walk
Random walk
Random walk
Random walk
Random walk
Random walk
Random walk
Random walk
Not a Random walk
Random walk
Random walk
Random walk
Random walk
Random walk
Random walk
Not a Random walk
Random walk
Random walk
Random walk
Not a Random walk
Random walk
Not a Random walk
Random walk
Random walk
Random walk
Not a Random walk
Random walk
Random walk
Random walk
Not a Random walk
Not a Random walk
Random walk
Not a Random walk
Random walk
Not a Random walk
Random walk
Random walk
Random walk
Random walk
Random walk
Random walk
Not a Random walk
37 | P a g e
TT
TWTR
TXT
VZ
WAT
WFC
WLTW
WM
WST
WYNN
XOM
ZBRA
0
0.83529
0
0.76075
0
0.79968
1 0.049842
1 0.0090252
0
0.18661
1 0.0038686
1
0.02177
0
0.31158
0
0.56085
0
0.33176
1 0.030756
0.20793
-0.30449
-0.25376
-1.9613
-2.6111
-1.3207
-2.8887
-2.2944
-1.0119
0.58158
-0.97058
-2.1602
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.96
1.0049
0.98568
0.99081
0.9143
0.88669
0.93905
0.85808
0.79273
0.93456
1.0174
0.97223
0.87261
Random walk
Random walk
Random walk
Not a Random walk
Not a Random walk
Random walk
Not a Random walk
Not a Random walk
Random walk
Random walk
Random walk
Not a Random walk
Table 8- LQB test
Ticker
AAPL
ABMD
ADP
ADSK
AMAT
AMD
AMT
ANSS
ANTM
APTV
ARE
ATO
AVGO
AZO
BAC
BLL
BWA
C
CBRE
CERN
CMCSA
CMS
COO
COST
CSCO
CTXS
h
pValue
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
stat
cValue
49705
31.41
51491
31.41
51379
31.41
50102
31.41
50757
31.41
48718
31.41
50363
31.41
49927
31.41
51024
31.41
40539
31.41
50888
31.41
51515
31.41
51208
31.41
50456
31.41
51217
31.41
50678
31.41
48564
31.41
49545
31.41
50275
31.41
50340
31.41
51263
31.41
51284
31.41
51000
31.41
50840
31.41
51221
31.41
48873
31.41
h analysis
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
p analysis
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
38 | P a g e
DISH
DRE
EBAY
ESS
EXPE
F
FB
FCX
FRT
GE
GM
GWW
HII
HOG
HOLX
HPE
HPQ
IEX
INTC
IPGP
IT
JKHY
JPM
KO
LNC
LRCX
MAA
MCHP
MKTX
MMC
MS
MSFT
MSI
MTD
MU
MXIM
NFLX
NLSN
NVR
ORCL
PAYX
PFE
RE
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
50726
50957
49999
50932
50691
45909
38823
50368
50393
50688
44605
49635
45406
48578
49990
20195
50644
51207
50083
50496
50998
50608
51435
50615
50537
50307
50623
50461
49428
50957
50726
50202
51262
51047
50575
50781
50562
44575
51172
50110
51382
51116
51419
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
39 | P a g e
RF
ROP
SIVB
SNA
STE
T
TDG
TFX
TRV
TT
TWTR
TXT
VZ
WAT
WFC
WLTW
WM
WST
WYNN
XOM
ZBRA
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
50896
50829
50869
51278
50550
50485
50799
50920
51412
37373
29618
50399
50719
51070
50543
50045
51494
49326
48748
47277
50038
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
31.41
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
Significant ARCH effects
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
serial correlation
Table 9-KStest
Ticker
AAPL
ABMD
ADP
ADSK
AMAT
AMD
AMT
ANSS
ANTM
APTV
ARE
ATO
AVGO
AZO
BAC
BLL
BWA
h
p
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
k
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
c
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
h analysis
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
40 | P a g e
C
CBRE
CERN
CMCSA
CMS
COO
COST
CSCO
CTXS
DISH
DRE
EBAY
ESS
EXPE
F
FB
FCX
FRT
GE
GM
GWW
HII
HOG
HOLX
HPE
HPQ
IEX
INTC
IPGP
IT
JKHY
JPM
KO
LNC
LRCX
MAA
MCHP
MKTX
MMC
MS
MSFT
MSI
MTD
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
41 | P a g e
MU
MXIM
NFLX
NLSN
NVR
ORCL
PAYX
PFE
RE
RF
ROP
SIVB
SNA
STE
T
TDG
TFX
TRV
TT
TWTR
TXT
VZ
WAT
WFC
WLTW
WM
WST
WYNN
XOM
ZBRA
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
2.20E-36
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.12546
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
0.026463
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Not a normal distribution
Table 10-TS-ADF test
Ticker
AAPL
ABMD
ADP
ADSK
AMAT
AMD
AMT
h
pvalue
stat
cValue
0 0.93569 -1.0449 -3.4141
0 0.63069
-1.923 -3.4141
0 0.062208 -3.3286 -3.4141
0 0.69756 -1.7879 -3.4141
0
0.164 -2.8972 -3.4141
0 0.98563 0.44793 -3.4141
0 0.84093 -1.4637 -3.4141
h analysis
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
42 | P a g e
ANSS
ANTM
APTV
ARE
ATO
AVGO
AZO
BAC
BLL
BWA
C
CBRE
CERN
CMCSA
CMS
COO
COST
CSCO
CTXS
DISH
DRE
EBAY
ESS
EXPE
F
FB
FCX
FRT
GE
GM
GWW
HII
HOG
HOLX
HPE
HPQ
IEX
INTC
IPGP
IT
JKHY
JPM
KO
0
0
0
0
1
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
1
1
1
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
1
0
0
1
0.92101
0.08315
0.11753
0.08866
0.005933
0.05863
0.21407
0.26186
0.48834
0.49519
0.19213
0.058049
0.21422
0.002695
0.004247
0.009272
0.5317
0.1609
0.71625
0.90325
0.002712
0.019057
0.009219
0.56326
0.42423
0.007337
0.24965
0.88243
0.97355
0.14182
0.14436
0.3728
0.69457
0.015639
0.86627
0.29151
0.14261
0.03383
0.4098
0.033532
0.41402
0.11596
0.002462
-1.1362
-3.2084
-3.0577
-3.181
-4.1367
-3.3524
-2.7646
-2.668
-2.2105
-2.1967
-2.8151
-3.3563
-2.7643
-4.4268
-4.2577
-3.9977
-2.1229
-2.9068
-1.7501
-1.2283
-4.4251
-3.7615
-3.9999
-2.0592
-2.34
-4.079
-2.6927
-1.3171
-0.677
-2.9691
-2.9602
-2.4439
-1.7939
-3.8267
-1.3784
-2.6081
-2.9663
-3.5606
-2.3692
-3.5639
-2.3607
-3.064
-4.4523
-3.4141
-3.4141
-3.4142
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4143
-3.4141
-3.4141
-3.4141
-3.4142
-3.4141
-3.4142
-3.4141
-3.4141
-3.4145
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
not a random process
not a random process
not a random process
not a random process
random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
random process
random process
random process
not a random process
not a random process
not a random process
not a random process
random process
random process
random process
not a random process
not a random process
random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
random process
not a random process
not a random process
not a random process
random process
not a random process
random process
not a random process
not a random process
random process
43 | P a g e
LNC
LRCX
MAA
MCHP
MKTX
MMC
MS
0 0.80275
0 0.32579
1 0.018999
1 0.03882
0
0.999
1 0.017228
0 0.12162
MSFT
MSI
MTD
MU
MXIM
NFLX
NLSN
NVR
ORCL
PAYX
PFE
RE
RF
ROP
SIVB
SNA
STE
T
TDG
TFX
TRV
TT
TWTR
TXT
VZ
WAT
WFC
WLTW
WM
WST
WYNN
XOM
ZBRA
0
0
0
0
1
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0.96591
0.37309
0.16781
0.2371
0.005888
0.65907
0.90997
0.11453
0.001
0.016336
0.015908
0.068352
0.48044
0.23071
0.29422
0.59368
0.70687
0.093939
0.26121
0.24716
0.14997
0.65668
0.47381
0.84875
0.007584
0.016038
0.9792
0.029384
0.33016
0.99892
0.54738
0.78858
0.6908
-1.5734
-2.5389
-3.7626
-3.5102
0.57896
-3.7964
-3.0412
0.77826
-2.4433
-2.8854
-2.718
-4.1386
-1.8656
-1.1953
-3.0698
-5.2271
-3.8134
-3.8215
-3.2901
-2.2265
-2.731
-2.6027
-1.9977
-1.7691
-3.1559
-2.6694
-2.6977
-2.9407
-1.8704
-2.2398
-1.4397
-4.068
-3.8191
-0.5853
-3.6111
-2.5301
0.38369
-2.0913
-1.604
-1.8016
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
not a random process
not a random process
random process
random process
not a random process
random process
not a random process
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4142
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4142
-3.4143
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
-3.4141
not a random process
not a random process
not a random process
not a random process
random process
not a random process
not a random process
not a random process
random process
random process
random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
not a random process
random process
random process
not a random process
random process
not a random process
not a random process
not a random process
not a random process
not a random process
44 | P a g e
Table 11-Vratio 60 day rolling window of stocks
Tickers
BAC
AAPL
GE
F
MSFT
AMD
INTC
CSCO
FB
PFE
MU
C
T
HPQ
CMCSA
WFC
FCX
JPM
TWTR
NFLX
EBAY
ORCL
RF
VZ
XOM
KO
MS
HPE
AMAT
GM
CBRE
LNC
AVGO
HOLX
MXIM
ADSK
DRE
DISH
TXT
CTXS
Efficiency of
the stock
96%
96%
94%
95%
95%
94%
91%
95%
95%
95%
97%
93%
95%
94%
96%
93%
95%
98%
98%
94%
93%
92%
93%
94%
96%
92%
93%
97%
97%
95%
94%
95%
98%
92%
95%
96%
98%
96%
93%
94%
45 | P a g e
NLSN
WYNN
CMS
EXPE
BLL
COST
LRCX
MCHP
AMT
ADP
WM
PAYX
ANTM
APTV
TRV
MMC
HOG
BWA
CERN
MSI
WAT
GWW
ARE
ATO
MAA
IT
WLTW
ROP
ABMD
SIVB
STE
ANSS
FRT
TDG
COO
SNA
IPGP
JKHY
IEX
ZBRA
TT
ESS
AZO
93%
97%
91%
93%
93%
96%
93%
91%
95%
95%
96%
96%
94%
94%
91%
94%
96%
95%
91%
94%
95%
94%
94%
97%
91%
94%
96%
95%
93%
94%
91%
95%
93%
95%
95%
94%
93%
90%
68%
96%
93%
93%
94%
46 | P a g e
HII
RE
WST
TFX
MKTX
MTD
NVR
Average of all the
stocks
94%
96%
90%
95%
94%
96%
95%
94%
Table 12- Ljung-Box Q-test for all stocks
Tickers
BAC
AAPL
GE
F
MSFT
AMD
INTC
CSCO
FB
PFE
MU
C
T
HPQ
CMCSA
WFC
FCX
JPM
TWTR
NFLX
EBAY
ORCL
RF
VZ
XOM
KO
MS
HPE
Percent of time not
autocorrelated
93%
97%
90%
91%
88%
92%
85%
96%
89%
91%
89%
91%
91%
91%
89%
88%
90%
89%
96%
93%
89%
87%
92%
90%
95%
94%
90%
89%
47 | P a g e
AMAT
GM
CBRE
LNC
AVGO
HOLX
MXIM
ADSK
DRE
DISH
TXT
CTXS
NLSN
WYNN
CMS
EXPE
BLL
COST
LRCX
MCHP
AMT
ADP
WM
PAYX
ANTM
APTV
TRV
MMC
HOG
BWA
CERN
MSI
WAT
GWW
ARE
ATO
MAA
IT
WLTW
ROP
ABMD
SIVB
STE
96%
93%
92%
94%
92%
90%
89%
89%
93%
93%
94%
91%
90%
97%
95%
88%
92%
94%
93%
89%
92%
92%
95%
88%
91%
92%
90%
91%
92%
93%
92%
91%
88%
91%
91%
96%
89%
91%
93%
90%
90%
89%
88%
48 | P a g e
ANSS
FRT
TDG
COO
SNA
IPGP
JKHY
IEX
ZBRA
TT
ESS
AZO
HII
RE
WST
TFX
MKTX
MTD
NVR
Average of all the
stocks
92%
93%
90%
90%
93%
97%
89%
41%
88%
91%
91%
93%
90%
93%
91%
90%
91%
95%
91%
91%
Table 13-Run test for all stocks
Tickers
BAC
AAPL
GE
F
MSFT
AMD
INTC
CSCO
FB
PFE
MU
C
T
HPQ
CMCSA
WFC
Percent of time data is
random
97%
97%
94%
97%
95%
95%
97%
93%
94%
96%
96%
94%
94%
94%
96%
94%
49 | P a g e
FCX
JPM
TWTR
NFLX
EBAY
ORCL
RF
VZ
XOM
KO
MS
HPE
AMAT
GM
CBRE
LNC
AVGO
HOLX
MXIM
ADSK
DRE
DISH
TXT
CTXS
NLSN
WYNN
CMS
EXPE
BLL
COST
LRCX
MCHP
AMT
ADP
WM
PAYX
ANTM
APTV
TRV
MMC
HOG
BWA
CERN
99%
97%
99%
97%
97%
95%
98%
99%
97%
93%
94%
94%
98%
96%
97%
98%
99%
96%
95%
95%
99%
99%
95%
94%
96%
98%
96%
96%
97%
97%
96%
96%
96%
97%
93%
94%
97%
99%
92%
92%
94%
92%
97%
50 | P a g e
MSI
WAT
GWW
ARE
ATO
MAA
IT
WLTW
ROP
ABMD
SIVB
STE
ANSS
FRT
TDG
COO
SNA
IPGP
JKHY
IEX
ZBRA
TT
ESS
AZO
HII
RE
WST
TFX
MKTX
MTD
NVR
Average of all the
stocks
94%
98%
94%
97%
99%
98%
92%
96%
93%
96%
97%
95%
96%
97%
94%
98%
95%
95%
94%
82%
95%
94%
92%
92%
95%
96%
97%
98%
97%
97%
93%
96%
51 | P a g e
52 | P a g e
8.2 Machine learning results
Table 14-Machine Learning Algorithms using Technical Analysis
count
mean
std
min
25%
50%
75%
max
count
mean
std
min
25%
50%
75%
max
LR
82
51%
3%
47%
50%
51%
52%
73%
LR
82
49%
9%
17%
48%
52%
54%
64%
LR
LR_L2
82
51%
3%
47%
50%
51%
52%
73%
LR_L2
82
49%
9%
17%
48%
52%
54%
64%
LR_L2
LDA
82
51%
3%
48%
50%
50%
51%
71%
LDA
82
53%
5%
19%
52%
54%
56%
62%
LDA
KNN
82
50%
3%
48%
49%
50%
50%
71%
KNN
82
54%
6%
20%
51%
54%
54%
54%
KNN
GB
82
50%
2%
46%
49%
50%
51%
69%
Accuracy
ABC
82
50%
3%
47%
49%
50%
51%
72%
RF
82
50%
3%
47%
49%
50%
51%
73%
XGB
82
50%
3%
48%
49%
50%
51%
71%
SVM
82
51%
3%
46%
50%
51%
52%
78%
SVMP
82
50%
3%
48%
49%
50%
51%
75%
SVMH
82
50%
2%
46%
49%
50%
51%
57%
mlp
82
50%
3%
47%
49%
50%
51%
75%
GB
82
55%
5%
24%
52%
55%
57%
66%
Precision
ABC
82
54%
6%
24%
51%
54%
57%
68%
RF
82
55%
5%
25%
53%
56%
58%
65%
XGB
82
53%
4%
26%
52%
54%
55%
61%
SVM
82
45%
11%
9%
37%
48%
53%
60%
SVMP
82
30%
8%
10%
26%
31%
35%
47%
SVMH
82
32%
8%
11%
26%
33%
37%
49%
mlp
82
39%
8%
16%
35%
39%
46%
55%
GB
Recall
ABC
RF
XGB
SVM
SVMP
SVMH
mlp
53 | P a g e
count
mean
std
min
25%
50%
75%
max
count
mean
std
min
25%
50%
75%
max
82
62%
22%
1%
42%
67%
80%
99%
LR
82
49%
15%
2%
38%
53%
59%
68%
82
62%
22%
1%
43%
67%
80%
99%
LR_L2
82
49%
15%
2%
39%
53%
59%
68%
82
49%
10%
22%
43%
49%
55%
69%
LDA
82
46%
7%
18%
42%
45%
50%
58%
82
38%
11%
14%
30%
38%
46%
64%
KNN
82
37%
8%
19%
31%
37%
44%
54%
82
37%
8%
21%
31%
37%
44%
62%
82
37%
9%
18%
30%
36%
43%
60%
82
36%
8%
18%
30%
35%
43%
58%
82
41%
6%
30%
37%
41%
46%
57%
82
62%
23%
4%
45%
63%
81%
100%
82
53%
15%
10%
41%
55%
61%
90%
82
52%
15%
20%
40%
49%
60%
90%
82
51%
13%
20%
43%
51%
58%
85%
GB
82
39%
6%
25%
35%
39%
42%
55%
F-Score
ABC
82
37%
6%
24%
33%
37%
42%
54%
RF
82
39%
6%
27%
35%
38%
42%
52%
XGB
82
43%
4%
30%
41%
43%
47%
52%
SVM
82
47%
15%
5%
38%
48%
58%
71%
SVMP
82
38%
10%
14%
31%
37%
45%
56%
SVMH
82
36%
9%
15%
28%
35%
43%
53%
mlp
82
39%
9%
13%
34%
41%
45%
61%
54 | P a g e
Economics and technical analysis
Table 15-Machine Learning Algorithms using Technical Analysis and Economic Indicators
count
mean
std
min
25%
50%
75%
max
count
mean
std
min
25%
50%
75%
max
count
LR
82
51%
3%
47%
50%
51%
52%
73%
LR
82
49%
9%
17%
48%
52%
54%
62%
LR
82
LR_L2
82
51%
3%
47%
50%
51%
52%
73%
LR_L2
82
49%
9%
17%
48%
52%
54%
62%
LR_L2
82
LDA
82
51%
3%
48%
50%
50%
51%
71%
LDA
82
53%
5%
19%
52%
54%
56%
62%
LDA
82
KNN
82
50%
3%
48%
49%
50%
50%
71%
KNN
82
54%
6%
20%
51%
54%
58%
68%
KNN
82
GB
82
50%
3%
47%
49%
50%
51%
70%
Accuracy
ABC
82
50%
3%
47%
49%
50%
51%
72%
RF
82
50%
3%
48%
49%
50%
51%
72%
XGB
82
50%
3%
48%
49%
50%
51%
71%
SVM
82
51%
3%
46%
50%
51%
52%
78%
SVMP
82
50%
1%
47%
49%
50%
51%
53%
SVMH
82
50%
2%
47%
49%
50%
51%
67%
mlp
82
50%
2%
48%
49%
50%
51%
63%
GB
82
55%
5%
24%
53%
55%
57%
64%
Precision
ABC
82
54%
6%
24%
51%
54%
57%
68%
RF
82
55%
5%
24%
53%
55%
57%
63%
XGB
82
53%
4%
26%
52%
54%
55%
61%
SVM
82
45%
11%
9%
37%
48%
53%
60%
SVMP
82
31%
9%
10%
26%
31%
36%
56%
SVMH
82
31%
8%
15%
25%
31%
37%
53%
mlp
82
40%
8%
18%
36%
41%
47%
54%
GB
82
Recall
ABC
82
RF
82
XGB
82
SVM
82
SVMP
82
SVMH
82
mlp
82
55 | P a g e
mean
std
min
25%
50%
75%
max
count
mean
std
min
25%
50%
75%
max
63%
22%
1%
41%
68%
80%
99%
LR
82
49%
15%
2%
38%
54%
59%
68%
63%
22%
1%
43%
68%
80%
99%
LR_L2
82
49%
15%
2%
38%
54%
59%
68%
49%
10%
22%
43%
49%
55%
69%
LDA
82
46%
7%
18%
42%
45%
49%
58%
38%
11%
14%
30%
38%
46%
64%
KNN
82
37%
8%
19%
31%
37%
44%
54%
37%
9%
20%
32%
37%
43%
61%
36%
9%
18%
30%
36%
42%
60%
36%
9%
22%
30%
35%
41%
58%
41%
6%
30%
37%
41%
46%
57%
62%
23%
4%
45%
63%
81%
100%
49%
16%
20%
38%
50%
62%
87%
51%
17%
12%
40%
50%
60%
90%
53%
13%
19%
44%
53%
62%
84%
GB
82
39%
6%
24%
35%
39%
43%
54%
F-Score
ABC
82
37%
6%
25%
33%
37%
42%
54%
RF
82
39%
6%
24%
34%
38%
43%
53%
XGB
82
43%
4%
30%
41%
43%
47%
52%
SVM
82
47%
15%
5%
38%
48%
58%
71%
SVMP
82
36%
10%
12%
28%
35%
42%
63%
SVMH
82
35%
9%
14%
28%
36%
41%
56%
mlp
82
40%
11%
16%
32%
42%
49%
59%
56 | P a g e