Prague University of Economics and Business Faculty of Economics
Field of Study: Economic Analysis
I S A RTIFICIAL I NTELLIGENCE B ETTER IN
F ORECASTING S TOCK M ARKET R ETURNS THAN C LASSICAL M ETHODS ?
Diploma Thesis
Author of the Thesis: Bc. Aleksandra Drašković Thesis Supervisor: Ing. Aleš Maršál, M.A., Ph.D.
Year: 2020
I hereby declare that this thesis has been composed solely by myself and that it has not been submitted, in whole or in part, in any previous application for a degree. Except where states otherwise by reference or acknowledgment, the work presented is entirely my own.
Aleksandra Drašković
Prague, December 18, 2020
Acknowledgements
Throughout the full process of conducting and writing this diploma thesis, I have received tremendous support. I would first like to thank my supervisor, Ing. Aleš Maršál, M.A., Ph.D., for his willingness to lead my thesis, his valuable inputs, advice, consultations, and expertise, and for the helpful and kind approach.
I would also like to thank Ing. Peter Trcka, Ph.D., for his valuable guidance with the empirical part of this thesis.
In addition, I would like to acknowledge and thank my family for unconditional support throughout all of my five years of studies, as well as while writing this diploma thesis.
Abstract
The thesis analyzes the issue of predictions of stock market returns. In particular, it concentrates on the discussion about which methods and models can predict yields in the foremost efficient manner. Apart from the initial introduction of AI and the investment theories, the study includes both an overview of statistical models (ARMA, ARIMA, ARCH, ARCH, NMA, TAR, and MSAR) and computationally intelligent models and approaches (such as artificial neural networks, support vector machines, fuzzy set theory, evolutionary algorithms, expert systems, singular spectrum analysis, naive Bayes, k-nearest neighbors, and hybrid models). The empirical part of the thesis aims to observe the issue of predicting stock returns while considering statistical and AI methods. It comprises of two sections. The first segment is dedicated to formulating and generating a systematic review. From the obtained results, it can be seen that about 73 % of all included studies assist in not rejecting the null hypothesis of the superior performance of models based on machine learning. The second subdivision is devoted to introducing own constructed models, namely ARIMA models and an LSTM model. Both types of models are initially introduced, then the models are formed, and subsequently, forecasts are generated. According to the results of the own construction of comparison models, the LSTM model, which belongs to the AI category, proved to be more efficient and accurate in its predicted future values. Therefore, the second part of the empirical research also does not reject the null hypothesis of the better performance of the AI methods.
Keywords: artificial intelligence, ANN, ARIMA, LSTM, stock market, forecast of financial time series, investment theories, systematic review, prediction of stock market returns
JEL classification: C53, E17, E27, E37, E47, G17
Abstrakt
Práce analyzuje problematiku predikce výnosů z akciových trhů. Zvláště se zabývá diskusí, jakými metodami a modely lze výnosy prognózovat nejlépe. Kromě počátečního představení umělé inteligence a investičních technologií, studie zahrnuje jak přehled statistických modelů (ARMA, ARIMA, ARCH, NMA, TAR, and MSAR), tak i výpočetně inteligentních modelů a metod (především pak přibližuje čtenáři metodu umělých neuronových sítí, metodu podpůrných vektorů, fuzzy teorii množiny, evoluční algoritmy, expertní systémy, analýzu singulárního spektra, naivní Bayesovský klasifikátor, algoritmus k-nejbližších sousedů a hybridní modely). Empirická část práce si klade za cíl sledovat problematiku predikce výnosů z akciového trhu při zohlednění užití statistických a AI metod. Empirický výzkum se skládá ze dvou částí. První segment je věnován formulaci a generování systematického přehledu. Ze získaných výsledků vyplývá, že přibližně 73 % všech zahrnutých studií napomohlo k nezamítnutí nulové hypotézy o lepším výkonu modelů založených na strojovém učení. Druhá část je věnována představení vlastních konstruovaných modelů, konkrétně modelů ARIMA a LSTM. Nejprve jsou představeny oba typy modelů, poté jsou vytvořeny modely a následně jsou generovány předpovědi. Podle výsledků vlastní konstrukce srovnaných modelů se model LSTM, který patří do kategorie AI, ukázal jako efektivnější a přesnější ve svých predikovaných budoucích hodnotách. Druhá část empirického výzkumu proto také nezamítá nulovou hypotézu o lepší výkonosti AI metod.
Klíčová slova: umělá inteligence, ANN, ARIMA, LSTM, akciový trh, předpověď finančních časových řad, investiční teorie, systematický přehled, predikce výnosů z akciových trhů
JEL klasifikace: C53, E17, E27, E37, E47, G17
Content
Introduction ... 1
1. Artificial Intelligence ... 4
1.1. AI in Finance ... 5
1.2. Types of AI ... 8
2. Theories and Models of Investment on Stock Returns ... 11
2.1. Random Walk Hypothesis ... 11
2.2. Efficient Market Hypothesis ... 12
2.3. Expected Utility Theory ... 13
2.4. Prospect Theory ... 14
2.5. Adaptive Market Hypothesis ... 15
2.6. Chaos Theory ... 16
2.7. Portfolio Theories ... 17
2.7.1. Markowitz Portfolio Selection (MPS) ... 17
2.7.2. Dow Theory ... 18
2.7.3. Behavioral Portfolio Theory (BPT) ... 18
2.7.4. Maslowian Portfolio Theory (MaPT) ... 19
2.7.5. Black-Litterman Model (BLM) ... 19
2.7.6. Dedicated Portfolio Theory (DPT) ... 20
2.7.7. Treynor-Black Model (TBM/TB) ... 20
2.7.8. Universal Portfolio Algorithm ... 20
2.7.9. Capital Asset Pricing Model (CAPM) ... 21
2.7.10. Arbitrage Pricing Theory (APT) ... 22
3. Forecasting Stock Market’s Returns ... 23
3.1. Statistical Methods ... 23
3.1.1. Linear models ... 24
3.1.1.1. Autoregressive Moving Average (ARMA) ... 24
3.1.1.2. Autoregressive Integrated Moving Average (ARIMA) ... 25
3.1.2. Nonlinear Models ... 27
3.1.2.1. Autoregressive Conditional Heteroskedasticity (ARCH) ... 27
3.1.2.2. Nonlinear Moving Average (NMA) ... 28
3.1.3. Regime-Switching Models ... 29
3.1.3.1. Threshold Autoregressive (TAR) ... 29
3.1.3.2. Markov-Switching Autoregressive (MSAR) ... 30
3.2. Computational Intelligence Methods ... 30
3.2.1. Artificial Neural Networks ... 34
3.2.1.1. Typology ... 37
3.2.1.2. Advantages of ANN ... 41
3.2.1.3. Disadvantages of ANN ... 41
3.2.2. Support Vector Machines (SVM) ... 42
3.2.3. Fuzzy Set Theory/Fuzzy Logic System (FLS) ... 43
3.2.4. Evolutionary Algorithms ... 43
3.2.5. Expert Systems (ES) ... 44
3.2.6. Singular Spectrum Analysis (SSA)... 44
3.2.7. Naive Bayes ... 45
3.2.8. k-Nearest Neighbors (KNN) ... 46
3.2.9. Hybrid Model (HM) ... 46
4. Empirical Research Regarding the Comparison of AI Methods and Statistical Methods ... 48
4.1. Hypothesis ... 48
4.2. Systematic Review ... 49
4.2.1. Model Specification ... 49
4.2.1.1. Benefits of Systematic Reviews and Meta-Analyses ... 51
4.2.1.2. Criticisms of Systematic Reviews and Meta-Analyses ... 51
4.2.1.3. Procedure ... 52
4.2.2. Related Research ... 52
4.2.3. Data ... 54
4.2.3.1. Included Studies ... 57
4.2.3.2. Performance Evaluation Indicators ... 62
4.2.4. Results ... 64
4.3. Construction of Own Models of Comparison ... 79
4.3.1. Data ... 79
4.3.1.1. The Training Subgroup of the Dataset ... 81
4.3.1.2. The Testing Subgroup of the Dataset ... 82
4.3.2. ARIMA ... 83
4.3.2.1. The Models ... 90
4.3.2.2. Summary ... 96
4.3.3. LSTM ... 97
4.3.3.1. The Model ... 98
4.3.4. Results of Comparison ... 100
Conclusion ... 102
References ... 106
Abbreviations ... 126
List of Tables ... 130
List of Figures ... 130
List of Equations ... 131
Appendices ... 133
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Introduction
“Some people call this artificial intelligence, but the reality is this technology will enhance us. So instead of artificial intelligence, I think we'll augment our intelligence.”
- Ginni Rometty
Artificial intelligence (AI) is a field of computer science that solves complicated tasks by adopting sophisticated algorithms. The capacity and complexity of artificial intelligence are constantly improving. The main goal of creating artificial intelligence is to bring it closer to the functioning of the human brain, especially by creating a memory, which would help to accelerate and enhance creations to solutions of various tricky problems and tasks, to endorse independent thinking, and encourage decision- making.
Artificial intelligence is significantly being employed in the financial market, especially while making forecasts of stock market profitability. For this reason, this diploma thesis focuses on addressing the question of whether the methods of artificial intelligence are better in forecasting returns of stock markets than the well-known "classical"
methods. Predictions of stocks, bonds, and values of indexes are essential for investors, as accurate forecasting may bring them high returns. Artificial intelligence thus makes it possible to apply complicated algorithms that have the potential to be more accurate in predictions. However, we are still dealing with the question of whether stock market returns are indeed predictable. Such kind of question remains one of the most important ones in the financial sector. For this reason, Chapter 2 further addresses and discusses investment theories, models, and hypotheses.
Although AI is in its beginning, it is increasingly becoming a sought-after area of interest, especially by investors, traders, banking institutions, and managers. Stephen Hawking (2016) once said: “The rise of powerful AI will be either the best or the worst thing ever to happen to humanity. We do not yet know which.” Hence, AI is perhaps the greatest invention of our modern society, but only time will tell whether AI is the destruction of humanity or the best invention that we have ever encountered.
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This study aims to address and further examine whether the null hypothesis, of the superior efficiency of the AI models over the traditional ones in predicting stock market returns, will be refuted or not (i.e., the 𝐻0: The AI models are better, more effective, and provide more accurate predictions of stock market returns than the classical (statistical) methods, the 𝐻1: The classical (statistical) methods are outperforming AI methods in forecasting stock market returns).
The essential objective of this study is to explain the concept of artificial intelligence, investment theories, and statistical and AI methods of predicting time series, as well as to conduct a systematic review and generate own models of comparison of statistical and AI approaches.
The first section of the thesis overviews a general description of AI. Furthermore, this study addresses the position of artificial intelligence in the financial sector, especially in the stock market earnings forecasts. The work also focuses on explaining the types of AI.
The second part of the study focuses on six traditional investment theories. Namely, the theory of random walk, the theory of efficient markets, the theory of expected utility, the prospect theory, the hypothesis of adaptive markets, and the theory of chaos. The last subdivision of this section is dedicated to explaining several portfolio theories.
The third chapter clarifies traditional and AI methods and approaches of forecasting stock market returns. Specifically, the work tackles the explanation of linear (ARMA, ARIMA models), nonlinear (ARCH, NMA models), and regime-switching (TAR, MSAR models) statistical models. Furthermore, this chapter outlines computational intelligence methods (ANN, supporting vector machine method, fuzzy set theory, evolutionary algorithms, expert systems, singular spectrum analysis, naive Bayes, k-nearest neighbors, and hybrid models).
The final fourth segment of the analysis, the empirical part, condenses empirical study regarding the research question of a comparison of AI and classical approaches in conducting forecasts in the financial market sector. The chapter is divided into two subchapters. First, a systematic review is constructed. This section includes a model specification, related research, introduction of data, included studies, and performance indicators, and lastly, the results section, where the outcomes are obtained and arranged in an extensive summary table. Second, an own comparison of AI and classical models
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on a chosen set of data is formed. Specifically, ARIMA models are designed to represent the classical statistical models, and an LSTM model is programmed to assess the AI approach of the research question. Both approaches are then compared to each other.
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1. Artificial Intelligence
What is AI? The abbreviation is derived from its original English word – Artificial Intelligence. AI is a relatively young area of scientific research. The term Artificial Intelligence first appeared in 1956, when John McCarthy organized a summer workshop called The Dartmouth Summer Research Project on Artificial Intelligence to which he invited scientists from various spheres. Before that, the name of the scientific field that persisted from its beginning was “thinking machines” (Marr, 2018). So far, not all definitions of artificial intelligence have been unified into one; thus, several definitions further explain the phenomena and the concept. The Encyclopedia Britannica states the description from B. J. Copeland (n. d.) that defines artificial intelligence as an area of computer science that deals with the creation of “intelligent” machines that solve complex tasks and problems, which should be able to work as intelligent beings. Another definition of AI is explained in the English Oxford Living Dictionary (Lexico, n. d.), as follows: “The theory and development of computer systems able to perform tasks that normally require human intelligence, such as visual perception, speech recognition, decision-making, and translation between languages.”.
According to Panesar (2019), AI is part of computer study and data science, which further comprises machine learning and data mining, as is displayed in Figure 1.
Figure 1: AI as Part of Computer Science
Source: (Panesar, 2019), own figure processing
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Artificial intelligence has origins in many domains. Namely, it has its foundations in science, technology, philosophy, mathematics, psychology, biology, and computer science (Bullinaria, 2005). According to Shi (2011) and Russell & Norvig (1995), the main goal of AI is to imitate and improve human thinking and intelligence with the help of artificial intelligence through intelligent machines (AI technologies).
Enforcing AI in business decision-making may generate many advantages. For instance, AI technology is much faster in carrying out decisions rather than humans. Such advanced technologies may conduct many inputs at once by completing them with a high level of accuracy. Furthermore, artificial intelligence does not suffer as much as humans do from decision fatigue, which means that AI machines can deliver the same results at any time. On the contrary, AI may be facing some barriers. For example, some outcomes which were obtained by sophisticated algorithms may lack accountability.
Therefore, one has to critically and cautionary evaluate the AI outputs. Moreover, AI technologies may be biased (such as data-driven bias, bias through interaction1, emergent bias, similarity bias, conflicting goals bias, or human bias since humans are creating algorithms) (Mohanty & Vyas, 2018).
This chapter aims to examine the history of artificial intelligence, to inspect in which areas of finance AI is used, and to define specific types of artificial intelligence.
1.1. AI in Finance
Artificial intelligence is vastly gaining its significance over time. In particular, AI has been becoming an attractive topic in the financial sector, especially in data mining, text recognition, and semantic analysis. In finance, AI helps in generating outputs while using automatic reading and text analysis, analyzing data, or predicting price level activity (Masarykova univerzita, n.d.). In this way, AI is employed, for instance, by AlphaSense Company (Schroer, 2019). Proof that the importance of AI is continuously deepening is the fact that about 85 % of all online payments in 2017 applied artificial intelligence
1 An example of bias through interaction may be the case of chatbot Tay, where through interaction with humans, Tay became racist and politically incorrect. Tay was developed by Microsoft in 2016. Originally, Tay was designed to interact with millennials. The more people would “talk” to Tay, the more he would learn. Not long after Tay was introduced, he started to tweet mean, racist, and political statements because humans were talking with him in such a manner. Eventually, Tay was shut down and replaced by Zo (Šrámek, 2018; Hunt, 2016).
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in its transactions (Shefrin, 2019). Moreover, according to Jeff Fraser (2017), the market of AI in the financial sector is “expected to grow from US-$1.3 billion in 2017 to US-$7.4 billion in 2022”.
The international Digital IQ Survey 2017 has conducted a remarkable study. Its results are captured in Figure 2 below. Around 52 % of all surveyed companies, with their major focus in the financial sector, claimed that they are investing a considerable amount of money in AI technologies. Around 2/3 of interviewed companies (66 %) stated to greatly invest in AI in the next three years, and about 72 % of respondents expect that AI will notably contribute to the development of business in the future (Kreutzer
& Sirrenberg, 2020).
Figure 2: Digital IQ Survey 2017
Source: (PWC, 2017), own figure processing
The benefits of using AI in the financial sector are various. For instance, improving and making the customer experience more advanced through chatbots are vastly being implemented in finance institutions (Maskey, 2018). Other benefits include recognizing the best investment opportunities, offering better conditions to both customers and companies themselves, or workflow automatization (OECD, 2019a; Kreutzer
& Sirrenberg, 2020). For example, Bank of America has evolved a robot and chatbot named Erica to help its clients with financial advice (Maskey, 2018). Customers can effortlessly speak or chat with Erica. Moreover, a company JPMorgan Chase has developed a contract intelligence platform that can, through image recognition
52%
66% 72%
0%
20%
40%
60%
80%
Substantial investments in AI
Will make substantial investments in three years AI will lead to significant business benefits
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and document scanning, evaluate contracts in a short amount of time (Kreutzer
& Sirrenberg, 2020). Nonetheless, artificial intelligence has been adopting as a tool for credit scoring, financial technology (FinTech), algorithmic trading, cost reduction in financial services, etc. (OECD, 2019a; Fliche & Yang, 2018). Interestingly, the fully automated process – dark processing – is being applied in the insurance industry.
Additionally, a German insurance company Deutsche Familienversicherung has found the use of virtual assistants in such a form that it is employing a virtual assistant Alexa in concluding contracts (Kreutzer & Sirrenberg, 2020).
AI is also generally employed to detect certain anomalies, such as identifying patterns of behavior that differ from standard manners, helping to investigate money laundering, inspecting illegal activities, or detecting security threats and frauds (Maskey, 2018).
Another sector in which AI can be a significant benefit is in market analysis, especially in making forecasts and recommendations, as well as in designing investment strategy formulations, peculiarly in further development and research of IT systems (Maskey, 2018). AI also enables better risk management (Fliche & Yang, 2018).
An example of such usage may be in asset management in which specific robot advisors are created with the purpose of helping particular investors with their targets and consecutive awareness of risks. For instance, Deutsche Bank has developed a robotic advisor ROBIN which helps with investing in Exchange-Traded Funds (ETFs) while taking into account the value of risk, initial investment, optional monthly investment contributions, and minimum investment horizons (Kreutzer & Sirrenberg, 2020; Deutsche Bank Group, n. d.).
Artificial intelligence creates appropriate algorithms that help with investment opportunities (e. g. algorithmic trading). AI systems thus obtain the necessary information about changing information in the market and can make recommendations to investors, who then, thanks to AI, make quick trades. Various institutions, especially banks, are broadly implementing systems with artificial intelligence support. It is estimated that by 2030, banks will save more than $ 1 trillion thanks to artificial intelligence (Maskey, 2018).
Presently, many companies and startups are using artificial intelligence, such as JPMorgan, Citibank, State Farm, and Liberty Mutual, or startups such as Zest Finance, WeCash Aire, and many others (OECD, 2019a).
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Predictions of stock market returns have been a prevalent focus of many economists and researchers for many years. The sole question of whether the returns can be predicted or not gave the impetus to create many investment theories and models. As the stock market is a very complex and often chaotic system, it may be difficult to predict its future values. That is why AI comes into the light in this field of economics to forecast the future behavior of those time series. Since AI technology is becoming more proficient and capable every day, it can operate with a vast amount of data and recognize patterns between various variables, it is natural that AI is being implemented in the prediction of price levels and stock market returns as well. The main benefit of AI in forecasting the stock market returns is that they can learn over time and therefore predict more accurate results. “…rather than telling the machine that predictors A, B and C define the response D, and this is etched in stone, we leave it up to the computer to figure out the relationships within the dataset and re-calculate them as new data comes in.” (I Know First Research, 2019).
1.2. Types of AI
Since AI is immensely based on imitating functions of the human brain, the degree to which AI technologies can or cannot copy those abilities determine different types of artificial intelligence. The essential qualifications which are observed while categorizing the types of AI are performance and versatility. Generally, the types of AI are compartmentalized into two classifications (Joshi, 2019a).
The first categorization is based on the similarities of AI to the human brain, and the possible capability to think and feel as humans do. Within this division, four main types are differentiated: reactive machines, limited memory machines, the theory of mind, and self-aware AI (Joshi, 2019a).
Reactive machines have limited abilities. They are classified as the oldest forms of AI.
This type of AI is not equipped with memory functionality, i.e., they cannot use past experiences while resolving problems in the present; therefore, they cannot learn over time while benefiting from the past. Frankly, such type of AI completes the task it was designed for, yet it does not do anything on top of that. The well-known AI machines
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Deep Blue2 and Google’s Alpha Go3 fall into this category (Joshi, 2019a; Panesar, 2019).
In general, reactive machines cannot use past values in order to form predictions. They simply observe the situation as it is in the present on the basis of which they make further forecasts. Moreover, such AI makes decisions in the same manner whenever it comes across the same situation (Hintze, 2016; Matthews, 2017). Therefore, this class of AI is not notably appropriate to be employed in the predictions of stock market returns.
For this reason, the second type of AI (limited memory) is applied.
The second respective group is limited memory. Besides having the same abilities as reactive machines, it can learn over time through its past experiences (e. g. deep learning), and additionally, it is able to make decisions on its own. Such types of machines are used in today’s applications and computer programs (such as virtual assistants, chatbots, etc.) that store a substantial amount of data (Joshi, 2019a).
The theory of mind is the third category within the first classification. It represents the next level of AI technology, where such machines will be able to completely understand human needs, emotions, and thoughts. Moreover, they will be able to communicate, interact, and learn from other beings and machines. Even though scientists and researchers are currently working on developing such types of artificial intelligence, it is not yet entirely invented (Joshi, 2019a; Mohanty & Vyas, 2018).
The last stage of AI evolution is self-awareness, where machines gain full consciousness (or self-awareness), and they will have their own emotions, thoughts, beliefs, etc. Such technologies will be very similar to the functioning of the human brain, which represents the ultimate peek of AI evolution (Burk & Miner, 2020; Mueller & Massaron, 2018; Joshi, 2019a)
There are three segments of the AI division within the second classification, namely, Artificial Narrow Intelligence (ANI), Artificial General Intelligence (AGI), and Artificial Super Intelligence (ASI) (Joshi, 2019a).
2 Deep Blue, a computer which was developed by IBM, is famous for its victory over a well-known chess player Garry Kasparov. The IBM computer and Garry Kasparov fought a total of six times. Kasparov won the first encounter, but in the second encounter in 1997, the situation turned around. In this round, the computer gained a historic victory over a man, and Deep Blue became a phenomenon (Thornhill, 2017;
Buchan, 2015).
3 AlphaGo is one of the most developed programs of all time. It was designed to be able to play an ancient Chinese board game called Go. AlphaGo played against Go’s champion Ke Jie and ultimately won the game (Šrámek, 2018; Agence France-Presse in Shanghai, 2017). AlphaGo was invented for the reason to be able to learn from its own and opponent's previous mistakes and experiences.
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ANI is a category of artificial intelligence that may be encountered in today’s world. It is analogous to reactive machines and limited memory, which was discussed above. Hence, ANI is defined by high accuracy, ability to store and work with a large amount of data, gaining excellent skills, the capacity to learn from own mistakes and experiences, and solving one task at a time wholly precisely. The following computer programs are included within this form of AI, for instance, AlphaGo from Google, which was created in 2015, or self-driving cars, virtual assistants, Spotify and its recommended playlists, etc. Interestingly, even Deep Blue is included in this segment. However, when comparing ANI with the human brain, the human mind is much more complex and can perceive multiple inputs at the same time, while having its own thoughts, ideas, emotions, etc. (OECD, 2019a).
A type of AI that is aimed to be developed in the near future is called AGI. By definition, AGI can solve multiple tasks at once, think on its own, make decisions, communicate, read, understand the emotions of others, and be able to think independently. Practically speaking, it will have functions as the human mind has, though, with higher accuracy and speed. It will also be able to store more information than the human brain can keep (Joshi, 2019b; OECD, 2019a).
The superintelligence that will, in the future, surpass the human mind, is identified as ASI.
The definition of ASI coincides with the term singularity. In substance, artificial intelligence will be more complex, capable, and powerful than the human brain. On one side, such development of technology may be immensely beneficial to humankind, though, from the other perspective, it may threaten the existence of life that we know today (Joshi, 2019a).
As we already know, AI technologies are equipped with progressive learning and reasoning abilities according to which such machines are then able to provide us with a large amount of data, decision-making, and outcomes. At present, two main branches further explain the progress of AI learning and reasoning. The first branch is dependent on “conditional instructions (transformation rules, and heuristics)”
(Mohanty & Vyas, 2018, p. 11). The second perspective is recognized as machine learning. Basically, it implies that the AI machine absorbs an enormous amount of data, in which it recognizes patterns and learns over time (Mohanty & Vyas, 2018).
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2. Theories and Models of Investment on Stock Returns
Several economists and researchers have been dealing with the question of whether the stock market returns are indeed predictable or not. In general, we can divide them into two major schools of thought. The first one assumes that the market is fully efficient, and all information is available; therefore, the returns are not predictable. This group includes, for instance, theories such as the random walk hypothesis or the efficient market hypothesis. The second school of thought takes the view that the systems are entirely chaotic. In reality, the stock market may behave like a mix of both of these inputs; thus,
“…markets are complex and chaotic systems and their behavior contains both a systemic and a random component. Therefore we can make a realistic stock market forecast, although it is precise only to a certain extent.” (I Know First Research, 2020).
Within this chapter, we will look more deeply into the numerous investment theories, models, and approaches. In particular, this chapter will put its focus on Random Walk Hypothesis, Efficient Market Hypothesis, Expected Utility Theory, Prospect Theory, Adaptive Market Hypothesis, Chaos Theory, and finally, several portfolio theories will be examined in larger detail.
2.1. Random Walk Hypothesis
The Random Walk Hypothesis (RWH or RW) has been supported by several economists.
The first to come up with this theory was the French broker Jules Regnault (1834-1894), who formulated the foundations of modern stochastic models of pricing behavior (Jovanovic & Le Gall, 2001). The theory of Random Walk is further associated with Louis Bachelier, who published the dissertation The Theory of Speculation in 1900, Alfred Cowels (1891-1984), and Paul Cootner (1930-1978), who with his book The Random Character of Stock Market Prices (1964) rediscovered the theory of Random Walk. Regnault focused mainly on stock market morale, Bachelier concentrated on stock market science, and Cowels aimed his attention on the practicality of the stock market (fondation maison des sciences de l’homme, n.d.). The title Random Walk was then popularized by Maurice Kendall & Bradford Hill with their publication The Analysis
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of Economic Time Series, Part 1 (1953), and Eugene Fama in 1965 with his study Random Walks in Stock Market Prices (1965).
According to the Random Walk theory, stock changes have the same distribution and are independent of each other. Thus, historical prices or trends cannot be adopted to predict future returns. Therefore, all stocks are random, and thus, we cannot predict them. The Random Walk theory further defines that in the long run, stock returns cannot be anticipated (Cootner, 1964).
Several research papers were written on this topic of determining whether the stock market returns follow the random walk or not. For instance, the paper of Chitenderu, Maredza, and Sibanda (2014) aimed at examining the predictability of the All Share Index on the Johannesburg Stock Exchange. The authors constructed the ARIMA(1,1,1) model.
Then, they formed tests of residuals in order to ascertain if the time series were behaving indeed according to the random walk, which was thereafter confirmed by the researchers.
On the other hand, many papers have proven that artificial neural networks were quite correct in predicting stock market returns, which accordingly gave evidence that stock market series do not follow the random walk. An example can be taken from a paper written by Chen (2003) in which the author manifested that the neural networks outperformed other methods, including the random walk hypothesis.
2.2. Efficient Market Hypothesis
The emergence of the theory of efficient markets is mainly linked to economists E. F.
Fama and P. A. Samuelson. However, it was not popularized until the book by Burton Malkiel (1973) A Random Walk Down Wall Street came to light. According to the efficient market hypothesis (EMH), prices are influenced only by objective information. Asset prices are a reflection of all the information available on the market, so stocks are always entirely valued on the market. As all the information that affects the price is independent of each other and unknown in advance, it is, therefore, not possible to predict prices (not even from historical data). As the market is continually changing, prices are always affected by new incentives; and hence, prices and the market are unpredictable (Lo, 2007; Degutis & Novickytė, 2014). Fama (1965) divides EMH into three main types: the weak version (all historical and present information on the market is included in the current prices), the semi-strong version (in addition
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to the weak version, the current prices comprise of public information of the asset), and the strong version (except the historical, present and public information, prices cover non-public4 information about the asset as well) (Vlček, 2016).
This theory has been heavily condemned by many economists. Notably, by Grossman (1976) and Grossman & Stiglitz (1980). Both of the economists believed, that perfectly efficient markets cannot occur given that one can never know all the information that significantly reflects the behavior of the market. They thought that if an investor can obtain all the information, she or he will then not have any motivation for trading;
therefore, no desire to try to achieve new information. It is thus crucial that the market is, to some extent, inefficient (Lo, 2007; Degutis & Novickytė, 2014).
The EMH has been tested and examined in many markets. In general, the researchers indicated that the emerging markets are not that efficient as the developed ones (Dima & Miloş, 2009). Moreover, Lo (1997) manifested several tests of market efficiency, such as: “random walk test, variance ratio tests, overreaction and under reaction, the winner-loser effect, price earnings ratios, the small firm effect, price book value ratios, the three-factor model, the January effect, the weekend effect, the earnings announcement drift, standardized unexpected earnings, the momentum effect, mean-reversion, calendar effects, the size effect and the value effect”
(Sofla, 2010, p. 14). Until this time, there is not a clear agreement about the EMH among scientists (Sofla, 2010).
2.3. Expected Utility Theory
The theory of expected utility (EUT) addresses the issue of predicting stock returns while assuming rational investors who can effortlessly and rationally evaluate all possibilities and then pursue their best decisions. Fundamentally, EUT portrays the choice of an agent who is enduring a risk (Dean, 2017). Agents compare expected utility outcomes (“the weighted sums obtained by adding the utility values of outcomes multiplied by their respective probabilities”) (Mongin, 1997, p. 1). As this theory assumes that the investor is rational; therefore, he targets at maximizing the expected utility of returns of his or her chosen portfolio (Sharpe, 2006). Sharpe (2006) introduced an algorithm that serves
4 Private or insider information.
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to detect the maximum expected utility portfolio. The procedure of such an algorithm is described below:
“
1. Find a feasible portfolio.
2. Determine the best possible two-asset swap.
3. If no such swap is feasible, terminate.
4. Otherwise, compute the best magnitude for the swap and revise the portfolio.
5. Repeat starting at step 2 until the condition in step 3 is met.
“ (Sharpe, 2006, p. 9).
This theory was refuted in 1979 in an empirical test by economists Kahneman and Tversky in Prospect Theory: An Analysis of Decision Under Risk (1979) (Grant & Van Zandt, 2007).
2.4. Prospect Theory
The prospect theory was developed by Daniel Kahneman and Amos Tversky in 1979 in which the authors address the individual’s choice between options with risks and with the probabilities of various outcomes. They explained that agents then choose according to the perceived gains of all options. The authors further defined that people tend to be more risk-averse for a positive prospect, and on the other hand, people incline to risk-seeking behavior when leaning to negative prospects (Kahneman & Tversky, 1979; University of Michigan Press, n. d.). Moreover, they emphasized that the loss would have a more substantial impact than the gain for an investor (Dichtl et al., 2016).
According to Barberis, Mukhejee, and Wang (2016), those stocks that have eminent prospect theory values now will manifest in lower returns in the future (the same holds for the opposite situation). This is explained with a clarification that those stock that poses higher prospect theory values now are more attractive for investors; thus, investors will buy more of them which will be demonstrated in their lower value in the future.
For instance, Zhong and Wang (2018) provided empirical evidence for bonds where they proved that the prospect value of a bond could be a future indicator of its forthcoming values. Moreover, the research of Nicholas Barberis, Lawrence Jin, and Baolian Wang (2019) proposed a model of forecasting the cross-section of average returns in order
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to explain some irregularities on the stock market while taking into account the prospect theory. The proposed model showed to be correct, for instance, when observing volatility and momentum but did not perform very well when taking into account value anomaly (Barberis et al., 2019, p. 2).
2.5. Adaptive Market Hypothesis
The Adaptive Market Hypothesis (AMH) was proposed in 2004 by Professor Andrew Lo in The Adaptive Market Hypothesis: Market Efficiency from an Evolutionary Perspective (2004). In essence, he sought to unify EMH with behavioral economics. He tried to combine the two by employing them in the financial environment (Lo, 2004). EMH emphasizes that one cannot buy undervalued stocks as companies are always listing them at their fair values. On the other hand, behavioral finance stresses that investors are not always behaving in an economically rational way, so the stocks do not have to be always valued fairly. This approach; therefore, considers the psychology and behavior of investors rather than their rationality. Lo (2004) believed that even though EMH and the rationality of investors are predominant, investors may occasionally succumb to behavioral tendencies, in such a manner, that investors may learn over time from their errors; and therefore, engage in formulating predictions which are based on their knowledge gained in the past (Liberto, 2019).
“The key point for AMH to be the harmonization of EMH and BF is investor’s ability to learn and to adapt to the market’s updated situation, as a result, financial markets can be wrong from time to time, but they learned, evolved to be right, until the next mistake.”
(Trung & Quang, 2019, p. 1).
Andrew Lo also emphasizes the occurrence of cycles in financial markets. Another essential point that contradicts traditional EMH is, according to the author, that AMH approves of the scope for arbitration opportunities that are variable over time.
Due to the existence of financial markets, arbitrage opportunities are necessary;
otherwise, the market would collapse (agrees with Grossman and Stiglitz (1980)). He also deals with the importance of the phenomena of innovation and survival of companies in the market, where he further labels it as the only important goal of companies (Lo, 2004).
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The AMH was examined and further tested in several studies. For example, Trung and Quang (2019) observed the AMH at the Vietnamese stock markets while adopting several tests, e.g., the variance ratio test, the automatic portmanteau test, or the generalized spectral test. The authors indeed confirmed the AMH in the observed period in the Vietnamese stock markets. Furthermore, Kim et al. (2011) likewise tested the AMH. The purpose of this study was to examine the AMH for the DJIA index.
According to the outcomes, the market proved to be inefficient over a crisis period and efficient when observing market failures. The results of Noda (2016) gave evidence for support of AMH, and the same holds for the study of Smith (2011).
2.6. Chaos Theory
The first to contribute to chaos theory in mathematics was the French mathematician Henri Poincaré (Blanchard et al., 1998). However, the first study on chaos theory was conducted by Edward Lorenz, who wanted to recreate the past weather sequence. He used a computer program in which he included 12 variables. However, he made one mistake.
He rounded the units to a smaller number than he should, and thus, he accidentally created a discrepancy, which caused an extensive deviation from reality in the future. In this way, Lorenz proved that a small alteration now could cause immense changes in the future.
The chaos theory serves to explain non-linear dynamic systems, for instance, financial markets. Therefore, Lorenz's theory is also applicable to financial markets (MIT News, 2008; Chang, 2008; Cottrell, 2014). Chaotic systems are exceedingly sensitive to initial perceptions, which may be unknown in advance. The chaos theory may seem to be random, but in fact, it is deterministic. For this reason, the chaos theory is applicable when predicting the short-run forecasts when the system is behaving deterministically rather than the long-term values (Cottrell, 2014; Sadil, 2016; Bau & Shachmurove, 2002).
“Even in this limited way, creating realistic stock market forecasts is surely possible and gives us the prospect to understand how market works and why big bubbles and big crashes happen” (I Know First Research, 2020). The effect when a small change today largely affects the future is also known as the butterfly effect (Sadil, 2016). As the chaotic systems are exceedingly complex, such structures have the ability to recognize patterns and use their memory (I Know First Research, 2020). The chaos systems are determined by chaos optimization, which may be done, for instance, by the logistic map as was done
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in the research of Dhanalaxmi et al. (2016). Furthermore, Bulusu et al. (2020) created a model based on chaotic and sentiment analysis for forecasting price levels on a stock market. The model proved to give significant predictions, which were generated for seven banks.
2.7. Portfolio Theories
Conclusively, several portfolio theories will be investigated throughout this subchapter.
Namely, Markowitz Portfolio Selection, Dow Theory, Behavioral Portfolio Theory, Maslowian Portfolio Theory, Black-Litterman Model, Dedicated Portfolio Theory, Treynor-Black Model, Universal Portfolio Algorithm, Capital Asset Pricing Model, and Arbitrage Pricing Theory.
2.7.1. Markowitz Portfolio Selection (MPS)
5Harry M. Markowitz6 concentrated on the analysis of portfolios, which comprises a great number of securities. He further references it as “a portfolio selection”.7 A good portfolio selection yields for an investor many investment opportunities. Therefore, such a portfolio shall fully satisfy the investor’s objectives and financial possibilities (Markowitz, 1959, 1952; Burke, 2020). An investor is putting all investments in one portfolio (De Brouwer, 2017). The analysis of the selection is affected by various types of information, such as past performance of securities and beliefs of the future outcomes of securities (Markowitz, 1959). Intuitively, MPS is based on expected returns and standard deviations of those portfolios. In this study, Markowitz explains that the portfolio is efficient if it yields maximum return with some given risk, or the portfolio may be also efficient by assuming that it gives a minimum risk with a given level of return. MPS further explains the efficient frontier. All portfolios that lie on this frontier are efficient ones. Those portfolios that are located under the frontier, are not optimal (the risk higher than their rate of return). In the interest to find an investor’s optimal portfolio, we need to find the best point of this frontier, which is located at the point
5 Also known as Modern Portfolio Theory and Mean-Variance theory.
6 H. M. Markowitz won the Nobel Prize in Economics in 1990.
7 H. M. Markowitz has pointed out several assumptions within this model.
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of tangency of the efficient frontier and the indifference curve (Markowitz, 1959, 1952;
Burke, 2020).8
Fundamentally, the MPS assumes that the historical sequence of such financial time series are normally distributed. In order to predict the future values of returns, the mean of the time series is employed and variance is used to assess the risk of the series (Freitas, et al., 2009). A study by Freitas, Souza, and Almeida (2006) aimed at comparing the Markowitz portfolio selection theory with the authors’ prediction-based forecasting model based on autoregressive neural networks. The results demonstrated that the researchers’ model outperformed the MPS approach.
2.7.2. Dow Theory
Charles Dow9, the founder and the editor of the Wall Street Journal, set the basis of the Dow Theory. According to Dow’s ideas10, the prices reflect all information that is obtainable on the market (it; therefore, corresponds to EMH). Dow further explains three main types of market trends and their movements: (i) the primary trend (it is the longest trend with a duration of months to even years), (ii) the secondary trend (which lasts for about some weeks and a couple of months), and (iii) the tertiary trend (with the very short duration with less than a week). By observing these different sorts of trends, investors can spot the right opportunities for investments. Another principle of this theory is that the market trend has three main phases (the accumulation phase, the public participation phase, and the distribution phase (Binance Academy, n.d.).
2.7.3. Behavioral Portfolio Theory (BPT)
The BPT was developed to make an alternative to Markowitz Portfolio Theory.
Behavioral Portfolio Theory “links two issues, the construction of portfolios and the design of securities.” (Shefrin & Statman, 2000). Portfolios within this theory are explained in layered pyramids (see Figure 35 in Appendix A), where each layer represents some goals. The bottom layer may portray the need to prevent financial
8 Markowitz in his paper used quadratic programming.
9 Dow also cooperated in creating the first stock index – Dow Jones Transportation Index (DJT) and the Dow Jones Industrial Average (DJIA).
10 As a matter of fact, Charles Dow did not write any paper on this topic (he wrote some editorials from which other authors extracted these ideas – as is William Hamilton, S. A. Nelson, and Robert Rhea.
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collapse, and the top layer shows the will to maximize returns (Shefrin & Statman, 2000).
Chang et al. (2018) observed in their research the BPT, and they put it into contrast with the mean-variance portfolio selection. The authors took into account the fear and hope levels of investors and the risk-seeking behavior of investors. According to the outcomes, the BPT proved that it can outperform the mean-variance approach.
2.7.4. Maslowian Portfolio Theory (MaPT)
The MaPT is directly supporting the BPT and the theory was based on Maslow’s description of human needs. Additionally, investors are not placing all their investments in one portfolio (as in MPS). They are rather focusing on several investments (De Brouwer, 2017). Such investments are consistent with multiple levels of priorities (goals), which are shown in the table below.
Table 1: Layers of Maslowian Portfolio Theory
Human needs Investments/MaPT
Physiological Needs Liquid/cash
Safety Needs Insurance, retirement savings
Love Needs Mixed portfolios for projects
Esteem Needs Mixed portfolios for projects
Self-Actualization Broker account
Source: (De Brouwer, 2017), own table processing
2.7.5. Black-Litterman Model (BLM)
The BLM was proposed by Black and Litterman in 1992. Its primary focus is on the asset allocation theory. Through the equilibrium analysis, the model attempts to calculate the returns of risky and uncertain outcomes. It further employs a Bayesian methodology to combine the estimates with investor’s beliefs about the assets. Additionally, investment bank Goldman Sachs is advocating for this model quite regularly (Bertsimas et al., 2012;
Black & Litterman, 1992). Ta & Vuong (2020) employed the ARIMA model to forecast the returns while adopting the BLM. Subsequently, the authors compared the results
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of BLM to the Markowitz portfolio selection, which indicated that the BLM approach gave better predictions than the MPS.
Since the BL model has come into focus, many other BL-type estimators were introduced, such as the MV-IO approach and the RMV-IO approach from Dimitris Bertsimas, Vishal Gupta, and Ioannis Ch. Paschalidis (2012), and many more.
2.7.6. Dedicated Portfolio Theory (DPT)
The DPT explains portfolios11 as a way how to guarantee secure and in a way predictable outcomes for future cash flow (the “dedicated” part of the name of the theory represents the security of those outcomes, which are in a way “dedicated” to future cash flows).
“The predictability is achieved by purchasing bonds and holding them to maturity, and then collecting coupon and redemption payments as income in retirement.”
(Huxley & Burns, 2018, p. 1; Leibowitz, 1986).
2.7.7. Treynor-Black Model (TBM/TB)
The TB model operates with the mean-variance criterion and with the specified market strategy, which is taken as the efficient evidence that the securities are priced efficiently while dividing it into the active and the passive portfolio. The TB model covers the portfolio of covered securities, which can be combined with the index in order to get the optimal portfolio (Kane et al., 2004; Brown, 2015).
2.7.8. Universal Portfolio Algorithm
Thomas Cover published in 1991 Universal Portfolios in which he explained the universal portfolio algorithm where he uses past data to predict future outcomes.
The algorithm of the universal portfolio is daily readjusted to keep track of the performances of investments. The portfolio is called universal because the investor buys a small amount of every stock in the market in order to reach the optimal strategy (Levy, 2000; Cover, 1991).
11 Also known as “cash matching“ portfolios.
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2.7.9. Capital Asset Pricing Model (CAPM)
The CAPM model, independently proposed by Jack Treynor (1961, 1962), William F.
Sharpe (1964), John Lintner (1965), and Jan Mossin (1966), portrays a solution for identifying the rate of return on risky assets (Kotzé, n. d.) and determining the pricing of a security or a portfolio. The model estimates the rate of return while observing risks;
thus, it helps investors in making decisions about maximizing outcomes and minimizing risks. The model can be written by Formula (1) depicted below (Sekreter, 2017).
𝐸(𝑅𝑖) = 𝑅𝑓+ 𝛽𝑖(𝐸(𝑅𝑚) − 𝑅𝑓) (1)
Where 𝑅𝑖 is the rate of a stock return, 𝑅𝑓 represents the risk-free rate, 𝛽𝑖 depicts the measurement of volatility of a stock which demonstrates a sensitivity (𝛽𝑖 = 𝑐𝑜𝑣(𝑅𝑖,𝑅𝑚)
𝑣𝑎𝑟(𝑅𝑚 ), and the 𝑅𝑚 means the rate of market return (Sekreter, 2017).
When beta is 0, then by looking at Formula (1), we can see that the expected rate of stock return is equal to the rate of risk-free return. Contrary, when beta is 1, then the expected rate of stock returns is the same as the expected rate of market return (Sekreter, 2017).
If beta is for example 1.5, then the expected return of an asset will be 15 % (Yiu, n. d.).
The model further presumes that investors concentrate their finance on heterogeneous portfolios; therefore, it only considers systematic risk (ACCA, n. d).
The CAPM considers in its applications only the fact, that the only market risk pays a risk premium (the idiosyncratic risk does not since one can easily eliminate it) (Yiu, n. d.).
The CAPM model has several extensions. For instance, we can name models of Lintner (1969) and Merton (1987) which included heterogeneous beliefs, a model with the possibility to include numerous time periods with various investment opportunities (Mayers, 1973), or the International Capital Asset Pricing Model, which contained country-specific needs and requirement of individual investors (Perold, 2004).
Intertemporal Capital Asset Pricing Model (ICAPM) serves as another example of an extension of classical CAPM. ICAPM was created by Robert Merton (1973) who created it as a multi-period model. Robert Merton based his model on consumer- investor behavior. It also takes into account investment opportunities over time and adds
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other time-varying factors. Merton, in his work, states that beta should not be the only component of risk evaluation12 (Bank, 2011; Merton, 1973; Sekreter, 2017).
The Consumption-Oriented Capital Asset Pricing Model (CCAPM) falls into the category of development of the original CAPM. The CCAPM explains that
“the expected excess return on any risky asset should be proportional to its marginal utility in consumption.” (Adegboye, 2017); thus, “securities with higher sensitivities of returns to movements in real consumption spending have more systematic risk and should have proportionately higher excess returns” (Breeden et al., 2015).
2.7.10. Arbitrage Pricing Theory (APT)
The APT evolved in 1976 and was introduced by S. A. Ross. Principally, the model works with the assumption that the returns can be calculated, only if there is no arbitrage on the market. In the model, there are several factors that incorporate market-wide risks where each of them has its beta coefficient (not only one as in the CAPM) (Sekreter, 2017; Kumar, 2016).
12 The Fama-French Three-Factor Model further analyzes the problem of determinants of risk.
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3. Forecasting Stock Market’s Returns
The stock market is one of the most significant indicators of a country's maturity.
The stock market may generate a considerable profit if the risks and future developments are correctly estimated. However, extensive returns carry high risks, so it is crucial to be able to predict the future of returns. Forecasting the stock market's returns thus remains one of the key areas of research and is an essential part of the investment plan.
Many economists, investors, and brokers have been forecasting the stock market's returns since the very beginning of stock market advancements. Over time, various methods and models have evolved. With the development of artificial intelligence, methods and models have become even more widespread. The dividend yield method was followed by, for example, Dow, Campbell (1987), Fama and French (1988), Hodrick (1992), Ang and Bekaert (2007), or Cochrane (2008). Campbell and Shiller (1988) and Lamont (1998) examined ratios of price and earnings. Economists who observed short-term interest rates in their publications were mainly Fama and Schwert (1977), Campbell (1987), Breen, Glosten, and Jagannathan (1989), Ang and Bekaert (2007).
Other economists think that historical price trends do not affect future prices (Ferreira & Santa-Clara, 2008; Klein, 2018).
The following chapter is devoted to the explanation of statistical models and computational intelligence methods.
3.1. Statistical Methods
The subsequent subchapter will take a closer look at the issue of linear, nonlinear, and regime-switching models. Even though numerous models have been explored and introduced over the years, within this thesis the focus will be put, in particular, on the ARMA, ARIMA, ARCH, and NMA models, as well as on the TAR and MSAR models.
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3.1.1. Linear models
Linear models are relatively easy to understand and implement; thus, that is the main reason they are often used (Adhikari & Agrawal, 2013). Autoregressive Integrated Moving Average (ARIMA) is a linear model for time series. It consists of three parts.
First, the autoregressive part - Auto Regression (AR). Second, the integrative part - Differencing (I), and third, the moving average part - Moving Average (MA). There are many combinations of these two time series models AR and MA. For instance, the Autoregressive Moving Average (ARMA) model is adopted for stationary time series data. The Autoregressive Fractionally Integrated Moving Average (ARFIMA) model is applied to generalize the ARMA and ARIMA models. Another variant of the ARMA model is employed for forecasting seasonally variant time series, and that is the Seasonal Autoregressive Integrated Moving Average (SARIMA) model. ARMA models and their alternative versions are derived from the Box-Jenkins Methodology (Adhikari & Agrawal, 2013).
3.1.1.1. Autoregressive Moving Average (ARMA)
The ARMA (p, q) model is constructed by two parts - AR (p) and MR (q) processes.
AR (p) indicates that in automatic regression, the values of a given time are regressive to their own delayed values, which is denoted by p in the model. Thus, p indicates the part of the time series that can be explained as a linear combination of past values (i.e., regression); and therefore, determines by how many periods the model goes backward.
For example, if the value of p = 1, we look only at the past period. If p = 2, we look at the past and pre-past period, etc. (Adhikari & Agrawal, 2013):
The model AR (p) can be written as follows:
𝑦𝑡 = 𝜙1𝑦𝑡−1+ ⋯ + 𝜙𝑝𝑦𝑡−𝑝+ 𝜀𝑡 (2)
The MA (q) model, or also known as the moving average, is denoted by q, which indicates the number of delayed values of the error term. Hence, it represents that part of the error of time series, which can be explained as a linear combination of past error terms.
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In essence, we inspect the number of periods in which the error term is repeating (Adhikari & Agrawal, 2013; Artl et al., 2002).
𝑦𝑡= 𝜀𝑡+𝜃1𝜀𝑡−1+ ⋯ + 𝜃𝑞𝜀𝑡−𝑞 (3)
If we combine both AR and MR models, we get the following equation (Adhikari & Agrawal, 2013):
𝑦𝑡 = 𝑐 + 𝜀𝑡+ ∑ 𝜑𝑡𝑦𝑡−1
𝑝
𝑡=1
+ ∑ 𝜃𝑗𝜀𝑡−𝑗
𝑞
𝑗=1 (4)
where 𝑦𝑡 is the value of the time series at time t, 𝑐 is a constant, 𝜀𝑡 represents the error term at time t, and the parameters 𝜑𝑡 a 𝜃𝑗 depict unknown parameters (Adhikari & Agrawal, 2013).
When identifying the correct model, one has to carry out the analysis of Autocorrelation Functions (ACF) and Partial Autocorrelation Functions (PACF) (Adhikari & Agrawal, 2013).
For instance, a study by Anaghi & Norouzi (2012) generated an ARMA model for the reason to forecast future stock market values. In total, the authors created six ARMA models.
3.1.1.2. Autoregressive Integrated Moving Average (ARIMA)
Due to the fact that many time series suffer from nonstationarity, the Autoregressive Integrated Moving Average (ARIMA (p, d, q)) model was formulated (Karamouz& Araghinejad, 2005; Valipour et al., 2012). In addition to ARMA, the I part of the model is introduced. ARIMA model is composed of the AR (p), which again denotes AutoRegression, I (d) indicates the Differencing, and MA (q) symbolizes the Moving Average. To be able to work with time series, the data must be stationary.
If they are not, we create a stationary time series from a non-stationary time series through
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the integration part of the model I (d). In a view of the ARMA model, the only component that is added to the model is the I (d), which helps to eliminate the trend. The value of d indicates the number of times the differencing is repeated. If d = 0, we model using the ARMA model. If d = 1, the difference between the two time series is reset.
Interestingly, the Random Walk theory is a variation of the ARIMA model, in which the values of p and q are zero13 so that only part of the integration remains (Adhikari & Agrawal, 2013). The ARIMA model is derived from the Box-Jenkins method.
𝑦𝑡 = 𝜙1𝑦𝑡−1+ 𝜙2𝑦𝑡−2+ ⋯ + 𝜙𝑝𝑦𝑡−𝑝+ 𝜀𝑡+ 𝜃1𝜀𝑡−1 + 𝜃2𝜀𝑡−2+ ⋯ + 𝜃𝑞𝜀𝑡−𝑞
(5)
Where 𝑦𝑡 is the value of the time series at time t, 𝜀𝑡 corresponds to the error term at time t, and parameters 𝜑 and 𝜃 are unknown (Adhikari & Agrawal, 2013).
The following Figure 3 explains how the ARIMA model is applied through the Box- Jenkins methodology through the three-step analysis of identification, estimation, and diagnosis checking (Adhikari & Agrawal, 2013).
Figure 3: The Box-Jenkins Method
Source: (Adhikari & Agrawal, 2013), own figure processing
13 It can be written as: ARIMA(0,1,0). Furthermore, the model AR can be reworded as ARIMA(p,0,0), or the MA model may be rephrased as ARIMA(0,0,q).
