SCHOOL OF ECONOMICS AND FINANCE

FACULTY OF COMMERCE, LAW AND MANAGEMENT

Testing The Adaptive Efficiency Of Bitcoin

Maromo Maredi

Supervisor

Prof Yudhvir Seetharam

A research report submitted to the Faculty of Law, Commerce and Management, University of the

Witwatersrand in partial fulfilment of the requirements for the degree of Master of Commerce (50%)

in Finance.

Johannesburg, South Africa

February 2022

Testing the Adaptive Efficiency of Bitcoin

Abstract

This research aims to investigate an alternative view of market dynamics referred to as the Adaptive

Markets Hypothesis which posits that an asset’s efficiency will change over time. As such, this

research will test whether Bitcoin is time-varyingly efficient. This will be accomplished in three

stages. Firstly, whether Bitcoin returns follow a random walk/martingale will be investigated. If they

do, that means that they cannot be predicted, thereby providing evidence of the weak-form market

efficiency. If they do not follow a random walk, however, the second phase of the investigation turns

to whether they can be modelled. The first attempt models the current Bitcoin return as a function of

its own lagged values, which is predicated the idea of all relevant information being reflected in

historical returns. The inadequacy of this model in its description of the returns generating process,

provides evidence that there is private information that historical returns do not reflect which impacts

returns. To account for this, the returns generating process is thus modelled using both historical

returns and exogenous lagged variables without need to specify the model’s functional form. If the

model performs better in some periods than in others, it can be inferred thus that Bitcoin is time-

varyingly efficient.

Declaration

I, Maromo Maredi, declare that this research paper is my own work and that I have correctly

acknowledged the work of others. It is submitted to fulfil the requirements for the degree of Master

of Commerce in Finance at the University of the Witwatersrand, Johannesburg. I declare that this

research paper has not been submitted for any other degree or examination in this or any other

institution.

Maromo Maredi

February 2022

iii

Dedication

To my mothers, Meta, Modiledi and Mohlapa Maredi and my late grandmothers, Shatadi

Matlejoane, Pheladi Matlejoane, Pheladi Maredi and Mateadi Maredi.

Acknowledgements

There are several who have supported me throughout the life of this project and have backed me to

the hilt. Prof Yudhvir Seetharam, words cannot express how grateful I am to you. You have been a

constant source of inspiration and advice throughout what has turned out to be an interesting and

wonderful journey. You challenged me to always strive for the best and never rest on my laurels. I

would also like to thank Dr Bevan Smith, whose advice and input I am ever thankful for.

I would also like to thank family. My mother, Mokgohloe Maredi, who is my role model and

confidante. I am all that I am because you never stopped encouraging me to believe in myself, even

(and perhaps, especially) when things became difficult. This research report is the product of the

many nights you called and found me working. To my sister, Octavia Maredi, you are the world’s

best older sister. Thank you for always having my back.

This project would also not have been impossible without the external funding from BankSETA, who

have, throughout my masters, ensured that my fees were not a factor in whether I could acquire further

training in a field I am deeply passionate about. In the same vein, I would also like to thank Prof

Hofmeyr and his team at Study Trust, who have throughout my undergraduate and honours studies,

provided me with funding.

Finally, I would like to thank my friends, Kgadile Masemola, Sanele Gumede, Thabisile Mahatlane

and Hayley Ncube for their constant support on this journey.

Table of Contents
Declaration ......................................................................................................................................................... ii

Dedication.......................................................................................................................................................... iii

Acknowledgements ........................................................................................................................................... iv

List of Figures................................................................................................................................................... vii

List of Tables ................................................................................................................................................... viii

Definition of abbreviations, terms and symbols ................................................................................................. x

1. INTRODUCTION ..................................................................................................................................... 1

1.1 Background ........................................................................................................................................ 1

1.2 Research gap and objectives .............................................................................................................. 2

1.2.1 Motivation of the study .............................................................................................................. 2

1.2.2 Research objective(s) and question(s) ....................................................................................... 3

1.2.3 Potential benefits ....................................................................................................................... 3

1.2.4 Hypotheses testing ..................................................................................................................... 4

1.3 Structure of the study ......................................................................................................................... 4

2 LITERATURE REVIEW .......................................................................................................................... 5

2.1 The Efficient Markets Hypothesis ..................................................................................................... 5

2.2 The Non-Assumption Issues with the Efficient Markets Hypothesis ................................................ 7

2.3 A dynamic approach to efficiency: The AMH .................................................................................. 9

2.4 Tests of the AMH in general cryptocurrency markets ..................................................................... 11

2.5 Tests of Bitcoin Efficiency .............................................................................................................. 12

2.6 A Review Of Return Forecasting Techniques ................................................................................. 14

2.7 The Efficient Markets Hypothesis ................................................................................................... 18

2.8 Summary of literature review .......................................................................................................... 19

3 DATA AND METHODOLOGY ............................................................................................................ 20

3.1 Data.................................................................................................................................................. 20

3.2 Methodology .................................................................................................................................... 20

3.2.1 Linearity Tests ......................................................................................................................... 22

3.2.2 Normality Tests ....................................................................................................................... 22

3.2.3 Stationarity Tests ..................................................................................................................... 23

3.3 Robustness Checks .......................................................................................................................... 24

3.4 Summary .......................................................................................................................................... 24

4 RESULTS ................................................................................................................................................ 25

4.1 Normality Tests ............................................................................................................................... 25

4.2 Stationarity Tests ............................................................................................................................. 26

4.3 Martingale Difference Hypothesis Tests ......................................................................................... 27

4.4 Linearity Tests ................................................................................................................................. 29

4.5 SETAR/ARIMA modelling without external regressors ................................................................. 31

4.6 Fourier Transform ........................................................................................................................... 38

4.7 Machine Learning Approaches to Modelling the RGP ................................................................... 48

4.8 Machine Learning Approaches to Modelling the RGP (FT) ........................................................... 53

4.9 NARX Modelling ............................................................................................................................ 59

4.9.1 External Regressors ................................................................................................................. 60

4.9.2 Daily and Weekly NARX Models ........................................................................................... 65

4.9.3 Daily and Weekly NARX Models (FT)................................................................................... 69

4.10 Summary .......................................................................................................................................... 73

5 CONCLUSION ....................................................................................................................................... 75

5.1 Limitations ....................................................................................................................................... 77

5.2 Areas of future research ................................................................................................................... 77

REFERENCES .............................................................................................................................................. 79

Appendix A: Further Bitcoin Analysis ............................................................................................................ 86

vii

List of Figures

Figure 4.1: Daily SETAR grid search ............................................................................................................. 31

Figure 4.2: Weekly SETAR grid search .......................................................................................................... 31

Figure 4.3: Daily SETAR Model Residuals .................................................................................................... 35

Figure 4.4: Weekly SETAR Model Residuals ................................................................................................. 35

Figure 4.5: Daily Model Residuals’ ACF ........................................................................................................ 36

Figure 4.6: Weekly Model Residuals’ ACF Plot ............................................................................................. 36

Figure 4.7: Daily AR Model Residuals ........................................................................................................... 46

Figure 4.8: Daily AR(4) Residuals’ ACF Plot ................................................................................................ 46

Figure 4.9: Weekly SARIMA Residuals ......................................................................................................... 47

Figure 4.10: Weekly SARIMA Residuals ACF .............................................................................................. 47

Figure 4.11: Daily LSTM Diagnostics ............................................................................................................ 49

Figure 4.12: Daily Residuals ACF Plot ........................................................................................................... 50

Figure 4.13: Weekly LSTM Diagnostics ......................................................................................................... 51

Figure 4.14: Weekly LSTM Residuals ACF Plot ............................................................................................ 51

Figure 4.15: Daily LSTM Diagnostics (FT) .................................................................................................... 54

Figure 4.16: Daily Residuals Plot .................................................................................................................... 55

Figure 4.17: Daily Residuals ACF Plot ........................................................................................................... 55

Figure 4.18: Weekly LSTM Diagnostics (FT) ................................................................................................ 57

Figure 4.19: Weekly Residuals Plot ................................................................................................................ 58

Figure 4.20: Weekly Residuals ACF Plot ....................................................................................................... 58

Figure 4.21: MSCI World Index...................................................................................................................... 61

Figure 4.22: Disaggregated MSCI World Index.............................................................................................. 61

Figure 4.23: Google Trends Monthly Data ..................................................................................................... 62

Figure 4.24: Disaggregated Bitcoin Trend Data .............................................................................................. 62

Figure 4.25: Federal Funds Rate...................................................................................................................... 63

Figure 4.26: Disaggregated Federal Funds Rate.............................................................................................. 64

Figure 4.27: US Annual GDP .......................................................................................................................... 64

Figure 4.28: Disaggregated US GDP .............................................................................................................. 65

Figure 4.29: Daily NARX Model .................................................................................................................... 66

Figure 4.30: Weekly NARX Residuals ........................................................................................................... 67

Figure 4.31: Daily NARX Residuals ACF ...................................................................................................... 67

Figure 4.32: Weekly NARX Residuals ACF ................................................................................................... 68

Figure 4.33: Daily NARX Residuals ............................................................................................................... 70

Figure 4.34: Weekly NARX Residuals ........................................................................................................... 71

Figure 4.35: Daily NARX Residuals' ACF ..................................................................................................... 71

Figure 4.36: Weekly NARX Residuals ACF ................................................................................................... 72

viii

List of Tables

Table 4.1: Daily Returns’ Normality Tests ..................................................................................................... 25

Table 4.2: Weekly Returns’ Normality Tests .................................................................................................. 25

Table 4.3: Daily Returns’ Stationarity Tests ................................................................................................... 26

Table 4.4: Weekly Returns’ Normality Tests .................................................................................................. 26

Table 4.5: Martingale Difference Hypothesis Tests - Daily Returns .............................................................. 27

Table 4.6: Hurst Exponent - Daily Returns ..................................................................................................... 28

Table 4.7: Martingale Difference Hypothesis Tests – Weekly Returns .......................................................... 29

Table 4.8: Hurst Exponents – Weekly Returns ............................................................................................... 29

Table 4.9: BDS Tests – Daily Returns ............................................................................................................ 30

Table 4.10: BDS Test - Weekly Returns ......................................................................................................... 30

Table 4.11: Best Daily SETAR Model Inputs ................................................................................................. 32

Table 4.12: Best Daily SETAR Model Inputs ................................................................................................. 32

Table 4.13: Daily SETAR Model .................................................................................................................... 33

Table 4.14: Weekly SETAR Model ................................................................................................................ 34

Table 4.15: Daily Residuals’ Normality Tests ................................................................................................ 37

Table 4.16: Hurst Exponent – Daily Residuals ............................................................................................... 37

Table 4.17: Weekly Returns’ Normality Tests ................................................................................................ 38

Table 4.18: Weekly Model Residuals Hurst Exponent ................................................................................... 38

Table 4.19: Martingale Difference Hypothesis Test – Daily Returns ............................................................. 39

Table 4.20: Hurst Exponent - Daily Returns ................................................................................................... 40

Table 4.21: BDS Test - Daily Returns ............................................................................................................. 40

Table 4.22: SETAR Test ................................................................................................................................ 41

Table 4.23: Martingale Difference Hypothesis - Weekly Returns .................................................................. 41

Table 4.24: Hurst Exponent - Weekly Returns ................................................................................................ 42

Table 4.25: BDS Test - Weekly Returns ......................................................................................................... 42

Table 4.26: Daily ARIMA Model ................................................................................................................... 43

Table 4.27: Weekly SARIMA Model .............................................................................................................. 43

Table 4.28: Daily Model Residual Normality Tests ........................................................................................ 44

Table 4.29: Daily Residuals Hurst Exponents ................................................................................................. 44

Table 4.30: Weekly Model Residuals Normality Tests ................................................................................... 45

Table 4.31: Weekly Residuals Hurst Exponents ............................................................................................. 45

Table 4.32: Daily LSTM Model ...................................................................................................................... 48

Table 4.33: Weekly LSTM .............................................................................................................................. 49

Table 4.34: Martingale Hypothesis Test - Daily Residuals ............................................................................. 52

Table 4.35: Daily Residuals Hurst Exponent .................................................................................................. 52

Table 4.36: Martingale Hypothesis Test – Weekly Residuals ......................................................................... 53

Table 4.37: Weekly Residuals Hurst Exponent ............................................................................................... 53

Table 4.38: Daily LSTM Model ...................................................................................................................... 54

Table 4.39: Martingale Hypothesis Test - Daily Residuals ............................................................................. 56

Table 4.40: Daily Residuals Hurst Exponent .................................................................................................. 56

Table 4.41: Weekly LSTM .............................................................................................................................. 57

Table 4.42: Martingale Hypothesis Test – Weekly Residuals ......................................................................... 59

Table 4.43: Daily Residuals Hurst Exponent .................................................................................................. 59

Table 4.44: Daily NARX Model ..................................................................................................................... 65

Table 4.45: Weekly NARX Model .................................................................................................................. 66

Table 4.46: Martingale Difference Hypothesis Tests - Daily Residuals ......................................................... 68

Table 4.47: Martingale Difference Hypothesis Test – Weekly Residuals ....................................................... 69

Table 4.48: Daily NARX Model ..................................................................................................................... 69

Table 4.49: Weekly NARX Model .................................................................................................................. 70

Table 4.50: Martingale Difference Hypothesis Tests - Daily Residuals ......................................................... 72

Table 4.51: Martingale Difference Hypothesis Tests – Weekly Residuals ..................................................... 73

Definition of abbreviations, terms and symbols

Adaptive Markets Hypothesis: An expanded view of market efficiency that makes provision for

changes in the market’s underlying conditions.

Bitcoin: An internet-based currency whose authenticity is governed by cryptographic evidence.

Cryptocurrency: A class of internet-based currencies that are backed by cryptographic evidence that

is presented to an online ledger. Their exchange is based on near-untraceable addresses.

Adaptive Efficiency: A view of efficiency from Lo (2004) that changes over time in line with changes

in the market’s conditions, including the level of competition and access to arbitrage opportunity.

This is used interchangeably with “time-varying efficiency” and “cyclical efficiency”.

Homo economicus: An idealized human being whose decision making is perfectly rational. Homo

economicus is often invoked in economic literature to simplify models.

Hurst Exponent: A measure of dependency in a time series. A Hurst Exponent higher than 0.50

indicates that the time series under investigation exhibits long-memory.

Long Short-Term Memory (LSTM): A class of artificial recurrent neural networks that is specialized

in processing input sequences and making a contextual prediction about the next step in the sequence.

They are often used in time series forecasting.

Martingale: A statistical process where the best predictor of the process’ next value is its current

value.

Mean Absolute Percent Error (MAPE) criterion: A measure of the predictive accuracy of a

(regression) model.

Recurrent Neural Network (RNN): A class of models that use their memory to enable them to process

input information and employ it in their prediction of an output.

Satisficing: A process where an individual does not pick the most optimal solution to a problem, but

instead chooses a satisfactory solution.

SETAR: A time series approach to modelling a process. It differs from the traditional autoregressive

models in its ability to capture regime switches.

1. INTRODUCTION

1.1 Background

Cryptocurrencies have been the subject of intense academic debate: are they a currency, an

asset, something that possesses the features of both, or an entirely different category of security

altogether? (Böhme et al., 2015) In contrast to conventional currencies backed by central

authorities, cryptocurrencies are digital currencies whose exchange is backed by cryptographic

evidence from a network of linked computers, which are collectively referred to as miners

(Selgin, 2015). Despite their introduction to the world as a currency, questions remain about

their effectiveness as a store of value and whether they will ever acquire mass acceptance as a

form of payment (Bariviera et al., 2017).

In a similar fashion, Bitcoin has attracted plenty of debate and scrutiny. The initial commentary

regarding characterised it as either a fad or tool for the exclusive use of those interested in

criminal activity (Rose, 2015). Bitcoin has also been accused of being a wholesale fraud that

is bound to collapse given its small market relative to other established currencies such as the

United States Dollar (USD) (Monaghan, 2017). Over time, that initial spirit of derision has,

however, given way to a more measured approach to the study of Bitcoin’s properties, such as

whether it is informationally efficient (Kristoufek, 2018a).

Fama (1970) defines an informationally efficient market as one in which the return of the asset

in the market reflects all available information. Efficiency can take any of the three distinctive

forms based on the information the asset incorporates: a weak, semi-strong as well as a strong

form. Weak-form efficiency, for example, describes a scenario where a security’s historical

return and returns data is reflected in the security’s current return.

Whether a single security (that exists alongside similar assets) can be tested for “market”

efficiency is a relevant point of discussion. While the term “market” is often associated with

an assortment of goods and services for sale, there can exist a market for a singular asset, as

evidenced by the market for, say, gold. To fully draw out the reductio ad absurdum, it can be

noted that the gold market is not made any less real by the fact that precious metals (and

therefore a market for precious metals) as a real category, exists. In sum, the existence of a

market for a category of assets does not preclude the existence of a market for a singular asset.

Moreover, the extant literature is awash in studies of singular assets, such as studies that test

whether gold is weak form efficient (Tschoegl, 1980).

In a similar vein, studies have examined whether Bitcoin is weak form efficient (Kristoufek,

2018a). While the conclusion of many of these studies has been that Bitcoin is not weak form

efficient in the manner envisaged by Fama (1970), namely that Bitcoin returns do not follow a

random walk/martingale (i.e., past information is useless in predicting the future Bitcoin

returns,) a new framework for understanding market efficiency has come into focus: the

Adaptive Markets Hypothesis (AMH). It posits that efficiency is not a binary state, where an

asset is either efficient or not. Instead, it argues that efficiency changes over time in reaction to

changes in liquidity, the number of market participants and competition. Therefore, while

Bitcoin may not fit the description of efficiency as envisaged by Fama (1970), it may well fit

the Lo (2004) view of efficiency. This study aims to investigate whether it does.

1.2 Research gap and objectives

1.2.1 Motivation of the study

This study will approach Bitcoin, the largest cryptocurrency by market capitalization, as an

asset. To this end, this study aims to address the market efficiency of Bitcoin from an Adaptive

Markets Hypothesis (AMH) perspective. In essence, it will test whether the market efficiency

of Bitcoin changes over time. Whilst tests of the informational efficiency of Bitcoin have been

conducted, this study aims to investigate whether Bitcoin is cyclically efficient using the robust

framework developed in Seetharam (2016).

Furthermore, several events have occurred which could have a significant impact on the

efficiency of Bitcoin. The world-wide spread of Covid-19 has seen a marked increase in interest

regarding cryptocurrencies, leading to Bitcoin’s market capitalization reaching over $700

billion as of May 2021 (Inman, 2020). The resultant financial climate — where relaxed fiscal

and monetary policies have been become a mainstay — has led many investors to look to

Bitcoin as a “safe haven asset” and a source of exposure to an asset class that is imperfectly

correlated with their portfolios (Shahzad et al., 2019).

Finally, cryptocurrency exchanges have also grown during the pandemic, with Coinbase (the

leading exchange) ultimately listing to extremely positive reception on the New York Stock

Exchange (NYSE) in April 2021 (Browning, 2021). The existence of these exchanges has led

to the mainstreaming of what was once a “rogue and subculture” asset by opening the door to

investors to easily acquire cryptocurrencies (Crabb, 2021).

This provides a ripe opportunity to study whether Bitcoin fits within the view of efficiency as

envisaged by the AMH.

1.2.2 Research objective(s) and question(s)

The objective of this research report is to test whether Bitcoin is informationally efficient in a

manner consistent with the AMH. The AMH predicts varying degrees of efficiency and

inefficiency over time; as such, there will be varying levels of asset return change predictability

(Lo, 2004). Due to changes in these factors over time, it is expected that the data will exhibit

time-varying levels of predictability. As such, this study will investigate whether Bitcoin has

varying levels of efficiency over time.

1.2.3 Potential benefits

While there have been many studies have tested whether the AMH holds with respect to

traditional asset such as stocks and futures, much less focus has been devoted to testing whether

it is a good explanation of market efficiency in the non-traditional asset classes such as

cryptocurrency. As a result, this study is aimed to address that gap and adapt models that have

shown success in showing cyclical market efficiency in traditional asset classes (using support

vector machines).

Secondly, given the elevated levels of attention and scepticism that Bitcoin and other

cryptocurrencies have received in both the academic literature and society in general, it is

imperative that further light is shed on Bitcoin. While this study does not delve into the

mechanics behind deciding whether to include cryptocurrencies in the portfolio formation

process, it nevertheless is aimed at demystifying the cryptocurrency world. Having a clearer

picture of the properties of cryptocurrencies allows for a more sober conversation regarding

them, particularly as they have gained more significance not just in academia, but in society in

general. El Salvador just became the first sovereign nation to adopt Bitcoin as legal tender,

other nations are also investigating whether they should launch virtual versions of their state

currencies.

1.2.4 Hypotheses testing

Primary Hypothesis

H0: Bitcoin efficiency is not cyclical.

H1: Bitcoin efficiency is cyclical.

Secondary Hypotheses

H0,A: Bitcoin returns do not follow a random walk/martingale.

H1,A: Bitcoin returns do follow a random walk/martingale.

H0,B: Bitcoin returns cannot be modelled via an autoregressive function with no exogenous

inputs.

H1,B: Bitcoin returns can be modelled by an autoregressive function with no exogenous inputs.

H0,C: Bitcoin returns cannot be modelled via an autoregressive function with exogenous inputs.

H1,C: Bitcoin returns can be modelled by an autoregressive function with exogenous inputs.

1.3 Structure of the study

This research is structured in five chapters and will proceed as follows: Chapter 2 surveys the

efficiency hypothesis literature, from the Efficient Markets Hypothesis (EMH) to its

continuation, the Adaptive Markets Hypothesis, and ends with an examination of the asset

return forecasting literature. Chapter 3 describes the data, variables, hypotheses and

methodology of the research report. This delves into the data and methods that test for whether

the Adaptive Markets Hypothesis is a good explanation of Bitcoin return behaviour as well as

presenting several models that are used to determine the return generating process. Chapter 4

shows the empirical results of the research report and provides a discussion of the results

Chapter 5 concludes the research report.

2 LITERATURE REVIEW

Whether one can predict returns of assets is a question whose answer possesses much value for those

asking it. This is because if return prediction is possible, then risk-free profits can be extracted by

exploiting one’s ability to predict returns. The academic literature is replete with articles that attempt

to answer this question, the dominant and novel approaches are discussed below:

2.1 The Efficient Markets Hypothesis

The Efficient Markets Hypothesis (EMH) is a description of market dynamics in a state of equilibrium

— an efficient market always fully reflects available information (Fama, 1970). The EMH is

subdivided into three forms based on the amount of information that is incorporated in the asset’s

return: the weak form, semi-strong form as well as its strong form (Lawrence et al., 2007). In its weak

form, the EMH describes a state in which all the historical returns and returns data of the asset is

incorporated into its return. The returns of the security therefore follow a random walk — returns are

independent and identically distributed, i.e., a return at one point in the time series does not affect

another and the returns are drawn from the same distribution.

This is in line with the argument advanced by Bachelier (1900) that market speculation should be a

“fair game”: the expected value of engaging in market speculation should be equal to zero, i.e., price

changes follow a martingale. An asset’s return series follows a martingale when the best predictor of

future returns, given the current information set, is the unconditional expectation of today’s price

(Escanciano & Velasco, 2006). As such, like the random walk view of efficiency, one cannot use past

information to predict future returns when the asset’s return series follows a martingale. While the

random walk and martingale hypotheses are distinct, Fama (1970) argues for their fungibility in

discussing weak-form efficiency as, in both cases, they predict that an asset “fully reflects” the

information set when the use of trading rules does not yield above buy-and-hold returns. As such, the

weak-form of the EMH negates the use of technical analysis, the use of historical data in predicting

future asset returns (Malkiel, 2003).

In the semi-strong form, all publicly available information is incorporated into the asset’s return

(Fama, 1970). As a result, the semi-strong form of the EMH would negate the use of fundamental

analysis — the analysis and use of financial information of the assets to predict stock returns (Malkiel,

2003). In its strong form, the EMH describes a state in which, all available information (both private

and public) is reflected in the asset’s return. This would negate the use of private information in the

prediction of future asset returns.

Empirical evidence for and against the EMH spans many asset classes in the literature. Grater and

Struweg (2015) examine logarithmic returns of stocks on the Johannesburg Stock Exchange (JSE)

and find evidence of a unit root, thereby rejecting the null of weak-form efficiency. In examining the

JSE for weak-form efficiency, however, Heymans and Santana (2018) present evidence that

contradicts the earlier evidence brought forth by Grater and Struweg (2015). Their findings suggest

that while the overall JSE is weak-form efficient, some of its sub-indices are not. This presents prima

facie evidence that runs counter to Samuelson’s dictum: the postulation that the market is “micro-

efficient” but not efficient at aggregate stock market level (Jung & Shiller, 2005).

Informational efficiency studies in other asset classes have also produced mixed results. Kristoufek

and Vosvrda (2014) find that while energy commodities (such as heating oil) are weak-form efficient,

agricultural commodities are not. Arouri et. al (2012) examined whether precious metal and energy

markets are weak-form efficient. Their results indicated that in the short-run, the null of efficiency is

rejected for these markets, while presenting mixed outcomes in the long-run.

While the EMH been widely regarded as the dominant model for describing market dynamics, it has

also attracted criticism. Its numerous and overly generous assumptions have been the centre of

controversy. It portrays human beings as infallible Bayesian agents, agents with well-calibrated

expectations that are updated in line with new information, who act in their own best interests.

(Sandroni, 2005) While this description is flattering given that we are human, it is demonstrably

wrong. For instance, Simon (1956) shows that instead of engaging in complete optimization, humans

instead “satisfice” (opt for solutions which may not be fully optimal but are nevertheless sufficient in

order to achieve whatever aims the individual has). Similarly, Tversky and Kahneman (1991)

formally demonstrate that humans exhibit biases such as loss aversion. The existence of these

limitations in human computational ability fundamentally undermines a realistic prospect of the EMH

holding in the real world (Lo, 2004). It stands to reason thus, that a model built on such faulty

assumptions can therefore be discarded.

Besides the computational limits that prohibit us from behaving in a similar fashion to the homo

economicus that inhabits the economic literature, further limits exist which similarly threaten the

EMH paradigm: limits to arbitrage (Thaler, 1999). While not explicitly stated, the EMH precludes

persistent arbitrage opportunities since informed speculators can exploit them, thus nullifying them

in the process (Alvarez-Ramirez et al., 2012). In practise however, due to lack of access to capital, a

clearly risk-free opportunity remains unexploited. As a premier example Thaler (1999) refers to

LTCM Capital and their Siamese-Twin (Royal Dutch/Shell) risk-free investment: due to dislocations

elsewhere in the market, they imploded before realizing a substantial return on rightfully predicting

that the Royal Dutch/Shell share return misalignment would be corrected. Similarly, Khandani and

Lo (2007) examine the 2007 hedge fund (statistical arbitrage) crunch and note dislocations in other

parts of the market than those arbitrageurs are invested in may severely impact their ability to execute

otherwise risk-free and profitable strategies. This shows that sometimes, external factors can inhibit

market participants’ ability to exploit risk-free opportunities. This should be viewed as a prima facie

threat to the viability of the EMH.

2.2 The Non-Assumption Issues with the Efficient Markets Hypothesis

A conundrum facing any proponent of the EMH is the arbitrage paradox. It follows from the weak-

form EMH that historical return data should therefore be of no use in forecasting future returns

(Lucas, 1978). Indeed, Lo (2004) argues that if the EMH is correct, then there is no room for arbitrage.

Given this, then no incentive would exist for investors to gather information that is useful in their

quest towards finding risk-free profits. Given that information-gathering agents are a prerequisite of

informationally efficient markets, the absence of arbitrageurs, would therefore result in the collapse

of the markets (Grossman & Stiglitz, 1976). This paradox (information gatherers are required and yet

are not rewarded with risk-free profits and therefore do not exist) illustrates that there is a fundamental

problem with the EMH.

In the same paradoxical vein, Lo (2004) jokes that a classical economist walking down a crowded

street would be forced to abandon a low-lying $100 because they would reason, in an EMH world,

that “if it were real, someone else would have picked it up”. If everyone applied this logic, the $100

would remain uncaptured because no one would pick it up, in the same way, markets would not be

efficient since arbitrage opportunities would remain unexploited if no arbitrageurs existed. Thus,

perfectly efficient markets are an impossibility.

Another fundamental issue that undermines the EMH is excess volatility. While returns should (in a

rational world) only occur due to the arrival of news, Shiller (1981) notes that the movement in

aggregate stock returns is in excess of what can be explained by the EMH. Thaler (1999) summarized

it thus: “bond and stock returns are more volatile than advocates of rational efficient market theory

would predict”. It is an indictment that a theory whose stated aim is to explain what Fama (1970)

termed the “accumulation of evidence” fails to explain evidence before it.

There are other predictive failures that have hampered the EMH: the random walk model predicts

that asset returns cannot be predicted using publicly available data. However, much evidence has

been adduced which shows that some widely known seasonal patterns persist in asset returns. Among

them is the (in)famous January effect where, returns of companies that previously performed poorly

in the previous 11 months drastically improve in January (Thaler, 1987).

There have been some attempts to explain the persistence of the January effect including the tax-loss

selling hypothesis - the notion that money managers sell-off “loser” stocks in order to realize their

losses on them with the intent of minimizing their tax burden (Haug & Hirschey, 2006). In other

academic circles, the “window dressing” hypothesis — money managers sell off “losers” in order to

have a portfolio of “winners” in time for their reporting period and pick up those assets after their

reporting period, causing improved demand and thus, superior returns — has also found favour

(Maxwell, 1998). There is evidence that the January effect is a persistent, global anomaly (Shen et

al., 2020). Without litigating the validity of each of its proposed explanations, one may recognize that

its mere existence threatens the EMH. It is a barefaced counterexample to the EMH: one can predict,

based on publicly available data that poorly performing shares will perform better in January.

Finally, many have argued that the EMH is not testable, and thereby falsifiable. Karl (2013) argues

that the line between science and pseudoscience is in whether the models can be falsified; and the

EMH given our inability to falsify it, is therefore relegated to the leagues of pseudoscience. To remedy

this, a more dynamic model that is inspired by complex systems, evolutionary psychology and

behavioural ecology is developed that harmonizes the EMH with these and other issues and is

(crucially) falsifiable: the AMH (Lo, 2004).

2.3 A dynamic approach to efficiency: The AMH

Instead of treating the frailties of the EMH as deviations from otherwise efficient markets, or

irrelevances to be rationalized away, Lo (2004) argues for a model of efficiency that also accounts

for the EMH anomalies, termed the Adaptive Markets Hypothesis (AMH). This model incorporates

various subjects inter alia; Behavioural Ecology, Complex Systems, Sociology and Psychology

(Seetharam, 2016).

At its point of departure, the AMH argues that humans have evolved to make decisions that maximise

the survival probability of their genetic material (Lo, 2004). In contrast, the current orthodox view in

Economics is that humans make decisions which are rationally optimal (Cox & Hayne, 2006). While

those views are not inherently mutually exclusive (the survival imperative is not manifestly opposed

in any way to rational optimization) there is some degree of difference between the choice architecture

in each model. This distinction is encapsulated in the idea that instead of making fully optimized

choices, humans are instead engaged in “satisficing” — optimization subject to constraints such as

time or computational ability (Simon, 1956). This can be distilled to “trying as best as humanly

possible to make the best decision” (Seetharam, 2016).

An evolutionary answer to how an individual begins to “satisfice” is converged upon through the

processes of random mutation and natural selection: The process begins when an individual attempts

to solve a problem by implementing a random solution, if it works (survives the selection process), it

is then kept, if it fails, then it is rejected (Lo, 2004). As a result, several permutations of random

solutions with refinements are selected for (or against) based on environmental pressures are

implemented until a satisfactory solution is converged upon. This solution, given a stable

environment, propagates and becomes entrenched. The individual, faced with a similar-enough

problem will utilize the satisfactory solution. Once the terrain is sufficiently distinct such that the

current solution is not applicable, a new solution needs to be found. The above description captures

how human beings (and agents in financial markets) learn.

Lo (2004) further argues that not changing tack in the face of ecological changes — continuing to

apply a previously-successful solution to an environment for which that solution is not satisfactory

— is tantamount to “irrationality”. In the financial context, this would emerge in such a context as,

say, the application of a trading rule past its useful shelf-life. Black (1986) argues that in the financial

jungle, funds are distributed away from those who mistake noise for a signal and trade on it (so-called

“irrational traders”) to information gatherers who trade on genuine signals. This creates an incentive

for information gathering and ensures that as much arbitrage is mopped up as possible, thus reducing

overall sub-optimal behaviour within the markets (Lo, 2004). It therefore stands to reason that

survival will be arrogated by those whose financial frameworks are “fit”, or in this case,

“satisfactory”.

In sum, the AMH holds that market efficiency is a function of the environment — the number of

market participants, the levels of competition among those participants and the availability of risk-

free opportunities (Lo, 2004). As such, the AMH conceives of efficiency in a dynamic fashion and

argues that it instead follows a cyclical process in line with changes in the environment. When

opportunities for profit are plentiful and competition is low, there is low efficiency and when harsher

conditions set in, market efficiency ticks up (Noda, 2016).

While the AMH and EMH both employ the term “efficiency”, what each hypothesis refers to is unique

to the specific hypothesis (Auret & Vivian, forthcoming). Thus, while Bitcoin may not be weak form

efficient in the manner described by the EMH (its returns follow a random walk), it may nevertheless

follow a cyclical process of efficiency, where at some points, it is efficient, and at others, it is not.

2.4 Tests of the AMH in general cryptocurrency markets

Given the novelty of cryptocurrencies, the literature that is aimed at determining its efficiency is also

recent. The initial set of studies aimed to directly test the Fama (1970) conception of market efficiency

by testing whether a cryptocurrency’s returns follow a random walk or martingale. The random walk

model consists of two hypotheses; firstly, that returns are identically distributed and that they are

independent. This is represented as: 𝑓(𝑟𝑗,𝑡+1|𝜙𝑡) = 𝑓(𝑟𝑗,𝑡+1). In this form, the random walk model

shows that the marginal, 𝑓(𝑟𝑗,𝑡+1|𝜙𝑡) , and conditional probability, 𝑓(𝑟𝑗,𝑡+1) , distributions of an

independent random variable are identical (Fama, 1970).

Given this formulation, the extant literature has tested whether an asset’s (either logarithmic or

differential) returns follows a random walk by examining whether its returns show significant

autocorrelation: if the returns process displays significant autocorrelation, the null of a random walk

can be rejected (Frennberg & Hansson, 1993). An asset whose returns do not follow a random walk

is therefore regarded as not being weak-form informationally efficient. In addition, the adaptive

markets hypothesis posits that assets have periods where they are efficient and periods in which they

are not (Lo, 2004). Whether cryptocurrencies are efficient in the manner envisaged by the AMH

hinges on whether they exhibit this time-varying efficiency.

In this regard, several multi-cryptocurrency studies have emerged. Kristoufek and Vosvrda (2019)

examined whether a collection of prominent cryptocurrencies are efficient by employing the

econophysics-based Efficiency Index which among other things, tests for, long range dependence and

entropy. Their findings suggested that all cryptocurrencies under investigation, including Litecoin,

Ripple and Bitcoin, were not weak-form efficient. They also found that there are periods in which the

cryptocurrencies were efficient, hence making a clear interpretation in favour of adaptive efficiency.

Similar multi-currency studies have also uncovered evidence that there are periods of efficiency and

non-efficiency in the cryptocurrency markets. Noda (2021) examined Bitcoin and Ethereum for time-

varying efficiency by employing a generalized least squares-based time-varying model and found that

they indeed exhibited varying degrees of efficiency over time. In the same vein, by examining the

results of a battery of tests that capture time-varying efficiency, Khursheed et al. (2020) found that

Litecoin, Bitcoin, Monero and Steller were indeed time-varyingly efficient. Interestingly, both Noda

(2020) and Khursheed et. al (2020) agreed that Bitcoin has, relative to its peer cryptocurrencies,

longer periods of efficiency.

2.5 Tests of Bitcoin Efficiency

The initial evidence has overwhelmingly been in favour of the conclusion that Bitcoin is not

informationally efficient. Urquhart (2016) found that Bitcoin is not weakly efficient through a set of

tests including the Ljung-Box and Variance-Ratio tests (these test whether there is autocorrelation in

the returns, and thereby test whether the returns follow a random walk). Similarly, Bariviera (2017)

examined Bitcoin returns’ Hurst exponent, this would be an indication of whether there is long term

memory in the returns, which would indicate that Bitcoin is not weak-form efficient. Prior to 2014,

the Hurst exponent often was above 0.5 — indicating that there is persistence in the returns — while

post-2014, its value creeped closer to 0.5, showing a lack of memory in the returns. Two conclusions

flow from these findings; firstly, prior to 2014, Bitcoin was not weak-form efficient and, crucially,

Bitcoin (in)efficiency is time-varying.

There has also been counterevidence provided on whether Bitcoin is efficient. In response to Urquhart

(2016), Nadarajah and Chu (2017) investigate whether Bitcoin is really weak-form inefficient. They

show that using an odd power transformation on Bitcoin returns, 𝑅𝑡
𝑚, where m is an odd integer, does

not lead to distortion or information loss. Thus, instead of examining Bitcoin returns, they examine

an odd power transformation of the returns to determine whether Bitcoin is weak-form efficient.

Adopting a similar level of methodological rigour, they employ several tests of efficiency to develop

a rich view of Bitcoin dynamics. Their results indicate that Bitcoin is, in fact, informationally

efficient.

It is worth reiterating that while the Nadarajah and Chu (2017) findings run counter to the current

consensus, further examination of Bitcoin weak-form efficiency by other academics did not vindicate

their results. Kristoufek (2018) employed the relatively novel Efficiency Index, developed by

Kristoufek and Vosvrda (2013) to examine whether Bitcoin was weak-form efficient. It captures both

local herding (through the fractal dimension) and the correlation structure of returns (through long

and short-term memory). The results indicated that between 2010 and 2017, Bitcoin was not weak-

form efficient.

While Lo (2004) outlines an alternative view of efficiency, no mathematical definition against which

assets can be tested against to determine whether they are ‘adaptively efficient’ is provided. Despite

this deficiency, several first-generation tests have been devised to test for whether assets are efficient

in a manner consistent with the EMH. These tests aim to examine whether efficiency is non-static.

(Alvarez-Ramirez et al., 2018) In essence, these tests examine whether a measure of efficiency (such

as the Hurst exponent) changes over time in line with evolving market dynamics.

The current consensus is that Bitcoin has periods where it is informationally efficient and periods

where it is not; a nod to the AMH concept of efficiency. In line with this consensus, Brauneis and

Mestel (2018) argue that the EMH is an inappropriate explanation of cryptocurrency returns. Indeed,

Kristoufek (2018) argues that the informational efficiency of Bitcoin is connected to the

environmental conditions of the market. Similarly, while the results from Urquhart (2016) indicated

that Bitcoin is not weakly efficient, they indicated that it was it was, however, “moving towards”

efficiency. This would support the Lo (2004) notion of efficiency, where efficiency can be moved to

and from based on the prevailing market dynamics.

Further evidence of time-varying efficiency has been uncovered outside the cryptocurrency markets.

By employing time-varying autoregressive (TV-AR) processes to model Japanese stock markets’

(TOPIX and TSE2) returns, Noda (2016) demonstrated that they exhibit various degrees of

informational efficiency over time. This is in line with the earlier work of Ito et al. (2014). They

employed a non-Bayesian time-varying vector autoregressive model (TV-VAR) to estimate the joint

degree of efficiency of the stock markets of the highly integrated G7 countries. Their results indicate

that there are times in which international stock markets are efficient and others in which they are

not.

Seetharam (2016) further outlines a rigorous framework for determining whether an asset is, instead,

time-varyingly efficient, in other words, is efficient in a dynamic manner consistent with the AMH.

It can be outlined as follows:

1) Test whether the returns follow a random walk. If they do, then the analysis stops here as the

asset is weak-form efficient. If they do not, we proceed to modelling the return-generating

process.

2) Test whether an autoregressive (AR) data generating process with no additional lagged

variables, save for lagged variables is suitable for modelling the returns generating process.

The Brock-Dechert-Scheinkman (BDS) test to determine whether to use an ARIMA-family

or STAR family to model the returns generating process. The STAR family of models are

preferred over ARIMA models where the BDS test finds that there is non-linear dependence

in the returns. If the chosen model is found to be suitable but still produces significant constant

or error terms, it implies that additional factors over and above historic returns affect

contemporaneous returns.

3) The return generating process (RGP) is modelled without specifying the functional form of

the model.

If this does not adequately describe the returns the returns-generating process, it implies that there is

some private information that affects the contemporaneous returns. A machine learning approach is

once more used to model the RGP with additional variables.

2.6 A Review Of Return Forecasting Techniques

To test whether a security is weak-form efficient, its return series is examined. If it follows a random

walk or Martingale (which, similar to a random walk, means that past information cannot be used to

predict future returns) process, it is deemed to be weak-form efficient. Failing which, an attempt at

modelling the return generating process can be made. From a statistical perspective, the Box-Jenkins

methodology is often used (Makridakis & Hibon, 1997). It is typified by the use of an autocorrelation

and partial autocorrelation plots of the returns and iteratively estimating an Autoregressive Integrated

Moving Average ― ARIMA (p,d,q) ― model based on each successive model’s diagnostics.

The process can be described as follows:

1) Generate an AR (1) model (the simplest of the ARIMA family) to attempt to model the time series:

𝑋𝑡 = 𝑐 + 𝜑𝑋𝑡−1 + 휀𝑡 (1)

Where 휀𝑡 represents a white-noise process with a mean = 0 and a constant variance = 𝜎𝜀
2.

2) Examine the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots

of the residuals for serial correlation.

3) If any serial correlation exists, use the ACF and PACF plots as guides for the ARMA order of the

model, differencing if necessary.

4) Once the new ARMA (p,q) model is fitted to the data, a new examination of the ACF and PACF

along with other model diagnostics are used to determine whether this model is appropriate for

forecasting. If it is, then it is adopted.

5) If not, then the process returns to step 3), where a new ARMA (p,q) model is generated until one

sufficiently describes the return generating process and thereby leaves no significant autocorrelation

in the residuals.

While this approach is often preferred in time series modelling, it has been shown to deliver inferior

performance to Smooth Transition Autoregressive (STAR) models when there is non-linear

dependence (Feng & Liu, 2003). As such, the decision tool regarding whether a STAR model is to be

used instead of the ARMA (p,q) model is the BDS test for nonlinear dependence (Baum et al., 2021).

Moreover, Crawford and Fratantoni (2003) argue that regime-switching models are a better fit for

markets that have exhibit boom-bust cycles and are non-linearly dependent. In concert, Seetharam

and Britten (2015) show that market cycles can be modelled using a regimes-switching approach such

as a STAR model. The process of finding the best STAR model to use entails using a grid search

approach, by typically homing in on a model with minimal pooled Akaike Information Criterion

(AIC).

While these approaches serve as a good forecasting baseline, they often have been shown to be

inferior in performance to machine learning approaches to forecasting: in the extant literature, there

has been a notable improvement from a model diagnostics perspective when using machine learning

approaches over purely statistical approaches in forecasting returns (Ding & Qin, 2020). Furthermore,

machine learning approaches have the advantage that they do not need to be pre-specified in a similar

fashion to the Box-Jenkins approach (Seetharam, 2016).

Despite their numerous advantages, machine learning approaches are also riddled with a rather severe

drawback: they are often “black-boxes” as they are not interpretable due to their structural complexity

(Rudin, 2019). While the field of interpretable machine learning algorithms is certainly blooming,

some evidence remains that the best (and therefore, most used) algorithms are often uninterpretable.

In fact, Adadi and Berrada (2018) argue that their high predictive accuracy is what ultimately leads

to their inability to be interpreted. As such, there may be some trade-off between accuracy and

interpretability.

It is to be noted, however, that machine learning approaches are not made equal, with respect to the

nature of the task at hand. The K-Nearest Neighbours (kNN) algorithm is almost exclusively

employed in classification tasks (Weinberger & Saul, 2009). In contrast, while Support Vector

Machines (SVMs) used in classification tasks, they have been demonstrated to be equally excellent

in time series forecasting. (Aggarwal et al., 2020)

Another often-used family of deep learning algorithms in time series forecasting, especially in

finance, are recurrent neural networks (RNNs). They, however, suffer from a “vanishing gradient”

problem that results in difficulties in the learning process. (Hochreiter, 1998) The “vanishing

gradient” problem refers to a scenario in which the gradients of a loss function tend to zero as more

layers (which often use a sigmoid activation function) are added thus increasing the difficulty of

training the network. (Li et al., 2019) Long-Short Term Memory (LSTM) are an augmented RNN

that does not suffer from this problem. (Bahad et al., 2019) Their ability to restrain gradients from

exploding or vanishing is attributable to their internal memory state cell’s ability to sustain a continual

error flow. Cao et al. (2019)

Moreover, unlike traditional RNNs, LSTMs filter a time series’ historical data through their three

gates and thereby extract useful information (Hochreiter & Schmidhuber, 1997). It is unsurprising

thus that LSTMs performance in time series prediction outshines that of their vanilla RNN

counterparts. McNally (2016) demonstrates that an LSTM network outperforms both a Bayesian

optimised RNN network and ARIMA model in time series prediction. In addition, LSTMs can be

used in combination with other methods in order to improve their predictive capabilities. Cao et

al. (2019) demonstrate that a CEEMDAN-LSTM — where the original time series is decomposed

into a finite number of intrinsic mode functions (IMF) prior to reconstructing the signal —

outperforms a vanilla LSTM in financial time series prediction. Li et al. (2019) also show that an

evolutionary attention-based LSTM (EA-LSTM) similarly outperforms a vanilla LSTM in time

series prediction. It avoids local minima traps by employing a competitive random search (CRS)

instead of a gradient-based approach to solve the focus layer weights. The R packages, “keras” and

“tensorflow” have machine learning capabilities and thus enable the use of Recurrent Neural

Networks, including LSTMs.

Seetharam (2016) found that non-linear autoregressive (NARX) models can both capture non-linear

dependency in asset returns. NARX models have found favour in engineering and other physical

sciences. Cadenas et al. (2016) show that NARX models are useful in predicting physical phenomena

such as wind speed. Similarly, Pisoni et al. (2009) demonstrate that air pollution (specifically above-

threshold pollution in cities) can be forecasted with the assistance of a NARX model. It is represented

as:

𝑦𝑡 = 𝐹(𝑦𝑡−1, 𝑦𝑡−2, 𝑦𝑡−3, … , 𝑢𝑡 , 𝑢𝑡−1, 𝑢𝑡−2, … ) + 휀𝑡 (2)

Where F is a neural network, such as a Support Vector Machine (SVM), 𝑦𝑡 represents the return

generating process and 𝑢𝑡−𝑘 and 𝑦𝑡−𝑘−1 represent the exogenous series and lagged values of the

returns, respectively.

The use of Support Vector Machines (SVMs) in predicting financial time series is well documented.

Kim (2003) demonstrates that SVMs outperform back-propagated neural networks in forecasting

financial index’ returns. He argues that this is attributable to SVMs being more generalizable due to

their implementation of the structural risk minimization principle. Cao and Tay (2001) also show that

SVMs outperform multi-layer perceptron networks in forecasting financial returns. The “mltsp” and

“e1071” packages have NARX model-building capabilities that allow one to use a NARX model

Given the demonstrated relative advantages of SVM and LSTM approaches to forecasting financial

time series, they are employed in the machine learning section of Bitcoin return forecasting. This is

done in sequential order; the LSTM approach is employed in univariate forecasting whereas the SVM

is used in the forecasting Bitcoin return with the help of external regressors.

2.7 The Efficient Markets Hypothesis

As with most field marred by controversy, there has been a clash of findings regarding assets’

efficiency. For example, while most studies such as Kristoufek (2018) reject the view that Bitcoin is

weak-form efficient, Nadarajah and Chu (2017) find that an odd-power transformation of Bitcoin

returns, 𝑅𝑡
𝑚, is weak-form efficient. Given the divergent findings in the literature, it is imperative thus

that attempts to examine the efficiency of Bitcoin returns be run on a parallel twin set to have results

that are robust to noise; a return series that has been transformed as well as a non-transformed Bitcoin

return series.

An often-used transformation is the Fourier Transformation. It is defined as:

𝐹𝑇(𝜔) = ∫ 𝑠(𝑡)𝑒−𝑗𝜔𝑡𝑑𝑡

∞

−∞

(3)

Where 𝐹𝑇(𝜔) is the Fourier Transform of the time series, 𝑠(𝑡), which itself has been decomposed

for the purposes of transforming it. (Alrumaih & Al-Fawzan, 2002) Applying a Fourier Transform to

a time series has an overall impact of “de-noising” it. (Jansen van Rensburg & van Vuuren, 2020) It

is thus regarded in extant literature as “cleaning” the data with the aim of having a better calibrated

model of the data-generating process. Having a filtered data set to compare against a non-transformed

data set also allows for a more robust set of results.

In cases where the return series is shown not to follow a martingale or random walk, an attempt to

model it can be done. Given that additional factors outside of historical returns can affect returns,

their use as exogenous regressors in modelling returns may prove to be of use. Vo et al. (2021)

demonstrate that US GDP data has a positive impact on Bitcoin returns. Similarly, Panagiotidis et. al

(2019) found that traditional stock markets have a strong relationship with Bitcoin returns. Their

results, however, contrast with those of Vo et al. (2021) in that they show that there is a weak

relationship between Bitcoin returns and the macroeconomy. Li et al. (2021) show that there is a

strong link between social media coverage and Bitcoin returns. In a similar vein of connecting the

public consciousness to Bitcoin returns, Dastgir et al. (2019) show that there is a bi-directional causal

link between Bitcoin returns and Google Trends searches. Given that these factors are consistently

shown to affect returns, Seetharam (2016) argues that they need to be incorporated into the return-

generating process model in the spirit of the Arbitrage Pricing Theory (APT).

2.8 Summary of literature review

This chapter walked through both the early views of efficiency, delineated between the random walk

and martingale views of weak-form efficiency, and delved into the issues that plague the Efficient

Markets Hypothesis. The focus was then shifted onto the Adaptive Markets Hypothesis and the

evidence in its favour in extant literature. A discussion on whether the general cryptocurrency

market’s dynamics are described better by the EMH or the AMH followed, followed by a focused

overview of the evidence of the AMH’s explanatory power with regards to Bitcoin dynamics. When

an asset is weak-form efficient (and by corollary, follow a martingale or random walk, their returns

cannot be forecasted). As such, literature regarding techniques used in return forecasting for assets

that are not weak-form efficient is entertained. To round off, a discussion on best practices in

forecasting was explored.

3 DATA AND METHODOLOGY

The data and methodology that are used to test the main and sub-hypotheses are described below:

3.1 Data

This study will use the daily closing return data of Bitcoin from Coindesk from Oct 2013 to Oct 2021.

This selection of dates is due to the availability of Bitcoin return data from Coindesk. The closing

return data is then transformed into a log return using the familiar formula:

Rt = ln(
𝑃𝑡

𝑃𝑡−1
) (4)

Where Pt and Pt-1 represent prices at time t and t-1 closing prices, respectively. The data was collected

at three different frequencies, viz., daily, weekly and monthly.

Macroeconomic, fundamental, and behavioural data was also obtained that will be included in the

exogenous model in a similar fashion to the Arbitrage Pricing Theory (APT). These variables include

US GDP data, the MSCI World Index, US Fed Funds rates, Bitcoin Google Trends Data. Like the

Bitcoin return data, the exogenous variable data will also be collected at three different frequencies

— daily, weekly and monthly. These variables were selected based on previous literature which has

demonstrated their capability in predicting Bitcoin returns. Vo et al. (2021) demonstrate that US GDP

data affects Bitcoin returns. Similarly, Demir et al. (2018) also showed that economic uncertainty has

a relationship with Bitcoin returns.

3.2 Methodology

The methodology is outlined (broadly) in the three steps below:

1) Test whether the returns follow a random walk using a battery of tests that includes the

Automatic Portmanteau (AQ), Generalized Spectral (GS) and Dominguez-Lobato (DL) tests.

If they do, then that is evidence in favour of the weak form of the EMH. If not, it can thus be

established whether modelling the returns is feasible.

2) A BDS test is then used to determine whether the returns follow a linear process, if they do

an autoregressive integrated moving average (ARIMA) process is used to model the return

generating process. If not, then the return generating process is modelled using a regime-

switching autoregressive (AR) process with no additional (exogenous) variables, save for the

lagged dependent variable (SETAR) is used. The general form of Threshold Autoregressive

Model (TAR) family of models, which SETAR models form a part of, are represented as:

𝑥(𝑡) = 𝜑𝑗(0) + ∑ 𝜑𝑗 + (𝑖)𝑥(𝑡 − 𝑖) + 𝑎𝑗(𝑡)

𝑛𝑗

𝑖=1

(5)

Where x(t), the variable’s current value is determined based on its own lagged values, x(t-i).

3) If this successful, this is evidence of the semi-strong form of the EMH (public information is

incorporated into the security’s return). If the model contains a significant constant or error

terms, that implies that significant additional factors over and above the lagged dependent

variable have an impact on the contemporaneous returns.

4) If there are significant additional factors that affect the returns contemporaneously, a Long

Short Term Memory network (LSTM) is utilized to model the data generating process. It is

used without additional variables. An LSTM has three sigmoid states that are known as the

forget gate, input gate and the output gate (Hochreiter & Schmidhuber, 1997). These are

represented as:

𝑓𝑡 = 𝜎(𝑥𝑡 ∗ 𝑈𝑓 + 𝐻𝑡−1 ∗ 𝑊𝑓) (6)

𝑖𝑡 = 𝜎(𝑥𝑡 ∗ 𝑈𝑖 + 𝐻𝑡−1 ∗ 𝑊𝑖) (7)

𝑜𝑡 = 𝜎(𝑥𝑡 ∗ 𝑈𝑜 + 𝐻𝑡−1 ∗ 𝑊𝑜) (8)

Where each gate receives, processes and passes the signal on to the other gates in the following

order: input(i), forget(f), which regulates which data to throw out and retain, and output (o).

5) If the model is inadequate to describe the return generating process, the enquiry moves to the

final step.

The second neural network, a non-linear autoregressive exogenous model (NARX), is employed. It

models the returns generating process through using both lagged returns and additional exogenous

variables that are incorporated in line with the Arbitrage Pricing Theory (APT). It is represented as,

𝑦𝑡 = 𝐹(𝑦𝑡−1, 𝑦𝑡−2, 𝑦𝑡−3, … , 𝑢𝑡 , 𝑢𝑡−1, 𝑢𝑡−2, … ) + 휀𝑡 (9)

where F is a neural network and 𝑦𝑡 represents the return generating process. 𝑢𝑡−𝑘 and 𝑦𝑡−𝑘−1

represent the exogenous series and lagged values of the returns, respectively. A neural network

approach in both regards is advantageous given that it allows for modelling of the DGP without the

requirement that the functional form of the model be specified in advance. It has been shown to be

applicable to modelling cyclical efficiency (Seetharam, 2016).

3.2.1 Linearity Tests

The Brock-Dechert-Scheinkman (BDS) test is used to examine whether a time series is linear. The

test statistic is given by:

𝑉𝑚 = √𝑇

𝐶𝑚,∈ − 𝐶1,∈
𝑚

𝑆𝑚,∈

(10)

Where the correlational integral, 𝐶𝑚,∈, captures the of repetition of a temporal pattern and 𝑆𝑚,∈ is the

standard deviation of √𝑇𝐶𝑚,∈ − 𝐶1,∈
𝑚 .

3.2.2 Normality Tests

As a measure to ensure that the results are robust, normality tests, which examine whether the data

exhibits normality, are presented below:

3.2.2.1 The Jarque-Bera Test

The Jarque-Bera test for normality is a goodness of fit test (Jarque & Bera, 1980). The test statistic,

JB, is a function of the measures of kurtosis (K) and skewness (S) computed from the sample. Where

the Jarque-Bera test statistic is given by:

𝐽𝐵 =

𝑛

6
∗ (𝑆2 +

(𝐾 − 3)2

4
)

(10)

Where the sample kurtosis is represented by 𝐾 =
𝑢4

𝑢2
2 , and the sample skewness, 𝑆 =

𝑢3

𝑢2
1.5. Where 𝑢2̂

and 𝑢3̂ are the theoretical second and third theoretical moments of a chi (𝜒2) distribution. Under

normality, the theoretical values of S = 0 and K = 3 (Thadewald & Büning, 2007). The null hypothesis

of this test is that the population is normally distributed. The null is rejected when the results of the

Jarque-Bera test in show a large 𝜒2 value and near-zero p-value.

3.2.2.2 The Shapiro-Wilk Test

Similarly, the null hypothesis of the Shapiro and Wilk (1965) test is that the population is normally

distributed. The Shapiro-Wilk test statistic is given by:

𝑆𝑊 =

(∑ 𝑎𝑖
𝑛
𝑥=1 𝑋(𝑖))

∑ (𝑋𝑖−𝑋𝑖)
2𝑛

𝑥=1

(11)

The null is rejected when the test’s associated p-value is near-zero.

3.2.3 Stationarity Tests

Two stationarity tests are presented below:

3.2.3.1 The Kwiatkowski-Phillips-Schmidt-Shin (KPSS)

The null hypothesis for the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test is that the data is

stationary around a deterministic trend. Its test statistic is given by:

𝐾𝑃𝑆𝑆 = 𝑇−2

∑ �̂�𝑡
2𝑇

𝑡=1

�̂�2

(12)

Where the test statistic, under a null of stationarity, follows a normal distribution. The null of

stationarity is rejected when the test’s associated p-value is less than 0.05.

3.2.3.2 The Augmented Dickey-Fuller (ADF) Test

The null hypothesis for the Augmented Dickey-Fuller test is that the data is not stationary. Its test

statistic is given by:

𝐷𝐹𝜏 =

𝛾

𝑆𝐸(𝛾)

(13)

The test statistic is compared against a critical value, where, if the computed value is more negative

than the critical value, then the null of a unit root is rejected.

3.2.3.3 The Phillips-Perron Test

The Phillips-Perron (PP) test is a non-parametric test that may also be used to examine whether a time

series has a unit root. Similar to the ADF test, the null hypothesis for the Phillips-Perron test is that

the data is not stationary. All PP test types used the model below to test for the null of a unit root:

𝑦𝑡 = 𝑐 + 𝛿𝑡 + 𝑎𝑦𝑡−1 + 𝑒(𝑡) (14)

Where yt, the time series’ current value is predicted based on its previous lagged value, yt-1 and an

innovation term, e(t).

3.3 Robustness Checks

In order to ensure that the results of this research report are robust to noise, a Fast Fourier Transform,

a variant of the Fourier Transform that requires much less computational costs than the traditional

version, is employed. Its use ultimately results in the de-noising of the data, effectively resulting in

an ability to compare an untransformed data set against a transformed one. Furthermore, several

stationary tests, normality and Martingale Difference Hypothesis tests are employed to ensure that

results are robust to the idiosyncrasies of the various tests. Finally, two different sets of machine

learning techniques are employed (with the caveat that they sit at different stages of the methodology,)

while not a precautionary measure per se, it nevertheless ensures that, given the black-box nature of

neural networks, that an alternative view of the return generating process can be entertained.

3.4 Summary

The methodology will be tested with four distinct steps on transformed and untransformed data:

1) Determine, via a battery of tests, whether returns follow a random walk/martingale, if they do,

then the enquiry stops at this level as they cannot be forecasted.

2) If they do not, they are subjected to linearity tests (beginning with the BDS test, and if its

results are inconclusive, a SETAR test) to test whether a linear (ARIMA-family) or non-linear

(SETAR) model should be used to model the return generating process. Once that decision is

made, the model is estimated via the various R packages and its residuals are tested to examine

model suitability, following which, the Hurst Exponent is used as an acid test to decide

whether to continue attempting to model the return generating process or end the enquiry at

this level.

3) If the Hurst Exponent results prompt further modelling, a machine learning approach (LSTM)

is used to attempt to model the return generating process. Its residuals are, similar to step 2,

analyzed and the Hurst Exponent is also used as a decision criterion to determine whether to

use a machine learning approach that incorporates exogenous regressors.

4) In the event that the computation of the Hurst Exponent calls for the incorporation of

exogenous regressors in the machine learning model, a NARX model is used. Its residuals are

analyzed and if found to be a good fit, this is taken as prima facie evidence of time varying

evidence, which machine learning approaches can capture, unlike their statistical approach

counterparts.

4 RESULTS

4.1 Normality Tests

The Jarque-Bera test for normality is a goodness of fit test (Jarque & Bera, 1980). The test statistic,

JB, is a function of the measures of kurtosis (K) and skewness (S) computed from the sample. The

null hypothesis of this test is that the population is normally distributed. The results of the Jarque-

Bera test in Table 1 show a large 𝜒2 value and near-zero p-value, indicating that the null of normality

can be rejected. Similarly, the null hypothesis of the Shapiro and Wilk (1965) test is that the

population is normally distributed. Its associated p-value is near-zero, also indicating that the null of

normality can be rejected.

Table 4.1: Daily Returns’ Normality Tests

Test Jarque-Bera Test Shapiro-Wilk Test

Test Statistic 5673.60 0.91

P-Value 0*** 0***

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

The results of the Jarque-Bera test in Table 2 show a large 𝜒2 value and near-zero p-value, indicating

that the null of normality can be rejected. Similarly, the Shapiro-Wilk test results indicate that since

the p-value is near-zero, the null of normality can be rejected.

Table 4.2: Weekly Returns’ Normality Tests

Test Jarque-Bera Test Shapiro-Wilk Test

Test Statistic 54.97 0.97

P-Value 0*** 0***

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level

4.2 Stationarity Tests

Table 3 below displays the results of three stationarity tests for daily returns. The null hypothesis for

the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test is that the data is stationary. The p-value is

greater than 0.05 (0.1) and thus, we fail to reject the null of stationarity. The Phillips-Peron and

Augmented Dickey-Fuller (ADF) tests’ null hypotheses is that the data is not stationary. The p-value

is less than 0.05 (0.01) and thus, we reject the null of non-stationarity in both instances. The three

tests converge on the conclusion that the data can be considered as stationary.

Table 4.3: Daily Returns’ Stationarity Tests

Test KPSS Phillips-Perron ADF

Test Statistic 0.07 -3202 -12.25

p-value 0.1* 0.01*** 0.01***

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

Table 4 below displays the results of three stationarity tests for weekly returns. The null hypothesis

for the KPSS is that the data is stationary. The p-value is greater than 0.05 (0.1) and thus, we fail to

reject the null of stationarity. The Phillips-Peron and ADF test’s null hypotheses is that the data is not

stationary. The p-value is less than 0.05 (0.01) and thus, we reject the null of non-stationarity in both

instances. The three tests agree that the data can be considered to be stationary.

Table 4.4: Weekly Returns’ Normality Tests

Test KPSS Phillips-Perron ADF

Test Statistic 0.06 -387.79 -8.38

p-value 0.1* 0.01*** 0.01***

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

4.3 Martingale Difference Hypothesis Tests

Whether the daily return series follows a martingale is tested via a battery of tests whose results are

displayed in Table 5 below. The automatic portmanteau (AQ) statistic asymptotically follows the 𝜒2

distribution with one degree of freedom. The null hypothesis of no return autocorrelation is rejected

at the 5% level of significance if the value of AQ is greater than 3.84. The AQ test results (0.36) thus

fail to reject the null of no return autocorrelation.

While the Automatic Portmanteau test only tests for linear dependence, the Generalized Spectral Test

(GS) investigates both linear and non-linear dependence. Both the Dominguez and Lobato test and

the Generalized Spectral test can detect non-linear dependence, however, the Generalized Spectral

Test has been shown to have more power against non-linear dependence (Salisu et al., 2016). The

Dominguez-Lobato test results have p-values of p = 0 and p = 0.1. This means that the null of a

martingale difference sequence (efficiency) cannot be rejected in one test but can be in the other,

indicating a conflicting view. The GS test returns a p-value = 0.2, also indicating that the null of

weak-form efficiency cannot be rejected. Taken together, the results of the AQ test, the GS and the

Dominguez-Lobato tests seem to reject both linear and non-linear dependence.

Table 4.5: Martingale Difference Hypothesis Tests - Daily Returns

Test AQ Test DL Test – Cp DL Test – Kp GS Test

t-statistic 0.37 0.36 1.11 0.2

p-value 0.54 0.1* 0*** NA

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

The Hurst Exponent is a long-range dependence estimator. Couillard and Davison (2005) argue that

the true value of the Hurst exponent for a finite time series cannot be computed. Furthermore, each

Hurst exponent computation method offers some mix of drawbacks and advantages; the rescaled

range analysis (R/S) Hurst, for example, was demonstrated by Davies and Harte (1987) to falsely lead

to the conclusion that there was long-term dependence when the time series had short-term correlation.

Annis and Lloyd (1976) derived the Empirical Hurst Exponent, whose rescaled range’s expected

value was given by:

𝐸(
𝑅

𝑆
)𝑛 =

Γ(
𝑛 − 1

2 )

√𝜋Γ(
n
2)

∑ √
𝑛 − 𝑟

𝑟

𝑛−1

𝑟=1

(15)

In response, Peters (1994) proposed the Corrected Empirical Hurst Exponent which avoided its

uncorrected predecessors’ overestimation of the rescaled range for small n values. Couillard and

Davison (2005) find, however, that the Empirical Hurst Exponent is useful in determining whether a

financial time series exhibited long-term memory. To get around the aforementioned problems,

several methods of calculating the Hurst exponent are used to get a more robust answer regarding

whether the time series has long-term memory.

The Hurst Exponents for the daily returns are in table 6 below. In contrast to the other inconclusive

Martingale Difference Hypothesis test results, however, the numerous methods of calculating the

Hurst exponent arrive at the same conclusion: they strongly indicate that there is persistence (long-

term memory) in the data. This would undermine weak form efficiency. This therefore means that

there is room to continue modelling the return generating process. The critical value for the Hurst

Exponent is 0.50, i.e., if the Hurst Exponent is greater than 0.5, then it can be concluded that there is

long-term dependence in the returns.

Table 4.6: Hurst Exponent - Daily Returns

Hurst

Exponent

Type

Simple R/S

Hurst

Exponent

Corrected R/S

Hurst

Exponent

Empirical

Hurst

Exponent

Corrected

Empirical

Hurst

Exponent

Theoretical

Hurst Exponent

Statistic

Value

0.54 0.61 0.64 0.61 0.53

Table 7 displays the results of the Martingale Difference Hypothesis Test results for the weekly log

returns. The automatic portmanteau (AQ) test’s null hypothesis of no return autocorrelation is rejected

at the 5% level of significance if the value of AQ is greater than 3.84. The AQ test results (3.33) thus

fail to reject the null of no return autocorrelation. The Dominguez-Lobato test results have p-values

= 0. This means that the null of martingale difference sequence (efficiency) can be rejected. Similarly,

the GS test returns a p-value = 0, also indicating that the null of weak-form efficiency can be rejected.

The results of the AQ test, the GS and the Dominguez-Lobato tests indicate that there is non-linear

dependence.

Table 4.7: Martingale Difference Hypothesis Tests – Weekly Returns

Test AQ Test DL Test – Cp DL Test – Kp GS Test

t-statistic 3.33 0.80 1.60 0

p-value 0.07* 0*** 0*** NA

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

Similar to the daily returns’ outcomes, Table 8 shows that the weekly returns’ different Hurst

exponents arrive at an overall conclusion of persistence (long-memory) in the data. This would

undermine weak form efficiency.

Table 4.8: Hurst Exponents – Weekly Returns

Hurst

Exponent

Type

Simple R/S

Hurst

Exponent

Corrected R/S

Hurst

Exponent

Empirical

Hurst

Exponent

Corrected

Empirical

Hurst

Exponent

Theoretical

Hurst Exponent

Statistic

Value

0.55 0.61 0.79 0.73 0.55

4.4 Linearity Tests

Finally, tables 9 and 10 display the results of the BDS test for the daily and weekly returns

respectively. The p-values of the BDS test for both daily returns are 0. This indicates that the data is

not linear. As a result, a non-linear model (such as a SETAR model) can be used to model the daily

return-generating process. Seetharam and Britten (2015) recommend using a grid-search to find the

optimal parameter values of the non-linear model to be used.

Table 4.9: BDS Tests – Daily Returns

Standard Normal =

[0.02] [0.04] [0.06] [0.08]

[2] 14.49 13.58 12.64 12.47

[3] 20.22 17.22 15.92 15.63

P-Value =

[0.02] [0.04] [0.06] [0.08]

[2] 0*** 0*** 0*** 0***

[3] 0*** 0*** 0*** 0***

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

Similar to the daily return results, the BDS test results for the weekly returns in Table 10 below

indicate that the return-generating process is non-linear as the p-values for the test are 0. To model

the return-generating process therefore, a grid-search approach is employed.

Table 4.10: BDS Test - Weekly Returns

Standard Normal =

[0.06] [0.011] [0.17] [0.23]

[2] 4.4 5.20 5.16 5.75

[3] 6.43 7.47 6.98 7.18

P-Value =

[0.06] [0.011] [0.17] [0.23]

[2] 0*** 0*** 0*** 0***

[3] 0*** 0*** 0*** 0***

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

4.5 SETAR/ARIMA modelling without external regressors

The grid search obtained the best SETAR parameters by minimizing the pooled Akaike Information

Criterion. This criterion attempts to create a balance between model-fit (by having as many

parameters as possible that are explain the data in the model) and parsimony (ensuring that the

minimal parameters necessary to model the data is included). The results of the grid search then used

as the inputs for the SETAR model. Figures 1 and 2 display a graphical representation of the grid

search.

Figure 4.1: Daily SETAR grid search

Figure 4.2: Weekly SETAR grid search

Tables 11 and 12 contain the saved SETAR model inputs for the daily and weekly return-generating

processes. Notably both daily and weekly SETAR grid searches resulted in 2 coefficients on the low

regime (mL) and a single high regime coefficient (mH).

Table 4.11: Best Daily SETAR Model Inputs

thDelay Th pooled-AIC mL mH

0 -0.03 -10515.48 2 1

Table 4.12: Best Daily SETAR Model Inputs

thDelay Th pooled-AIC mL mH

0 -0.11 -678.69 2 1

Table 13 captures the results of the daily model. The generated model is a SETAR with two regimes

(a high and low regime) where most of the points (84,99%) are captured by the high regime. Notably,

𝜙𝐿1 and 𝜙𝐿2 and the constant are statistically significant. To examine whether the model is a good fit,

the residuals are examined. In general, autocorrelation in the residuals indicates that the model may

not be a good fit.

Table 4.13: Daily SETAR Model

SETAR Model – Daily Returns

Low-Regime Coefficient Value P-Value

Constant – L -0.01 0.09*

𝜙𝐿1 -0.17 0***

𝜙𝐿2 -0.16 0***

Constant – H 0.01 0.23

𝜙𝐻1 0.04 0.12

Model Fit and Diagnostics

AIC MAPE Threshold

-18874 144.80% -0.03

Low-Regime Point Proportion High-Regime Point Proportion

15% 85%

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

Table 14 displays the results the weekly return SETAR model. The generated model is a SETAR with

two regimes (a high and low regime) where most of the points (72,88%) are captured by the low

regime. Notably, 𝜙𝐿1 and 𝜙𝐻1 and the constant are statistically significant. To examine whether the

model is a good fit, the residuals are examined.

Table 4.14: Weekly SETAR Model

SETAR Model – Weekly Returns

Low-Regime Coefficient Value P-Value

Constant – L -0.03 0.10

𝜙𝐿1 -0.28 0.04**

𝜙𝐿2 0.03 0.74

Constant – H 0.002 0.73

𝜙𝐻1 0.33 0***

Model Fit and Diagnostics

AIC MAPE Threshold

-1874 161.10% -0.03

Low-Regime Point Proportion High-Regime Point Proportion

27.12% 72.88%

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

A time series plot of the residuals seems to show volatility bunching (non-constant standard

deviation) but a zero mean in Figures 3 and 4 below. This indicates that the residuals approximate,

but do not follow a white noise process, more formal tests are required to arrive at a conclusion,

however.

Figure 4.3: Daily SETAR Model Residuals

Figure 4.4: Weekly SETAR Model Residuals

An examination of the residuals’ ACF will aid in helping decide whether there is any serial correlation

in the model’s residuals. The correlogram is show in Figure 5 below. The ACF plot of the daily

SETAR model’s residuals indicates that there are a few edge-cases of statistically significant

autocorrelation in the residuals. There are also at least two lags at which there is clearly significant

autocorrelation. This may indicate that there is some further information that needs to be added into

the model (and thus, this may not be the best “final” model of the return generating process).

Figure 4.5: Daily Model Residuals’ ACF

While the weekly ACF plot in Figure 6 shows even fewer edge-cases of statistically significant

autocorrelation in the residuals, there is at least one lag at which there is clearly significant

autocorrelation. This may indicate that there is some further information that needs to be added into

the model. In addition to a visual inspection of the ACF plots, formal statistical are employed to test

for normality in the residuals.

Note: The blue bands indicate the line of significant, at lags where it is crossed, it can be inferred that there is

significant autocorrelation in the time series under investigation.

Figure 4.6: Weekly Model Residuals’ ACF Plot

The normality test results of the daily model residuals are shown in Table 15. The Jarque-Bera test

returns a near-zero p-value, indicating that the null of normality in the residuals can be rejected. The

Shapiro-Wilk test also has a near-zero p-value, also indicating that the null of normality in the

residuals can similarly be rejected. The errors are tested for normality to examine whether the

tentative model provides a good explanation of the underlying data generating process. In the same

vein, the examination of the stationarity of the residuals is important insofar as it aids in the avoidance

of spurious regressions. (Wooldridge, 2012)

Table 4.15: Daily Residuals’ Normality Tests

Test Jarque-Bera Shapiro-Wilk

Test Statistic 5626.20 0.91

p-value 0*** 0***

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

The Hurst exponent results in Table 16, can aid in deciding whether to use more advanced approaches

to model the return generating process. Since the Hurst Exponent > 0.5, it therefore is rational to more

advanced approaches to model the data generating process.

Table 4.16: Hurst Exponent – Daily Residuals

Hurst

Exponent

Type

Simple R/S

Hurst

Exponent

Corrected R/S

Hurst

Exponent

Empirical

Hurst

Exponent

Corrected

Empirical

Hurst

Exponent

Theoretical

Hurst Exponent

Statistic

Value

0.55 0.62 0.65 0.61 0.53

Table 17 displays the results of the normality tests of the weekly model residuals. In similar fashion

to the daily residuals’ test results, the Jarque-Bera test returns a near-zero p-value, indicating that the

null of normality in the residuals can be rejected. The Shapiro-Wilk test also has a p-value less than

5%, also indicating that the null of normality in the residuals can similarly be rejected.

Table 4.17: Weekly Returns’ Normality Tests

Test Jarque-Bera Shapiro-Wilk

Test Statistic 43.36 0.97

p-value 0*** 0***

Note: * Significant at the 10% level, ** Significant at the 5% level, *** Significant at the 1% level.

Table 18 displays the weekly residuals’ Hurst exponents. While the weekly residuals’ Hurst exponent

is closer to 0.5 than the daily residuals’ Hurst exponent, it nevertheless is greater than 0.5. Taking the

Hurst exponent as a decision criterion regarding whether to continue modelling, the conclusion is

therefore that for both the daily and weekly time series, more advanced attempts to model the return

generating process must be employed.

Table 4.18: Weekly Model Residuals Hurst Exponent

Hurst

Exponent

Type

Simple R/S

Hurst

Exponent

Corrected R/S

Hurst

Exponent

Empirical

Hurst

Exponent

Corrected

Empirical

Hurst

Exponent

Theoretical

Hurst Exponent

Statistic

Value

0.53 0.61 0.80 0.74 0.55

Before that, an examination of whether an application of a Fourier Transform to the data will yield a

different conclusion (regarding whether to use more advanced approaches to model the return

generatin