Critical comparison of statistical and deep learning models applied to the New Zealand Stock Market Index

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Front Matter

UNITEC INSTITUTE OF TECHNOLOGY

DOCTORAL THESIS

Critical Comparison of

Statistical and Deep Learning Models Applied to the New Zealand Stock Market Index

Wajira Dassanayake

Supervisor: Associate Professor Iman Ardekani

A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Computing in

School of Computing, Electrical, and Applied Technology, New Zealand

April 2022

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Abstract

Financial markets enable buyers and sellers to trade financial instruments (stocks, bonds, foreign currencies, and derivatives) and improve capital allocation. These markets play a pivotal role in facilitating the interactions between those who seek capital and those who are prepared for capital investments, allowing market participants to transfer risks and stimulate economic growth. Financial time series are inherently dynamic, interdependent, and highly sensitive to many factors. These time series contain deterministic and stochastic characteristics, and many interrelated factors influence them. Accurate predictions of financial time series benefit various market participants to generate wealth through the right trading strategies and other stakeholders to enhance funds. However, due to their inherent complexities, financial time series prediction is considered one of the most challenging problems in data mining.

This thesis employs popular and efficient time series prediction models, reformulates them and implements them to analyse stock market index movements. This scientific exploration uses two widely used classical forecasting techniques [Auto-Regressive Integrated Moving Average (ARIMA) and Holt Winter's Exponential Smoothing (HWES)] and efficient deep learning (DL) [Long Short-Term Memory (LSTM)] network. The predictive precision of the reformulated models will be empirically tested using Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root Mean Square Error (RMSE).

Four research questions are meticulously examined to close the identified empirical research gaps in the time series prediction models applied to the New Zealand stock market.

Once the redesigned models are sufficiently trained, they are implemented as prediction models on selected stock market indices. Several statistical and econometric tests are executed to substantiate my research findings.

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The models' predictive precisions are assessed on the New Zealand stock market (NZX 50) Index in different samples, including a sample consisting of the year 2020 devastated by the COVID-19 pandemic. The predictive efficacy of the revised models is further investigated when each model was applied to produce ten-by-ten slots of cumulative incremental prediction intervals. Subsequently, the reformulated models' forecasting precisions are further appraised when the models are applied to a different stock market index (Australian stock market (ASX 200 Index). This research, thus, leads to several empirical and practical contributions.

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Dedication

Life is full of revelations combined with serendipity, and my family is the tower of my strength.

Their love and encouragement persuaded me to complete this thesis.

Thus, this thesis is dedicated to my family.

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Acknowledgements

A few extraordinary people have offered a great deal of support during my doctorate journey.

First and foremost, I must express my sincere appreciation to Associate Professor Iman Ardekani, whose expertise was invaluable in completing my thesis. His wisdom, gentle navigation, insightful feedback, and guidance influenced my work to a higher standard. A special acknowledgement must go to Professor Chandimal Jayawardena, who provided extended support during the journey of my thesis. I also appreciate Associate Professor Hamid Sharifzadeh's advice and assistance in completing this thesis.

I want to honour authors whose work I have referred to in my thesis, and their imperative thoughts, suggestions and years of research experiences were the foundation pillars of my research.

I gratefully acknowledge Associate Professor Marcus Williams and Dr Kerry Kirkland for their continuous support. Special acknowledgements must go to my employer, Unitec Institute of Technology and the School of Applied Business for supporting me in obtaining the Doctor of Computing degree for many years. I also want to appreciate my colleagues at Unitec School of Applied Business for their collegiality, encouragement and professionalism.

A heartfelt appreciation must go to Narmada Gamage and Lakshitha (Abba) Dassanayake. Without their prompt support, my journey could have been more laborious. Also, I must acknowledge Tishni Fernando's generosity which helped me format the long tables.

I must express my profound respect and admiration to my late parents for their significant role in my life. My father persuaded me to become an academic with a high degree of integrity, and my mother, nurtured me, directed me and developed ethical and spiritual values. Without my late father's strategic plans, I would not have ended my career as an academic. Towards the

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final stages of thesis writing, my mother's demise disrupted my concentration and thesis writing. Yet, my determination paved the way to complete this thesis. I admire my two sisters (Indrani and Nuresha) and two brothers-in-law (late Gamini and Janaka) for their love and affection.

Finally, I must express my heartfelt gratitude and appreciation to my beloved wife Rajika and the children. Rajika has generously allowed me to have quality research time with a gentle nudge to complete this thesis at the earliest opportunity. Harshangi, Tharindu, Lakshitha and Reshika are my lovely children; your affection, encouragement, and commitment to succeed in your life made my doctoral journey less rugged.

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Publications from this research

Conference papers:

• W. Dassanayake, I. Ardekani, N. Gamage, C. Jayawardena, and H. Sharifzadeh,

"Effectiveness of Stock Index Forecasting using ARIMA model: Evidence from New Zealand," ICAC 2021.

• W. Dassanayake, and C. Jayawardena, "Determinants of Stock Market Index Movements;

Evidence from New Zealand Stock Market," National Conference on Technology and Management, Colombo, Sri Lanka (Vol. Volume 6), 2017.

• W. Dassanayake, "Effectiveness of Stock Index Forecasting Using ARIMA Model:

Evidence from New Zealand," Unitec Research Symposium, 2020.

• W. Dassanayake, "Effectiveness of Stock Index Forecasting Using LSTM Model: Evidence from New Zealand" Unitec Research Symposium 2020.

Journal paper:

• W. Dassanayake, I. Ardekani, C. Jayawardena, H. Sharifzadeh, and N. Gamage,

"Forecasting Accuracy of Holt-Winters Exponential Smoothing: Evidence from New Zealand," New Zealand Journal of Applied Business Research (Vol. 17 (1)), 2020.

Book chapter:

• W. Dassanayake, C. Jayawardena, I. Ardekani, and H. Sharifzadeh, "Models Applied in Stock Market Prediction: A Literature Survey" Unitec ePress Research Report Series (Vol.

1), 2019.

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List of Figures

Figure 4.1 The architecture of Long Short-Term Memory (LSTM) cell ... 63

Figure 6.1 Sample period one: NZX 50 Index (2009 – 2017) ... 84

Figure 6.2: Sample period two: NZX 50 Index (2007 – 2017) ... 84

Figure 6.3: Sample period three: NZX 50 Index (2007 – 2020) ... 84

Figure 6. 4: Sample period one: Log return of NZX 50 Index (2009 – 2017) ... 85

Figure 6. 5: Sample period two: Log return of NZX 50 Index (2007 – 2017) ... 85

Figure 6.6: Sample period three: Log return of NZX 50 Index (2007 – 2017) ... 85

Figure 6.7: HWES (Model 1) Actual versus Prediction - Period one (2009 – 2017) ... 94

Figure 6.8: HWES (Model 2) Actual versus Prediction - Period one (2009 – 2017) ... 95

Figure 6.9: HWES (Model 1) Actual versus Prediction - Period two (2007 – 2017) ... 96

Figure 6.10: HWES (Model 2) Actual versus Prediction - Period two (2007 – 2017) ... 97

Figure 6.11: HWES (Model 1) Actual versus Prediction - Period three (2007 – 2020) ... 98

Figure 6.12 HWES (Model 2) Actual versus Prediction - Period three (2007 – 2020) ... 99

Figure 6.13: ACF of the differenced NZX 50 Index ... 103

Figure 6.14: PACF of the differenced NZX 50 Index ... 103

Figure 6.15: ARIMA (Model 1) Actual versus Prediction – Period one (2009 – 2017) ... 107

Figure 6.16: ARIMA (Model 2) Actual versus Prediction – Period one (2009 – 2017) ... 108

Figure 6.17: ARIMA (Model 1) Actual versus Prediction – Period two (2007 – 2017) ... 109

Figure 6.18: ARIMA (Model 2) Actual versus Prediction – Period two (2007 – 2017) ... 110

Figure 6.19: ARIMA (Model 1) Actual versus Prediction – Period three (2007 – 2020) ... 111

Figure 6.20: ARIMA (Model 2) Actual versus Prediction – Period three (2007 – 2020) ... 112

Figure 7.1: Steps of LSTM Model Development ... 125

Figure 7.2: Univariate LSTM model training error for the NZX 50 Index (sample period one: upward movement) ... 128

Figure 7.3: Univariate LSTM model training error for the NZX 50 Index (sample period two: nonlinear movement without COVID-19) ... 128

Figure 7.4: Univariate LSTM model training error for the NZX 50 Index (sample period three: nonlinear movement with COVID-19) ... 129

Figure 7.5: Univariate LSTM Actual versus Prediction – Period one (2009 – 2017) ... 131

Figure 7.6: Univariate LSTM Actual versus Prediction – Period two (2007 – 2017) ... 132

Figure 7.7: Univariate LSTM Actual versus Prediction – Period three (2007 – 2020) ... 133

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Figure 7.8: Multivariate LSTM model training error for the NZX 50 Index (sample period

one: upward movement) ... 136

Figure 7.9: Multivariate LSTM model training error for the NZX 50 Index (sample period two: nonlinear movement without COVID-19) ... 136

Figure 7.10: Multivariate LSTM model training error for the NZX 50 Index (sample period three: nonlinear movement with COVID-19) ... 137

Figure 7.11: Multivariate LSTM Actual versus Prediction – Period one (2009 – 2017) .... 139

Figure 7.12: Multivariate LSTM Actual versus Prediction – Period two (2007 – 2017) .... 140

Figure 7.13: Multivariate LSTM Actual versus Prediction – Period three (2007 – 2020) .. 141

Figure 8.1: RMSE results of each model tested in incremental forecasting ... 156

Figure 8.2: MAE results of each model tested in incremental forecasting ... 157

Figure 8.3: MAPE results of each model tested in incremental forecasting ... 157

Figure A.1: Logarithmic time plots at level ... 207

Figure A.2: Logarithmic time plots at first difference ... 208

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List of Tables

Table 5.1: Vector Error Correction Model (VECM) ... 77 Table 6.1: Unit Root Test results for the natural logarithm of NZX 50 Index-Upward Trend (2009 – 2017) ... 87 Table 6.2: Unit Root Test results for the natural logarithm of NZX 50 Index- Nonlinear Trend without COVID-19 pandemic period (2007 – 2017) ... 88 Table 6.3: Unit Root Test results for the natural logarithm of NZX 50 Index- Nonlinear Trend including the COVID-19 pandemic period (2007 – 2020) ... 89 Table 6.4: Performance comparisons of the two alternative HWES models for the tested periods ... 91 Table 6.5: Performance comparisons of the two alternative ARIMA models for the tested periods ... 105 Table 7.1: Features related to Univariate-LSTM model ... 126 Table 7.2: Features related to Multivariate-LSTM model ... 135 Table 8.1: Comparisons of RMSE, MAE, and MAPE of the assessed models on three sample periods ... 149 Table 8.2: Comparison of the two most efficient models: Percentage difference of Univariate- LSTM from the Multivariate-LSTM ... 152 Table 8.3: Comparison of Deep Learning (DL) models and statistical models ... 152 Table 8.4: Percentage difference of the average of DL models from the average of Statistical models ... 152 Table 8.5: Comparison of the tested models applied to ASX 200 Index and NZX 50 Index 159 Table 8.6: Order of model preference when the analysis is conducted on the NZX 50 Index ... 160 Table 8.7: Order of model preference when the analysis is conducted on the ASX 200 Index ... 161

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Glossary

ANN Artificial Neural Network

FOREX Foreign Exchange Market

DJIA Dow Jones Industrial Average Index

NZX 50 S&P/NZX 50 (New Zealand's benchmark stock index) ASX 200 S&P/ASX 200 (Australia's benchmark stock index) COVID-19 Coronavirus disease

MERS Middle East respiratory syndrome SARS Severe acute respiratory syndrome

HBM Hybrid model

AI Artificial Intelligence

CS Classical Statistical

HWES Holt Winter's Exponential Smoothing ARIMA Autoregressive Integrated Moving Average

MAE Mean Absolute Error

MAPE Mean Absolute Percentage Error

RMSE Root Mean Square Error

MAD Mean Absolute Deviation

MSE Mean Square Error

R² Coefficient of Determination

Adj-R² Adjusted Coefficient of Determination AIC Akaike Information Criterion

SIC Schwarz Information Criterion BIC Bayesian Information Criterion

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FPE Final Prediction Error Criterion HQC Hannan-Quinn Information Criterion

WRST Wilcoxon Rank-Sum test

RelMAE Relative Mean Absolute Error

MASE Mean Absolute Scaled Error

VAR Vector Autoregressive

VECM Vector Error Correction Model

TSD Time Series Decomposition

GC Granger Causality

S&P 500 Standard and Poor's 500 (Benchmark stock index in the United States)

FTSE/ASE-20 This index comprised the top 20 companies in Greece FTSE/ASE-mid40 This index comprised the top 40 companies in Greece FTSE 100 Financial Times Stock Exchange 100 Index

Nikkei 225 Nikkei is a stock market index for the Tokyo Stock Exchange S&P 100 The S&P 100 index is a stock market index in the United States NASDAQ National Association of Securities Dealers Automated

Quotation

NASDAQ 100 The NASDAQ 100 is an index on the NASDAQ stock market

NZSE New Zealand Stock Exchange

NYSE New York Stock Exchange

BSE Bombay Stock Exchange

BSE Sensex Bombay Stock Exchange Sensitive index NSE National Stock Exchange of India Limited

VOC Dutch East India Company

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ASE Amsterdam Stock Exchange

BRSE Brussels Stock Exchange

Paris Bourse Paris Stock Exchange

LSE London Stock Exchange

AMEX American Stock Exchange

NZX New Zealand's Exchange

NZSE 40 NZSE 40 Capital Index

NZSX NZX Equity Market

NZDX NZX Debt Market

NZCX NZX Equity Derivatives

FSM Fonterra Shareholders Market S&P/NZX S&P/NZX family of Indices S&P DJI S&P Dow Jones Indices

Nifty Midcap 50 Top 50 firms listed on the National Stock Exchange of India SET Stock Exchange of Thailand Index

SSE Shanghai Stock Exchange

CAPM Capital Asset Pricing Model

APT Arbitrage Pricing Theory

EMH Efficient Market Hypothesis

T-bill Treasury Bills

ES Exponential Smoothing

AR Autoregressive

MA Moving Average

ARMA Autoregressive Moving Average

SARIMA Seasonal Autoregressive Integrated Moving Average

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ARCH Autoregressive conditional heteroskedasticity

GARCH Generalised Autoregressive Conditional Heteroskedasticity

GDP Gross Domestic Product

NZD/USD New Zealand Dollar/United States Dollar Exchange Rate AUD/NZD Australian Dollar/New Zealand Dollar Exchange Rate AUD/USD Australian Dollar/United States Dollar Exchange Rate AUD/JPY Australian Dollar/Japanese Yen Exchange Rate AUD/GBP Australian Dollar/Great British Pound Exchange Rate AUD/SGD Australian Dollar/Singapore Dollar Exchange Rate AUD/CHF Australian Dollar/Swiss Franc Exchange Rate

USD/NZD The United States Dollar/New Zealand Dollar Exchange Rate FED United States Federal Reserve

ECB European Central Bank

ONR US Office of Naval Research

SES Simple Exponential Smoothing

STES Smooth Transition Exponential Smoothing

RW Random Walk

EA Evolutionary algorithm

ES Evolutionary strategy

GA Genetic Algorithm

FL Fuzzy logic

ES Evolution Strategies

AES Adaptive Evolution Strategy

ABC Artificial Bee Colony

ABC-RNN Recurrent Neural Network-based Artificial Bee Colony

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MLP Multi-Layer Perceptron

DANN Dynamic Artificial Neural Network

FI Fractional Integrated

ESWA Exponentially Weighted Moving Average

HW Holt-Winters

ETS Exponential smoothing state space method EMD-HW Empirical Mode Decomposition Holt-Winters EWMA Exponentially Weighted Moving Average

SV Stochastic Volatility

ACF Autocorrelation Function

PACF Partial Autocorrelation Function

NN Neural Network

FFNN Feed-Forward Neural Network

SVR Support Vector Regression

BPNN Back-Propagation Neural Network

TDNN Time-Delay Neural Network

PNN Probabilistic Neural Networks

EWH Equal Weight Hybrid

ARFIMA Autoregressive Fractional Integrated Moving Average

FTS Fuzzy Time Series

ETS Exponential Smoothing State Pace

LAPSO Learning Automata Particle Swarm Optimisation PFTS Polynomial Fuzzy Time Series

MARS Multivariate Adaptive Regression Splines

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Wavelet-MARS- SVR

Wavelet Transform, Multivariate Adaptive Regression Splines and Support Vector Regression

RBF-NN Radial Basis Function Neural Network

LAR Linear Autoregressive

DWT Discrete Wavelet Transform

SWT Stationary Wavelet Transform

DL Deep Learning

SDL Supervised Deep Learning

ML Machine Learning

SSDL Semi-Supervised Deep Learning

LSTM Long Short-Term Memory

RNN Recurrent Neural Network

DNN Deep Neural Networks

CNN Convolutional Neural Networks

GRU Gated Recurrent Units

GAN Generative Adversarial Networks

DRL Deep Reinforcement Learning

USDL Unsupervised Deep Learning

SC Sparse Coding

RBM Restricted Boltzmann Machines

AE Autoencoders

VG Vanishing Gradients

CNN-LSTM Convolutional Neural Network and Long Short Term Memory LTSM-RNN Long Short Term Memory Recurrent Neural Network

LRC Logistic Regression Classifier

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RAF Random Forest

BLSTM Bidirectional Long Short-Term Memory SLSTM Stacked Long Short-Term Memory ARMA-

GJRGARCH

Autoregressive Moving Average and Glosten Jagannathan Runkle Generalised Autoregressive Conditional

Heteroskedasticity

OLM Ordinary Linear Mode

GLM Generalised Linear Model

ELSTM Embedded Layer LSTM

AELSTM Automatic Encoder based LSTM

DBN Deep Belief Networks

DBN-MLP Deep Belief Networks-Multi-Layer Perceptron UREC Unitec Research Ethics Committee

I(d) Integrated of an order d

y_t Observed value of the NZX 50 Index in period t ln y_t Natural logarithm of the NZX 50 Index in period t ln INF_t Natural logarithm of New Zealand inflation in period t ln EXC_t Natural logarithm of New Zealand exchange rate in period t ln INT_t Natural logarithm of New Zealand interest rate in period t ln S&P_t Natural logarithm of the S&P 500 Index in period t

ETC_t−1 Error Correction Term

ε_t′s Serially uncorrelated error terms

ESSR Residual sum of squares of the restricted regression ESSU Residual sum of squares of the unrestricted regression l_t Overall smoothed level of the time series at time t

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b_t Smoothed multiplicative trend at time t s_t Smoothed seasonal index at time t

p Number of autoregressive terms

q Number of lagged forecast errors in the prediction equation d Number of nonseasonal differences needed for stationarity

c_t Cell state at t

c_t−1 Last cell state

h_t Hidden state

h_t−1 Last hidden state

σ Sigmoid activation function

Element-wise product Concatenation operation

f_t Output vector of the forget gate

w_f and b_f Set of trainable parameters of the forget gate

i_t Input gate vector

w_i and b_i Trainable parameters for the input gate c̃_t A vector of new candidate values w_c and b_c Trainable parameters

o_t Output gate

w₀ and b₀ Trainable parameters of the output gate ŷ_t Predicted y_t values

n Sample size

σ

̂_e² Residual sum of squares

OLS Ordinary Least Squares

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SOLS Static Ordinary Least Squares

d.f. Degrees of freedom

C1 Error correction term of the cointegrated model C2 to C11 Short-run causality

IDE Integrated development environment API Application Programming Interface

NaN Not a number

Adam Adaptive Moment Estimation

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Chapter 1: Introduction and problem description

"The stock market became an exciting place, like a gambling casino, but tied to business reality rather than mere amusement, and it was a place where investors could diversify and limit their risks. It, therefore, was highly effective in attracting capital for the enterprise."

Robert J. Shiller, Nobel Prize winner in Economic Sciences, 2013

1.1 Introduction and overview

Financial markets play a fundamental role in economic development. Many empirical studies revealed that well-developed financial markets positively contribute to economic growth and development [1]–[6]. Financial markets refer to any marketplace where the marketable securities are traded, and these markets include stock markets, bond markets, Foreign Exchange Market (FOREX), derivatives markets, and others.

Stock (equity, share) markets are dynamic marketplaces where participants trade stocks (ownership claims on businesses) through organised exchanges or over-the-counter markets.

The stock market is considered one of the most fascinating, sophisticated, and complex financial markets whose movements are influenced by many interconnected factors.

Globalisation and international financial market integration intensify further complexities in the stock market trading behaviour. In general, the stock market prices are influenced by the domestic economic environment, government policies, the motivation and psychology of individual and institutional investors, global financial condition, domestic and international political environment changes, and the degree of integration with other markets. Stock markets provide opportunities for market participants to create wealth through investment gains,

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dividend incomes, diversification benefits, ownership stakes, and tax deferrals. Stock markets tend to transfer funds from the impatient to the patient traders. Some market participants intend to exploit the market situation to make a profit through strategic prediction. As such, accurate forecasts of stock price movements become vital to private and institutional investors, speculators, arbitrageurs, hedgers, brokers, dealers, and government organisations to make informed decisions. These would generate effective trading strategies at the right time to earn profits with minimum associated risks.

1.2 Background to the problem

The stock market prices are inherently dynamic, volatile, sensitive, nonlinear, and nonparametric, and the interdependent factors concurrently influence them. Because of these characteristics, accurate prediction of stock market prices and indices becomes a complex and challenging endeavour; nevertheless, millions of dollars are traded on the global stock market.

These rewards and challenges make financial time-series prediction a fascinating field for researchers, analysts, and traders. These time series contain deterministic and stochastic features and mostly exhibit nonlinear attributes (trends, seasonal and cyclical patterns, random walks, high volatility).

In the last few decades, the world has experienced several financial and monetary crises; Black Tuesday (1929), Black Monday (1987), Asian Financial Crisis (1997-1998), Subprime Mortgage Crisis (2007-2010), COVID-19 pandemic (2019 to date) are a few of these catastrophes ([7], [8]). Most of these had originated in one (several) markets, and their volatility shockwaves rapidly spread across global financial markets resulting in financial contagion. The commonality of these crises seems to be that stock prices are separated from economic reality ([7], [8]). The existence of similar effects makes accurate stock market prediction more challenging.

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Fundamental and technical analyses are the two main analytical methods and trading philosophies used in financial time series prediction ([9]–[12]. Fundamental factors, such as micro and macroeconomic influences, company financials and management strategies, company and industry dynamics, political environment, and so on, impact a security's intrinsic (fair) value. Therefore, fundamental analysts employ numeric information of the fundamental factors to predict a security's inherent value. On the other hand, the technical analysis relies entirely on past market data and typically evaluates the historical price trends to predict the stock market price movements. Therefore, the technicians utilise charts and modelling techniques to capture the historical patterns and trends to predict stock market price movements. Technical analysts usually employ advanced statistical methods, soft computing techniques, and hybridised models. Also, market sentiment and spontaneous breaking news are linked to the stock market movement. Investor sentiments could create asset mispricing ([13]

and [14]). Investors who believe the stock prices rise over time combined with the perception of positive news would generate optimistic effects on the stock markets, thereby emerging the bullish market sentiments. The market participants with a pessimistic opinion on the stock markets with negative news result in bearish market sentiments. Investors' sentiment is vital for contrarian investors who trade in contrast to the prevailing emotions. Fundamental and technical forces are interlinked and eventually influence the investors' decision-making on marketable securities.

1.3 Motivation

Primary motive:

Financial time series prediction is considered one of the most challenging problems in data mining [15]. Time series of stock market prices and indices contain deterministic and stochastic features; thus, they could primarily present nonlinear attributes. Because of these

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characteristics, accurate prediction of such time series becomes a complex and challenging endeavour and turns out to be one of the central topics in Finance. The demanding nature of the stock market index prediction and the identified gaps stated in Section 3.4 (Chapter 3) motivated me to research the efficient prediction models for New Zealand stock market index movements. NZX 50 Index is recognised as the benchmark index [16] of the New Zealand equity market; thus, NZX 50 Index is used as the primary time series index for my investigations.

Subsequent Motive:

In the midst of my research, the world experienced Coronavirus (COVID-19) pandemic in early 2020, which created unparalleled destruction to the world economies and financial markets.

Due to the impact of the COVID-19 pandemic, Dow Jones Industrial Average (DJIA) Index plummeted by 6,400 points, equivalent to an approximate drop of 26% in March 2020 [17];

similarly, the New Zealand stock market (NZX 50) Index has fallen approximately 780 points, by about 8.5 percent in March 2020, its most significant fall in one session since the financial market crash in 1987 [18]. Additionally, [19] highlighted the complexity and extreme difficulty of accurate stock market prediction during the global financial crisis, similar to the COVID-19 pandemic. This predicament influenced and motivated me to expand my research endeavours;

thus, I extended my sample period to include the year 2020, a unique and extraordinary year shaken by the COVID-19 pandemic. Therefore, the sample I eventually investigate is from 2007 to 2020, which is further divided into three sub-samples, namely period 1: 2009 to 2017 (refer to Figure 6.1); period 2: 2007 to 2017 (refer to Figure 6.2) and period 3: 2007 to 2020 (refer to Figure 6.3). Each reformulated model is applied to the NZX 50 Index and tested in three subperiods to make effective comparisons. This investigation enables me to assess the tested models' forecasting efficacies, especially during the COVID-19 pandemic.

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Additionally, the reformulated models will be further scrutinised by applying them to the Australian stock market (ASX 200 Index [20], the benchmark index in the Australian stock market) to ascertain the redesigned models' predictive efficacies when applied to different stock markets.

Consequently, my research will be a vital and distinctive investigation that enables me to comparatively evaluate the predictive capabilities of the reformulated models when applied to the New Zealand financial market (NZX 50 Index) in three subsamples, appraise these models' efficacies in cumulative incremental prediction intervals, assess these models' predictive efficiencies when they are applied to a different stock market (ASX 200 Index) and, most importantly, evaluate how effective these reformulated models are when they experiment on a sample that was devastated by the global COVID-19 pandemic.

1.4 Problem Statement

The predicament I plan to address in this research is centred around the time series of stock market index prediction. Predicting the always challenging directions, movements, and future values of financial time series is one of the central topics in data mining. Thus, researchers, academics, portfolio managers, investment experts etc., are always keen to find solid stock market forecasting models. The capability of a Classical Statistical (CS) model, an Artificial Intelligence (AI) model, or a Hybrid Model (HBM) to accurately predict the future values of the stock price series and or stock market index series could bring several benefits, either individual or collective.

In this thesis, I employ HWES and ARIMA as the two preferred statistical forecasting models and LSTM as the more suitable and efficient DL model for time series prediction. I reformulate them based on the salient time series characteristics present in the historical time series of the NZX 50 Index. Then, the redeveloped HWES, ARIMA, and LSTM models will

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be applied to the NZX 50 Index [16] to determine their predictive precisions under different test conditions. Firstly, the forecasting precisions of the redesigned models are assessed in three sample periods where the samples exhibit diverse time series characteristics. One of the selected sample periods includes data during the COVID-19 pandemic; thus, the investigation enables the comparison of predictive efficiencies of the models before COVID-19 and during the COVID-19 pandemic. Secondly, the forecasting precisions of the reformulated models will be further investigated when they are applied to generate cumulative incremental prediction intervals. Subsequently, the explicitly redesigned models for the NZX 50 Index [16] will be applied to the ASX 200 Index [20] to investigate how the redesigned models perform when applied to different time series. The rationale for NZX 50 Index selection is explained in Section 2.2. An extensive literature review and the principle of parsimony are used to rationalise the selection of HWES, ARIMA, and LSTM models.

1.5 Research Objective

This thesis intends to utilise popular and efficient time series prediction models, reformulate them, and apply them to analyse the New Zealand stock market (NZX 50) Index movements.

I employ two widely used classical forecasting techniques and a deep learning model to achieve this objective. The two classical statistical forecasting techniques are Auto-Regressive Integrated Moving Average (ARIMA, [21]) and Holt Winter's Exponential Smoothing (HWES, [22]–[24]). The selected deep learning model is Long Short-Term Memory (LSTM, [25]) networks, a model that financial time series analysts have extensively used. The rationale for selecting models (HWES, ARIMA, and LSTM) was the principle of parsimony, predictive efficacy, popularity, reputation and testimony of the literature review on time series prediction models performed in Chapter 3. These are broadly discussed in Chapters 3, 4, 6, 7 and 8.

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Applying reformulated HWES, ARIMA and LSTM models in the New Zealand financial markets is significant as this study creates an opportunity to compare their predictive efficacies in different test conditions. The predictive accuracy of these models will be empirically tested using Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root Mean Square Error (RMSE). The reformulated models will be effectively trained and tested to forecast the New Zealand stock market (NZX 50) Index movements.

Four specific sub-objectives are described below to accomplish the above research objective.

1. The first sub-objective is to conduct a critical investigation of the influential fundamental determinants of the NZX 50 Index. From a fundamental analysis perspective, Cointegration analysis, Vector Error Correction (VECM [26], [27]) and Granger Causality (GC [28]–[31]) tests will be performed to determine the influential determinants of the NZX 50 Index.

2. The second sub-objective is to employ two conventional statistical forecasting methods (ARIMA and HWES) to reformulate ARIMA and HWES prediction models and apply the revised models to forecast the NZX 50 Index movements. A technical analysis approach to the stock market index prediction will be carried out.

3. The third sub-objective is to utilise an efficient Deep Learning architecture (LSTM) to redevelop Univariate LSTM and Multivariate LSTM prediction models and apply the revised models to predict the NZX 50 Index movements. A technical analysis approach to the stock market index prediction will be carried out.

4. The final sub-objective of this study is to perform a comparative analysis of the forecasting accuracies of HWES, ARIMA and LSTM in varying time windows and different testing methods to ascertain the effectiveness of the evaluated models in time series forecasting.

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1.6 Research Questions

The following four research questions encapsulate my research interest in addressing the above research objectives:

1. What are the critical fundamental determinants of the NZX 50 Index movements?

2. How can effective forecasting models based on HWES and ARIMA methodologies be devised and applied with high precision to the NZX 50 Index prediction?

3. How can an efficient univariate LSTM forecasting model and a multivariate LSTM forecasting model be formulated and applied to forecast the NZX 50 Index movement with a high degree of predictive efficacy?

4. Considering all the models tested in different sample periods and scrutiny processes, is it possible to identify a superior forecasting model? Is the recognised model consistently outperforming other tested models in all the testing procedures? Can the redeveloped models efficiently handle the impact of the COVID-19 pandemic?

1.7 Contributions

Implementation of the redesigned models on the NZX 50 Index prediction in different samples led to several contributions, summarised below.

1. I have comprehensively reviewed the literature on the linear and nonlinear models applied in financial time series prediction. This survey provides a base for the financial time series analysts to evaluate the salient characteristics of the prediction models, the pros and cons of the applied models, and their influence on similar scholarly impetus.

2. Secondly, the fundamental analysis-based study discovers that there is no statistically significant long-run causality from the New Zealand inflation rate, the exchange rate, the

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interest rate, and the S&P 500 Index on the NZX 50 Index. However, our study reveals that the S&P 500 Index has a significant short-run Granger Causality (GC) to the NZX 50 Index.

3. The third contribution from this thesis is related to the extensive empirical application and comparative evaluation of the reformulated HWES, ARIMA and LSTM models to the financial time series prediction. The implemented steps adhered to the data science value chain and created an appropriate and consistent platform for comparing the forecasts effectively. First, I derived sensible features in the training datasets and then standardised these elements during the pre-processing to enable model training. Next, I evaluated and reformulated the HWES, ARIMA and LSTM models and tested them to determine the ideal NZX 50 Index prediction models. The executed experiments are performed to obtain robust and generalised findings.

4. Fourthly, this research enables us to compare the foresting precisions of the reformulated HWES, ARIMA and LSTM models tested on the NZX 50 Index in different sample periods.

This investigation discovered that Univariate and Multivariate LSTM are two superior models for the NZX 50 Index prediction as they consistently and significantly outperform the ARIMA and HWES by a considerable margin.

5. Fifthly, the study enables us to compare the predictive efficacies of the redesigned models in three different time horizons, which include a period of upward trend (from 22/01/2009 to 31/12/2017); a period that exhibits a nonlinear trend (from 1/1/2007 to 31/12/2017); a nonlinear period which includes the COVID-19 lockdown period (01/01/2007 - 31/12/2020). For a robust comparison, training and testing samples for each time frame remained constant.

6. Sixthly, this research enables us to compare the reformulated models' adaptability and predictive efficacies in different time series. In this investigation, the reformulated models,

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which were explicitly designed for the NZX 50 Index, are applied to the ASX 200 Index forecasting to establish whether the reformulated models are effective prediction models in other time series. The same reformulated models were used for both NZX 50 Index and ASX 200 Index for robust comparisons.

7. Finally, this research also allows us to investigate the forecasting accuracies of the redesigned HWES, ARIMA and LSTM models when tested on a sample devastated by the COVID-19 pandemic. Including data affected by the COVID-19 pandemic enables comparing the results before and during the pandemic.

8. Six journal and conference presentations and publications are generated from this thesis. A list of the outputs is available on Page vii in the front matter.

1.8 The organisation of the thesis

This thesis is organised into nine chapters. An introduction with the research objectives and contribution is presented in Chapter 1. A contextual background that provides a summary of the stock market and an overview of different schools of thought in stock market prediction is described in Chapter 2. A systematic review of the literature performed in financial time series prediction is surveyed in Chapter 3. Chapter 4 outlines and rationalises the research methodologies applied to answer the research questions stipulated in Chapter 1. Addressing research objective (research question) (i), Chapter 5 presents the results of the study conducted to determine the long-run and short-run dynamic relationship between selected macroeconomic variables and the NZX 50 Index. Vector Error Correction (VECM [26], [27]) and Granger Causality (GC [28]–[31] tests are employed as the basis of the fundamental analysis. Chapter 6, which investigated the research objective (research question) (ii), provides the results of HWES [22]–[24] and ARIMA [21] models when the models were applied to the NZX 50 Index.

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Chapter 7, which assessed the research objective (research question) (iii), presents the LSTM [25] models' results when applied to the NZX 50 Index. Addressing the research objective (research question) (iv), Chapter 8 provides the findings of the comparative analysis of the tested models. Chapter 9 concludes the thesis and examines the limitations of this research and the avenues for further investigations.

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Chapter 2: Contextual Background

"Everyone can rise above their circumstances and achieve success if they are dedicated to and passionate about what they do."

Nelson Mandela, Former president of South Africa

2.1 Introduction

This chapter aims to provide background information to certain content presented in this thesis.

This chapter starts with a brief overview of the stock market and subsequently introduces the financial time series forecasting.

2.2 Overview of Stock Markets

A stock market is a forum where the market participants (buyers and sellers) can trade stocks (ownership claims of businesses) through organised exchanges or over-the-counter markets.

These markets create a platform for the participants to generate wealth through investment gains, dividend incomes, diversification benefits, ownership stakes and tax deferrals. Although the two terms are interchangeably used, a stock exchange is considered a subset of a stock market.

2.2.1 New Zealand Stock market

The New Zealand stock market development dates back to 1866 when the first stock exchange was inaugurated in Dunedin for stock trading with gold mines during the sudden influx of gold in the 1870s. The services of this equity exchange have positively contributed to the development of banking, insurance, shipping, and freezing industries. In the early 1870s, more local exchanges were established in Otago in 1868, Auckland in 1872, Wellington in 1882, Christchurch in 1900, etc. In 1915 Stock Exchange Association of New Zealand was

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established to bring the independently ran regional exchanges under one entity. With further refinements to the regulatory framework, operation and monitoring processes impacted forming the New Zealand Stock Exchange (NZSE) in 1983. All the regional exchanges were merged into this exchange. Subsequently, NZSE adapted to technology, shut down the regional trading floors, and established screen trading. In 2002, NZSE became a limited liability company, and in 2003 New Zealand Stock Exchange Limited changed its name to the New Zealand's Exchange (NZX) [32].

2.2.1.1 New Zealand's Exchange (NZX)

New Zealand's Exchange (NZX) is the registered securities exchange in New Zealand. NZX is the responsible authority for enforcing the rules and guidance required for efficient operations of the NZX markets [namely NZX Equity Market (NZSX), NZX Debt Market (NZDX), NZX Dairy Derivatives, NZX Equity Derivatives (NZCX), and Fonterra Shareholders' Market (FSM)] and continuous monitoring for smooth functioning of the markets [33]. The group of S&P/NZX indices evaluates the performance of the listed companies in the NZX. NZX holds a strategic partnership with S&P Dow Jones Indices (S&P DJI). As a result, New Zealand's S&P/NZX indices are closely aligned with the global suite of indices of S&P DJI, which enhances international recognition and global comparison. According to the agreement between NZX and S&P Dow Jones Indices (S&P DJI), S&P DJI is responsible for estimating, maintaining, and licensing the S&P/NZX family of indices [16].

2.2.1.2 New Zealand Stock Index

A stock (market) index is designed and constructed to measure changes in a stock market (or a subset of a stock market). The stock index usually signifies the investors' sentiment towards a set of stocks in the economy and their general opinion of the economy. The S&P/NZX 50 (NZX 50) Index is considered the primary barometer of the New Zealand stock (equity) market.

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The NZX 50 Index is the single most-quoted Index in the New Zealand stock market and covers almost 90% of the New Zealand stock market capitalisation. In March 2003, the Index was introduced as the NZSX 50 Index, replacing the NZSE 40 Index as the headline index. The NZSE 40 Index comprised 40 most prominent and liquid companies in the New Zealand stock market, and this was used as the benchmark index before the formation of the NZX 50 Index.

The NZSX 50 Index subsequently renamed the NZX 50 Index in 2005. The NZX 50 Index is designed to measure the overall performance of the 50 leading listed stocks on the NZX Equity Market (NZSX) by float-adjusted market capitalisation. This Index is recognised as New Zealand's main benchmark index. Consequently, the NZX 50 Index [16] is used in this thesis.

2.2.2 Brief History

The Dutch East India Company (VOC), which was formed in March 1602 for the spice trade in East India, is documented as the first publicly traded joint-stock company in the world and issued its stocks and bonds on the Amsterdam Stock Exchange (ASE, [34]). Even though ASE was established in 1602, ASE merged with Brussels Stock Exchange (BRSE, established in 1801) and Paris Stock Exchange (Paris Bourse, founded in the early 19th century) in 2000 to form Euronext. Therefore, the London Stock Exchange (LSE), formed in 1698, is considered the oldest active stock market exchange in the world, followed by the New York Stock Exchange (NYSE in 1792), the American Stock Exchange (AMEX in 1849), the Bombay Stock Exchange (BSE in 1875) and National Association of Securities Dealers Automated Quotation (NASDAQ in 1971). Stock exchanges in the early days used an open-outcry strategy in trading;

however, with the development of technology, most exchanges moved into computer systems to replace floor traders. However, NYSE is one of the rarest stock market exchanges that still uses an open-outcry trading strategy.

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Over the last few decades, trading on stock markets has increased exponentially. Every country has its stock market and trades trillions of dollars worth of stocks, bonds, and other marketable securities daily in broader global stock markets. There are more than 60 leading stock exchanges around the globe with varying market capitalisations whose total value is more than USD 78 trillion. There are sixteen stock exchanges in the world, each with more than USD one trillion of market capitalisation. In 2021, the largest stock exchange in the world was the New York Stock Exchange (NYSE) which has a market capitalisation of approximately USD 22.9 trillion. [35]).

2.3 Financial time series forecasting

A scientific exploration conducted by [15] to ascertain the most challenging endeavours in data mining research found that sequential and time-series data mining is one of the top 10 complex problems. [15] pointed out that this is primarily because time series data are contaminated by noise with high dimensionality. Learning meaningful information from chaotic sequential data for accurate prediction is always demanding. A univariate time series is a set of chronologically ordered data represented in the form y = [y1, y2,.., yn]^T, where yt is the data at time step t.

Modelling and forecasting financial time series is one of the central topics in the field of Finance. Forecasting models could be reformulated when the models are implemented on the historical observations of univariate or multivariate time series. Researchers and practitioners have attempted to establish financial time series forecasting models accommodating different statistical models, soft computing techniques, and Hybrid Models (HBMs). Many attempts have been carried out to analyse stock market price prediction. The methods can be broadly clustered into fundamental analysis or technical analysis. These are extensively reviewed and presented in Chapter 3.

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2.3.1 An introduction to fundamental analysis

Fundamental analysis is a cohesive approach accommodating the underlying forces of the economy, industry, and an individual company's economics to predict the future movements of the selected stock/index prices [36]. The fundamental analysis depends on various types of statistics (company, industry, economic, financial, managerial, social, news data) and uses a systematic approach to analyse them to determine whether to buy (under-priced) or sell (overpriced) security. Fundamentalists, thus, endeavour to predict the intrinsic value of selected security based on its related economic, financial, and company-specific or industry-specific qualitative and quantitative factors. Intrinsic value, according to fundamentalists, is the actual (true) value of a security based on the law of demand and supply. If the current market price of an asset is above its intrinsic value, it suggests that the asset should be sold because the market is overpriced. If the intrinsic value of an asset is above its current market price, it indicates that the asset should be bought because the market is under-priced.

The Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) devise the theoretical foundation of the relationship between fundamental macroeconomic factors and stock market returns. The economic fundamentals may include employment, production indices, growth, inflation rate, monetary policy, fiscal policy, exchange rate, interest rates, international trade, supply and demand conditions, and so forth. The fundamental analysis also considers financial statements (balance sheet, income statement, cash flow statement), assets, and liabilities of the company being assessed. Other fundamentals include correlated company-specific or industry-specific factors (treasures' reports, auditors' reports, business models, competitive advantages, corporate governance, management policies, competition, regulations, business cycles, industry growth, market share, and the rest). The fundamentalists then employ statistical models to delve into the interlinkages between the

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fundamental factors and the security selected and to transmit a cumulative effect of the fundamental factors to forecast the intrinsic value of the chosen security (index).

An appraisal of fundamental analysis-based research conducted on the financial time series is presented in Chapter 3.

2.3.2 An introduction to technical analysis

Technical analysis is the art of tracking the actual historical values of the time series data to capture the historical trends and then forecast the probable future trends based on the visualised history. The technicians utilise charts and modelling techniques to capture the historical trends to predict stock market price movements [36]. One of the strengths of technical analysis is its versatility to any time dimension and trading medium. The chartists can use intraday ultra-high frequency data (i.e., second-by-second, minute-by-minute) for "day trading" purposes as well as low frequent data (daily or even less frequent) for "trend trading" purposes. The chartists can then use the acquired information to enter a long (short) position when an asset trends upward (downward).

The charting approach can be applied to different types of securities (such as stocks, bonds, options, futures, interest rates, swaps, indices), different time frequencies (ultra-high, high, or low), as well as other markets (domestic or international). The technical trading mechanism has recently been split into two more comprehensive experts; the traditional chartist and the quantitative technicians, although there is considerable overlap. The primary forecasting tool for traditional chartists is historical graphs, but they may use quantitative techniques to supplement the analysis and recommendations. By contrast, the quantitative technicians who snapshot the historical trends use statistical, mathematical, and soft computing techniques to formulate a scientifically sound mechanical trading algorithm. Subsequently, they programme the algorithms to computer systems for mechanistic buy or sell decisions and

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aim to achieve a globally optimal forecasting performance. These computer technicians can be further classified into black-box technicians and white-box technicians.

The former employs a quantifiable investment approach where mathematical formulae outline the choices, and these models intend to predict the future based on the analysed historical patterns. On the other hand, the latter adopts an internal approach and attempts to eliminate hidden prediction errors through automated optimisation hybrid techniques and programmer introspection.

A review of the technical analysis-based research performed on the time series of stock price and stock index prediction is presented in Chapter 3.

2.4 Conclusion to this chapter

This chapter intends to provide a synopsis of the stock market and introduce the financial time series forecasting methods. A detailed discussion on the financial time series forecasting is described in Chapter 3.

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Chapter 3: Appraisal of Financial Time Series Forecasting

"The primary goal of forecasting is to identify the full range of possibilities facing a company, society, or the world at large."

Paul Saffo, Professor of Engineering and Technology Forecaster

This chapter presents a systematic review of the financial time series prediction literature.

Many different techniques and methodologies that have been used to predict stock market movements are classified in this chapter. Section 3.1 provides a brief introduction, while Section 3.2 evaluates the fundamental analysis-related literature in Finance. A comprehensive review of the technical analysis based literature in the financial time series prediction domain is presented in Section 3.3. Section 3.4 provides a summary and the direction for research.

3.1 Introduction

Over the past several decades, many researchers have attempted to predict the movements of stock market prices, bond market prices, financial derivative prices, foreign exchange rates, financial market indices, etc. These prediction methodologies can be broadly categorised into two groups:

• fundamental analysis and

• technical analysis.

Fundamental analysis evaluates stock market prices based on overall macroeconomic conditions, the management strategies of the company and its industry, and political environments. Thus, the fundamentalists employ numeric information about macroeconomic, financial, and other related factors to predict the intrinsic value of stocks.

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The technical analysts instead have total reliance on past market data and typically evaluate the historical price trends to predict stock market price movements. Therefore, the technicians use charts and modelling techniques to capture the historical trends to predict stock market price movements. Technical analysts usually employ advanced CS methods, soft computing techniques, and HBMs, combining different statistical methods, soft-computing methods, or statistical techniques with soft-computing techniques to produce better forecasting performances.

Early research on stock market prediction, such as that of [37]–[40], advocated and supported the Efficient Market Hypothesis (EMH). EMH postulates that the stock market prices reflect all information; thus, it is immediately reflected in the stock prices when new information transpires. According to EMH, it is impossible to beat the market; therefore, neither the technical analysts nor the fundamental analysts can effectively generate an excess return. EMH presume it is impossible to create a reliable prediction model for stock market price movement. However, subsequent researchers such as [41]–[45] have challenged the validity of EMH. They managed to justify that the stock market prices can be predicted with a reasonable degree of accuracy.

3.2 Fundamental Analysis

Fundamental analysis is a cohesive approach accommodating the underlying forces of the economy, industry, and or financial information of an individual company to predict the future movements of the selected stock (Index) prices. The fundamental analysis depends on various types of statistics (company, industry, economic, financial, managerial, social, news data). It systematically analyses the underlying factors to determine whether to buy (under-priced) or sell (overpriced) security. Fundamentalists, thus, endeavour to predict the intrinsic value of selected securities based on their related economic, financial, and company/industry-specific

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qualitative and quantitative factors.If the intrinsic value of an asset is above its current market price, this suggests that the asset should be bought because the market is under-priced.

The theoretical foundation of the relationship between macroeconomic variables and stock market returns is explained by the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT). Sharpe [46] and Lintner [47] established the CAPM model, which is the origin of the asset pricing theories. The CAPM provides a coherent framework to explain the risk-return trade-off of the investments. The CAPM measures the relationship between non-diversifiable risk and the expected return for assets and postulates that investors need to be compensated in two ways, time value of money and market risk. Although the model provides an intuitively good prediction about risk and return trade-off, the empirical findings do not support the model.

APT, formulated by Ross [48] and expanded by [49, 50], is considered a better testable alternative to the CAPM. APT is a general theory of asset pricing which assumes that each asset's (stock's) return to an investor is influenced by multiple independent (macroeconomic/risk) factors. [51] discovered that empirical data support the APT against CAPM. [52] found that the multifactor APT is a superior model for explaining cross-sectional variation in asset returns.

Most of the subsequent studies based on the APT have evaluated the short-term relationship between several macroeconomic variables and the return of an asset or a stock market index. For example, [28], [29], [38], [39], [53]–[55] and so on used the APT model.

Fama [39] is one of the pioneering researchers to evaluate the relationship between stock market returns and macroeconomic variables. He discovered that the real stock market returns are positively related to the real macroeconomic variables (such as the real rate of return on capital and output). Chen, Roll & Ross [53] examined how stock market prices responded to

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selected macroeconomic variables and paved the way for the principle of a long-term equilibrium relationship between macroeconomic variables and stock prices. They used a spread of interest rates, inflation rate, industrial production and spread between high and low- grade bonds as determinants of the stock market return. They found that industrial production and changes in risk premium are crucial determinants, and inflation is identified as a relatively insignificant determinant. [55] studied how selected variables such as industrial production, default spread, term structure, T-bill rate, dividend yield, Gross National Product (GNP) and consumption influence the stock returns. He uncovered that the default spread, term spread, one-month T-bill rate, lagged industrial production growth rate, and the dividend-price ratio could be important in determining future stock market returns.

Limited research has been carried out in the New Zealand Stock Market context.

Adopting a GARCH model, [56] evaluated the relationship between the NZX 50 Index and selected macroeconomic variables. [56] used real Gross Domestic Product (GDP), world stock market index, US government bond yield, real interest rate, the nominal exchange rate (NZD/USD), inflation rate, and ratio of New Zealand government debt to GDP as the macroeconomic determinants to analyse the NZX 50 Index. This study discovered that the NZX 50 Index positively relates to real GDP and the world stock market index. However, NZX 50 Index is negatively associated with the nominal exchange rate (NZD/USD), the ratio of government debt to GDP, inflation rate and US government bond yield. [57] evaluated the relationship between the long-run real stock prices and inflation for sixteen industrialised countries, including New Zealand. Using the methodology of King and Watson [57], [58]

found no evidence of inflation eroding the real value of stock prices due to a long-run inflation neutrality effect on the real stock prices. [59] assessed the existence of stock market volatility in the long run. Their study focused on twelve industrial countries, including New Zealand and eleven emerging markets, and found a positive association between stock market volatility and

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monetary policy volatility. They also discovered that a fixed exchange rate regime influences a lower stock market volatility than a flexible exchange rate. [60] examined how the FOREX volatility can affect the Pacific Basin stock markets, including New Zealand. They found that stock market volatility is more sensitive to equity's bad news and more responsive to currency depreciation. [61] evaluated how the volatility spillovers due to exchange rate changes have affected the stock market returns in Ne