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This paper provides a review of the Fractal Market Hypothesis (FMH) focusing on financial times series analysis. In order to put the FMH into a broader perspective, the Random Walk and Efficient Market Hypotheses are considered together with the basic principles of fractal geometry. After exploring the historical developments associated with different financial hypotheses, an overview of the basic mathematical modelling is provided. The principal goal of this paper is to consider the intrinsic scaling properties that are characteristic for each hypothesis. In regard to the FMH, it is explained why a financial time series can be taken to be characterised by a 1/t1−1/γ scaling law, where γ>0 is the Lévy index, which is able to quantify the likelihood of extreme changes in price differences occurring (or otherwise). In this context, the paper explores how the Lévy index, coupled with other metrics, such as the Lyapunov Exponent and the Volatility, can be combined to provide long-term forecasts. Using these forecasts as a quantification for risk assessment, short-term price predictions are considered using a machine learning approach to evolve a nonlinear formula that simulates price values. A short case study is presented which reports on the use of this approach to forecast Bitcoin exchange rate values.

Research background: Covid-19 has affected the global economy and has had an inevitable impact on capital markets. In the week of February 24?28, 2020, stock markets crashed. The index FTSE 100 decreased 13%, while the indices DJIA and S&P 500 fell 11?12%, the biggest drop since the 2007?2008 financial and economic crisis. It is therefore of interest to test the random walk hypothesis in developed capital markets, European and also non-European, in order to understand the different predictabilities between them. Purpose of the article: The aim is to analyze capital market efficiency, in its weak form, through the stock market indices of Belgium (index BEL 20), France (index CAC 40), Germany (index DAX 30), USA (index DOW JONES), Greece (index FTSE Athex 20), Spain (index IBEX 35), Ireland (index ISEQ), Portugal (index PSI 20) and China (index SSE) for the period from December 2019 to May 2020. Methods: Panel unit root tests of Breitung (2000), Levin et al. (2002) and Hadri (2002) were used to assess the time series stationarity. The test of Clemente et al. (1998) is used to detect structural breaks. The tests for the random walk hypothesis follows the variance ratio methodology proposed by Lo and MacKinlay (1988). Findings & Value added: In general, we found mixed confirmation about the EMH (efficient market hypothesis). Taking into account the conclusions of the rank variance test, the random walk hypothesis was rejected in the case of stock indices: Dow Jones, SSE and PSI 20, partially rejected in the case indices: BEL 20, CAC 40, FTSTE Athex 20 and DEX 30, but accepted for indices: IBEX 35 and ISEQ. The results also show that prices do not fully reflect the information available and that changes in prices are not independent and identically distributed. This situation has consequences for investors, since some returns can be expected, creating opportunities for arbitrage and for abnormal returns, contrary to the assumptions of random walk and information efficiency.

In this study, we examine the random walk hypothesis for two well-known Indian indices, BSE (Sensex) and NSE (Nifty), by considering country-specific political events (parliamentary elections). Two pertaining questions were studied. First, efficiency with respect to weak form, semi-strong form, and strong form; and second, the random walk pattern of the return by applying the new variance ratio tests. We use 21 years of daily closing stock price data of both the National Stock Exchange (NSE) Nifty Index and the Bombay Stock Exchange (BSE) Sensex Index. The hypothesis is tested by using both conventional and new variance ratio tests: The Lo–MacKinlay variance ratio test, the Chow–Denning test, and Wright’s Rank and Sign test. All three tests report that the return does not follow the random walk for the full sample, suggesting the possibility of making gains by exploiting various investing strategies. It was found that both Indian indices follow the random walk hypothesis during the phase of parliamentary elections. This study contributes to the existing literature on the Efficient Market Hypothesis (EMH).

We study the informational efficiency of the Saudi stock market (SSM), while accounting for corporate governance change, based on single, multiple, and variance ratio-based WALD tests and runs test. The main findings indicate that when the whole period is considered, the random walk hypothesis is rejected, but when divided into two sub-periods separated by the pre-corporate governance and the period marked by corporate governance change, the analysis demonstrates sub-period improvement in weak-form efficiency for the examined series. Robustness of results is verified by analysis using sector indices, which point to market efficiency. Interestingly, Hurst Exponent estimates evidence long-range dependence which suggests the predictability of stock prices and the prospect of speculative opportunities.

This article examines the martingale difference hypothesis (MDH) and the random walk hypothesis (RWH) for nine conventional and nine Islamic stock indices: Asia-Pacific, Canadian, Developed Country, Emerging, European, Global, Japanese, UK, and United States. It investigates whether Islamic stock indices are more, less, or as efficient as their conventional counterparts. We test four sub-periods of bullish and bearish stock markets, together with the financial meltdown and its recovery, over the period 1997-2012. We use the Escanciano and Lobato's (2009) automatic portmanteau test (AQ) and Deo's (2000) test for the MDH. We also apply the automatic variance ratio test (AVR) developed by Choi (1999) and Kim (2009) for the RWH. Over the period from 1997 to 2012, we find that three conventional indices (Europe, Japan, and UK) are efficient, but that none of the Islamic indices are efficient in these markets. During the recent financial crisis, our results indicate slightly more efficiency for the Islamic indices than their conventional counterparts. Our study finds that overall the conventional indices are more efficient than their Islamic counterparts. Nevertheless, during periods of general downturns the Islamic indices have shown the same level of efficiency as their counterparts. Furthermore, it appears that during the last two sub-periods under study, the Islamic indices have moved toward efficiency, displaying the same level of efficiency as their counterparts.

Deviations of asset prices from the random walk dynamic imply the predictability of asset returns and thus have important implications for portfolio construction and risk management. This paper proposes a real-time monitoring device for such deviations using intraday high-frequency data. The proposed procedures are based on unit root tests with in-fill asymptotics but extended to take the empirical features of high-frequency financial data (particularly jumps) into consideration. We derive the limiting distributions of the tests under both the null hypothesis of a random walk with jumps and the alternative of mean reversion/explosiveness with jumps. The limiting results show that ignoring the presence of jumps could potentially lead to severe size distortions of both the standard left-sided (against mean reversion) and right-sided (against explosiveness) unit root tests. The simulation results reveal satisfactory performance of the proposed tests even with data from a relatively short time span. As an illustration, we apply the procedure to the Nasdaq composite index at the 10-minute frequency over two periods: around the peak of the dot-com bubble and during the 2015–2106 stock market sell-off. We find strong evidence of explosiveness in asset prices in late 1999 and mean reversion in late 2015. We also show that accounting for jumps when testing the random walk hypothesis on intraday data is empirically relevant and that ignoring jumps can lead to different conclusions.

This study re-examined the traditional research topic in finance—the efficient market hypothesis (EMH) within the context of a nonlinearity unit root test using daily data sourced on the Nigerian economy. This was premised on two key motivations. First, the study observed that most of the existing studies on EMH are based on an aggregate stock index and the presence of heterogeneity of listed firms on the floor of the exchange can make the results obtained from aggregate stock price-based tests misleading due to spuriousness. To overcome this, the study tested the validity of EMH at a sectorial level. The second motivation centers on the observed nonlinear property of the time series data used in the existing literature. The study first examines the unit root properties of the data by applying the Harvey, Leybourne, and Xiao (2008) methods. The results indicate rejections of the null hypothesis for all the series, an indication that stock market indices in Nigeria are nonlinear and asymmetric in nature. This suggests that results obtained from linear based models could be biased. In order to achieve more accurate results, the study applied the ESPAR model and observed that, overall, there is an abundance of evidence to show that the Nigerian stock exchange is mean reverting, thus investors are advised to adopt a contrarian investment strategy to maximize the opportunities in the market.

The Cowles-Jones test for sign dependence is one of the earliest tests of the random walk hypothesis, which stands at the beginning of modern empirical finance. The test is still discussed in popular textbooks and used in research articles. However, the Cowles-Jones test statistic considered in the literature requires that the upward probability of the market or asset under consideration be specified under the null hypothesis, which is only very rarely possible. If the upward probability is estimated in advance, the resulting test is undersized (even asymptotically). This note considers a corrected Cowles-Jones test statistic which does not require the upward probability to be specified under the null. It turns out that the asymptotic variance is greatly simplified as compared to the uncorrected test. The corrected test is illustrated with an application to daily returns of the Dow Jones Industrial Average index and monthly returns of the MSCI Emerging Markets index. It is shown that the corrected and uncorrected tests can lead to opposite conclusions.

This study investigates the effect of COVID-19 on the random walk behavior of the emerging Islamic stock market over the period from October 9, 2019, to August 11, 2020. Two Islamic stock indices, namely the FTSE Bursa Malaysia Hijrah Shariah Index and the FTSE Bursa Malaysia EMAS Shariah Index, are examined in this study. The unit root test and variance ratio test are employed to examine the random walk hypothesis. The results of the study showed that (i) Islamic stock indices did not follow a random walk process before and during COVID-19, as well as for the entire period, thus suggesting that the emerging Islamic stock market is not weak form efficient, and (ii) returns for Islamic stock indices were much higher and more volatile during the period of COVID-19. The findings of this study have important implications. First, the stationarity of the return series suggests that it is possible to gain excess return, thus offering investors the opportunity to diversify investment risk. Second, the findings of inefficient weak form Islamic stock indices suggest that stock market regulators may undertake policies to develop the stock market and improve information dissemination in order to promote more efficient resource allocation.

The purpose of this study is to examine the market efficiency of cryptocurrencies, specifically at a weak level. The study focuses on six prominent cryptocurrencies selected based on their significant market capitalization: Bitcoin (BTC), Tether (USDT), Ethereum (ETH), Binance Coin (BNB-USD), Ripple (XRP-USD), and Cardano USD (ADA-USD). The analysis utilizes unit root, Ljung–Box, variance ratio, runs, and the Brock–Dechert–Scheinkman (BDS) tests to assess different aspects of market efficiency. The data spans from September 2017 to April 2023, encompassing a wide time frame to capture potential shifts in market behavior. The results of all the tests, except the BDS test, indicate that the tested cryptocurrencies' markets are inefficient. However, the BDS test yielded different results, suggesting that BTC and ETH exhibit market efficiency compared to the other cryptocurrencies. This discrepancy indicates that the BDS test may be capturing different aspects of the time series behavior. The practical implication is that investors and market participants should exercise caution and consider the varying levels of efficiency when making decisions regarding these cryptocurrencies. Also, investors should consider a range of factors, including technical and fundamental analyses, when making investment decisions in a dynamic and evolving market.