Comparing Market Efficiency in Developed, Emerging, and Frontier Equity Markets: A Multifractal Detrended Fluctuation Analysis

Abstract

In this article, we investigate the market efficiency of global stock markets using the multifractal detrended fluctuation analysis methodology and analyze the results by dividing them into developed, emerging, and frontier groups. The static analysis results reveal that financially advanced countries, such as Switzerland, the UK, and the US, have more efficient stock markets than other countries. Rolling window analysis shows that global issues dominate the developed country group, while emerging markets are vulnerable to foreign capital movements and political risks. In the frontier group, intensive domestic market issues vary, making it difficult to distinguish similar dynamics. Our findings have important implications for international investors and policymakers. International investors can establish investment strategies based on the degree of market efficiency of individual stock markets. Policymakers in countries with significant fluctuations in market efficiency should consider implementing new regulations to enhance market efficiency. Overall, this study provides valuable insights into the market efficiency of global stock markets and highlights the need for careful consideration by international investors and policymakers.

1. Introduction

The efficient market hypothesis (EMH) posits that information in the stock market is promptly incorporated into stock prices. According to this theory, current stock prices already encompass all available information. Consequently, stock prices are subject to random movements and cannot be deemed undervalued or overvalued. Fama divides the efficient market hypothesis into three hypotheses based on the range of related information. A weak-form efficient market is a market that reflects historical information in the current price, and excess profits can be obtained if current or future information is used. The semi-strong-form efficient market reflects current information in addition to past information and can generate excess profits if future information is obtained. The strong-form efficient market hypothesis states that all market participants know all information, including future undisclosed information, and that no information analysis can generate excess profits.

Determining the specific efficient market hypothesis that governs the stock market holds significant implications for comprehending the information reflected in current stock prices. This understanding plays a vital role in effective risk management by facilitating the prediction of future stock price fluctuations. Furthermore, it facilitates effective resource allocation, leveraging the prevailing market prices for efficient growth. In cases where the market does not strictly adhere to a strong-form efficient market, valuable insights can be gained to guide decision making concerning the specific information and information analysis methods necessary for obtaining excess profits.

As market efficiency is important, many studies have been conducted on this topic. For example, Horta et al. [1] showed that financial crises significantly influenced most stock markets, and that markets lost efficiency during the subprime crisis. Hull and McGroarty [2] investigated the relationship between efficiency and market development. Rizvi et al. [3] revealed a relatively high efficiency ranking in developed markets in the short term and a medium efficiency ranking in the long term. Charfeddine and Khediri [4] researched the weak-form efficiency of the Gulf Cooperation Council stock markets, and concluded that while the GCC market has different degrees of efficiency over time, it is also improving over time. Ali et al. [5] compared the efficiency of conventional stock markets and their Islamic counterparts, and the result was that developed markets were relatively more efficient.

Two main methodologies have been employed to investigate the EMH. The first approach utilizes the Hurst–Mandelbrot–Walis R/S statistics, a statistical methodology. Based on this method, Horta et al. [1] discovered that stock price increments in the pre-crisis era align more closely with the random walk paradigm compared to the post-crisis period. Similarly, Hull and McGroarty [2] noted that more advanced emerging markets display lower levels of long-term persistence. The second methodology employed is the multifractal detrended fluctuation analysis (MF-DFA). By using the MF-DFA approach, Rizvi et al. [3] emphasized the efficient performance of Islamic markets during crises, while Horta et al. [1] suggested deviations that are indicative of long memory and reverting patterns. Charfeddine and Khediri [4] ranked Qatar as the most efficient market, while Bahrain and Oman were considered less efficient. Furthermore, Ali et al. [5] provided rankings of the efficiency of stock markets in non-Islamic, BRICS, and developed countries, as well as their corresponding Islamic counterparts.

This study investigated the efficiency of the stock markets of several countries around the world. Market efficiency was calculated using stock indices from 60 countries, and the results were analyzed by dividing them into developed, emergent, and frontier groups (MSCI market classification, https://www.msci.com/our-solutions/indexes/acwi accessed on 1 May 2023). We describe the stock index data in Section 3.

We employed the MF-DFA method to calculate market efficiency. Recently, this method has been widely used in research on the market efficiency hypothesis and multifractality of financial assets (e.g., Miloş et al. [6], Choi [7], Yin and Wang [8], Pak and Choi [9], Gaio et al. [10]). Furthermore, we performed both static and rolling window analyses in order to assess the market efficiency of global stock markets. Through this method, we analyzed market efficiency for the entire sample period and time-varying market efficiency.

This study contributes to the financial literature in two ways. First, the market efficiency of various countries around the world shows some differences between the developed, emerging, and frontier groups. Rather than focusing on individual countries, this research adopted a group perspective to analyze stock market efficiency. While studies on market efficiency in developed countries have been extensively reported, the attention given to emerging and frontier countries has been relatively limited. Previous research on emerging or frontier countries has primarily compared and analyzed market efficiencies among individual countries or a small number of countries (e.g., Caraiani [11], Arshad et al. [12], Aslam et al. [13], and Nargunam and Lahiri [14]). In contrast, this study aims to overcome these limitations by incorporating as many stock markets as possible, spanning developed, emerging, and frontier countries. This approach provides a comparative advantage in terms of the number of stock markets considered compared to previous studies (e.g., Rizvi et al. [3], Ali et al. [5], Lee et al. [15], and Aslam et al. [16]).

Second, we methodologically investigated how market efficiency in individual countries changes over time using rolling window analysis. In particular, we analyzed and compared the degree of response of individual markets to systematic risks, such as the 2008 global financial crisis and COVID-19 pandemic, while several studies have examined the impact of the COVID-19 pandemic on stock market efficiency (e.g., Aslam et al. [17], Mensi et al. [18], and Saadaoui [19]), the range of stock markets considered in these studies was narrower than that of our research. By employing rolling window analysis, this study provides valuable insights into the evolving nature of market efficiency in individual countries over time.

Overall, this study contributes to the financial literature by examining stock market efficiency from a group perspective and incorporating a wide range of stock markets from developed, emerging, and frontier countries. Furthermore, it enhances the understanding of how market efficiency changes within individual countries over time by investigating their responsiveness to systematic risks. The findings of this study have implications for investors, policymakers, and researchers seeking to comprehend and navigate the dynamics of global financial markets.

The remainder of this paper is organized as follows. In the following section, we introduce previous studies on market efficiency. Section 3 describes global stock market data and provides a preliminary statistical analysis. In Section 4, we briefly review the MF-DFA method. Section 5 presents the results of static and rolling window analyses. Section 6 presents the summary and concluding remarks.

2. Literature Review

2.1. Global Stock Markets

Many studies have compared national stock market efficiencies. Studies on efficiency suggest that most developed markets are more efficient than emerging markets. Lim [20] defined the sequence of efficiency of the US, Korea, Taiwan, Japan, Thailand, Philippines, Brazil, Mexico, India, Indonesia, Malaysia, Chile, and Argentina, with emerging and developed markets using H-statistic. The relationship between market efficiency and market development was investigated by Hull and McGroarty [2]. Their study employed the Hurst–Mandelbrot–Walis R/S statistic and the Jarque–Bera test to analyze markets categorized as either advanced or secondary. The findings indicated that advanced markets demonstrate higher levels of efficiency in comparison to secondary markets. Using MF-DFA, Rizvi et al. [3] showed that developed markets (e.g., the US, Japan, and Hong Kong) are more efficient, excluding Malaysia, Indonesia, and Turkey, which have Islamic and developed markets (e.g., Bahrain, Bangladesh, and Egypt). Further, Ali et al. [5] revealed that developed markets are the most efficient, followed by BRICS markets. In addition, every Islamic stock market, except Russia, Jordan, and Pakistan, is more efficient than its conventional counterparts, such as Turkey, the USA, and China. Anagnostidis et al. [21] investigated the effects of the 2008 financial crisis on Eurozone stock markets, using GHE (generalized Hurst exponent). They classified them into three groups according to how adversely they were affected by the crisis.

2.2. Time-Varying Market Efficiency

Several studies have investigated the evolution of market efficiency over time. For example, Rizvi and Arshad [22], using MF-DFA, examined the efficiency of East Asian stock markets during booms and busts. The study concluded that economic booms affected each market differently, with a decrease in efficiency during recession periods and an increase during boom periods. Interestingly, only Singapore’s market efficiency increased during these events. In another comprehensive examination of financial crises, Horta et al. [1] studied the impact of the 2008 and 2010 crises on stock markets. Their findings indicated the presence of market efficiency dynamics and financial contagion during these periods. Moreover, the study observed that developed markets were minimally affected by financial crises, while less-developed markets experienced significant impact.

Many studies have been conducted to examine the fluctuations in market efficiency levels. Smith [23] studied European emerging and developed stock markets and used variance ratio tests of the martingale hypothesis. They revealed that each country has different levels of efficiency; the most efficient market is Turkey and the least efficient markets are Malta and Ukraine. They also suggested that the global financial crisis had little effect on weak-form efficiency markets, such as Greece and Russia. Furthermore, according to the changing efficiency over time, the relative efficiency also changes. While the market with the largest improvement is Romania, that with the smallest improvement is Croatia. Sensoy [24] showed that the MENA stock markets have different levels of long-term dependence by using the rolling window technique. They also suggested that the most efficient markets, such as Turkey and Israel, reduce the capital costs of less-efficient markets, such as Iran and the UAE.

3. Data Description

In our analysis, we utilized a range of global stock indices to capture market performance. Specifically, we considered stock price indices from 23 developed markets, 22 emerging markets, and 15 frontier markets. The data for these indices were sourced from Reuters, providing a comprehensive view of market trends and movements. Our analysis covered the period from January 2007 to June 2022, allowing us to examine the performance of these markets over a substantial time period. We display the log returns of the global stock indices in three groups in Figure 1, Figure 2 and Figure 3. Furthermore,

Table 1. Descriptive statistics of global stock indices in three groups. ${}^{‡}$ indicates rejection of the null hypothesis at the $1\%$ significance level. Note: Skew., Kurt., and J.-B. refer to skewness, kurtosis, and Jarque–Bera statistics, respectively.

Group	Country	Mean	Std.Dev.	Skew.	Kurt.	J.-B.
Developed	Austria	−0.0003	0.0286	−2.23	23.97	34,312.3 ${}^{‡}$
	Australia	0.0001	0.0181	−1.37	14.29	12,225.4 ${}^{‡}$
	Belgium	−0.0001	0.0227	−2.49	33.61	66,593.6 ${}^{‡}$
	Canada	0.0003	0.0186	−3.59	54.18	172,326.9 ${}^{‡}$
	Switzerland	0.0001	0.0178	−0.89	8.43	4291.6 ${}^{‡}$
	Germany	0.0005	0.0236	−1.46	13.40	10,865.4 ${}^{‡}$
	Denmark	0.0008	0.0207	−1.45	17.40	17,969.6 ${}^{‡}$
	Spain	−0.0004	0.0243	−0.64	5.78	2030.5 ${}^{‡}$
	Finland	0.0001	0.0229	−1.01	8.93	4844.8 ${}^{‡}$
	France	0.0001	0.0228	−1.2	10.69	6959.5 ${}^{‡}$
	United Kingdom	0.0001	0.0193	−1.54	19.23	21,893.2 ${}^{‡}$
	Hong Kong	0.0001	0.0250	−0.33	7.19	3015.5 ${}^{‡}$
	Ireland	−0.0002	0.0267	−1.98	19.56	23,003.4 ${}^{‡}$
	Israel	0.0005	0.0182	−1.35	11.96	8688.4 ${}^{‡}$
	Italy	−0.0005	0.0262	−0.99	7.53	3508.5 ${}^{‡}$
	Japan	0.0003	0.0234	−1.13	11.26	7612.8 ${}^{‡}$
	Netherlands	0.0002	0.0223	−2.58	33.46	66,148.6 ${}^{‡}$
	Norway	0.0004	0.0254	−2.58	36.18	77,071.1 ${}^{‡}$
	New Zealand	0.0007	0.0135	−1.35	17.00	17,098.1 ${}^{‡}$
	Portugal	−0.0004	0.0230	−0.90	7.33	3295.8 ${}^{‡}$
	Sweden	0.0005	0.0212	−1.45	12.99	10,230.5 ${}^{‡}$
	Singapore	0.0001	0.0197	−1.36	26.08	39,688.4 ${}^{‡}$
	United States	0.0007	0.0195	−2.44	30.52	55,108.3 ${}^{‡}$
Emerging	Brazil	0.0006	0.0275	−1.35	16.90	16,904.2 ${}^{‡}$
	Chile	0.0005	0.0208	−1.51	21.89	28,191.9 ${}^{‡}$
	China	0.0010	0.0381	−0.84	6.52	2622.6 ${}^{‡}$
	Czech Republic	−0.0001	0.0225	−2.11	24.99	37,082.7 ${}^{‡}$
	Egypt	0.0002	0.0285	−0.85	9.12	4970.4 ${}^{‡}$
	Greece	−0.0012	0.0352	−0.56	5.68	1941.4 ${}^{‡}$
	Indonesia	0.0010	0.0225	−1.31	19.56	22,467.4 ${}^{‡}$
	India	0.0010	0.0243	−1.46	19.31	22,025.8 ${}^{‡}$
	South Korea	0.0004	0.0216	−2.00	23.92	33,942.8 ${}^{‡}$
	Mexico	0.0004	0.0191	−0.62	9.94	5792.5 ${}^{‡}$
	Malaysia	0.0002	0.0144	−1.60	19.73	23,069.1 ${}^{‡}$
	Peru	0.0003	0.0277	−1.07	20.70	25,000.7 ${}^{‡}$
	Philippines	0.0005	0.0231	−1.83	30.66	55,014.2 ${}^{‡}$
	Poland	−0.0004	0.0251	−1.33	11.32	7812.2 ${}^{‡}$
	Qatar	0.0004	0.0229	−0.10	12.68	9282.7 ${}^{‡}$
	Saudi Arabia	0.0003	0.0219	−1.30	11.51	8030.4 ${}^{‡}$
	Thailand	0.0007	0.0209	−1.87	24.35	35,040.5 ${}^{‡}$
	Turkey	−0.0020	0.1294	−34.34	1239.47	88,866,224 ${}^{‡}$
	Taiwan	0.0005	0.0219	−1.00	13.11	10,152.5 ${}^{‡}$
	U.A.E.	−0.0002	0.0286	−0.91	19.27	21,636.1 ${}^{‡}$
	South Africa	−0.0002	0.0371	−1.29	24.93	36,242.1 ${}^{‡}$
Frontier	Bahrain	−0.0001	0.0103	−1.06	9.22	5171.3 ${}^{‡}$
	Estonia	0.0005	0.0196	−1.32	18.41	19,949.7 ${}^{‡}$
	Croatia	−0.0003	0.0196	−1.19	23.81	33,047.3 ${}^{‡}$
	Iceland	−0.0007	0.0433	−25.71	818.68	38,803,863 ${}^{‡}$
	Jordan	−0.0009	0.0236	−4.71	103.63	624,502.1 ${}^{‡}$
	Kenya	−0.0010	0.0181	−1.39	17.91	18,944.9 ${}^{‡}$
	Lithuania	0.0004	0.0185	−1.55	36.21	76,200.1 ${}^{‡}$
	Morocco	0.0001	0.0151	−2.14	34.50	69,721.5 ${}^{‡}$
	Mauritius	0.0004	0.0149	−2.33	66.04	252,860.5 ${}^{‡}$
	Oman	−0.0002	0.0164	−0.41	14.93	12,911.4 ${}^{‡}$
	Romania	0.0003	0.0254	−1.41	14.45	12,514.1 ${}^{‡}$
	Serbia	−0.0005	0.0234	−0.92	28.09	45,734.8 ${}^{‡}$
	Slovenia	−0.0002	0.0250	−0.45	101.78	597,553.8 ${}^{‡}$
	Tunisia	0.0008	0.0102	−1.82	22.05	28,837.1 ${}^{‡}$
	Vietnam	0.0002	0.0269	0.01	7.56	3302.1 ${}^{‡}$

shows the summary statistics for the log returns of global stock indices. Based on the information provided in the above table, it can be observed that all three groups exhibit a bias towards the left side of the distribution. This indicates that the data do not follow a normal distribution. However, Vietnam is biased toward the right, showing lower-than-average profits. The average kurtosis is 19, 76, 84 (rounded from the first decimal place) for each group, and the emerging and frontier groups are greatly affected by extreme values, showing relatively unstable returns. In particular, Turkey has a value of 1239.4757 in the emerging group and Iceland has a value of 838.6851 among the frontier group, which is extreme within the group.

Larger kurtosis values tend to increase J-B statistics. The following are the top five countries in each group: Canada, Norway, Belgium, the Netherlands, and the US in the developed group; Turkey, the Philippines, the Czech Republic, South Africa, and Thailand in the emerging group; and Iceland, Jordan, Slovenia, Mauritius, and Lithuania in the frontier group.

Comparing the Jarque–Bera statistics for each group, there are very large outliers such as Turkey, where the median is 1225.4, emerging is 22,025.8, and frontier is 30,942.2, showing that developed countries are relatively trying to follow the normal distribution, making the most stable return. Spain has the smallest value which can be considered stable, followed by Hong Kong.

4. Multifractal Detrended Fluctuation Analysis

The multifractal detrended fluctuation analysis (MF-DFA) method was used to assess the multifractal characteristics of the financial time series and rank market efficiency. It can be executed in five steps, as outlined by [25] and summarized by [26].

Let $\{x_k,k=1,\cdots ,N\}$ be a time series, where N is its length.

• Step 1. Determine the profile $Y\left(i\right)(i=1,2,\cdots ,N)$

  $$\begin{array}{c}\hfill Y\left(i\right)=\sum _{k=1}^i\left(x\left(k\right)-\overline{x}\right),\end{array}$$

  (1)

  where

  $$\begin{array}{c}\hfill \overline{x}=\sum _{k=1}^{N}x\left(k\right)/N.\end{array}$$

  (2)
• Step 2. Divide the profile $\left\{Y\right(i\left)\right\}(i=1,2,\cdots ,N)$ into $N_s\equiv int(N/s)$ non-overlapping segments of equal length s. To ensure that the entire sample is covered, the same procedure is repeated starting from the end of the sample. By doing so, a total of $2N_s$ segments are obtained:

  $$\begin{array}{ccc}& & {\left\{Y[(\nu -1)s+i]\right\}}_{i=1}^s,\nu =1,2,\dots ,N_s\hfill \end{array}$$

  (3)

  $$\begin{array}{ccc}& & {\left\{Y[N-(\nu -N_s)s+i]\right\}}_{i=1}^s,\nu =N_s+1,N_s+2,\dots ,2N_s.\hfill \end{array}$$

  (4)
• Step 3. Calculate the local trend for each of the $2N_s$ segments. To estimate the local trend in each segment, a least-squares fitting polynomial is used. Once the local trend has been determined, the variance is calculated accordingly.

  $$\begin{array}{c}\hfill F^2(s,\nu )=\left\{\begin{array}{cc}\frac{1}{s}\sum _{i=1}^s{\left\{Y[(\nu -1)s+i]-{\widehat{Y}}_{\nu }^m\left(i\right)\right\}}^2,\hfill & \nu =1,2,\dots ,N_s\hfill \\ \frac{1}{s}\sum _{i=1}^s{\left\{Y[N-(\nu -N_s)s+i]-{\widehat{Y}}_{\nu }^m\left(i\right)\right\}}^2,\hfill & \nu =N_s+1,N_s+2,\dots ,2N_s.\hfill \end{array}\right.\end{array}$$

  (5)
  The fitting polynomial with order m in segment $\nu$ is denoted as ${\widehat{Y}}_{\nu }^m\left(i\right)$. Typically, a linear ($m=1$), quadratic ($m=2$), or cubic ($m=3$) polynomial is used to estimate the local trend in each segment, as reported in previous studies ([27,28,29]). However, in this study, in order to avoid overfitting and simplify the calculation process, a linear polynomial ($m=1$) is employed, as suggested in [30,31].
• Step 4. Average over all the segments. Then, we obtain the q-th order fluctuation function:

  $$\begin{array}{c}\hfill F_q\left(s\right)=\left\{\begin{array}{cc}{\left[\frac{1}{2N_s}\sum _{\nu =1}^{2N_s}{\left(F^2(s,\nu )\right)}^{q/2}\right]}^{1/q},\hfill & q\ne 0\hfill \\ exp\left[\frac{1}{4N_s}\sum _{\nu =1}^{2N_s}ln\left(F^2(s,\nu )\right)\right]\hfill & q=0.\hfill \end{array}\right.\hfill \end{array}$$

  (6)
• Step 5. Determine the scaling behavior of the fluctuation functions. To determine if a long-range power law correlation exists, the log–log plots of $F_q\left(s\right)$ are compared for each value of q. If the series exhibits long-range power law correlation, $F_q\left(s\right)$ will increase as s becomes large. The power law relationship can be expressed in the following form.

  $$\begin{array}{c}\hfill F_q\left(s\right)\propto s^{h\left(q\right)},\end{array}$$

  (7)

  where $h\left(q\right)$ represents the generalized Hurst exponent. Equation (7) can be written as $F_q\left(s\right)=a·s^{h\left(q\right)}+b$. After taking the logarithms of both sides,

  $$\begin{array}{c}\hfill log\left(F_q\left(s\right)\right)=h\left(q\right)·log\left(s\right)+c,\end{array}$$

  (8)

  where c is a constant.

The value of the exponent $h\left(q\right)$ depends on q. If $h\left(q\right)$ is independent of q, then the time series is monofractal; otherwise, it is multifractal. When $q=2$, $h\left(2\right)$ is equivalent to the Hurst exponent ([32]); thus, $h\left(q\right)$ is referred to as the generalized Hurst exponent. $h\left(2\right)=0.5$ indicates that the time series are uncorrelated and follow a random walk process, suggesting that the market is weakly efficient ([32,33]). If $0.5<h\left(2\right)$, the time series is long-range dependent, meaning that an increase (decrease) is more likely to be followed by another increase (decrease). By contrast, $h\left(2\right)<0.5$ indicates a non-persistent series, where an increase (decrease) is more likely to be followed by a decrease (increase).

According to Reference [25], the relationship between $h\left(q\right)$ and multifractal scaling exponent $\tau \left(q\right)$ can be expressed as follows:

$$\begin{array}{c}\hfill \tau \left(q\right)=qh\left(q\right)-1.\end{array}$$

(9)

To estimate multifractality, a Legendre transform was used to convert q and $\tau \left(q\right)$ into $\alpha$ and $f\left(\alpha \right)$, respectively, using the following equations:

$$\begin{array}{c}\hfill \alpha =\frac{d}{dq}\tau \left(q\right),f\left(\alpha \right)=\alpha \left(q\right)q-\tau \left(q\right),\end{array}$$

(10)

where $f\left(\alpha \right)$ is the multifractal or singularity spectrum and $\alpha$ is the singularity strength.

In addition, a definition of the width of the multifractal spectrum $\Delta \alpha$ is provided in References [34,35,36].

$$\begin{array}{c}\hfill \Delta \alpha =max\left(\alpha \right)-min\left(\alpha \right).\end{array}$$

(11)

A wider multifractal spectrum implies a higher degree of multifractality. The next section presents the empirical results pertaining to the multifractality of the average return series for all sectors.

5. Empirical Results

This section provides a static and rolling window analysis of the market efficiencies of the three groups.

5.1. Static Analysis

Figure 4, Figure 5 and Figure 6 show the log−log plots of $F_q\left(s\right)$ compared with s for all average sector return series during the GFC and COVID-19 pandemic for $q=-10,-9,\dots ,9,10$ corresponding to the curve from the bottom to the top when the polynomial order $m=1$.

The Hurst exponent $H\left(q\right)$ for developed, emerging, and frontier countries is shown in Figure 7, Figure 8 and Figure 9. As shown, the generalized Hurst exponent of the index return series decreases as q increases from $-10$ to 10, indicating that the return series of all markets have clear multifractal characteristics. For $q=2$, the generalized Hurst exponent $h\left(q\right)$ is equal to the Hurst exponent. Most Hurst exponents differed from 0.5, providing evidence against the presence of the random-walk behavior. Furthermore, the Hurst exponents of all markets in the frontier group are greater than 0.5. In other words, the average return series for all the markets in the frontier group are persistent. In particular, the stock markets of Switzerland, the United Kingdom, and the United States have Hurst exponents close to 0.5, indicating that their stock markets are close to the efficiency market.

We rank the efficiency of each stock market using the measure $\Delta \alpha$ in

Table 2. The width of the multifractal spectrum $\Delta \alpha$ for the developed, emerging, and frontier groups.

Developed		Emerging		Frontier
Country	$\Delta \mathbf{\alpha }$	Country	$\Delta \mathbf{\alpha }$	Country	$\Delta \mathbf{\alpha }$
Austria	0.5278	Brazil	0.404	Bahrain	0.3762
Australia	0.4531	Chile	0.6333	Estoina	0.4434
Belgium	0.4315	China	0.4274	Croatia	0.4039
Canada	0.5514	Czech Republic	0.4204	Iceland	0.5567
Switzerland	0.4183	Egypt	0.4296	Jordan	0.7454
Germany	0.4921	Greece	0.5321	Kenya	0.6379
Denmark	0.3635	Hungary	0.6306	Lithuania	0.2978
Spain	0.5384	Indonesia	0.4228	Morocco	0.6059
Finland	0.3852	India	0.3645	Mauritius	0.6464
France	0.5453	Korea	0.3497	Oman	0.2073
United Kingdom	0.443	Mexico	0.3681	Romania	0.2951
Hong Kong	0.3024	Malaysia	0.3257	Serbia	0.2197
Ireland	0.3125	Peru	0.4241	Slovenia	0.2665
Israel	0.3101	Philippines	0.4638	Tunisia	0.5804
Italy	0.684	Poland	0.3303	Vietnam	0.3494
Japan	0.5402	Qatar	0.4415
Netherlands	0.3639	Saudi Arabia	0.5248
Norway	0.3343	Thailand	0.5434
New Zealand	0.5209	Turkey	1.0386
Portugal	0.4261	Taiwan	0.1773
Sweden	0.4222	United Arab Emirates	0.3078
Singapore	0.3291	South Africa	0.3442
United States	0.4544

. Table 2 lists the efficiency rankings of the three groups. Although the rankings changed according to these two criteria, the higher and lower ranks remained the same. Furthermore, Figure 10, Figure 11 and Figure 12 plot the multifractal spectra of all the index returns. In several subfigures, a phenomenon called knotting phenomena occurs, in which the curve is twisted near the peak of the curve. This phenomenon can be attributed to several factors, including the scaling range, irregular fluctuation functions observed at large scales, and the non-monotonic behavior exhibited by the estimated generalized Hurst index function according to previous studies (Jiang et al. [27], Zhou et al. [37], Gao et al. [38]).

According to Table 2, within the developed group, the stock market of Hong Kong is the most efficient, while the stock market of Italy is the least efficient. Within the emerging group, Taiwan and Turkey have the most- and least-efficient stock markets, respectively. Within the emerging group, Oman and Jordan have the most- and least-efficient stock markets, respectively, in the frontier group. Among all countries, Taiwan and Turkey have the highest and lowest stock market efficiencies, respectively.

Since the static analysis does not reflect changes in the market efficiency of a stock market over time, we investigate the dynamics of market efficiency using the rolling window analysis in the following section.

5.2. Rolling Window Analysis

In this section, we provide the dynamics of $\Delta \alpha$ for each stock-index return using rolling window analysis. This approach has been used in several studies on MF-DFA methods (Wang et al. [39], Sensoy and Tabak [40], Gajardo and Kristjanpoller [41], and Aloui et al. [42]). In this study, we chose a window length of 400 days to include the GFC period in our calculation and avoid severe fluctuations. In particular, we test the robustness of our spillover analysis results by comparing different rolling window sizes (300, 400, and 500 days). Figure A1–Figure A3 display the $\Delta \alpha$ time series for each rolling window length. In the subfigure, the $\Delta \alpha$ shows a similar fluctuation pattern under each variation of rolling window length. Therefore, our choice of widow size is reasonable for our investigation.

Figure 13, Figure 14 and Figure 15 illustrate the dynamics of $\Delta \alpha$ for the three groups. We summarize the key features by developed, emerging, and frontier groups as follows.

The fluctuations in the efficiencies of developed markets differ from country to country; however, they generally exhibit similar patterns. The key characteristics of these markets can be summarized as follows:

First, developed markets are highly sensitive to global issues. The dynamics of market efficiency often change rapidly, as observed during significant events such as the 2008 global financial crisis and the 2020 COVID-19 pandemic. The United States, with its large stock market and diverse range of market participants, is particularly susceptible to various issues. Notable changes in market efficiency were observed during events such as the 2010 European fiscal crisis and the 2018 US–China trade dispute. The United Kingdom, on the other hand, is influenced by changes in the energy industry and the financial sector (“15 February: Energy Firms Furthermore, Banks Lead Buoyant Market”, Forbes; “UK stocks close higher following Truss energy plan”, “Miners and energy stocks help FTSE 100 end volatile week higher”, Reuters).

Second, each developed country has its own unique factors that strongly influence its stock market dynamics. For instance, Japan is greatly affected by exchange rate and interest rate fluctuations due to its economic policy, as demonstrated during the 2013 Abenomics (Fukuda [43]) and the 2016 Negative Interest Rate and Yield Curve Control initiatives (Kawamoto et al. [44]).

Third, similar dynamics can be observed based on trade relations between countries, governance, and institutions. For example, countries within the North American Free Trade Agreement (NAFTA) such as Canada and the United States exhibit comparable patterns. Similarly, European countries within the EU share certain dynamics, and countries like England, Australia, and New Zealand, belonging to the Commonwealth of Nations, may display similar market behavior. Emerging markets exhibit similar trends to developed markets, but the dynamics of market efficiency within individual countries in this group show significant variations. Several factors contribute to these variations:

First, financial policies play a crucial role in shaping market efficiency dynamics. China, for example, has strict trading regulations and limited foreign investment, resulting in relatively low fluctuations in market dynamics during events such as the COVID-19 pandemic. Previous studies have also highlighted these observations (Hu et al. [45], Petry [46]). Similarly, in Egypt, stock returns are significantly influenced by EGP/USD exchange rates, which are impacted by various exchange rate systems (Ahmed [47]).

Second, compared to developed markets, emerging markets are more influenced by foreign capital. For instance, South Korea tends to follow foreign investment trends (Kim and Jo [48]), which greatly affect corporate dividend policies to attract foreign institutional investors (Kang et al. [49]). In Taiwan, stocks with high foreign ownership outperform those with low foreign ownership (Huang and Shiu [50]). In India, there is a positive relationship between foreign direct investment, foreign institutional investment, and stock market indices (Nagpal [51]).

Third, geopolitical and political risks are prevalent in this group. The risk of war exists in several regions, including South Korea, North Korea, Taiwan, and China. Conflicts also frequently occur in the Middle East and South America. Furthermore, significant fluctuations in market dynamics were observed during events such as the 2011 Egyptian Revolution and the 2013 military coup in Egypt.

Frontier markets often exhibit unique characteristics compared to developed and emerging markets, making it challenging to identify similar dynamics within the group. These distinctions arise due to the following factors:

First, frontier markets tend to experience significant impacts from crises, resulting in high levels of risk compared to other markets. Although it is difficult to pinpoint specific periods of rapid fluctuations beyond the 2008 financial crisis and the 2020 COVID-19 pandemic, it is worth noting that Iceland was greatly affected by the crisis (Reuters, “Iceland’s biggest taken over and all shares halted”).

Second, frontier markets are heavily influenced by specific industrial and regional issues. The key factors that affect their stock markets and subsequent dynamic fluctuations vary widely, largely due to their reliance on domestic markets. (Meziani [52]).

6. Discussion and Concluding Remarks

We investigated the market efficiency of each stock market using global stock market index data. To this end, we used the MF-DFA methodology and conducted static and rolling window analyses. We analyzed the results by categorizing them under developed, emerging, and frontier groups.

According to the static analysis results, Switzerland, the UK, and the US, known as financially advanced countries, have more efficient stock markets than other countries. Based on the several previous studies (Lim and Brooks [53], Ali et al. [5], and Bouoiyour et al. [54]), it has been observed that stock markets in developed countries demonstrate a higher degree of efficiency when compared to their emerging counterparts. This efficiency gap can be attributed to the incomplete nature of financial systems in emerging economies (Mookerjee and Yu [55], Islam et al. [56]), in contrast to the more advanced counterparts. On the other hand, advanced countries have relatively efficient stock markets, which can be explained by factors such as robust economic growth rates, larger market sizes, and the presence of open financial markets (Zunino et al. [57], Tongurai and Vithessonthi [58]). In order to minimize the disparity in stock market efficiency between developed and emerging countries, scholars have put forth proposals for laws pertaining to foreign investment, financial liberalization, and de-politicization (Vo [59], Rejeb and Boughrara [60], Goodell et al. [61]).

The notable findings of the rolling window analysis are as follows: First, markets cannot be insulated from global issues such as the 2008 financial crisis and the COVID-19 pandemic. These issues dominate, particularly in the developed country group, including the United States and the United Kingdom. Asian countries in this group, such as Japan, are impacted differently due to their strong domestic markets. Similar dynamics are also found in countries that are part of international trade blocs, such as NAFTA, the EU, and the Commonwealth. Second, emerging markets are vulnerable to foreign capital movement, and financial policies related to foreign investment heavily influence them. This is evident in the cases of China and Egypt, for example. Emerging markets also tend to follow foreign investment trends, as observed in South Korea, Taiwan, and India. Additionally, geopolitical risk is a critical issue, including the risk of war in South Korea, North Korea, Taiwan, and China, as well as frequent conflicts in the Middle East, South America, and Egypt. Third, distinguishing similar dynamics within the frontier group is difficult due to the varying intensive domestic market issues. Some frontier markets have been significantly affected by global issues, as was the case in Iceland during the 2008 financial crisis. Therefore, investment in frontier group markets should be approached with careful consideration compared to the other groups.

These findings have important implications for international investors and policymakers. First, international investors can establish investment strategies based on the degree of efficiency of individual stock markets. In other words, data on long-term memory and the degree of persistence over time can be utilized by global investors to gain an edge in the market and achieve above-average returns. However, the efficient nature of stock markets implies that consistently outperforming it through individual stock selection or timing is challenging. Therefore, investors may opt for a passive investment approach, such as investing in index funds that strive to match the market’s performance. Second, while the efficient market hypothesis assumes that the market is self-regulating and does not require government intervention, discovering market inefficiencies may require more regulations to safeguard investors and preserve market stability. Consequently, policymakers in countries with significant variations in market efficiency should consider implementing new regulations to enhance market efficiency.

As an additional research topic, we propose to analyze the market efficiency of the three groups of countries from a behavioral finance perspective. The efficient market hypothesis assumes that all investors have access to the same information and make rational decisions based on this information. However, behavioral finance research has shown that investors often make irrational decisions based on emotions, biases, and heuristics. Therefore, the proposed research can help inform our understanding of how individual markets function and investors make decisions. Furthermore, we suggest employing clustering analysis for $\Delta \alpha$ time series data. This approach can help identify distinct patterns in $\Delta \alpha$ movements within each country or confirm the similarity of these patterns through cluster analysis. By conducting such future studies, we can enhance the robustness of our findings and gain further insights from this research.