The adaptive markets hypothesis: Insights into small stock market efficiency

ABSTRACT In this paper, we explore whether the adaptive markets hypothesis (AMH) describes the efficiency of the Finnish stock market better than the efficient markets hypothesis (EMH) does. Building on this, we also test how small market size and market liberalization impact the efficiency of the Finnish stock market and examine the relationship between market volatility and return in this market. We conduct this study by applying the subsample analysis and the rolling window analysis to the daily returns of the OMXH25 index and by measuring the efficiency through three linear and two nonlinear predictability tests. The results of our study strongly support the AMH. They also suggest that small market size alone does not make a market less efficient; opening a market to foreign investors improves its efficiency after a delay; and the correlation between market volatility and return varies over time in the Finnish stock market, being usually negative. These findings mostly contradict the traditional investment paradigm.


I. Introduction
A well-known debate in finance is that between the followers of the efficient markets hypothesis (EMH) and the proponents of behavioural finance; the EMH states that markets are efficient, but behavioural finance argues the opposite (Malkiel 2003).Lo (2004) developed the adaptive markets hypothesis (AMH), which states that the EMH and behavioural finance are just two sides of the same coin.According to the AMH, market efficiency varies over time, fluctuating between highly efficient and inefficient.
The AMH's view of stock market efficiency has been much researched.See, for example, Mateus and Hoang (2021) for frontier markets, Lim (2007) for emerging and developed markets, and Urquhart and McGroarty (2016) for the most established markets.Although market size is seen as a significant factor in market efficiency (Kavussanos and Dockery 2001), there is little AMH-related research from the perspective of a small stock market.
The aim of this study is to examine the efficiency of small stock markets by examining whether the AMH explains the efficiency of a small, developed stock market, such as the Finnish stock market, better than the EMH.The EMH offers a solid benchmark for the AMH as it is the most used market efficiency-related hypothesis.We also challenge the view that small market size renders a market less efficient (Abdmoulah 2010;Hong 1978;Tolvi 2003), shed light on the issue of whether opening a stock market to foreign investors has any effect on its efficiency, and provide a new view of how the relationship between market volatility and return varies in the Finnish stock market.
For the purposes of this study, Finland's stock market is an excellent subject for three reasons.First, in terms of market capitalization, Finland is listed among the lowest third of developed countries (De La Cruz, Medina, and Tang 2019).This is where our paper differs from related studies (Ghazani and Araghi 2014;Lekhal and El Oubani 2020;Obalade and Muzindutsi 2020), which have examined the AMH in small emerging countries.A low degree of market development has a clear impact on market efficiency (Harvey 1995); therefore, these studies do not formulate a direct view of how market size affects market efficiency.Second, we study the Nasdaq OMXH25 index.This established index focuses on the 25 most traded stocks in the Finnish stock market, which helps minimize the thin trading bias.Third, Finland has witnessed a significant, rapid increase in the foreign ownership of listed firms.Therefore, Finland is an interesting country for studying how increased foreign ownership has affected market efficiency.
We test the weak-form efficiency in the Finnish stock market with three tests that measure the linear predictability of stock returns and two tests that measure their nonlinear predictability.We use the term inefficiency interchangeably with return predictability and do not take a stand on other phenomena.Additionally, we study return predictability primarily from a statistical viewpoint and do not expect the results to help in straightforward trading.However, the results of this study can serve as a basis on which to develop different investment strategies for small stock markets.
The main tests of this paper contribute to AMH literature in two ways.To begin with, we add to the literature by focusing on the OMXH25 index.Direct AMH tests focusing on this index have not been conducted in prior research, but the results of Pätäri and Vilska's (2014) paper provide support for testing whether the efficiency of the Finnish stock market varies in a way consistent with the AMH.They studied moving average trading strategies in the OMXH25 index and obtained consistent results with the AMH.Moreover, to the best of our knowledge, this is the first study to use the subsample and rolling window analyses in tandem for examining the predictability of stock returns with traditional linear and nonlinear tests.This practice overcomes the shortcomings of each type of analysis (Urquhart and McGroarty 2014).However, in market efficiency-related studies, this approach has thus far been adopted only in studies of calendar anomalies and the Hurst exponent (Akhter and Yong 2021;Hiremath and Narayan 2016;Urquhart and McGroarty 2014;Xiong et al. 2019).
Building on the main tests, we developed three additional tests that contribute to the AMH literature in several aspects.First, we add to the literature by examining whether a small stock market can sometimes be more efficient than the US stock market in terms of return predictability.Under the framework of the classical EMH, researching that is useless.Also, traditional studies already suggest that small markets are generally less efficient than large ones.However, under the AMH framework, market efficiency is directly related to how investors adapt to varying market conditions (Lo 2012).Market conditions vary at different periods of time in each market, and so does the predictability of each market (Urquhart and McGroarty 2016).Therefore, we believe that there are no impediments for a small market being, at times, more efficient than a large one.
Second, we add a fresh perspective to the debate on stock market liberalization where others state that market liberalization to foreign investors increases market efficiency (Kim and Singal 2000;Song, Douthett, and Jung 2003), and others have found no evidence of this phenomenon (Kawakatsu and Morey 1999;Lagoarde-Segot and Lucey 2008).The novelty of our perspective comes from the way we process data.Rather than analysing one subsample from a period preceding the market liberalization date, and another subsample from the period immediately following the liberalization, we take the latter subsample starting at a time when foreign investors own a significant portion of the market.We also use rolling window analysis to capture the gradual increase in the level of foreign ownership and market efficiency.Thus, we can investigate how the presence of foreign investors really affects market efficiency instead of studying a temporary moment after market liberalization, when, at least in Finland, foreign investors have mainly bought shares and domestic investors sold them (Ali-Yrkkö and Ylä-Anttila 2003).Third, we complement Lo's (2012) research, where he finds a time-varying and usually negatively correlated relationship between geometrically compounded returns and volatility in the US stock market.We study whether this kind of phenomenon can be found in a small stock market.
Based on previous studies, we hypothesize the following.First of all, several studies such as Urquhart and Hudson (2013) and Ghazani and Araghi (2014) show that stock market efficiency varies in a way consistent with the AMH but not the EMH.We expect to find similar results in the Finnish stock market.Second, findings from previous studies (e.g.Jefferis and Smith 2005;Kavussanos and Dockery 2001) suggest that small size makes a market less efficient.
Therefore, we hypothesize that the Finnish stock market is generally less efficient than the US stock market.Third, literature (e.g.Laopodis 2003;Vaihekoski 1997) presents conflicting views on whether market liberalization improves market efficiency.Hence, we anticipate that market liberalization either has a positive impact or no impact on the Finnish stock market's efficiency.Lastly, Lo (2012) shows that, as the traditional view suggests, the US stock market has a positive risk-reward relationship when looking at historical averages.However, when time variation of returns is taken into account using the rolling window approach, the correlation between market return and volatility is mainly negative.We expect a similar time varying and negative correlation between return and volatility in the OMXH25 index.
The main result of this study is that the AMH provides a better description of the efficiency of the Finnish stock market than the EMH does.With additional tests, we show that a small market size does not render a market less efficient in itself.Instead, we demonstrate that a small, developed market can sometimes, even for years, be more efficient than a large, developed and mature market.We also show that opening a market to foreign investors does not have an immediate effect on its efficiency, but the efficiency increases after a delay.Finally, we point out that, in Finland, the relationship between market volatility and return is time-varying and usually negatively correlated.
This paper is organized as follows.In Section II, we introduce the methodology.Our aim is to present the tests and their differences as clearly as possible, for in the AMH literature, less attention has been paid to how these common tests complement each other.We do this, for example, by using the same symbol for each similar variable in every test formula, as well as by displaying the asymptotic distribution of each test and a flowchart of the research methodology.In Section III, we present and discuss the results of this study.Additional test results can be found in Section IV.Section V concludes this study and discusses the implications that arise from the test results.

Data
The data for our study are taken from Refinitiv Eikon Datastream.It consists of the daily returns of the OMX Helsinki 25 (OMXH25) index from 4 May 1988 to 28 February 2019.The daily returns, R t , are calculated by: where P t ¼ ln p t ð Þ and P tÀ 1 ¼ ln p tÀ 1 ð Þ are the natural logarithms of the index prices on the trading days, t and t À 1, respectively.These definitions of R t and P t will be used in all equations presented in this paper.Dividends are not included in the data.
The data are analysed through the subsample and rolling window analyses.Both methods have their pros and cons.The subsample analysis contains a smaller number of subsamples, which enables an easy-to-read table presentation of the test results, their p-values and different parameter choices.The downside of these subsamples is that they can fail to capture some important aspects of the time-related fluctuations, particularly when their placement does not match underlying phenomena.The rolling window analysis does not suffer from this problem but offers detailed information throughout the data.Its numerous and overlapping subsamples can be captured in a graph that shows how a market's efficiency varies through time.However, the high number of subsamples limits a comprehensive presentation of the test results.Therefore, only the p-values of one lag length are usually displayed.We follow previous literature in choosing the test parameters and by showing only the p-values of the rolling window analysis.While our study is unaffected, it should also be noted that any aggregated analysis of the rolling window results would give less weight to the observations within the first and last subsamples.All in all, applying the subsample and rolling window analyses in tandem provides the most robust results and overcomes their own shortcomings, as Xiong et al. (2019) point out.
With both methods, the data are divided into two-year-long subsamples.Only the first subsample in the subsample analysis has a length of two years and 10 months.Each subsample, of which there are 15 in total, contains approximately 500 observations.In the rolling window analysis, the subsample moves forward one month at a time.For example, the first subsample has observations from 1 May 1988 to 30 April 1990, and the second subsample has observations from 1 June 1988 to 31 May 1990.The total number of subsamples in the rolling window analysis is 347.The results of the subsample and the rolling window analyses are presented in the same figures, so their conformities are easy to see.
Descriptive statistics for the OMXH25 index are presented in Figure 1.In particular, the kurtosis shows that the logarithmic returns of the OMXH25 index do not follow a normal distribution.When the Jarque-Bera statistic is calculated from the kurtosis and skewness, it can be seen that the critical values defined by Thadewald and Büning (2007) are exceeded in every subsample, except in the time period from 1 March 1993 to 28 February 1995.Except for that period, the normal distribution can be rejected at a significance level of 1%.

Tests and preprocessing
Figure 2 presents the flowchart of the research methodology.The efficiency of the Finnish stock market is tested using five different tests that examine the predictability of returns.First, three tests are used to measure linear autocorrelation in returns.Then, two tests are used to measure nonlinear autocorrelation in returns.The null hypothesis of each test is that the returns are not predictable.The returns are whitened before conducting the nonlinear tests so that linear autocorrelations can be removed.The whitening process also serves to remove possible spurious autocorrelations caused by nonsynchronous trading (Hong and Lee 2005).This paper filters out the linear autocorrelations using an autoregressive AR q ð Þ model.The correct q-value within each subsample is selected by the Akaike information criterion (AIC) developed by Akaike (1974).The Ljung-Box test (Ljung and Box 1978) is used to evaluate how well the linear autocorrelations are removed from the returns.
A prerequisite for independent returns is that a time series possesses a unit root.In other words, a time series should be a difference-stationary process.Therefore, before conducting the main tests, three unit root tests are used to determine the stationarity of the returns series.These tests are the augmented Dickey-Fuller (ADF) test (Said and Dickey 1984); the Phillips-Perron (PP) test (Phillips and Perron 1988); and the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test (Kwiatkowski et al. 1992), which measure underlying pure random walk process, random walk process with drift, and random walk process with drift and deterministic trend, respectively.Many studies have surprisingly concluded that a market is or is not efficient based on the unit root tests.However, unit root tests are not designed to test return predictability, as Campbell, Lo and MacKinlay (1997) and Rahman and Saadi (2008) have noted.Therefore, this study uses them only to measure whether the preconditions of market efficiency are filled.
The results of the subsample analysis are summarized in Figure 5.Following Urquhart and Hudson's (2013) example, the AMH is supported if at least three phases of efficiency are found, and switch in efficiency can indicate an early-stage adaptive market.In contrast, the EMH is supported if the market stays efficient throughout the whole sample, or the market becomes more efficient over time, as Lo (2005) notes.

Ljung-Box test
The Ljung-Box test is used to test the absence of linear autocorrelations through joint null hypothesis of zero autocorrelation coefficients up to order q.Autocorrelation coefficient at lag s, ρ s ð Þ measures the degree of correlation between the current return and the s-th previous return and is a natural extension of the correlation coefficient between two random variables (Campbell, Lo, and Mackinlay 1997).Ljung-Box, LB q ð Þ combines these up to q lags and is calculated as follows: where χ 2 q ð Þ is the chi-squared distribution with q degrees of freedom, n is the number of observations, Cov R t ; R tÀ s ð Þ is the autocovariance of returns at lag s, and Var R t ð Þ is the variance of returns.This paper follows Campbell, Lo and Mackinlay (1997) and evaluates the autocovariance and the variance by the sample moments.

Runs test
The runs test by Wald and Wolfowitz (1940) is a nonparametric test that does not assume normal distribution of time series.As such, it suits the stock markets well because, as Mandelbrot (1963) already showed, financial series often deviate from normality.Here, a run is defined as a sequence of positive or negative returns.The observed number of runs, r is close to the expected number of runs, E r ð Þ if the returns are generated by a random process.The Z-value of r is calculated as follows: where N 0; 1 ð Þ is the standard normal distribution, n a n b is the product of the number of observations that are above and below the returns series mean.The standard normal Zstatistic is used to test whether the actual number of runs is consistent with the hypothesis of independence of the series.Positive (negative) Z-values show that the number of runs is greater (fewer) than expected.

Variance ratio test
Since the study of Lo and MacKinlay (1988), variance ratio tests have been extensively used to measure market efficiency.Variance ratio tests are based on the finding that if stock returns follow a random walk, then the expected value of variance ratio at each individual lag interval is equal to unity.Like the Ljung-Box's quadratic summation, the variance ratio with a holding period of k, VR k ð Þ can be written as a special linear combination of the first k À 1 autocorrelation coefficients with steadily decreasing weights: where the variances can be expressed by the unbiased estimators derived by Lo and MacKinlay (1988) and Campbell, Lo and Mackinlay (1997).Under the null hypothesis VR k ð Þ ¼ 1. Building on this, Lo and MacKinlay (1988) developed two tests, one robust under heteroscedasticity and another robust under homoscedasticity.As heteroscedasticity is usually seen in stock returns and as the same can be seen in Figure 1, this paper uses the former test.This heteroscedasticity-consistent test statistic is calculated as follows: where μ is the unbiased estimator of mean.This study follows the common example of Lo and MacKinlay (1988) in choosing the values for the holding periods, k, which are 2, 4, 8 and 16.Engle (1982) suggested the Lagrange Multiplier (LM) test to detect autoregressive conditional heteroscedasticity (ARCH).The ARCH-LM test is calculated from an auxiliary test regression.The null hypothesis is that there is no evidence of ARCH effects in the data.The ARCH-LM test is calculated as follows:

ARCH-LM test
where e's are the residuals from the pre-whitening AR model, q is the highest order lag, a's are the model fitting parameters, v t is the white noise error process, and R 2 is the coefficient of determination.The test statistic used in this paper is the usual Fstatistic for the regression on the squared residuals.

BDS test
The BDS test by Brock et al. (1996) is a nonparametric test based on the correlation integral from chaos theory (Grassberger and Procaccia 1983).For sample sizes of 500 or larger, which is the sample size used in this study, the BDS test provides good finite-sample performance and good rejection power against a broad class of alternative hypotheses (Brock, Dechert, and Scheinkman 1996).The null hypothesis is that the data-generating processes are independent and identically distributed.The BDS statistic can be calculated as follows: where k is the model embedding dimension that determines the embedded rolling window size of the correlation integral C k; ε ð Þ, and ε is the model metric bound that sets the largest distance between pairs of observations to count as correlating with each other.In effect, the correlation between all possible pairs of embedded rolling windows is inferred by their k-dimensional distance.For the full description of these correlation integrals refer to Brock et al. (1996).Finally, σ k; ε ð Þ represents the standard deviation of B k; ε ð Þ.In our study, we adopt the same ranges for k ¼ 2 À 10 as Hsieh (1989), and ε ¼ 0:5 À 1:5σ as Hsieh and LeBaron (1988), where σ is the standard deviation of the observations.

III. Results and discussion
In this section, we present and discuss the results of the five market efficiency tests, and compare the findings of the subsample analysis and the rolling window analysis.
Before conducting the main tests, we performed three unit root tests.According to the ADF and the PP tests, the data are non-stationary and the first differences of the data are stationary.The KPSS test, which has stationarity as its null hypothesis, confirms these results.Therefore, it can be stated that the preconditions of market efficiency (Rahman and Saadi 2008) are met.See Table S1 in Supplemental Material for more details.
Figure 3 presents the results of the linear tests.The results of the Ljung-Box test show that according to lag 1 of the subsample analysis, the Finnish stock market has gone through six different stages of efficiency, so it can be deemed adaptive.The rolling window analysis confirms these results and reports more specifically that the Finnish stock market was not efficient until 1995.According to the Z-values of the subsample analysis of the runs test, the market has gone through four stages, so it is deemed adaptive.These results are in line with the subsample analysis of lag 1 of the Ljung-Box test, with the exception that the runs test did not find any inefficiencies from 1 March 2013 to 28 February 2015.However, the rolling window analysis of the runs test found linear autocorrelations near the 5% significance line during that period, and some inefficiencies were found just before 2010 and 2011.The test statistics of the subsample analysis of the variance ratio test demonstrate that when the holding period, k is 2, the market goes through six stages of efficiency.This once again provides evidence for the AMH.The p-values of the first-order variance ratio and Ljung-Box tests closely resemble each other, which is not surprising since they both reduce to the first-order autocorrelation coefficient test when the data are nearly homoscedastic.
Before running the nonlinear tests, we successfully whitened the data (see Table S2 in Supplemental Material for more details).Figure 4 presents the results of the nonlinear tests.The subsample analysis of the ARCH-LM test shows that nonlinear correlations are strongly present before 1993.However, the results suggest that the period from 1 March 1993 to 28 February 1995 was efficient, indicating that the inefficiencies during that period were caused by linear autocorrelations.The results suggest that returns have gone through three stages of efficiency, which is proof of adaptive markets.The rolling window analysis provides more detailed results, showing that just the latter part of the period from 1 March 1993 to 28 February 1995 was actually efficient.The rolling window analysis also finds short efficient periods that the subsample analysis was not able to recognize.
Similar to the ARCH-LM test, the BDS test suggests that the period from 1 March 1993 to 28 February 1995 was efficient.In addition, the BDS test finds efficient periods in the subsamples from 1 March 2001 to 28 February 2003 and from 1 March 2013 to 28 February 2015 with every dimension and embedding dimension.The rolling window analysis of the BDS test shows longer efficient periods than the rolling window analysis of the ARCH-LM test.What is notable about the results of the BDS test is that all three major economic crises that occurred in Finland during the time period analysed are surrounded by nonlinear dependencies.Those crises are the early 1990s depression in Finland, the dot-com bubble in the late 1990s, and the financial crisis of 2007-2008.Overall, the nonlinear ARCH-LM and BDS tests found periods of efficiency and inefficiency, which is consistent with the AMH.
To ensure the robustness of our findings, we performed all the linear and nonlinear tests also for the period from March 2020 to February 2022, which includes the COVID-19 pandemic.The results are consistent with those from our latest reported period (March 2017 to February 2019), demonstrating that our results hold even in the context of a global pandemic.

Summary of the test results
For a summary of the test results from the subsample analyses, see Figure 5. Overall, these results  a) shows Ljung-Box (LB) test results for lags q ¼ 1; 3; 5, runs test Z-statistics, and variance ratio (VR) test results at k ¼ 2; 4; 8; 16.Significance levels 1%, 5%, and 10% are marked as a , b , and c , respectively.The graphs (b) display the rolling window analysis with p-values for the Ljung-Box test at lag 1, runs test, and variance ratio test at k ¼ 2. Dashed lines represent 5% and 10% significance levels, indicating market inefficiency below these lines.Circles denote p-values of the subsample analysis.
strongly suggest that, compared with the EMH, AMH better explains the efficiency of the Finnish stock market.The EMH requirements would have been met had the market stayed efficient throughout the whole sample or had its efficiency increased (Urquhart and Hudson 2013).In none of the tests did either occur.Instead, both linear and nonlinear tests found efficient and inefficient periods.
A glance at the subsample side of Figure 5 might lead to an erroneous conclusion: that the results of these linear and nonlinear tests have a strong negative correlation.However, the results of the rolling window analysis demonstrate that this correlation is weak.The same analysis also highlights that linear inefficiencies only occur for short periods, while nonlinear inefficiencies occur most of the time.These findings underline that rolling window analysis is useful for achieving a more detailed picture of market efficiency.They also reveal the rapid changes in market efficiency.
The subsample analysis of every test demonstrates that, in 1988-1993, the Finnish stock market was inefficient.Thereafter, the results of the linear tests align with each other.The only exception in their results is the period from 1 March 2013 to 28 February 2015.The  a) presents results of the ARCH-LM test at the number of lags, q ¼ 2; 4; 6; and the BDS test with embedded dimensions, k and metric bounds, ε as values of the sample's standard deviation.Significance levels of 1%, 5%, and 10% are marked as a , b , and c , respectively.The graphs (b) display the rolling window analysis with p-values for the ARCH-LM test at lag 4 and the BDS test with k ¼ 2 and ε ¼ 1σ.Dashed lines represent 5% and 10% significance levels, indicating market inefficiency below these lines.Circles denote p-values of the subsample analysis.
results of the nonlinear tests are mostly in line with each other, but the ARCH-LM test revealed more inefficiencies.

Market size's impact on market efficiency
Our idea of testing whether a small market could sometimes be more efficient than the largest stock market in the world is based on two findings.First, we have demonstrated that the efficiency of the Finnish stock market varies over time.Second, Lo (2004) has shown that because of the variation in efficiency, the US stock market was periodically more efficient in the 1950s than in the early 1990s.Building on these insights, we compared the efficiency of the Finnish and US stock markets simply by applying the Ljung-Box test for both OMXH25 and Standard & Poor's 500 (S&P500) indices.Figure 6 shows that a small market, in terms of the first-order autocorrelation, can sometimes be more efficient, even for years, than the world's largest and most mature stock market.
This finding is not in line with the principles of the EMH.Also, it contradicts the literature which explains market inefficiencies by market size or states that a small size makes a market less efficient (Jefferis and Smith 2005;Kavussanos and Dockery 2001).In our study, we focus on the OMXH25 index, which consists of the 25 most traded stocks of a developed stock market, so our findings do not oppose the view that market inefficiencies can be explained by the state of market development, thin trading or low liquidity.However, when these other factors are excluded, we find no evidence for the statement that a small market size would, in itself, render a market less efficient.
Nevertheless, our finding is consistent with the AMH statement that market efficiency can vary in any market due to changes in investor groups or market conditions (Lo 2012).Therefore, even a large mature market can be inefficient at times. Figure 6 demonstrates that the US stock market was more efficient than the Finnish stock market before the 1993 market liberalization in Finland, but the US stock market was inefficient for years after the beginning of the financial crisis of 2007-2008.During that time, the Ljung-Box test did not find the first-order autocorrelation in the returns of the Finnish OMXH25 index.
A juxtaposition with other papers reveals that the other tests in this study bring similar results to the Ljung-Box test.For example, Urquhart and McGroarty (2016) investigated the efficiency of the world's major stock markets with the same sample size as that of our study.The results from their BDS test do not indicate that the major stock markets would be noticeably more efficient than the Finnish stock market.Instead, efficiency varied in every market they studied, and it is easy to see that there are periods when the Finnish stock market is more efficient than the major markets.Likewise, there are periods when the major stock markets are more efficient than the Finnish stock market.

Market liberalization's impact on market efficiency
Our results, combined with the history of foreign ownership in the Finnish listed companies (Ali-Yrkkö and Ylä-Anttila 2003; Jakobsson and Korkeamäki 2014;Vaihekoski 1997), provide an insightful angle by which to examine how a market's liberalization to foreign investors can affect its efficiency.Finland is highly suitable for the study of this phenomenon, as the abolishment of all restrictions on foreign investors in January 1993 led to a rapid increase in foreign ownership.The earliest steps in the liberalization process can be traced back to 1984, but they did not have a significant effect on foreign ownership (Vaihekoski 1997).
According to the AMH, changes in investor groups significantly affect a market's efficiency (Lo 2005).Therefore, those time periods in which foreigners own just a small portion of the market, and then a significantly larger portion of the market are of interest.We look at periods from 1 May 1988 to 28 February 1993 and from 1 March 1995 to 28 February 2001 from our subsample analysis, when foreigners owned, on average, 6.3% and 55% of the market capitalization of the Finnish listed companies, respectively.The skipped subsample from 1 March 1993 to 28 February 1995 is not very useful for measuring foreign investors' effect on market efficiency, for it represents the rapid transitory period when the level of foreign ownership rose from a very low level to close to its current levels.
The subsample side of Figure 7 shows that the efficiency of the Finnish stock market clearly increased after the market was liberated.This suggests that the presence of foreign investors, which are mainly sophisticated institutions (De La Cruz, Medina, and Tang 2019), improves market efficiency.The graph from the rolling window analysis demonstrates that the Finnish stock market did not become efficient immediately after market liberalization.Instead, it exceeded the 10% threshold level for efficiency for the first time at the point where investors owned 33% of the market.Refer to Section II for the benefits of using both the subsample and the rolling window analyses.
The results of the runs and variance ratio tests of this paper (Figure 3) show even more clearly that the Finnish stock market became efficient around 1995.This supports the argument that market liberalization to foreign investors increases market efficiency with a delay.Nonlinear tests provide weaker evidence for improved efficiency (Figure 4).However, both nonlinear tests found a peak in efficiency a year after the market was liberalized.The ARCH-LM test also detected a peak in efficiency just before the market was liberalized.We do not take a strong position on the findings of the nonlinear tests, but propose that the temporarily higher level of efficiency a year after the liberalization date might be caused by increased competition in the market.The peak in efficiency before the market liberalization may result from the market's reaction to the announcement of the market opening.
Overall, our results suggest that a permanent positive change in market efficiency occurs with a delay after a market has been opened to foreign investors.This delay might explain the conflicting results in the literature, where some studies propose that an increase in foreign ownership improves market efficiency (Kim and Singal 2000;Song, Douthett, and Jung 2003), while other studies have found no evidence of it (Groenewold and Ariff 1998;Kawakatsu and Morey 1999).In said literature, the graduality and delayed effects of market liberalization are usually considered.Still, when subsample analysis is used, it is common to start the post-liberalization time period from the liberalization date.This practice focuses on the transition period after the reform instead of the actual effects of the reform.Therefore, we argue that in the examination of foreign ownership's impact on market efficiency, the post-liberalization time period should start from the moment when foreigners own a significantly larger portion of the market.In addition, the rolling window analysis can be used to demonstrate the gradual change in the level of foreign ownership and market efficiency.The graph in Figure 7 illustrates the importance of this issue by showing that analysing two short time periods from right before and after the market liberalization would have led to contrary results.
While our results show that this small market's efficiency varies, the risk-return relationship is also expected to vary over time.To test this, we created a graph (Figure 8) to see how the relationship varies over time in the Finnish stock market.
Figure 8 shows that the relationship between standard deviations and geometrically compounded returns of the OMXH25 index is timevarying and often negative; the correlation between them is − 26%.This implies that when the market is unstable, investors move into safer assets, leading to better returns for safer assets and worse returns for riskier assets.This is the opposite of what the EMH predicts and cannot be explained by the leverage effect (Hasanhodzic and Lo 2019).However, this finding is in line with the AMH and complements Lo's (2012) paper, in which he demonstrates the time-varying and usually negative correlation between market volatility and returns in the US stock market.The table (a) presents the subsample analysis, comparing Ljung-Box test results at lags q ¼ 1; 3; 5; 10 to various foreign ownership statistics.Here, a indicates significance at the 1% level.The graph (b) displays the liberalization date, the level of foreign ownership, and the rolling window analysis with p-values for the first-order Ljung-Box test.Vertical dashed lines represent the omitted subsample and horizontal dashed lines represent 5% and 10% significance levels, indicating market inefficiency below these lines.Foreign ownership data are derived from Vaihekoski (1997), Ali-Yrkkö andYlä-Anttila (2003), and Jakobsson and Korkeamäki (2014).

V. Conclusion and implications
In this study, we investigated the AMH in the Finnish stock market.Building on the main results of this study, we also tested whether a small market size makes a market less efficient, whether market liberalization to foreign investors increases its efficiency and how the risk-reward relationship varies in the Finnish stock market.Our main conclusion is that the AMH provides a better explanation than the EMH for the efficiency of the Finnish stock market.This finding indicates that the practical implications of the AMH (Lo 2012) can also be applied in the Finnish stock market.
With the three additional tests, we conclude the following.First, a small market size alone does not render a market less efficient.Instead, a small stock market can stay on a higher efficiency level than a large, mature stock market, even for years.Also, the opposite can happen.From the investors' perspective, this implies that geographical diversification reduces the risk of being solely invested in a market that is going through an inefficient period.Alpha hunters can use this information the opposite way.Just switching the target index between the US and Finnish stock markets can help find inefficiencies.The limitation of this study is that it only examined one small market, although an index with the same characteristics as the Nasdaq OMXH25 is not easy to find.In future research, it would be interesting to see an efficiency comparison between several small and large markets.In such a comparison, thin trading bias should be controlled methodologically.We see no impediments to even emerging markets being, at times, more efficient than the most established markets, for under the AMH framework, any market can go through highly inefficient periods due to changes in investor groups or market conditions (Lo 2012).
Second, we show that the efficiency of the Finnish stock market changed when new investors entered the market.This is exactly what the AMH proposes.In this case, the Finnish stock market became more efficient after all restrictions on foreign ownership were abolished.However, we point out that while the rise in foreign ownership did not happen immediately after the market liberalization, neither did the increase in efficiency.This highlights the importance of considering the actual effects of market liberalization when determining the dates of the time periods under investigation.From a regulatory perspective, our findings suggest that a noticeable permanent improvement in market efficiency should not be expected until foreign investors, which are mainly institutions, own a significant portion of the market capitalization of a market.
Third, variations in market efficiency can lead to variations in the relationship between market volatility and return.In Finland, the correlation between these two is actually negative.This finding opens possibilities for new investment products and strategies.For example, investors in a small volatile market could benefit from an investment strategy that invests passively but contains risks actively.A base for this kind of strategy could be rooted in Lo's (2016) example of an index fund that contains no alpha but uses active risk management to maintain the target level of volatility.

Funding
We would like to thank the Finnish Cultural Foundation (grant 65211974) for funding this research.

Figure 1 .
Figure 1.Descriptive statistics of the daily returns of the OMXH25 index.Note. a represents the significance of the Jarque-Bera statistic at the 1% level.

Figure 3 .
Figure 3.The results of the linear tests.The table (a) shows Ljung-Box (LB) test results for lags q ¼ 1; 3; 5, runs test Z-statistics, and variance ratio (VR) test results at k ¼ 2; 4; 8; 16.Significance levels 1%, 5%, and 10% are marked as a , b , and c , respectively.The graphs (b) display the rolling window analysis with p-values for the Ljung-Box test at lag 1, runs test, and variance ratio test at k ¼ 2. Dashed lines represent 5% and 10% significance levels, indicating market inefficiency below these lines.Circles denote p-values of the subsample analysis.

Figure 4 .
Figure 4. Results of the nonlinear tests.The table (a) presents results of the ARCH-LM test at the number of lags, q ¼ 2; 4; 6; and the BDS test with embedded dimensions, k and metric bounds, ε as values of the sample's standard deviation.Significance levels of 1%, 5%, and 10% are marked as a , b , and c , respectively.The graphs (b) display the rolling window analysis with p-values for the ARCH-LM test at lag 4 and the BDS test with k ¼ 2 and ε ¼ 1σ.Dashed lines represent 5% and 10% significance levels, indicating market inefficiency below these lines.Circles denote p-values of the subsample analysis.

Figure 5 .
Figure 5. Summary statistics.The table (a) presents the test results of the subsample analysis, where I stands for inefficiency and E stands for efficiency in the daily returns of the OMXH25 index.The statistical significance required for inefficiencies is 5%.The graph (b) displays the p-values of the rolling window analysis.Dashed lines denote 5% and 10% significance levels, indicating market inefficiency below these lines.

Figure 6 .
Figure 6.Comparison between the results of the Ljung-Box test in the OMXH25 and S&P500 indices.The solid lines present the p-values of the rolling window analysis at lag 1, where the dashed lines denote significance at the 5% and 10% levels.The market is defined as inefficient below these lines.

Figure 7 .
Figure7.The results of the Ljung-Box test in the OMXH25 index and foreign ownership of the market capitalization of the Finnish listed companies.The table (a) presents the subsample analysis, comparing Ljung-Box test results at lags q ¼ 1; 3; 5; 10 to various foreign ownership statistics.Here, a indicates significance at the 1% level.The graph (b) displays the liberalization date, the level of foreign ownership, and the rolling window analysis with p-values for the first-order Ljung-Box test.Vertical dashed lines represent the omitted subsample and horizontal dashed lines represent 5% and 10% significance levels, indicating market inefficiency below these lines.Foreign ownership data are derived fromVaihekoski (1997), Ali-Yrkkö andYlä-Anttila (2003), andJakobsson and Korkeamäki (2014).