Does the U.S. extreme indicator matter in stock markets? International evidence

We propose a new predictor—the innovation in the daily return minimum in the U.S. stock market ( (cid:31) MIN US )—for predicting international stock market returns. Using monthly data for a wide range of 17 MSCI international stock markets during the period spanning over half a century from January 1972 to July 2022, we find that (cid:31) MIN US have strong predictive power for returns in most international stock markets: (cid:31) MIN US negatively predicts the next-month stock market returns. The results remain robust after controlling for a number of macroeconomic predictors and conducting subsample and panel data analyses, indicating that (cid:31) MIN US has significant predictive power and it outperforms other variables in international markets. Notably, (cid:31) MIN US demonstrates excellent predictive power even during the periods driven by financial upheavals (e.g., Global Financial Crisis and European Sovereign Debt Crisis). Both panel regressions and out-of-sample tests also support the robust predictive performance of (cid:31) MIN US . The predictive power, however, disappears during the non-financial crisis caused by COVID-19 pandemic, which is originated from the health sector rather than the financial sector. The results provide a new perspective on U.S. extreme indicator in stock market return predictability.


Introduction
Stock markets provide firms an effective channel to raise funds for both current operations and future developments, whilst investors have opportunities to obtain high returns from the markets.Since stock markets are replete with risks and opportunities, stock return predictability remains an area of active research in the field of finance.With the increase in global trade and cross-regional capital flows, international markets have become more interconnected than ever; see Singh (2007Singh ( , 2016a) ) for the surveys.It is, therefore, crucial to go beyond narrow single market predictability, and examine the inter-market relationship in international stock markets.Many researchers in the past have claimed that the financial markets are not perfectly efficient and that investors are not rational, by showing that the stock trend is predictable (Poterba and Summers 1988;Fama and French 1992;Campbell et al. 1997;Bae et al. 2003;Hood and Malik 2013;Barillas and Shanken 2018;Baker et al. 2020;Zerbib 2022).The literature documents many factors closely connected to stock markets, and among these, the factors related to risks are particularly important (see e.g., Bali et al. 2011;Chiang and Hughen 2017;Nartea et al. 2017;Baker et al. 2020;Huang et al. 2022).This is because the relationship between risk and returns is studied by almost every participant in the stock market-the trade-off between risk and returns is fundamental in investments.
A variety of risk-related factors have been used in stock return prediction, and researchers have achieved good prediction results; these factors include market beta, leverage, liquidity effect, momentum, volatilities, spread, and some macroeconomic variables (Rapach et al. 2005;Gharghori et al. 2009;Bali et al. 2011;Huang et al. 2012;Chung and Kim 2017;Nartea et al. 2017;Herrera and Clements 2018;Bogousslavsky 2021;Borup and Schütte 2022;Liu and Matthies 2022).Among these, extreme value factors seem to be able to better explain the risks in the stock market.The proposition of extreme value theory (EVT) makes it possible to analyse extreme returns.The EVT is a branch of statistics that addresses extreme deviations from the mean of a probability distribution (Singh et al. 2013;Jin and Sui 2022;Jin and Sui 2022).Longin (1996) and Embrechts et al. (1999) introduced it to financial studies.The EVT indicates that the extreme historical patterns can be considered as "shock" factors, and the "shock" tends to have strong predictive power for the future stock prices, especially during market turbulences (e.g., Huang et al. 2012;Corradin and Maddaloni 2020;Liu and Matthies 2022).
The extreme patterns of the historical returns, as the special states of volatilities, are directly linked with tail risks and returns, and thus they can be considered as the primary factor in predicting future returns.The empirical studies have supported this hypothesis in many international stock markets (Bae et al. 2003;Muchnik et al. 2009;Boyer et al. 2010;Bali et al. 2011;Annaert et al. 2013;Barberis et al. 2016;Jiang et al. 2018;Baker et al. 2020;Bogousslavsky 2021).Bali et al. (2011) and Kelly and Jiang (2014), considering different fluctuations of stock prices, split the extreme value into two indicators: the maximum (MAX) and the minimum (MIN) of daily stock returns in the past month.They reveal a significantly negative relationship between MAX and the expected stock returns.The extreme indicators used by Jiang et al. (2018) also exhibit short-term predictive power.However, Forand and DeRubeis (2014) and Nartea et al. (2017) question the significant results of the extreme indicators, as they cannot find robust evidence for the predictive power of these indicators in many other stock markets.
Under the internationally integrated financial environment, the shift of certain factors in the U.S. stock market can be used as predictive factors for the performance of the international stock markets.The market anomalies or the "shock" identified in the U.S. stock market can be observed in the stock markets of other countries (Jegadeesh and Titman 1993;Fama and French 1996;Rouwenhorst 1999;Grundy and Martin 2001;Titman et al. 2004;Bali et al. 2011;Kelly and Jiang 2014;Corradin and Maddaloni 2020;Xu et al. 2023).Researchers have discovered the lead-lag relationship between the U.S. stock market and the international stock markets.Rapach et al. (2013) show that the lagged U.S. returns are a useful prediction tool for the returns in other industrialised countries.In fact, these lagged U.S. returns perform better in predicting stock returns than the countries' own nominal interest rates and dividend yields.However, Siliverstovs (2017) tests the robustness of the U.S. indicators in different phases of the business cycle (recession and expansion) and finds that the Rapach et al. (2013) results are fragile.He finds that the predictive power of lagged U.S. return is significant only during the recession phase of the business cycle.
The synchronised movements theory also supports the predictability in international stock markets.In the highly integrated world economy with more potent channels of transmission, the effectiveness of coordination has become increasingly more relevant (Akın and Kose 2008).With the rise in direct and indirect linkages from the increasingly tight connection among financial markets, the global equity correction shows significant synchronisation (Otto et al. 2001).The previous research also argued that the economic shock plays an important role in shaping international volatility.Common international shocks, common transmission channels, and country-specific shocks form the synergy of economic cycles in international financial markets.One of the main channels for these shocks is the spill-over effect that spreads through international trade.The U.S. is the world's largest trading exporter, and shocks originating from its market can widely spread to the international market and lead to synchronised periodic fluctuations in market capitalisations across different financial markets in the world (Norrbin and Schlagenhauf 1996;Otto et al. 2001;Beltratti and Morana 2010;Yetman 2011;Singh 2016b;Mikhaylov et al. 2022;Wen et al. 2022;Moiseev et al. 2023).The international markets, therefore, show predictability under such conditions.Yetman (2011) argues that higher frequency business cycles are strongly synchronised during recession periods, but typical "decoupling" could occur during low frequency business cycles.Therefore, the predictability of international financial markets should be strong during recession periods, such as global financial crisis (GFC), but weak during decoupling period, such as the COVID-19 pandemic period, when most countries introduced ubiquitous lockdowns and restrictive measures around the world.
This study aims to extend previous research (Bali et al. 2011;Rapach et al. 2013;Barberis et al. 2016;Ghosh et al. 2017;Jiang et al. 2018;Bogousslavsky 2021;Liu and Matthies 2022) and provide a new perspective on the use of extreme measure factor of the U.S. market to predict future stock returns in other international markets.Unlike previous studies, we focus on the change in expansion or contraction of the extreme minimum return (the extreme loss) of the daily closing price of the stock index in each month, which is the price difference in the daily return minimum, and we denote it as MIN .We use this innovation of the extreme minimum in the U.S. stock market ( MIN US ) as an indicator to predict future returns in the international stock markets.
The framework of behavioural finance theory suggests that the stock market is replete with irrational investors (Berardi 2022).The investment patterns in the markets tend to exhibit loss aversion rather than risk aversion.The changes in extreme losses, therefore, can affect investment decisions more than other extreme factors, and also impact future stock returns.The investors are biased towards the possibility of extreme stock market conditions (Forand and DeRubeis 2014;Bakshi et al. 2018).Under such scenarios, investors' irrationality will lead to lower-level diversification of portfolios to obtain skewed returns.In the international financial markets, due to the leadership of the U.S. stock market, when the extreme losses of the U.S. stock market expand ( MIN US decreases), then there is an occurrence of a loss signal, and the risk-averse investors prefer to transfer assets to other valuable portfolios during a recession to hedge risks (Hood and Malik 2013;Berardi 2022).The short-term capital transfers (capital flight) may occur between international stock markets to avoid potential losses.
The investors' pessimistic sentiments and herding behaviour in the wake of imperfect (asymmetric) information induce more and more irrational cross-market investors to begin selling U.S. stocks and turning to invest in other stock markets, driving up the prices in other stock markets in the short-term horizon.Therefore, theoretically there should be a significant negative correlation between MIN US and future stock returns in non-U.S.stock markets.However, because of the leadership of the U.S. stock market in the international stock markets, its economic downturn will also gradually spread to other markets and cause their stock prices to fall.Thus, from a longer-term perspective, the trend of the U.S. stock market and other stock markets should be similar.Therefore, MIN US is expected to have weak predictive power in the longer-term horizons.
The study focusses on three empirical questions: Does MIN US have robust predictive power in international stock markets?Does MIN US outperform other macroeconomic control variables in international stock markets?What is the relationship between MIN US and stock market returns during economic recessions-both NBER-based and major recession periods?These research questions form the basis of the hypotheses tested in the empirical section.
The empirical results suggest that MIN US has strong predictive power for stock mar- ket returns in most international stock markets: MIN US negatively predicts the next- month stock market return.The results remain robust after (i) controlling for a number of macroeconomic predictors, (ii) conducting panel data analysis, and (iii) performing robustness checks.Notably, MIN US demonstrates excellent predictive power even dur- ing the times of financial upheavals (Global Financial Crisis and European Sovereign Debt Crisis).The predictive power, however, disappears during the unprecedented nonfinancial crisis caused by COVID-19 pandemic.
Our study contributes to the existing body of literature on two counts.First, previous studies have focused mainly on the extreme value of individual stocks, and especially in the U.S. stock market.Our study extends the analysis to stock market index predictions, and covers a comprehensively wider set of 17 developed stock markets and historically a longer period of time spanning over half a century from January 1972January (1972:01) :01) to July 2022 (2022:07).The study follows time-series approach and takes a country-by-country account of the evidence.The findings based on our comprehensive coverage in terms of both space and time could be generalised to represent the behaviour of international financial markets.Second, the previous research has used the extreme indicators in level, whether MIN or MAX, and the predictive power of these indicators for stock market returns remains controversial (Rapach et al. 2013;Nartea et al. 2017;Huang et al. 2022).This study extends previous research and proposes a novel indicator, MIN US , which not only shows strong predictive power in international stock markets but also outperforms other extreme indicators.
The remainder of this study is organised as follows.Section "Data and model specification" discusses the data and methodology used in the study.Section "Empirical analysis" reports and analyses the empirical results.Section "Conclusions" concludes the study.The research process of the study is summarised in terms of a self-explanatory visualisation (Fig. 1).
International Standard Country (MSCI) index in most countries in the DataStream.Since one of the objectives of the study is to test the predictive power of extreme indicators in economic turmoil periods, the overall sample period is sub-divided into the sets of both NBER-based U.S. recession periods and the major recession periods.The data for the NBER-based U.S. recession periods are sourced from the Federal Reserve Economic Data (FRED) website; the recession period is represented by "1", and by "0" otherwise.The major recession periods include: (1) the period of Asian financial crisis (AFC), which extended from June 1997 to December 1999; (2) the period of global financial crisis (GFC), which extended from June 2007 to July 2009; (3) the period of European Sovereign Debt Crisis (ESDC), which stretched from the beginning of 2009 to the end of 2013; and (4) the period of COVID-19 pandemic which stretched from December 2019 to July 2022.The ESDC period is chosen to examine the impact of economic turmoil on the stock markets, especially the European markets.
To conduct the analysis of the relationship between the extreme returns and the future stock market returns, this study uses the data on MSCI Indexes for a comprehensive set of 17 stock markets of the industrialised countries.The MSCI Indexes are widely used by researchers in analysing the international stock return predictability.Apart from its ease of access, the MSCI is a reliable index for conducting comparative analysis.The data on MSCI are sourced from Refinitiv DataStream.Rapach et al. (2005Rapach et al. ( , 2013) ) use data on the stock markets of 10 countries in their sample and these markets include Australia, Canada, France, Germany, Italy, Japan, the Netherlands, Sweden, Switzerland, and the United Kingdom.Some studies have included a different set of 7 stock markets in their samples and these markets include Austria, Belgium, Denmark, Hong Kong, Norway, Singapore and Spain (e.g., Annaert et al. 2013;Walkshausl 2014).Unlike these studies, this study conducts an encompassing analysis and covers a comprehensive set of 17 stock markets of the MSCI developed countries and a longer time span of 50 years.Besides, this study also uses the overall MSCI World stock index, which can reflect the general situation of international stock markets.We also consider MSCI indexes for 17 other markets, including 5 MSCI young developed markets and 12 MSCI developing markets, which have data available over a time span of 30 years for robustness check.
The monthly log stock returns are used as the dependent variable for regression testing.The stock returns are calculated by summing log daily returns of the MSCI indexes in each market; denominated in both U.S. dollars and local currencies.Previous research uses the extreme indicators as the independent variable (Bali et al. 2011;Rapach et al. 2013;Barberis et al. 2016;Ghosh et al. 2017;Jiang et al. 2018;Barro and Liao 2021;Bogousslavsky 2021;Liu and Matthies 2022).The extreme value takes the highest or the lowest daily stock return of the MSCI index in every month in each market.This study extends the previous research and instead uses the innovation (first-level difference) of negative extreme return.These are defined as follows. (1) where MIN m t refers to the negative extreme returns of market m at month t; ∀R d,t refers to every daily stock return in month t.MIN m t refers to the innovation of negative extreme return at month t.
We follow Xu et al. (2023) to examine the effects of several other macroeconomic factors on stock returns, such as the ratio of dividend to stock price (DP), rate of inflation (IFL), 1-month Treasury bill rate (TB), money supply (M3) and crude oil price (OIL), and compare their predictive power with MIN US t .Plugging multi-indicators as control variables can also alleviate the model mis-specification bias and enhance the explanatory power of the model.
The descriptive statistics of the monthly log returns of each stock market (18 dependent variables) and the key independent variable, MIN US t , are reported in Table 1.The table shows that the mean values of dependent variables are smaller than their median values, and the values of the skewness are mostly less than zero, indicating that the stock market returns are skewed to the left (except for the United Kingdom, which has a positive skewness), with long or fat tails.However, the independent variables are skewed to the right.Therefore, the assumption of the efficient market hypothesis (EMH) is considered invalid, as none of the stock returns are normally distributed.The standard deviations are relatively high, which indicates that the variables spread widely from the mean.The pairwise correlation of stock market returns and the U.S. extreme indicator suggests that MIN US is negatively correlated with all the selected stock returns in international markets (see Table 2).

Table 1 Descriptive statistics of key variables [monthly returns of MSCI market indexes]
(1): "Std" refers to the standard deviation.(2): "25% and 75%" stand for the percentiles of the statistics.(3): MIN US stands for the innovation in the daily return minimum in the U.S. stock market.( 4): ρ 1 is the first-order autocorrelation coefficient

Market
Mean

Model specification
Previous research has examined the relationship between the U.S. extreme indicator in level and stock market returns (Bali et al. 2011;Rapach et al. 2013;Barberis et al. 2016;Ghosh et al. 2017;Jiang et al. 2018;Barro and Liao 2021;Bogousslavsky 2021;Liu and Matthies 2022).This study extends the previous research and instead uses the novel indicator to predict the subsequent stock returns.The benchmark prediction model is specified as follows.
where α refers to the constant term;β 1 and β 2 refer to the parameters of the associated variables; R i,t−1 denotes lagged own returns of the countries; MIN US t−1 denotes the innovation of extreme indicators of the U.S. market at month t−1; and ε i,t represents an error term.
Model ( 3) is extended to control for the effects of alternatively the (i) macroeconomic factors and (ii) economic recessions on stock market returns.Model (3) augmented to control for the effects of macroeconomic factors is specified as follows.
where X t−1 denotes a vector of control variables in period t-1; β X is a vector of the parameters of the control variables, including R i,t−1 , TB, DP and OIL.
Model (3) augmented with a slope dummy to capture the effects of economic recessions on stock market returns is specified as follows.
where D 0 represents a dummy variable, which takes value "1" in the recession period, and value "0" otherwise.β D is the coefficient of the interaction variable.
Model ( 5) is estimated separately for the NBER-based recession periods and for the major recession periods, such as AFC, GFC and ESDC, and the COVID-19 pandemic, to have a comprehensive account of the evidence.
We use Python software to develop programming code for data processing and model estimation.Specifically, we organise and analyse the time-series data through Python libraries, such as Numpy and Pandas.We also use the Matplotlib library to visualise the test results.

Empirical analysis
The U.S. extreme indicatorthe benchmark model Previous studies have shown that the U.S. stock market demonstrates a lead-lag relationship with other stock markets, and they have claimed that the extreme shocks occurred in the U.S. market also affect other markets (Rapach et al. 2013;Nyberg and Ponka 2016;Ghosh et al. 2017;Shue and Townsend 2021;Huang et al. 2022).Our study uses MIN US as an indicator to predict future returns in the international stock markets.Table 3 shows the results of regression testing for 17 developed markets, in addition to the World market.It shows that MIN US has strong predictive power in 10 markets, (3) including Austria, Canada, Germany, Hong Kong, the Netherlands, Norway, Sweden, Switzerland, the United Kingdom and the overall World market.The coefficients of the MIN US for the above markets are significant at the 5% level, with R-squared sta- tistics of more than 0.5%.The coefficients of MIN US , β 2 , are negative for all the 18 selected markets, which indicates an explicit negative relationship.The U.S. extreme negative returns thus show a good predictive ability in international stock markets.The coefficients of their own past returns ( R t−1 ) are insignificant for the stock markets of all the countries, except for Belgium.This shows that in the international stock markets, MIN US can significantly influence other stock markets and take a dominant position, and its influence is even stronger than the trend of other markets themselves.
We also conduct the similar regression tests in 17 other stock markets (including young developed markets and developing markets), the results are shown in Table 9 of the "Appendix".We find that the significance of MIN US disappears in almost all these markets.This may be due to the unique characteristics and less integration of these markets with the U.S. market.The previous studies reported that the connection of the U.S. and Asian stock markets may not be as strong as perceived.Liu and Pan (1997) report that the U.S. has been influential in transmitting the returns and volatilities in the Asian region, but this transmission is unstable and is affected by the contagion effect.Ng (2000) suggests that the regional spill-overs from Japan are more significant than the global spill-overs from the U.S. market.Moreover, the relatively shorter time span of the regressions may also be the reason why the predictive power of the U.S. extreme indicator is not statistically significant in these markets.
We also test the correlation between each market's own extreme minimum innovation ( MIN own ) and future returns.We find that MIN own have predictive power only in few markets, and it does not outperform the own past return.So MIN own , as the indicator, is not as good as MIN US in predicting returns in international stock mar- kets. 1 Moreover, when it comes to the longer-term time-horizons, including 3-month, 6-month, 12-month and 24-month horizons, the predictive ability of the extreme indicators (both the MIN US and MIN own ) does not show the signs of persistence.It means that extreme indicators are different from some other traditional forecasting indicators in that they have significant predictive power only in short horizons, and they cannot affect the stock market in the long run. 2 These results are similar to those obtained in previous studies (Jiang et al. 2018;Bena et al. 2022;Johnson et al. 2022).

The U.S. extreme indicator and the control variables
The results of the regressions with additional control variables (including R own , TB, DP and OIL) are reported in Table 4.The table provides the coefficients and t-statistics for both MIN US and control variables, along with adjusted R-squared values.The results demonstrate that the coefficients of MIN US barely change and they remain significant across 10 markets.This indicates that MIN US remains robust in the international stock markets, even with the control variables included in the model.While R own , TB and OIL have limited correlations in few markets, there is insufficient evidence that these indicators correlate with international stock returns, and therefore they have weaker predictive power than MIN US .The coefficient of DP is also significant in 10 markets, suggesting this to be a good predictor in international stock markets.
The comparison suggests that the model with control variables has stronger explanatory power, as the adjusted R 2 values are generally higher (see Table 4), as compared to the model without control variables (Table 3).This indicates that the addition of control 1 The test results are reported in Table 10 of "Appendix".
2 Persistent predictors such as dividend yield (Fama & French 1988) tend to have strong predictive power over longhorizon returns.By contrast, MIN US has a relatively low autocorrelation coefficient.Like technical trading indicators, it is a short-term indicator, thus playing no role in long-horizon return predictability.The longer-term horizons tests results are not reported, but are available upon request.
variables significantly improves the goodness-of-fit of the model.The coefficients of all constants in 17 markets are reduced and have insignificant t-statistics (Table 4).

The U.S. extreme indicator and the recession periods
The extreme value theory suggests that the extreme factors tend to be more important, with the occurrence of extreme economic conditions (Lettau 2003;Barberis and Huang 2008;Colacito et al. 2021;Perotti and Rola-Janicka 2022).We draw on these studies and    examine the predictive power of extreme indicators during recession periods.We augment the benchmark regression model (3) with a slope dummy to control for the effects of recession periods, and we specify this as model ( 5).We estimate model ( 5) separately for the NBER-based recession periods as well as for the major recession periods, such as AFC, GFC, ESDC and the COVID-19 pandemic.

NBER-based recession periods
Table 5 reports the results of the regression model ( 5) estimated for the NBER-based recession periods.Surprisingly, unlike the conclusions of previous studies, MIN US  does not exhibit a significant predictive ability during recession periods.The coefficient of the slope dummy for recession is not significant even at the 10% level in all selected markets.However, the coefficient of MIN US remains significant in several markets even after plugging the dummy variable in the regressions.This indicates that the predictive power of MIN US is significant whether in recession periods or other periods in international stock markets.Therefore, unlike some other indicators that have predictive power only during recession periods (e.g., Ghosh et al. 2017;Wen and Li 2020;Colacito et al. 2021;Bena et al. 2022;Borup and Schütte 2022), MIN US shows strong predictive power not only during recession periods, but also during the whole sample period of 50 years.

Major crises periods
After testing the predictive power of MIN US during the NBER-based recession peri- ods, we extend the analysis and dig deeper into sub-sample tests, using the extreme episodes of financial upheavals, such as AFC, GFC and ESDC, and the COVID-19 pandemic.These crises are documented to have significant influence on the global economy (Ghosh et al. 2017;Colacito et al. 2021;Augustin et al. 2022).The results of the model estimated for these extreme episodes of financial upheavals and COVID-19 pandemic are reported in Table 6.These results demonstrate the superior predictive power of MIN US during the GFC (July 2007 to June 2009) for all the international stock markets.The coefficient of the U.S. extreme indictor is significant in all the 17 markets, indicating that the predictive performance of MIN US in the period of the GFC exceeds its perfor- mance in the full-sample period.
During the GFC period, the R 2 of the model estimated using MIN US as the predictor is higher than 10% in 16 of the 18 markets, stronger than in other periods.For investigating why the MIN US has such extreme strong predictive performance in GFC, we trace the trend of MIN US and three representative stock markets (including Hong Kong, the United Kingdom and World indexes) in the GFC period (Fig. 2).It can be observed from Fig. 2 that there are similar trends in these markets, and this may be due to the synchronised movements occurring in international stock markets.By contrast, MIN US shows an obvious opposite trend, which presents a strong negative correlation between MIN US and international stock markets.As mentioned earlier, the U.S. mar- ket has a leading position in the international stock markets and the information shock of U.S. market spreads and affects other markets, thus indicating the forecasting ability of MIN US .Rapach et al. (2013) note that at the extreme moment in history, such as the GFC, the impact of information shocks will be amplified, thereby improving the predictive ability of MIN US in the international stock markets.
We also follow Augustin et al. (2022) and Bena et al. (2022) and conduct another sub-sample regression for the period of the European Sovereign Debt Crisis (ESDC), which was also quite widespread globally, especially in the European region, from the beginning of 2009 to the end of 2013.The ESDC is considered as the important catastrophic event that have global influences (Chaplinsky and Haushalter 2010; Ari 2014;  Barro and Liao 2021).The test results are similar to those in the GFC period, where MIN US still has robust predictive ability, especially in the European countries.The results even show a slightly higher adjusted-R 2 than in the GFC period.The coefficient of MIN US is not significant only in the case of Japan, which may be because the Japanese market is not directly affected by the ESDC.
In the sub-sample period of COVID-19 (December 2019 to July 2022), different from the above results, MIN US shows significant results only in four (France, Italy, Japan and Spain) out of 17 markets.And these results are quite intuitive.The global financial crises can be understood as the result of the frictions in financial sector.By contrast, the COVID-19 crisis originated as a health sector shock, different from previous financial and economic crises, which created enormous uncertainty for financial markets for an extended period of time (Bao and Huang 2021).
During the COVID-19 pandemic, the government interventions effectively curbed the spread of the virus in the world.The stock market and even all the international financial markets are affected by such government-control (Papanikolaou et al. 2021).The central banks, such as the U.S. Federal Reserve, have actively intervened in the market through monetary easing, slashing interest rates, and introducing large-scale supportive financing programs to improve liquidity to stimulate the economic activities and avoid a full-scale financial crisis (Eichenbaum et al. 2021).However, the interventions during COVID-19 further increased the central banks' footprints in the markets; these patterns may lead to many suspecting that there is a disconnection between financial markets and the real economy.Therefore, there is a lack of correlation between a large number of indicators in financial markets and economic trends in the COVID-19 period.This implies that it is difficult to conduct asset pricing tests without considering the role played by the central banks.
Under the traditional financial crisis, the flow of capital between international financial markets is unrestricted.During the COVID-19 pandemic, the connection between the country markets has weakened or even been interrupted, resulting in the breakdown of the lead-lag relationship in international markets, which makes it more difficult to predict stock returns using cross-market variables.Many studies have reported poor results in examining stock return predictability in the COVID-19 period (e.g., Augustin et al. 2022;Wen et al. 2022).The sub-sample analysis undertaken for the period of the AFC (June 1997 to December 1999) suggests that the predictive ability of MIN US seems weaker, as compared to its predictive ability during the GFC and the ESDC each (Table 6).The main drivers of the AFC of 1997 seem to have been the problems of the Asian markets' input-oriented developmentalism as well as the ineffective state intervention, and these are not directly related to the role of the U.S.
We follow Augustin et al. (2022) and further test the predictive ability of MIN US in 17 other markets during the sub-sample periods of AFC, GFC, ESDC and COVID-19.The results suggest that the predictive performance of MIN US is similar to that in the developed markets.During GFC, MIN US demonstrates strong predictive ability in most of the markets, while during ESGC it shows weak predictive ability.This may be because most selected markets are from the Asian rather than from the European region. 3Second, we do the sub-sample tests in four additional recession periods.The results suggest that the predictive power of MIN US is still significant in the interna- tional stock markets during the oil shock recession period, the world debt crisis period, and the Nordic banking crisis period; it is insignificant only during the Dot-com bubble period (see "Appendix" Table 12).The Dot-com bubble crisis is driven by the internet industry, which only affects the relevant sectors in the U.S. stock marketsit does not have a serious impact on the international financial markets.We also perform two common sub-sample tests to further assess the robustness of our results.We split the entire time span into two periods-first period stretches from January 1972 to December 1999 and the second period spans from the new millennium January 2000 to July 2022.The results remain similar to those obtained for (i) the NBERbased recession periods and (ii) the extreme episodes of financial upheavals, such as AFC, GFC and ESDC, and the COVID-19 pandemic periods.These results reinforce the robust predictive ability of MIN US .4

Robustness check
In the international stock markets, stock returns not only have serial-correlations in time-series, but also have inter-market correlations.The significant results obtained from time-series analysis may be insufficient to support the strong predictive power of MIN US without considering the inter-market influences in multiple stock markets.We extend the analysis further and estimate panel regressions to control for both cross-sectional and time effects, and test the robustness of MIN US .The panel data analysis is useful to analyse two-dimensional (cross-sectional and longitudinal) effects and it can improve estimation efficiency and inference power, compared to one-dimensional timeseries or cross-sectional regressions.We consider 5 panel regression methods: fixed effects model, random effects model, pooled Ordinary Least Squares (OLS) with fixed market dummies, pooled OLS with fixed year dummies, and pooled OLS with standard errors clustered by both market and month following Petersen (2009).
The results of the (i) model with a single predictor, MIN US , and the (ii) model with additional control variables, including R own , TB, DP and OIL, are reported in Table 7.These results show that the coefficient of MIN US is significant in all 5 panel tests, indi- cating that MIN US has robust predictive power in international stock markets, even after considering the correlations both over time and across markets.After plugging four control variables in the model, MIN US remains significant and its t-statistic is above 1.96 in all 5 regressions.Some of the other control variables, however, do not show the same level of significance in all the panel regressions.Therefore, MIN US outperforms other indicators as a predictor in panel regressions as well, again verifying its robustness.
For the rigour of testing, we use the out-of-sample R-squared ( R 2 os ) statistic of Welch and Goyal (2008) to perform the out-of-sample test (OOS) on MIN US and examine whether the predictors have robust predictive performance.The OOS prediction test is computed as follows.
where R 2 os is the out-of-sample R-squared; T is the total period, counted in months; K is the monthly number for out-of-sample testing; r is the in-sample forecast of stock returns within a certain month span (window), r 1 is the out-of-sample forecast of stock returns in the remaining time period, and r 2 is the arithmetic average of stock returns.The study uses two kinds of OOS tests, fixed rolling window test and expanding window test.The window selection of OOS should follow the criteria that the window span selection should be long enough to include enough initial data to get a reliable regression estimate.According to Welch and Goyal (2008), OOS tests, especially long-time span OOS tests, usually need to start after 20 years of data (240-month window), or choose half of the entire sample time span as the window for testing.Since the time span of our study is 50 years (600 months), it is appropriate to use 240-month and 300-month window in the OOS tests.We, therefore, perform the OOS tests of MIN US , includ- ing 240-and 300-month expanding windows and rolling windows, and the test results are reported in Table 8.
The OOS results of expanding windows show that the U.S. extreme indicator demonstrates strong out-of-sample predictive power in both 240-month and 300-month windows.Its R 2 os is positive in 16 of the 18 developed markets and is significant at the 10% level in 9 markets.This significant result is consistent with and even stronger than the in-sample test results (Table 3), verifying the robustness of MIN US .The rolling window OOS results, although, are weaker than those in the expanding window, they still show positive R 2 os in most markets, indicating that the predictive performance of MIN US are higher than the historical average level over the past 50 years in international markets.

Conclusions
This study has investigated the relationship between the extreme indicators and stock market returns for a comprehensive set of 17 advanced international stock markets using monthly data for half a century from 1972:01 to 2022:07.The sample span includes the sets of both NBER-based economic recessions and several extreme episodes of financial upheavals, such as AFC, GFC and ESDC, and the COVID-19 pandemic periods.The study proposes a novel extreme value indicator, the innovation in the daily return minimum in the U.S. stock market ( MIN US ), to predict future returns in international stock markets.We conduct a series of investigations on the relationship between MIN US and the international stock market returns.We find that MIN US has a significant negative correlation with returns in most stock markets.The results remain robust after controlling for a number of macroeconomic predictors and conducting subsample and panel data analyses.The predictive power of MIN US remains strong and it outperforms other variables in the international markets.
The results obtained from the model augmented with a slope dummy to capture the effects of economic recession suggest that the dummy variable has insignificant correlation with stock returns, indicating the weak predictive power of MIN US during recessions.However, MIN US remains significantly correlated with stock returns and it outperforms the recession dummy.MIN US shows very strong predictive power for international stock returns even during the periods driven by extreme financial upheavals such as the GFC and the ESDC.The significance of MIN US , however, disappears during the COVID-19 pandemic period.This could be ascribed to the unique nature of the COVID-19 pandemic, which stemmed from the health sector, as compared to financial crises which originated from the financial sector.We conduct the robustness checks by using both (1) panel regressions to consider inter-market correlations in the international stock return prediction and (2) out-of-sample tests to estimate the predictive performance of the indicator.Both sets of results successfully verify the robustness of MIN US .The findings of the study have important implications for the researchers, the policy makers, and the finance practitioners.The results provide a new perspective on the U.S. extreme indicator in stock market return predictability.The MIN US outperforms other extreme indicators in predicting stock prices in international stock markets, and it should be considered as an important factor in the asset pricing and risk management.The study also shows the significance of the U.S. stock market in the non-U.S.stock markets.

( 1 )
: MIN US refers to the extreme minimum innovation of the U.S. stock market.(2): R i,t−1 refers to the own past return of the markets respectively.(3): TB refers to the prices of one-month Treasury bill.(4): DP refers to the ratio of dividend to stock price.(5): Oil refers to the crude oil price.(6) Figures in parentheses are t

Fig. 2
Fig. 2 U.S. Extreme Indicator and Stock Returns in GFC period

Table 3
Predictive Test Results with the U.S. Extreme Indicator (1): R i,t−1 refers to the own past return of the markets.(2):MIN US refers to the extreme minimum innovation of the U.S.

Table 4
Predictive Test Results with the U.S. Extreme Indicator and Control Variables

Table 5
Predictive test results with the U.S. extreme indicator [under NBER-based recession periods] (1): MIN US refers to the extreme minimum innovation of the U.S. stock market.(2) : R i,t−1 refers to the own past return of the markets respectively.(3): Dummy refers to the dummy variable that stands for the recession periods.(4) Figures in parentheses are t-values.(5) *** and ** indicate statistical significance at 1% and 5% level, respectively

Table 6
Predictive test results with the U.S. extreme indicator [under extreme recessionary conditions]

Table 8
Out-of-sample test results with the U.S. extreme indicator OS refers to the out-of-sample R-Squared of the tests.(2) P Val is the probability under the assumption of no effect (null hypothesis)

Table 9
Predictive test results with the U.S. extreme indicator-other markets (1): R i,t−1 refers to the own past return of the markets respectively.(2): MIN US refers to the extreme minimum innovation of the U.S. stock market.(3) Figures in parentheses are t-values.(4) *** and ** indicate statistical significance at 1% and 5% level, respectively

Table 10
Predictive test results with the own market extreme indicator (1): R i,t−1 refers to the own past return of the markets respectively.(2): MIN own refers to the extreme minimum innovation of the own stock market.(3) Figures in parentheses are t-values.(4) *** and ** indicate statistical significance at 1% and 5% level, respectively

Table 11
Predictive test results with the U.S. extreme indicator-developing markets [under extreme recessionary conditions] (1): MIN US refers to the extreme minimum innovation of the U.S. stock market.(2):AFC refers to the sample period of June 1997 to December 1999 (Asian Financial Crisis).(3):GFC refers to the sample period of July 2007 to June 2009 (Global Financial Crisis).(4):ESDC refers to the sample period of January 2009 to December 2013 (European Sovereign Debt Crisis).(5):COVID refers to the sample period of December 2019 to July 2022, that COVID-19 crisis originates as a health shock.(6)Figures in parentheses aret-values.(7) ***, ** and * indicate statistical significance at 1%, 5% and 10% level, respectively