Abstract— Our research goal is to apply fractal analysis to financial time series to gain insights on the type of market, the persistence of the market type, i.e. the presence of long term memory, and to identify if a possible regime change is in the making (by using the Hurst exponent as a proxy for volatility) in order to have an edge for the application of 2 basic trading strategies: momentum trading and mean reversion trading. In addition, we want to verify the applicability of the Fractal Market Hypothesis (FMH) vs the Efficient Market Hypothesis (EMH). The principal method of fractal analysis we will employ is the R/S statistic, introduced by Hurst in 1951, to estimate the fractal dimension (and the Hurst exponent, H), in order to classify time series in 3 main buckets: mean reverting (low volatility, anti-persistent) for H < 0.5, trending (higher volatility, series has long term memory, or persistence) for H > 0.5, and for H = 0.5 (or close to it), the EMH regime. Index Terms— Fractal Analysis, Hurst Exponent, R/S Statistic, Regime Shifts. ——————————  —————————— 1 INTRODUCTION NE of the key factors contributing to profitability and competitive advantage when participating in financial markets is the ability to mathematically represent the fluc- tuations in market prices and volatility as accurately and real- istically as possible. Mandelbrot (1) stipulates that financial market time series directly inform, influence and impact the de- cisions and actions of market participants. As such, over many decades, various theories and hypotheses have been set forth to attempt to model these movements and market fluctuations, so as to identify trends and patterns that may exist, taking ad- vantage of them for the purposes of prediction. One example of these theories is the Efficient Market Hypothesis, proposed and studied from as far back as the early 20th century. A related, but distinct, theory is the Random Walk Hypothesis. Another ex- ample is the Fractal Market Hypothesis, proposed by Benoit Mandelbrot in his seminal paper (1). According to (2), beyond the regular fluctuations of markets, another aspect of market movement that is of interest to partic- ipants is the analysis and/or prediction of regime shifts. Re- gime shifts can be detected where sudden, and sometimes per- manent, swings in market patterns occur as a result of factors that may be external to the markets themselves. As such, it is important to be able to analyse historical regime shifts, identi- fying when they happen, how they affect market characteris- tics, and statistically modeling these shifts. Again, like with reg- ular market movements, being able to identify current regime shifts and/or predict future ones is crucial input to the making of profitable trading decisions (2). Hence, this will be the basis of our research. One of the most commonly employed techniques of detect- ing regimes, and hence shifts from one regime to another, is through the use of the Hurst exponent, first used by Harold Hurst to measure long-range dependence of time series in hy- drology (3), and later applied to financial time series by Man- delbrot. Once the value of the Hurst exponent is evaluated, the range in which it falls determines whether there is a regime shift, as well as what kind of regime is prevailing. Two common market regimes, the trending market regime and the mean-re- verting market regime, are the basis of the two popular choices in trading strategy: the mean-reversion trading strategy and the momentum trading strategy. While these regimes do not neces- sarily exist concurrently in the same time series, it is possible for this to be the case, depending on the timeframe and time scale of the time series being analyzed (2). Hence, the length of the time series is in fact a key parameter for the analysis. Our ultimate goal through this research is to investigate whether the Hurst exponent can be considered a useful and ex- ploitable indicator of regime shifts for different sectors of US financial markets, as well as to investigate the extent to which it is applicable. We seek to draw meaningful conclusions re- garding whether using the Hurst exponent is equitably accurate across sectors, or if it is better suited to some sectors more than others. We are also interested in how the applicability of the Hurst exponent differs between large cap and small cap indices. Our enumerated objectives are the following: 1. Identify appropriate market data. 2. Develop and apply tests to identify the price and re- turn properties of the sectors under study. 3. Identify the most appropriate and accurate tech- nique of estimating the Hurst exponent. 4. Identify regimes and regime shifts per individual time series. 5. Test the validity of indications by Hurst exponent. For the first objective, we identified the S&P 500 as a whole as the large cap index of interest, as well as all 11 of its sectors disparately. We also chose to study the Russell 2000 index for the purposes of analysis of a small cap index. The second objec- tive was accomplished through the completion of comprehen- sive exploratory data analysis (EDA) on our market data. O ———————————————— Brooke O.Nyatogo,Enrico A. Elia, Marissa W. A. Mutare Fractal Analysis of US Financial Markets: The Hurst Exponent as an Indicator of Regime Shifts  Enrico A. Elia Email: elia.enrico@gmail.com  Marissa W. A. Mutare Email: marissa.mutare2018@gmail.com  Brooke O. Nyatogo Email: brookeochiengnyatogo@gmail.com All three authors studied master’s degree program in financial engineering at Worldquant University and graduated in 2020.