Stock Market Stationarity

Marina Ferent
marina.ferent@orpheusindices.com
Abstract
Stationarity tests are used to detect mean reversion in a certain dataset. Mean Reversion processes suggest a non-random behavior in a time series (Lo and MacKinley, 1988). Previous research has focused on studying mean reversion at stock price level (Debondt and Thaler, 1985; Lindemann et al., 2004) and considers stationary assumptions to be restrictive for a financial time series (Lo and MacKinley, 1988). The authors look at the concept of stationarity at group specific level as previously defined by the ‘Mean Reversion Framework’. The group approach allows for a different interpretation of stationarity, as it overcomes limitations of stationary tests on time series and the problems regarding trend and difference stationarity when it comes to finite data (Cochrane, 1987). The groups approach to look at stationarity also offers an easy way to prove the co-existence of non-random and random behavior in a group (stock market). The stationarity trends are defined as the percentage number of components that exhibit stationarity at the Value, Core or Growth bin levels in the ‘Framework’. The trends observed were consistent and showcase a duration dependency. More than 50% of all the components in the three bins exhibit stationarity, suggesting that the ‘Framework’ is a good proxy for complex and natural systems, which express both random and non-random behavior. The authors combine the absolute trends of bin transformation with the stationarity trends to draw parallels to understand how reversion and divergence co-exists in the ‘Framework’. Stationarity at a group level strengthens the case of markets as a complex system and ‘Framework’ as a good proxy to understand behavior of such systems.
The Mean Reversion Framework
The framework as explained in ‘Mean Reversion Framework’, Pal, 2015 is the relative price performance proxy consisting of five bins. The bottom 0-20 percentile bin is considered value, 80-100 percentile is considered growth and the rest of the core (middle) bins are considered transition bins 20-40, 40-60 and 60-80. These rankings are referred to as derived rankings.

 
The illustration below explains how the ‘Framework’, consists of both reversion and divergence (as presented in the above-cited paper). The V value bin (light grey) and G growth bin (dark grey) components have a tendency to move above the relative 50 percentile mean ranking and below 50 rankings respectively. An average of nearly 50% value and growth components move above and below the 50 mean. This was the Galtonian mean reversion. It’s the section on the right hand side marked ‘Divergence’ that completes the Galtonian observation. The number of components in the 40-60 Core ranking move toward 0-20 Value bin at the bottom extreme and 80-100 Growth at the top extreme. This case of divergence illustrates that just like reversion, there are components in the proxy that illustrate divergence from the mean. The summary illustration suggests that the proxy experiences both reversion and divergence, a classic expression in all natural systems.
Momentum and Reversion
The framework is a new way of looking at natural systems, group behavior. This paper extends the idea of ‘Momentum and Reversion’ (Pal, 2015) by re-confirming the extreme reversion nature as observed in the Framework and showcasing the transformation of reversion into momentum and vice versa.
The 2015 ‘Momentum and Reversion’ paper has brought into discussion:
1: Momentum and reversion (M&R) is not just a price return characteristic but a natural behavior and hence could be observed in case of relative percentile performance rankings;
2: Behavior of a group of components defined by the mean reversion framework instead of asset price level showcased the idea of extreme(s) (boundaries) reversion, where relative outliers tend to revert back to relative mean;
3: In the context of dynamic systems, reversion and divergence is a universal behavior for natural systems;
4: Momentum and reversion coexist and transform into each other.
Reversion is not just a price return characteristic but also a natural behavior and hence can be observed in case of derived rankings also. ‘Reversion’ is a universal behavior observed across regions, bins and time durations. Mean Reversion Indexing (Pal, 2012) has focused on studying the reversion behavior of the worst performers in a group of assets. Stationarity tests were applied to the respective losers exhibiting reversion in the following three years (2008-2011), moving from below 20 relative percentile ranking to above 50.
Current model limitations and observations
Building on the study in Mean Reversion Indexing paper, this study uses a dataset comprising of five regions. The results further strengthen the case of a need for stock market proxy to understand universal behavior in stock market systems. Tests indicate dynamic behavior of the framework. A cyclical behavior in reversion is witnessed across various groups for different group sizes. Smaller groups exhibit higher degrees of stationarity.
First; DeBondt and Thaler (1985) studied the worst losers and best winner portfolios without a relative context. Their study ignored the rest that lies between worst and best. Understanding behavior of a group requires studying all the segments of the market i.e. best, worst and rest. By testing the entire group defined as value, growth and core, the study extends the idea beyond worst losers and best winners. The composite view allows for complete study of group behavior hence extending the work of Debondt and Thaler.

Second; Previous studies have focused more on sets of data rather than on understanding behavior over various holdings (durations) (Lo and MacKinley, 1988; Lindemann et al., 2004). This study shows that both momentum and reversion behavior have a duration dependency. As durations increase, reversion transforms into momentum and vice-versa. This confirms the co-existence of both momentum and reversion inside the framework.

Third; The previous statistical methods used to detect reversion have focused on studying behavior at asset price level rather than on a group level. The current paper illustrates the behavior of a group of components defined by the mean reversion framework. This approach is particularly beneficial in addressing the possible limitations of the unit root statistical test, as mentioned by Lo and MacKinley, 1988, “Whether or not the assumption of stationarity is a restrictive one for a financial time series is still an open question”.
Fourth; The question of restrictiveness of stationarity as a statistical measure arises from the fact that financial data is characterized by unstable volatility, time varying volatility. One would expect that a component in any bin in the framework would see it’s risk changing over time, as value component moves towards growth; core move towards value or growth; or a growth component move towards value. The dynamic nature of a stock market behavior over time translates into different variances, a behavior which does not satisfy stationarity’s condition of a constant second moment.
Fifth; As opposed to return series where stock returns could vary without bounds, the proposed group ranking offers the advantage of fixed variance inside the framework (i.e. no stock could have a ranking lower than 0 or higher than 100). So whenever there is a shock in the price series, the same shock has a considerably lower or limited impact on the rankings due to the normalized group ranking. Even though the second moment is not constant in time it is significantly less sensitive to changes inside the framework than in the absence of it, allows for a superior model to run stationarity tests to search for reversion patterns in a group.
Sixth; Continuing on Lo and MacKinley’s concern related to unstable volatilities – “there is ample evidence of changing variances in stock returns over periods longer than five years”, the same statement though limited would be true for our derived rankings in the framework. As such, a natural expectation would be to obtain decreasing values of group stationarity percentage, especially as time durations increase marking an environment with unstable volatility. This redundancy of stationarity over longer periods of time owing to time varying volatilities is one of the major limitations of current studies on stationarity. The absence of a framework to study group behavior worsens the ability of the stationarity as a statistical measure to study reversion.
Seventh; The focus towards group based stationary studies vs. time series based stationarity offers a critical trade off. On one side the framework gives up the characteristics associated with an asset price i.e. an individual time series, but on the other it makes up for the less noise at a relative group level allowing for an understanding of group behavior using the stationarity measure.

Eighth; The stationarity stats offers relevant comparisons to the absolute trend stats (Momentum and Reversion, Pal, 2015). Since the stationarity tests is a theoretical measure of reversion compared to the study of absolute bin trends, the drop in stationarity percentage stats compared to absolute trend stats was expected. However, the surprise was an overall increasing trend in both stats. The tendency to see an overall increasing stationarity over time was a peculiar trend suggest that not only the absolute stats were correct in identifying the dynamic group behavior but also that this behavior was indeed owing to the reversion happening at a component level. Intra bins components were expressing mean reversion and the group behavior was indeed a composite of reverting and non-reverting behavior.
Ninth; The five regions tested showcased decline in stationarity percentages at some point, but by extending the duration it could be inference that the decline could be durational inflexion point as a declining trend is shortly followed by an increasing one. If the stationarity tests would have been strongly affected by the time varying variances, the stationarity stats could have continued to record lower readings with increasing durations. The proposed percentile derived ranking framework may thus overcome the limitation of unit root tests encountered in the price series case.
Tenth; Joint covariance stationarity also attempts to study covariance across assets, but it still has the time series limitation. Study of group stationarity assumes influence across components in a group. Again the duration dependent behavior indicates that looking at group stationarity maps a group behavior better than studying stationarity at a time series level.
Eleventh; Trend and difference stationarity classifications are relevant on a time series and may remain like that considering the dynamic nature of markets. Our case of looking at broad stationarity seems like a needed simplification to understand market dynamics as a group. When components are interacting at multi-durational level identifying the stochastic and trend behavior in an individual time series, is like shooting in the dark. The simplification hence overcomes some of the limitations and failures of stationarity as a statistical measure for mean reversion.
Dataset
Five stock markets; S&P100, FTSE100, NIKKEI225, STOXX50 and STOXX50Banks were tested for daily data over five time durations (1, 3, 5, 10 and 14 years). 14 rolling periods for each period were generated and an average of 70 portfolios for each of the five time durations were generated and tested.
Research methodology
The research employed the Augmented-Dickey Fuller (ADF) unit root test. Standard Dickey-Fuller (DF) tests for the null hypothesis i.e. the presence of a unit root in the time series. DF is based on the fact that if a time series (in the present case, the ranking series) possesses mean reversion (i.e. stationarity), then the current ranking should be similar to the previous period. The extended version of DF i.e. ADF has the same purpose as the standard one, but it controls for the existence of serial correlation by including lagged changes of the variable. A critical point here is deciding upon the number of lags introduced in the model as too many lags may affect and decrease the sample power of the test. While in case of introducing too many lags, the size of the test is affected. As such, the loaded test used the Bayesian-Information Criterion (BIC) in order to discriminate between the models and choose the optimum number of lags to be used.
The authors calculated the overall percentage of stationary processes (i.e. mean reverting stock percentile ranking series) for each group tested along with the percent stationarity for each bin in the framework.
The percent number of components of the bins exhibiting stationary processes is recorded over time.  The results argue in favor of the existence of stationarity even in the case of price percentile rankings. This we refer to as ‘Group Stationarity’. All the market groups tested witnessed higher than 50% stationary components for different durations. Detailed observations are listed below.
Results and discussion
Group size sensitivity
Smaller groups exhibit higher group stationarity percentages, emphasizing the faster tendency of group components to move from one extreme to another. As group size decreases, the distance between extremes is lowered and the tendency becomes more obvious. The impact of group size on the group stationarity numbers is a subject of further study.
‘Reversion’ behavior
‘Value stationary’ plus ‘Growth stationary’ encompass the ‘Reversion’ behavior i.e. the value and growth are expected to revert towards core and even further in the respective opposite directions. This means towards growth for reverting value and towards value for reverting growth. This is in line with the reversion behavior highlighted in the previous paper on ‘Momentum and Reversion’.
Growth generally showcased higher stationarity percentages in the first periods compared to value components. This might suggest that growth is more prone to reversion than value is and thus, growth is more likely to lose in ranking position compared to a likely rise in value ranking. Holding growth for longer periods is a riskier strategy, because falling rankings indicate underperformance. In the same time, value holding becomes a source of improved outperformance.
‘Divergence’ behavior
‘Core stationary’ is the ‘Divergence’ behavior i.e. the core is expected to diverge out and contribute to value and growth bins. Out of the three bins, core exhibits the highest stationarity percentages. This might suggest that divergence is more prone to transform compared to the value and growth bins because of the bidirectional nature, i.e. witness exiting components both upwards towards growth and downwards towards value. The symmetry of the core bin, which is in the middle of the group, drives its transformation process.
Earlier research on the subject of stationarity has ignored the importance of core within the intra group behavior. It is the divergence at the portfolio level, which keeps the repetitive cycle running. 54% of the core exhibits stationary behavior suggesting a divergence from its original state, pushing in both upwards and downwards direction towards the value and growth respectively and transforming the relative position of components intra group.
The core acts as a buffer. On one side it pushes components to take the place of reverting components (stocks), and on the other side the components in reversion from a growth bin are either pushed into further negative momentum or core gives a push back to the respective component (stock) back upwards into growth. A similar process happens when value is reverting up. The core either assists the component (stock) to gather further positive momentum or pushes it back into growth. Meanwhile it also replenishes the value bin by pushing new components to replace the old ones. The cycle of momentum – reversion -momentum continues to repeat.
Stationarity confirms transformation
Duration analysis showcases increasing stationarity percentages with increased holding periods. As the absolute trend suggested that bins are always transforming, it’s the stationarity process that is a statistical proof of the dynamic process of components exiting the respective bins, but also how new components replenish empty positions, like a natural system. Authors observed an upward trend of group stationarity inside the framework suggesting the emergence of momentum from reversion. This is how stationarity reconfirms the idea of momentum – reversion transformation. After reverting from their initial bin placement, the extremes would continue their upward (value) or downward (growth) trends attaining their momentum. Another observation made is the presence of inflexion points, which suggest a possibility of reverting behavior inside the stationarity results.
Stationarity stats vs. bin movement stats
The graph below illustrates the stationarity stats against the bin movement absolute stats; the latter are the results driven by the Momentum and Reversion (Pal, 2015c) research paper.  Stationarity behavior confirms the dynamic nature of bins, persistence of reversion, emergence of momentum and transformation of momentum into reversion before the process starts again. Stationarity stats and absolute stats in bin movement exhibit similar bin characteristics namely higher divergence percentages, and growth transformation numbers are always higher than value transformation numbers across durations.
Absolute stats are consistently higher than stationarity stats. It seems that the stationary percentages do not go beyond a certain level, a case of higher highs. This rising pattern challenges the conventional expectation for stationarity numbers to vane with increasing time. An increasing trend could suggest two things. First; the framework is an appropriate way to study stock markets systems, natural systems. Second; Group stationarity is a better measure to understand stationary at a group level. An inability to explain the statistical reason for the rising pattern also indicates the limitations of the statistical test as previously discussed. Still, the two series of results are in line; the group stationarity is a good approximation of the bin movement stats. Thus, authors consider unit root tests as good proxies for predicting mean reverting and divergence behaviors in the Framework.
Duration dependency in stationarity
The stationarity has a duration dependency. Unit root tests suggest that duration gives stationarity its behavior; tests showcase increased stationarity percentages with increasing durations, a drop in stationarity numbers in time followed by a rise again.
Portfolios exhibit dynamic structure with increased duration; in three years, more than half of the portfolio’s structure is changed (i.e. more than 50% of the portfolio stocks exhibit group stationary behavior) (Reversion + Divergence).
Conclusion
Stock markets are natural dynamic systems which can’t be understood using conventional statistical techniques. When components are interacting at multi-durational level identifying the stochastic and trend behavior in an individual time series, is like shooting in the dark. Stationarity is a key statistical measure used to understand mean reversion. It fails when put into test for increasing durations. If we need to understand group behavior, the first thing we need to do is define a composite group. The mean reversion framework defined a proxy ranking and explained how momentum and reversion transformed into each other. This paper built on that idea and introduced a new concept called group stationarity, which on one side gave up on asset specific understanding but made up for mapping group behavior. The stationarity numbers reconfirmed the absolute statistics trends discussed in the previous studies and how momentum and reversion was indeed transforming into each other. Group stationarity also illustrated how statistical measures can have different meaning all together if looked at from a group perspective. The challenges we face today are linked to a lack of a composite framework. If indeed the mean reversion framework does well on defining a proxy and explaining how momentum and reversion are transforming into each other, a host of new interpretations using other statistical measures can strengthen our case that stock market systems can be mapped and understood. This should open new ways to invest and manage market risk.
Bibliography
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