Cointegration Archives - QUANTITATIVE RESEARCH AND TRADING

December 15, 2020

Volatility Forecasting in Emerging Markets

The great majority of empirical studies have focused on asset markets in the US and other developed economies. The purpose of this research is to determine to what extent the findings of other researchers in relation to the characteristics of asset volatility in developed economies applies also to emerging markets. The important characteristics observed in asset volatility that we wish to identify and examine in emerging markets include clustering, (the tendency for periodic regimes of high or low volatility) long memory, asymmetry, and correlation with the underlying returns process. The extent to which such behaviors are present in emerging markets will serve to confirm or refute the conjecture that they are universal and not just the product of some factors specific to the intensely scrutinized, and widely traded developed markets.

The ten emerging markets we consider comprise equity markets in Australia, Hong Kong, Indonesia, Malaysia, New Zealand, Philippines, Singapore, South Korea, Sri Lanka and Taiwan focusing on the major market indices for those markets. After analyzing the characteristics of index volatility for these indices, the research goes on to develop single- and two-factor REGARCH models in the form by Alizadeh, Brandt and Diebold (2002).

Cluster Analysis of Volatility
Processes for Ten Emerging Market Indices

The research confirms the presence of a number of typical characteristics of volatility processes for emerging markets that have previously been identified in empirical research conducted in developed markets. These characteristics include volatility clustering, long memory, and asymmetry. There appears to be strong evidence of a region-wide regime shift in volatility processes during the Asian crises in 1997, and a less prevalent regime shift in September 2001. We find evidence from multivariate analysis that the sample separates into two distinct groups: a lower volatility group comprising the Australian and New Zealand indices and a higher volatility group comprising the majority of the other indices.

Models developed within the single- and two-factor REGARCH framework of Alizadeh, Brandt and Diebold (2002) provide a good fit for many of the volatility series and in many cases have performance characteristics that compare favorably with other classes of models with high R-squares, low MAPE and direction prediction accuracy of 70% or more. On the debit side, many of the models demonstrate considerable variation in explanatory power over time, often associated with regime shifts or major market events, and this is typically accompanied by some model parameter drift and/or instability.

Single equation ARFIMA-GARCH models appear to be a robust and reliable framework for modeling asset volatility processes, as they are capable of capturing both the short- and long-memory effects in the volatility processes, as well as GARCH effects in the kurtosis process. The available procedures for estimating the degree of fractional integration in the volatility processes produce estimates that appear to vary widely for processes which include both short- and long- memory effects, but the overall conclusion is that long memory effects are at least as important as they are for volatility processes in developed markets. Simple extensions to the single-equation models, which include regressor lags of related volatility series, add significant explanatory power to the models and suggest the existence of Granger-causality relationships between processes.

Extending the modeling procedures into the realm of models which incorporate systems of equations provides evidence of two-way Granger causality between certain of the volatility processes and suggests that are fractionally cointegrated, a finding shared with parallel studies of volatility processes in developed markets.

Download paper here.

October 13, 2020

Modeling Asset Volatility

I am planning a series of posts on the subject of asset volatility and option pricing and thought I would begin with a survey of some of the central ideas. The attached presentation on Modeling Asset Volatility sets out the foundation for a number of key concepts and the basis for the research to follow.

Perhaps the most important feature of volatility is that it is stochastic rather than constant, as envisioned in the Black Scholes framework. The presentation addresses this issue by identifying some of the chief stylized facts about volatility processes and how they can be modelled. Certain characteristics of volatility are well known to most analysts, such as, for instance, its tendency to “cluster” in periods of higher and lower volatility. However, there are many other typical features that are less often rehearsed and these too are examined in the presentation.

Long Memory
For example, while it is true that GARCH models do a fine job of modeling the clustering effect they typically fail to capture one of the most important features of volatility processes – long term serial autocorrelation. In the typical GARCH model autocorrelations die away approximately exponentially, and historical events are seen to have little influence on the behaviour of the process very far into the future. In volatility processes that is typically not the case, however: autocorrelations die away very slowly and historical events may continue to affect the process many weeks, months or even years ahead.

Volatility Direction Prediction Accuracy

There are two immediate and very important consequences of this feature. The first is that volatility processes will tend to trend over long periods – a characteristic of Black Noise or Fractionally Integrated processes, compared to the White Noise behavior that typically characterizes asset return processes. Secondly, and again in contrast with asset return processes, volatility processes are inherently predictable, being conditioned to a significant degree on past behavior. The presentation considers the fractional integration frameworks as a basis for modeling and forecasting volatility.

Mean Reversion vs. Momentum
A puzzling feature of much of the literature on volatility is that it tends to stress the mean-reverting behavior of volatility processes. This appears to contradict the finding that volatility behaves as a reinforcing process, whose long-term serial autocorrelations create a tendency to trend. This leads to one of the most important findings about asset processes in general, and volatility process in particular: i.e. that the assets processes are simultaneously trending and mean-reverting. One way to understand this is to think of volatility, not as a single process, but as the superposition of two processes: a long term process in the mean, which tends to reinforce and trend, around which there operates a second, transient process that has a tendency to produce short term spikes in volatility that decay very quickly. In other words, a transient, mean reverting processes inter-linked with a momentum process in the mean. The presentation discusses two-factor modeling concepts along these lines, and about which I will have more to say later.

Cointegration
One of the most striking developments in econometrics over the last thirty years, cointegration is now a principal weapon of choice routinely used by quantitative analysts to address research issues ranging from statistical arbitrage to portfolio construction and asset allocation. Back in the late 1990’s I and a handful of other researchers realized that volatility processes exhibited very powerful cointegration tendencies that could be harnessed to create long-short volatility strategies, mirroring the approach much beloved by equity hedge fund managers. In fact, this modeling technique provided the basis for the Caissa Capital volatility fund, which I founded in 2002. The presentation examines characteristics of multivariate volatility processes and some of the ideas that have been proposed to model them, such as FIGARCH (fractionally-integrated GARCH).

Dispersion Dynamics
Finally, one topic that is not considered in the presentation, but on which I have spent much research effort in recent years, is the behavior of cross-sectional volatility processes, which I like to term dispersion. It turns out that, like its univariate cousin, dispersion displays certain characteristics that in principle make it highly forecastable. Given an appropriate model of dispersion dynamics, the question then becomes how to monetize efficiently the insight that such a model offers. Again, I will have much more to say on this subject, in future.

September 29, 2020

Robustness in Quantitative Research and Trading

What is Strategy Robustness? What is its relevance to Quantitative Research and Trading?

One of the most highly desired properties of any financial model or investment strategy, by investors and managers alike, is robustness. I would define robustness as the ability of the strategy to deliver a consistent results across a wide range of market conditions. It, of course, by no means the only desirable property – investing in Treasury bills is also a pretty robust strategy, although the returns are unlikely to set an investor’s pulse racing – but it does ensure that the investor, or manager, is unlikely to be on the receiving end of an ugly surprise when market conditions adjust.

Robustness is not the same thing as low volatility, which also tends to be a characteristic highly prized by many investors. A strategy may operate consistently, with low volatility in certain market conditions, but behave very differently in other. For instance, a delta-hedged short-volatility book containing exotic derivative positions. The point is that empirical researchers do not know the true data-generating process for the markets they are modeling. When specifying an empirical model they need to make arbitrary assumptions. An example is the common assumption that assets returns follow a Gaussian distribution. In fact, the empirical distribution of the great majority of asset process exhibit the characteristic of “fat tails”, which can result from the interplay between multiple market states with random transitions. See this post for details:

http://jonathankinlay.com/2014/05/a-quantitative-analysis-of-stationarity-and-fat-tails/

In statistical arbitrage, for example, quantitative researchers often make use of cointegration models to build pairs trading strategies. However the testing procedures used in current practice are not sufficient powerful to distinguish between cointegrated processes and those whose evolution just happens to correlate temporarily, resulting in the frequent breakdown in cointegrating relationships. For instance, see this post:

http://jonathankinlay.com/2017/06/statistical-arbitrage-breaks/

Modeling Assumptions are Often Wrong – and We Know It

We are, of course, not the first to suggest that empirical models are misspecified:

“All models are wrong, but some are useful” (Box 1976, Box and Draper 1987).

Martin Feldstein (1982: 829): “In practice all econometric specifications are necessarily false models.”

Luke Keele (2008: 1): “Statistical models are always simplifications, and even the most complicated model will be a pale imitation of reality.”

Peter Kennedy (2008: 71): “It is now generally acknowledged that econometric models are false and there is no hope, or pretense, that through them truth will be found.”

During the crash of 2008 quantitative Analysts and risk managers found out the hard way that the assumptions underpinning the copula models used to price and hedge credit derivative products were highly sensitive to market conditions. In other words, they were not robust. See this post for more on the application of copula theory in risk management:

http://jonathankinlay.com/2017/01/copulas-risk-management/

Robustness Testing in Quantitative Research and Trading

We interpret model misspecification as model uncertainty. Robustness tests analyze model uncertainty by comparing a baseline model to plausible alternative model specifications. Rather than trying to specify models correctly (an impossible task given causal complexity), researchers should test whether the results obtained by their baseline model, which is their best attempt of optimizing the specification of their empirical model, hold when they systematically replace the baseline model specification with plausible alternatives. This is the practice of robustness testing.

Robustness testing analyzes the uncertainty of models and tests whether estimated effects of interest are sensitive to changes in model specifications. The uncertainty about the baseline model’s estimated effect size shrinks if the robustness test model finds the same or similar point estimate with smaller standard errors, though with multiple robustness tests the uncertainty likely increases. The uncertainty about the baseline model’s estimated effect size increases of the robustness test model obtains different point estimates and/or gets larger standard errors. Either way, robustness tests can increase the validity of inferences.

Robustness testing replaces the scientific crowd by a systematic evaluation of model alternatives.

Robustness in Quantitative Research

In the literature, robustness has been defined in different ways:

as same sign and significance (Leamer)
as weighted average effect (Bayesian and Frequentist Model Averaging)
as effect stability We define robustness as effect stability.

Parameter Stability and Properties of Robustness

Robustness is the share of the probability density distribution of the baseline model that falls within the 95-percent confidence interval of the baseline model. In formulaeic terms:

Robustness is left-–right symmetric: identical positive and negative deviations of the robustness test compared to the baseline model give the same degree of robustness.
If the standard error of the robustness test is smaller than the one from the baseline model, ρ converges to 1 as long as the difference in point estimates is negligible.
For any given standard error of the robustness test, ρ is always and unambiguously smaller the larger the difference in point estimates.
Differences in point estimates have a strong influence on ρ if the standard error of the robustness test is small but a small influence if the standard errors are large.

Robustness Testing in Four Steps

Define the subjectively optimal specification for the data-generating process at hand. Call this model the baseline model.
Identify assumptions made in the specification of the baseline model which are potentially arbitrary and that could be replaced with alternative plausible assumptions.
Develop models that change one of the baseline model’s assumptions at a time. These alternatives are called robustness test models.
Compare the estimated effects of each robustness test model to the baseline model and compute the estimated degree of robustness.

Model Variation Tests

Model variation tests change one or sometimes more model specification assumptions and replace with an alternative assumption, such as:

change in set of regressors
change in functional form
change in operationalization
change in sample (adding or subtracting cases)

Example: Functional Form Test

The functional form test examines the baseline model’s functional form assumption against a higher-order polynomial model. The two models should be nested to allow identical functional forms. As an example, we analyze the ‘environmental Kuznets curve’ prediction, which suggests the existence of an inverse u-shaped relation between per capita income and emissions.

Note: grey-shaded area represents confidence interval of baseline model

Another example of functional form testing is given in this review of Yield Curve Models:

http://jonathankinlay.com/2018/08/modeling-the-yield-curve/

Random Permutation Tests

Random permutation tests change specification assumptions repeatedly. Usually, researchers specify a model space and randomly and repeatedly select model from this model space. Examples:

sensitivity tests (Leamer 1978)
artificial measurement error (Plümper and Neumayer 2009)
sample split – attribute aggregation (Traunmüller and Plümper 2017)
multiple imputation (King et al. 2001)

We use Monte Carlo simulation to test the sensitivity of the performance of our Quantitative Equity strategy to changes in the price generation process and also in model parameters:

http://jonathankinlay.com/2017/04/new-longshort-equity/

Structured Permutation Tests

Structured permutation tests change a model assumption within a model space in a systematic way. Changes in the assumption are based on a rule, rather than random. Possibilities here include:

sensitivity tests (Levine and Renelt)
jackknife test
partial demeaning test

Example: Jackknife Robustness Test

The jackknife robustness test is a structured permutation test that systematically excludes one or more observations from the estimation at a time until all observations have been excluded once. With a ‘group-wise jackknife’ robustness test, researchers systematically drop a set of cases that group together by satisfying a certain criterion – for example, countries within a certain per capita income range or all countries on a certain continent. In the example, we analyse the effect of earthquake propensity on quake mortality for countries with democratic governments, excluding one country at a time. We display the results using per capita income as information on the x-axes.

Upper and lower bound mark the confidence interval of the baseline model.

Robustness Limit Tests

Robustness limit tests provide a way of analyzing structured permutation tests. These tests ask how much a model specification has to change to render the effect of interest non-robust. Some examples of robustness limit testing approaches:

unobserved omitted variables (Rosenbaum 1991)
measurement error
under- and overrepresentation
omitted variable correlation

For an example of limit testing, see this post on a review of the Lognormal Mixture Model:

http://jonathankinlay.com/2018/08/the-lognormal-mixture-variance-model/

Summary on Robustness Testing

Robustness tests have become an integral part of research methodology. Robustness tests allow to study the influence of arbitrary specification assumptions on estimates. They can identify uncertainties that otherwise slip the attention of empirical researchers. Robustness tests offer the currently most promising answer to model uncertainty.

August 4, 2020

Daytrading Index Futures Arbitrage

Trading with Indices

I have always been an advocate of incorporating index data into one’s trading strategies. Since they are not tradable, the “market” in index products if often highly inefficient and displays easily identifiable patterns that can be exploited by a trader, or a trading system. In fact, it is almost trivially easy to design “profitable” index trading systems and I gave a couple of examples in the post below, including a system producing stellar results in the S&P 500 Index.

http://jonathankinlay.com/2016/05/trading-with-indices/

Of course such systems are not directly useful. But traders often use signals from such a system as a filter for an actual trading system. So, for example, one might look for a correlated signal in the S&P 500 index as a means of filtering trades in the E-Mini futures market or theSPDR S&P 500 ETF (SPY).

Multi-Strategy Trading Systems

This is often as far as traders will take the idea, since it quickly gets a lot more complicated and challenging to build signals generated from an index series into the logic of a strategy designed for related, tradable market. And for that reason, there is a great deal of unexplored potential in using index data in this way. So, for instance, in the post below I discuss a swing trading system in the S&P500 E-mini futures (ticker: ES) that comprises several sub-systems build on prime-valued time intervals. This has the benefit of minimizing the overlap between signals from multiple sub-systems, thereby increasing temporal diversification.

http://jonathankinlay.com/2018/07/trading-prime-market-cycles/

A critical point about this system is that each of sub-systems trades the futures market based on data from both the E-mini contract and the S&P 500 cash index. A signal is generated when the system finds particular types of discrepancy between the cash index and corresponding futures, in a quasi risk-arbitrage.

Arbing the NASDAQ 100 Index Futures

Developing trading systems for the S&P500 E-mini futures market is not that hard. A much tougher challenge, at least in my experience, is presented by the E-mini NASDAQ-100 futures (ticker: NQ). This is partly to do with the much smaller tick size and different market microstructure of the NASDAQ futures market. Additionally, the upward drift in equity related products typically favors strategies that are long-only. Where a system trades both long and short sides of the market, the performance on the latter is usually much inferior. This can mean that the strategy performs poorly in bear markets such as 2008/09 and, for the tech sector especially, the crash of 2000/2001. Our goal was to develop a daytrading system that might trade 1-2 times a week, and which would perform as well or better on short trades as on the long side. This is where NASDAQ 100 index data proved to be especially helpful. We found that discrepancies between the cash index and futures market gave particularly powerful signals when markets seemed likely to decline. Using this we were able to create a system that performed exceptionally well during the most challenging market conditions. It is notable that, in the performance results below (for a single futures contract, net of commissions and slippage), short trades contributed the greater proportion of total profits, with a higher overall profit factor and average trade size.

Conclusion: Using Index Data, Or Other Correlated Signals, Often Improves Performance

It is well worthwhile investigating how non-tradable index data can be used in a trading strategy, either as a qualifying signal or, more directly, within the logic of the algorithm itself. The greater challenge of building such systems means that there are opportunities to be found, even in well-mined areas like index futures markets. A parallel idea that likewise offers plentiful opportunity is in designing systems that make use of data on multiple time frames, and in correlated markets, for instance in the energy sector.Here one can identify situations in which, under certain conditions, one market has a tendency to lead another, a phenomenon referred to as Granger Causality.

March 31, 2020

Correlation Cointegration

In a previous post I looked at ways of modeling the relationship between the CBOE VIX Index and the Year 1 and Year 2 CBOE Correlation Indices:

http://jonathankinlay.com/2017/08/modeling-volatility-correlation/

The question was put to me whether the VIX and correlation indices might be cointegrated.

Let’s begin by looking at the pattern of correlation between the three indices:

If you recall from my previous post, we were able to fit a linear regression model with the Year 1 and Year 2 Correlation Indices that accounts for around 50% in the variation in the VIX index. While the model certainly has its shortcomings, as explained in the post, it will serve the purpose of demonstrating that the three series are cointegrated. The standard Dickey-Fuller test rejects the null hypothesis of a unit root in the residuals of the linear model, confirming that the three series are cointegrated, order 1.

Vector Autoregression

We can attempt to take the modeling a little further by fitting a VAR model. We begin by splitting the data into an in-sample period from Jan 2007 to Dec 2015 and an out-of-sample test period from Jan 2016 to Aug 2017. We then fit a vector autoregression model to the in-sample data:

When we examine how the model performs on the out-of-sample data, we find that it fails to pick up on much of the variation in the series – the forecasts are fairly flat and provide quite poor predictions of the trends in the three series over the period from 2016-2017:

Conclusion

The VIX and Correlation Indices are not only highly correlated, but also cointegrated, in the sense that a linear combination of the series is stationary.

One can fit a weakly stationary VAR process model to the three series, but the fit is quite poor and forecasts from the model don’t appear to add much value. It is conceivable that a more comprehensive model involving longer lags would improve forecasting performance.

February 11, 2020

Why Statistical Arbitrage Breaks Down

An extract from my book, Quantitative Research and Trading, to be published in 2019.

Why Statistical Arbitrage Trades Break Down

October 23, 2018October 23, 2018

Developing Long/Short ETF Strategies

Recently I have been working on the problem of how to construct large portfolios of cointegrated securities. My focus has been on ETFs rather that stocks, although in principle the methodology applies equally well to either, of course.

My preference for ETFs is due primarily to the fact that it is easier to achieve a wide diversification in the portfolio with a more limited number of securities: trading just a handful of ETFs one can easily gain exposure, not only to the US equity market, but also international equity markets, currencies, real estate, metals and commodities. Survivorship bias, shorting restrictions and security-specific risk are also less of an issue with ETFs than with stocks (although these problems are not too difficult to handle).

On the downside, with few exceptions ETFs tend to have much shorter histories than equities or commodities. One also has to pay close attention to the issue of liquidity. That said, I managed to assemble a universe of 85 ETF products with histories from 2006 that have sufficient liquidity collectively to easily absorb an investment of several hundreds of millions of dollars, at minimum.

The Cardinality Problem

The basic methodology for constructing a long/short portfolio using cointegration is covered in an earlier post. But problems arise when trying to extend the universe of underlying securities. There are two challenges that need to be overcome.

The first issue is that, other than the simple regression approach, more advanced techniques such as the Johansen test are unable to handle data sets comprising more than about a dozen securities. The second issue is that the number of possible combinations of cointegrated securities quickly becomes unmanageable as the size of the universe grows. In this case, even taking a subset of just six securities from the ETF universe gives rise to a total of over 437 million possible combinations (85! / (79! * 6!). An exhaustive test of all the possible combinations of a larger portfolio of, say, 20 ETFs, would entail examining around 1.4E+19 possibilities.

Given the scale of the computational problem, how to proceed? One approach to addressing the cardinality issue is sparse canonical correlation analysis, as described in Identifying Small Mean Reverting Portfolios, d’Aspremont (2008). The essence of the idea is something like this. Suppose you find that, in a smaller, computable universe consisting of just two securities, a portfolio comprising, say, SPY and QQQ was found to be cointegrated. Then, when extending consideration to portfolios of three securities, instead of examining every possible combination, you might instead restrict your search to only those portfolios which contain SPY and QQQ. Having fixed the first two selections, you are left with only 83 possible combinations of three securities to consider. This process is repeated as you move from portfolios comprising 3 securities to 4, 5, 6, … etc.

Other approaches to the cardinality problem are possible. In their 2014 paper Sparse, mean reverting portfolio selection using simulated annealing, the Hungarian researchers Norbert Fogarasi and Janos Levendovszky consider a new optimization approach based on simulated annealing. I have developed my own, hybrid approach to portfolio construction that makes use of similar analytical methodologies. Does it work?

A Cointegrated Long/Short ETF Basket

Below are summarized the out-of-sample results for a portfolio comprising 21 cointegrated ETFs over the period from 2010 to 2015. The basket has broad exposure (long and short) to US and international equities, real estate, currencies and interest rates, as well as exposure in banking, oil and gas and other specific sectors.

The portfolio was constructed using daily data from 2006 – 2009, and cointegration vectors were re-computed annually using data up to the end of the prior year. I followed my usual practice of using daily data comprising “closing” prices around 12pm, i.e. in the middle of the trading session, in preference to prices at the 4pm market close. Although liquidity at that time is often lower than at the close, volatility also tends to be muted and one has a period of perhaps as much at two hours to try to achieve the arrival price. I find this to be a more reliable assumption that the usual alternative.

The risk-adjusted performance of the strategy is consistently outstanding throughout the out-of-sample period from 2010. After a slowdown in 2014, strategy performance in the first quarter of 2015 has again accelerated to the level achieved in earlier years (i.e. with a Sharpe ratio above 4).

Another useful test procedure is to compare the strategy performance with that of a portfolio constructed using standard mean-variance optimization (using the same ETF universe, of course). The test indicates that a portfolio constructed using the traditional Markowitz approach produces a similar annual return, but with 2.5x the annual volatility (i.e. a Sharpe ratio of only 1.6). What is impressive about this result is that the comparison one is making is between the out-of-sample performance of the strategy vs. the in-sample performance of a portfolio constructed using all of the available data.

Having demonstrated the validity of the methodology, at least to my own satisfaction, the next step is to deploy the strategy and test it in a live environment. This is now under way, using execution algos that are designed to minimize the implementation shortfall (i.e to minimize any difference between the theoretical and live performance of the strategy). So far the implementation appears to be working very well.

Once a track record has been built and audited, the really hard work begins: raising investment capital!

October 10, 2018October 11, 2018

Cointegration Breakdown

The Low Power of Cointegration Tests

One of the perennial difficulties in developing statistical arbitrage strategies is the lack of reliable methods of estimating a stationary portfolio comprising two or more securities. In a prior post (below) I discussed at some length one of the primary reasons for this, i.e. the lower power of cointegration tests. In this post I want to explore the issue in more depth, looking at the standard Johansen test Procedure to estimate cointegrating vectors.

Johansen Test for Cointegration

Start with some weekly data for an ETF triplet analyzed in Ernie Chan’s book:

After downloading the weekly close prices for the three ETFs we divide the data into 14 years of in-sample data and 1 year out of sample:

We next apply the Johansen test, using code kindly provided by Amanda Gerrish:

We find evidence of up to three cointegrating vectors at the 95% confidence level:

Let’s take a look at the vector coefficients (laid out in rows, in Amanda’s function):

In-Sample vs. Out-of-Sample testing

We now calculate the in-sample and out-of-sample portfolio values using the first cointegrating vector:

The portfolio does indeed appear to be stationary, in-sample, and this is confirmed by the unit root test, which rejects the null hypothesis of a unit root:

Unfortunately (and this is typically the case) the same is not true for the out of sample period:

More Data Doesn’t Help

The problem with the nonstationarity of the out-of-sample estimated portfolio values is not mitigated by adding more in-sample data points and re-estimating the cointegrating vector(s):

We continue to add more in-sample data points, reducing the size of the out-of-sample dataset correspondingly. But none of the tests for any of the out-of-sample datasets is able to reject the null hypothesis of a unit root in the portfolio price process:

The Challenge of Cointegration Testing in Real Time

In our toy problem we know the out-of-sample prices of the constituent ETFs, and can therefore test the stationarity of the portfolio process out of sample. In a real world application, that discovery could only be made in real time, when the unknown, future ETFs prices are formed. In that scenario, all the researcher has to go on are the results of in-sample cointegration analysis, which demonstrate that the first cointegrating vector consistently yields a portfolio price process that is very likely stationary in sample (with high probability).

The researcher might understandably be persuaded, wrongly, that the same is likely to hold true in future. Only when the assumed cointegration relationship falls apart in real time will the researcher then discover that it’s not true, incurring significant losses in the process, assuming the research has been translated into some kind of trading strategy.

A great many analysts have been down exactly this path, learning this important lesson the hard way. Nor do additional “safety checks” such as, for example, also requiring high levels of correlation between the constituent processes add much value. They might offer the researcher comfort that a “belt and braces” approach is more likely to succeed, but in my experience it is not the case: the problem of non-stationarity in the out of sample price process persists.

Conclusion: Why Cointegration Breaks Down

We have seen how a portfolio of ETFs consistently estimated to be cointegrated in-sample, turns out to be non-stationary when tested out-of-sample. This goes to the issue of the low power of cointegration test, and their inability to estimate cointegrating vectors with sufficient accuracy. Analysts relying on standard tests such as the Johansen procedure to design their statistical arbitrage strategies are likely to be disappointed by the regularity with which their strategies break down in live trading.

October 9, 2018

Successful Statistical Arbitrage

I tend not to get involved in Q&A with readers of my blog, or with investors. I am at a point in my life where I spend my time mostly doing what I want to do, rather than what other people would like me to do. And since I enjoy doing research and trading, I try to maximize the amount of time I spend on those activities.

As a business strategy, I wouldn’t necessarily recommend this approach. It’s just something I evolved while learning to play chess: since I had no-one to teach me, I had to learn everything for myself and this involved studying for many, many hours alone.

By contrast, several of the best money managers are also excellent communicators – take Roy Niederhoffer, or Ernie Chan, for example. Having regular, informed communication with your investors is, as smarter managers have realized, a means of building trust and investor loyalty – important factors that come into play during periods when your strategy is underperforming. Not only that, but since communication is two-way, an analyst/manager can learn much from his exchanges with his clients. Knowing how others perceive you – and your competitors – for example, is very useful information. So, too, is information about your competitors’ research ideas, investment strategies and fund performance, which can often be gleaned from discussions with investors. There are plenty of reasons to prefer a policy of regular, open communication.

As a case in point, I was surprised to learn from comments on another research blog that readers drew the conclusion from my previous posts that pursuing the cointegration or Kalman Filter approach to statistical arbitrage was a waste of time. Apparently, my remark to the effect that researchers often failed to pay attention to the net PnL per share in evaluating stat. arb. trading strategies was taken by some to mean that any apparent profitability would always be subsumed within the bid-offer spread. That was not my intention. What I intended to convey was that in some instances, this would be the case – some, but not all.

To illustrate the point, below are the out-of-sample results from a research study applying the Kalman Filter approach for four equity pairs using 5-minute data. For competitive reasons I am unable to identify the specific stocks in each pair, which result from an exhaustive analysis of over 30,000 pairs, but I can say that they are liquid large-cap equities traded in large volume on the US exchanges. The performance numbers are net of transaction costs and are based on the assumption of a 5-minute delay in execution: meaning, a trading signal received at time t is assumed to be executed at time t+5 minutes. This allows sufficient time to leg into each trade passively, in most cases avoiding the bid-offer spread. The net PnL per share is above 1.5c per share for each pair.

While the performance of none of the pairs is spectacular, a combined portfolio has quite attractive characteristics, which include 81% winning months since Jan 2012, a CAGR of over 27% and Information Ratio of 2.29, measured on monthly returns (2.74 based on daily returns).

Finally, I am currently implementing trading of a number of stock portfolios based on static cointegration relationships that have out-of-sample information ratios of between 3 and 4, using daily data.

October 1, 2018

ETF Pairs Trading with the Kalman Filter

I was asked by a reader if I could illustrate the application of the Kalman Filter technique described in my previous post with an example. Let’s take the ETF pair AGG IEF, using daily data from Jan 2006 to Feb 2015 to estimate the model. As you can see from the chart in Fig. 1, the pair have been highly correlated over the last several years.

Fig 1. AGG and IEF Daily Prices 2006-2015

We now estimate the beta-relationship between the ETF pair with the Kalman Filter, using the Matlab code given below, and plot the estimated vs actual prices of the first ETF, AGG in Fig 2. There are one or two outliers that you might want to take a look at, but mostly the fit looks very good.

Fig 2 – Actual vs Fitted Prices of AGG

Now lets take a look at Kalman Filter estimates of beta. As you can see in Fig 3, it wanders around a lot! Very difficult to handle using some kind of static beta estimate.

Fig 3 – Kalman Filter Beta Estimates

Finally, we compute the raw and standardized alphas, being the differences between the observed and fitted prices , i.e. Alpha(t) = AGG(t) – b(t)* IEF(t) and kfAlpha(t) = (Alpha(t) – mean(Alpha(t)) / std(Alpha(t) I have plotted the kfAlpha estimates over the last year in Fig 4.

Fig 4 – Standardized Alpha Estimates

The last step is to decide how to trade this relationship. You might, for example, trade the portfolio in proportion to the standardized deviation (i.e. the size of kfAlpha(t)). Alternatively, you might set a threshold level, say +/- 1 Sd, and trade the portfolio when kfAlpha(t) exceeds this the threshold. In the Matlab code below I use the particle swarm method to maximize the likelihood. I have found this to be more reliable than other methods.

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