Thanks, Jonathan.

I read through your two Kalman filter papers and I found them interesting. Good analysis. Your approach is similar to Ernie Chan's chapter on Kalman filter in the book I mentioned. I believe his "measurement prediction error", e(t), is the same as your alpha(t).

You've hit on a major challenge in applying the Kalman filter, namely, how to determine the noise variances/covariances, R and Q. Most coders seem to use values determined by trial-and-error. However, if you're interested, I've come up with a derivation based on the observed measurement errors for calculating R (what I call ve), and the observed variation in beta(t) for calculating Q (what I call vw). This necessitates an iterative approach -- using initial estimates for Q, R, and the initial state and state covariance, implementing the Kalman filter, calculating new estimates, and so on, until the estimates converge to stable values.

This approach is more math intensive, but it allows generalization beyond pairs to trading cointegrated portfolios of 3 or more financial instruments. (I prefer trading a portfolio of ETFs.) I've posted my method here on the Quantitative Finance area of the StackExchange website.

I've been trading a portfolio of 3 ETFs using this algo for three months and nearly all of my trades have been profitable. I use weekly data to calculate the z-score and I usually get one or two "buy" signals a week with a holding time that seems to vary from a few hours to two weeks, partly depending upon where I choose my "sell" level. After a couple dozen trades, profit per trade has been in the range 0.5% to 3%, except for one trade where I had a loss of about 1%. This should give me a good return over the next year, if I can maintain that performance. I'm still fine tuning the algo, especially with regard to where to set "buy" and "sell" limits.

Here's a plot of the z-score that my code produces:

This plot is only showing the last two years of weekly data but I use anywhere from 5 - 10 years of data in my algo. I'm displaying weekly data because I update my Kalman filter parameters every weekend. However, I actually calculate and plot the z-score in real-time during trading hours (using weights from the Kalman filter). When I hit a z-score of say, 1 (-1), I put on a short (long) portfolio position. I close the position when the z-score returns back to zero (or perhaps a little beyond zero). It's too early to say how profitable this will be over the long term. When I have more data perhaps I'll post my total returns.

I hope this is helpful.

Jonathan,

I wrote my Kalman Filter routine in Mathematica, from scratch. This way I know exactly what it does.

Regarding your questions:

1) My backtesting showed average yearly returns (AYRs) in the 30% - 40% range (over a 6-year period), with a maximum drawdown under 10%. This was with fixed entry/exit limits, and 100% of my cash in and out. However, in live trading, what I do is put on 50% of my position when I cross one limit, another 25% when I cross another limit, etc., so that I reduce my drawdown if I get a large excursion in the statistic (say, 2 or 3 standard deviations), while capturing some returns on the smaller excursions (1 standard deviation). I really feel that I need to see how my track record goes with live trading. That's what counts.

2) I wrote a small routine to download real-time ETF data from nasdaq.com. Basically, I use the Mathematica URLRead function and screen scrape for the real-time quote. I use the Mathematica Dynamic function to do this, and update the plot and recommended positions, automatically once per minute. Real-time 1-minute data is good enough for my purposes. I enter the orders manually on a multi-order trading screen. I've got a system that keeps the lag to a few seconds. Again, good enough for my purposes.

3) Yes, GARCH can be useful to show changes in volatility. I haven't implemented that. However, I've recently applied the Mathematica functions HiddenMarkovProcess and FindHiddenMarkovStates to detect and display a shift from a low volatility state to a high volatility state (and vice-versa) in my statistic. It's mainly for informational purposes. (I basically highlight areas of the plot with white or light-gray background, depending on whether I'm in a low-volatility state or a high volatility state.) It may affect when I place my trades. Too early to say yet. A big issue for me is how best to display the information so that I can easily and quickly react and trade when needed.

Hi Per. I got an email today notifying me of your post to this discussion. I've been trading this algorithm during this entire time, as well as having a lot of email exchanges with Jonathan Kinlay regarding how to implement both the Johansen code and the Kalman filter in a manner similar to Chan. The algorithm was working well for me until about 2 months ago when one of the components of my triplet started rising in price in a way that appears to violate the cointegration. The result was that I waited 9 weeks for mean reversion. The portfolio finally mean-reverted, but by that time I had taken a significant hit on my profits. As a result my total trading profit over this time is much smaller than it was before this event -- around 10% or so over the past 9 months.

The past 2 months are clearly an outlier as compared to past algorithm performance. I can tell that simply by looking at the variation of the z-score over the past 10 years. The average half-life for mean-reversion of the z-score was 1.17 weeks. On a few occasions it took as long as 4 or 5 weeks to mean revert. (Remember, I don't make a profit until the portfolio value -- and thus the z-score -- mean-reverts.) Over the past 2 months or so, it took 9 weeks for my portfolio to mean revert. Is the cointegration breaking down, or was this just a one-time statistical fluke?

The problem is that when I re-calculate the Kalman filter parameters each weekend, the z-score often shifts in such a way that losses can accumulate if the mean reversion is delayed for too long. For example, on the close Friday, I may show a z-score of around 1.0. Over the weekend I re-calculate the Johansen test + Kalman filter parameters, and then when I run the algo on Monday morning, the z-score is significantly lower (in the range 0.5 - 0.8), even if the market is essentially unchanged. Thus, even if mean reversion occurs that day, I don't make as much profit as I'd hoped. This isn't a big deal if the portfolio mean reverts within 3 weeks -- I still make a profit. But if mean reversion doesn't happen for more than 3 or 4 weeks, I may end up with a loss. Waiting 9 weeks for mean reversion accumulated a 12% loss, which was more than half my profit over the past 9 months.

I'm still trading with this algorithm, but I'm waiting for a higher z-score before I "pull the trigger" on my trades, in order to increase the probability of a quicker mean-reversion. This means I may miss some trades, but that's OK. Also, I scale in my buys. (Partial buy at, say, z-score = 1.0, and another partial buy at z-score = 1.5, etc.)

If you're interested in how I implement the Kalman filter -- which is significantly different than Chan -- I wrote a detailed post on StackExchange - Quantitative Finance:

Does Chan use the wrong state transition model in his Kalman filter code?

Using a careful analysis, I argued in my post that Chan uses the wrong state transition matrix, i.e., the identity matrix, in his Kalman filter. I showed how to calculate the correct state transition matrix for a cointegrated portfolio, as well as how to initialize the Kalman filter using an adaptive tuning method. I received positive feedback from the readers of the forum, and one reader emailed Ernie Chan. This precipitated an email exchange between Ernie and I. Ernie basically said that his treatment was meant to be more general and so he didn't assume cointegration. I didn't want to argue, so I let it go. (My method actually works when there's no cointegration -- you basically get the identity matrix solution that Chan uses in that case.) Chan replied that "a stationary [cointegrated] portfolio can be more profitable", and that "your analysis may be correct". I didn't press him any further. I know what I did is correct, and none of the readers of my StackExchange post criticized my analysis. I received a fair number of up-votes.

I hope this is helpful.

Amanda has correctly anticipated the direction I was headed in i.e to show that regardless of how small the size of the OOS period relative to the IS period, the Johansen procedure by itself is unable to produce a cointegrating vector capable of yielding a portfolio price process that is stationary out of sample. But her iterative Kalman Filter approach is able to cure the problem.

I don't want to gloss over this finding, because it is very important. 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 researchers 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.

For a more detailed discussion of the problem see this post: Why Statistical Arbitrage Breaks Down

I was hitherto unaware of any methodology for tackling this problem, which is why Amanda's discovery is so important. As she demonstrates in her latest post, the iterative Kalman Filter approach is capable of producing a stationary out of sample process, based on the initial estimates of the cointegrating vector derived from the Johansen procedure.

In fact, Amanda's discovery is important in two fields of econometric research: cointegration theory and the theory of Kalman Filters in modeling inter-asset relationships where, as with the Johansen procedure, KF models have traditionally suffered from difficulties associated with nonstationarity in the out of sample period.

It's a tremendous achievement.

So, despite the fact that Amanda has leapt ahead to the finish line, I shall continue to plod along because, firstly, only by implementing the methodology can I be sure that I have properly and fully understood it and, secondly, as one discovers as one progresses in the field of quantitative research, fine details are often very important. So I am hoping that Amanda will provide additional guidance if I stray too far off piste in the forthcoming exposition.

Before we delve into the Kalman Filter model, its worth pointing out that 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):

IterateCointegrationTest[data_, n_] := 
  Module[{isdata, osdata, JT, isportprice, osportprice},
   isdata = Take[data, All, n];
   osdata = Drop[data, 0, n];
   JT = JohansenTest[isdata, 2, 1];
   isportprice = JT[[2, 1]].isdata;
   osportprice = JT[[2, 1]].osdata;
   {UnitRootTest[isportprice], UnitRootTest[osportprice]}];

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:

ListLinePlot[
 Transpose@
  Table[IterateCointegrationTest[stockprices, 52*14 + i], {i, 1, 50}],
  PlotLegends -> {"In-Sample", "Out-of-Sample"}]

A problem with out-of-sample testing is that market structure can shift so that relationships (such as cointegration) may start to break down. One way to try to minimize this effect is to update your Johansen coefficients more frequently. In backtesting, I update the Johansen coefficients weekly, being careful to use only past data to calculate the current portfolio weights at any time point. (I think this is called "walk forward".) This reflects how I actually use the function in practice. In effect, my out-of-sample period is always one time step. This gives better backtest results, but because I'm avoiding look-ahead bias, it's valid. That's what I did in the backtest I described in a previous reply. You can even track the trace/eigen-statistics over time to make sure that the cointegration is not falling apart.

Also, the Kalman filter dynamically adjusts the Johansen weights so that the weighted price series is more stationary.

Amanda - Your help regarding the implementation of the Kalman filter would be greatly appreciated and I fully understand that you don't want to publish your code - I wouldn't either! I'm just about to finalize the index arbitrage backtesting and I'll let you know whether there is any value to be gained. Then I'll start working on your Kalman idea, expecting to get stuck rather soon(!). So if you don't mind, I'll contact you again once I'm on the move with that.

Per

Amanda - thanks for taking your time to write such elaborate answer. To me this is extremely interesting and I had a similar experience beginning of this year in one of my cointegration baskets with European stock index futures, where the basket wandered off on a really long adverse excursion before eventually reverting at a loss. I realized that this became a quite lengthy post so I apologize for that in advance.

There are a few core concepts associated with this type of trading that I'm constantly working on in addition to refining the mathematical procedure of constructing a stationary portfolio. It would be interesting to here your view on these as well:

1. Selecting the ETFs? Are you using the same ETFs or do you continuously screen for new combinations with potentially better cointegration statistics? I have relied on a basket with the same set of stock index futures, reasoning that the European economies are fundamentally interlinked at some level and indeed this can be validated statistically for extended periods. But not always…and there's the problem.

2. Arbitrage between the ETF and its constituents? This is also briefly described by Chan, but of course any practical implementation comes with a heap of issues not covered in the book. I alluded to this in my first post and I think it is a quite interesting approach. The point here is that the ETF is perfectly cointegrated with its portfolio of weighted constituents by construction and not by a hidden set of underlying factors et c. The task and the challenge here is to find a subset of constituents with high enough cointegration properties in combination with sufficient variance to overcome the transaction cost. I'm exploring this approach again for stock indices and their constituents, where I periodically reconstruct the constituent subset basket. I would imagine it to be quite straight forward to apply your existing model to this approach as well?

3. Combining strategies? Wether they are statistical flukes or cointegration breakdowns, one of the cures for painful or even catastrophic drawdowns is to maintain a portfolio of different strategies with limited correlation. I tend to focus more on methods to construct a portfolio of cointegratin baskets than going all in on one or a few of them. What are your thoughts in this?

Regarding Chan's book, I think the depth of your research is on a completely different level. You are rebuilding the concepts from scratch, finding and solving issues not even mentioned in the book. There are clearly shortcuts and maybe even mistakes in the book, but in all fairness I think Ernie's doing a great job explaining the basic concepts and setting the scene for further investigation.

Your post on StackExchange is what led me to the Wolfram site in the first place. I'm already working on incorporating your procedure in Python.