Are there any papers about cointegration consisting of time series of more than two assets?

Are there any papers about cointegration consisting of time series of more than two assets ? I wonder if there could be any trading strategy for three assets case.

Technically, if you do PCA on the yield curve (live dangerously!, do it in levels), the first two PCAs are nonstationary. The third is questionable. Note that this is a perfectly valid way of looking for stochastic common trends (see Madalla-Kim, Unit Roots, Cointegration and Structural Change for refs). The fourth principal component is stationary by most measures.

FYI, stochastic common trends is the state-space analog of Cointegration. In cointegration, we use Johansen for testing. In Stoch-common trends, one uses Nyberg, which switches H0 and H1, just like for unit root testing one uses ADF, but for mean-reversion testing, one uses KPSS.

The fourth principal component requires the use of four assets to replicate it. Since PCs are ordered by their modes (first PC=level=no zeros, second PC=slope=one zero, third PC=curvature=two zeros,..) the fourth PC=three zeros and best corresponds to the relationship between one slope on another.

A proxy we used to use for this was 2s-30s slope vs 5s-10s slope. These slopes are generally cointegrated although as with most relationships, the beta can shift. We used to call this the condor back when we covered this trade pre-crisis.

It turns out, that finding the cointegrating relationship is not very robust. Doing the regression of say 2s on 5s,10s and 30s in levels is not robust. The stderrs are too high. Regressing 2s30s slopes on 5s10s slope performs much better OOS.

This was traded pre-crisis, but I don't think it's a very viable trade for most given that there are too many transaction costs.

A similar slope vs slope, well-behaved, trade would be close-maturity futures slopes, so EDH8-EDM8 vs ERH8-ERM8).

It is pretty straightforward to come up with multi-asset portfolios in rates which one can make sense of, because each product is only modestly different from other products. In fact, the factor mimicking portfolio for levels or for slope or for curvature are all multi-asset portfolios.

You might be asking about equities, however, and, unlike rates, I do not personally know whether it makes sense for a triplet of assets to be mean-reverting, since it is a little less intuitive.

• Empirically in trading-butterflies (3leg) and condors (4leg) trades in fixed income. More legs = more transaction costs so there are limits.
– NBF
May 31, 2018 at 5:16

The Johansen test can be used to test for cointegration among $n$ assets.

https://en.wikipedia.org/wiki/Johansen_test

The original paper: Johansen, Søren (1991). "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models". Econometrica. 59 (6): 1551–1580

• yes, but I wonder if there are any empirical studies using more then two cointegrated assets (cointegrated as whole) ?
– Qbik
Nov 18, 2017 at 20:48
• Yeah, there are empirical studies of trading more than two cointegrated assets. (e.g., a rates condor trade involving say 2yr, 5yr, 10yr, 30yr swaps). What some studies show is that restricting degrees of freedom (i.e., don't use 3 or 4 coefs, but just 2), perform much better out of sample. In the case of condors, 30y - 2y = a + b*(10y - 5y) + eps set up as an OLS works better than the whole Johansen machinery.
– NBF
Jun 21, 2018 at 14:03