Two time series are cointegrated if a linear combination has constant mean and standard deviation. In other words, the two series never stray too far from one another. Cointegration is a useful technique for studying relationships in multivariate time series, and provides a sound methodology for modelling both long-run and short-run dynamics in a financial system.

Example

Suppose you have two stocks Si and Si and you find that Si — 3 Si is stationary, so that this combination never strays too far from its mean. If one day this 'spread' is particularly large then you would have sound statistical reasons for thinking the spread might shortly reduce, giving you a possible source of statistical arbitrage profit. This can be the basis for pairs trading.

Long answer

The correlations between financial quantities are notoriously unstable. Nevertheless correlations are regularly used in almost all multivariate financial problems. An alternative statistical measure to correlation is cointegration. This is probably a more robust measure of the linkage between two financial quantities but as yet there is little derivatives theory based on the concept.

Two stocks may be perfectly correlated over short timescales yet diverge in the long run, with one growing and the other decaying. Conversely, two stocks may follow each other, never being more than a certain distance apart, but with any correlation, positive, negative or varying. If we are delta hedging then maybe the short timescale correlation matters, but not if we are holding stocks for a long time in an unhedged portfolio. To see whether two stocks stay close together we need a definition of stationarity. A time series is stationary if it has finite and constant mean, standard deviation and autocorrelation function. Stocks, which tend to grow, are not stationary. In a sense, stationary series do not wander too far from their mean.

Testing for the stationarity of a time series Xt involves a linear regression to find the coefficients a, b and c in

If it is found that a > 1 then the series is unstable. If —1 < a < 1 then the series is stationary. If a = 1 then the series is non-stationary. As with all things statistical, we can only say that our value for a is accurate with a certain degree of confidence. To decide whether we have got a stationary or non-stationary series requires us to look at the Dickey-Fuller statistic to estimate the degree of confidence in our result. So far, so good, but from this point on the subject of cointegration gets complicated.

How is this useful in finance? Even though individual stock prices might be non stationary it is possible for a linear combination (i.e. a portfolio) to be stationary. Can we find k^ with J2N= 1 ki = 1, such that

is stationary? If we can, then we say that the stocks are cointegrated.

For example, suppose we find that the S&P500 index is cointegrated with a portfolio of 15 stocks. We can then use these fifteen stocks to track the index. The error in this tracking portfolio will have constant mean and standard deviation, so should not wander too far from its average. This is clearly easier than using all 500 stocks for the tracking (when, of course, the tracking error would be zero).

We don't have to track the index, we could track anything we want, such as e02t to choose a portfolio that gets a 20% return. We could analyze the cointegration properties of two related stocks, Nike and Reebok, for example, to look for relationships. This would be pairs trading. Clearly there are similarities with MPT and CAPM in concepts such as means and standard deviations. The important difference is that cointegration assumes far fewer properties for the individual time series. Most importantly, volatility and correlation do not appear explicitly.

Another feature of cointegration is Granger causality which is where one variable leads and another lags. This is of help in explaining why there is any dynamic relationship between several financial quantities.

References and Further Reading

Alexander, CO 2001 Market Models. John Wiley & Sons Ltd

Engle, R & Granger, C 1987 Cointegration and error correction: representation, estimation and testing. Econometrica 55 251-276

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