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Cointegration

It is unfortunate that many market practitioners still base their analysis of the relationships between markets on the very limited concept of correlation. Trying to model the complex interdependencies between financial assets with so restrictive a tool is like trying to surf the internet with an IBM AT. It is, therefore, rather gratifying to see that more sophisticated models are now being applied to analyse relationships between financial assets. This chapter concerns an important development in this field; it introduces a multivariate time series model for the dynamic co-dependencies that are often found in financial markets.

Trying to model the complex

interdependencies between financial assets with so restrictive a tool is like trying to surf the internet with an IBM AT

Cointegration refers not to co-movements in returns, but to co-movements in asset prices (or exchange rates or yields). If spreads are mean-reverting, asset prices are tied together in the long term by a common stochastic trend, and we say that the prices are cointegrated. Since the seminal work of Engle and Granger (1987) cointegration has become the prevalent tool of time series econometrics. Every modern econometrics text covers the statistical theory necessary to master the practical application of cointegration, Hamilton (1994), Enders (1995) and Hendry (1996) being among the best sources. Cointegration has emerged as a powerful technique for investigating common trends in multivariate time series, and provides a sound methodology for modelling both long-run and short-run dynamics in a system.

Cointegration is a two-step process: first any long-run equilibrium relationships between prices are established, and then a dynamic correlation model of returns is estimated. This error correction model (ECM), so-called because short-term deviations from equilibrium are corrected, reveals the Granger causalities that must be present in a cointegrated system. Thus cointegration may be a sign of market inefficiency, but it can also be the result of market efficiency - as, for example, is the cointegration between spot and futures prices.

The basic building blocks for time series analysis were described in Chapter 11. Some of the empirical data that were applied there are also applied in this chapter, but this time to models of non-stationary processes to capture long-run common features such as a common stochastic trend. The first section introduces cointegration, the relationship between cointegration and correlation, and the implications of cointegration for common trends in

If spreads are mean-reverting, asset prices are tied together in the long term by a common stochastic trend, and we say that the prices are cointegrated

financial data. Then §12.2 describes how to test for cointegration, with empirical examples of term structures, spot and futures prices and equity indices. Models for the dynamic relationships between returns in cointegrated systems are introduced in §12.3, with examples on equity indices and commodity spot and futures prices. A large number of cointegration models have been applied in financial markets, and this literature is surveyed in §12.4.

The relationship between the mean and the variance of portfolio returns is a corner-stone of modern portfolio theory. However, returns are short-memory processes (Granger and Hallman, 1991) and so investments that are based on the characteristics of returns alone cannot model long-run cointegrating relationships between prices. Section 12.5 examines how cointegration, applied to investment analysis, presents a modern and powerful alternative to mean-variance analysis.

A common stochastic trend is just one of many common features that multivariate time series could possess. Section 12.6 presents some empirical evidence on other common features, such as common volatility patterns and common autocorrelation properties.

12.1 Introducing Cointegration

Although empirical models of cointegrated financial time series are commonplace in the academic literature, the practical implementation of these models in systems for investment analysis or portfolio risk is still in its early stages. This is because the traditional starting point for asset allocation and risk management is a correlation analysis of returns. In standard risk-return models the price data are differenced before the analysis is even begun, and differencing removes a priori any long-term trends in the data. Of course these trends are implicit in the returns data, but when standard risk-return models are used, it is not possible to base any investment decision on the presence of established common trends in the data. The fundamental aim of cointegration analysis, on the other hand, is to detect any common stochastic trends in the price data, and to use these common trends for a dynamic analysis of correlation in returns. Thus cointegration analysis is an extension of the simple correlation-based risk-return analysis that was described in Chapter 7.

Correlation is based only on return data, but a full cointegration analysis is based on the raw price, rate or yield data as well as the return data. Price, rate and yield data are not normally stationary, in fact they are usually integrated of order 1 (denoted 1(1), see §11.1.3). And since it is normally the case that log prices will be cointegrated when the actual prices are cointegrated it is standard, but not necessary, to perform the cointegration analysis on log prices.1

When standard risk-return models are used, it is not possible to base any investment decision on the presence of established common trends in the data

Also the error correction models that are described in §12.3 have a more natural interpretation when log prices are used.

12.1.1 Cointegration and Correlation

A set of 1(1) series are termed cointegrated if there is a linear combination of these series that is stationary. So in the case of just two integrated series:

x and are cointegrated if x, ~ 1(1) but there exists a such that x - ay ~ 1(0).

The definition of cointegration given in Engle and Granger (1987) is far more general than this, but the basic definition presented here is sufficient for the purposes of this chapter.

Cointegration and correlation are related but different concepts. High correlation does not imply high cointegration, and neither does high cointegration imply high correlation. In fact cointegrated series can have correlations that are quite low at times. For example, a large and diversified portfolio of stocks in an equity index, where allocations are determined by their weights in the index, should be cointegrated with the index (§12.5). Although the portfolio should move in line with the index in the long term, there will be periods when stocks that are not in the portfolio have exceptional price movements. Following this, the empirical correlations between the portfolio and the index may be rather low for a time. Another example where cointegration exists without high correlation is given in the next section.

Cointegration and correlation are related but different concepts. High correlation does not imply high cointegration, and neither does high cointegration imply high correlation

The converse also holds true: returns may be highly correlated without a high cointegration in prices. An example is given in Figure 12.1, with 8 years of daily data on US dollar spot exchange rates of the German Mark (DEM) and the Dutch guilder (NLG) from 1986 to 1992. Their returns are very highly correlated, in fact the unconditional correlation coefficient over the whole period is 0.9642. The rates themselves also appear to be moving together. The spread is very stable indeed, and in fact they appear to be cointegrated, which is highly unusual for two exchange rates (Alexander and Johnston, 1992, 1994).

But suppose that extremely small, low variance, daily incremental returns are added to the spread, to create the NLG plus series that is also shown in Figure 12.1. The NLG plus is clearly not cointegrated with the DEM. They are not tied together by a stationary spread, instead they are diverging more and more as time goes on. But the correlation between the returns to NLG plus and the DEM is still very high, at 0.9620.

Thus high correlations can easily occur when there is cointegration and when there is no cointegration. That is, correlation tells us nothing about the long-term behavioural relationship between two markets: they may or may not be moving together over long periods of time, and correlation is not an adequate tool for measuring this.