# Cointegration

Juan J. Dolado, Jesus Gonzalo,

and Francesc Marmol

A substantial part of economic theory generally deals with long-run equilibrium relationships generated by market forces and behavioral rules. Correspondingly, most empirical econometric studies entailing time series can be interpreted as attempts to evaluate such relationships in a dynamic framework.

At one time, conventional wisdom was that in order to apply standard inference procedures in such studies, the variables in the system needed to be stationary since the vast majority of econometric theory is built upon the assumption of stationarity. Consequently, for many years econometricians proceeded as if stationarity could be achieved by simply removing deterministic components (e. g. drifts and trends) from the data. However, stationary series should at least have constant unconditional mean and variance over time, a condition which hardly appears to be satisfied in economics, even after removing those deterministic terms.

Those problems were somehow ignored in applied work until important papers by Granger and Newbold (1974) and Nelson and Plosser (1982) alerted many to the econometric implications of nonstationarity and the dangers of running nonsense or spurious regressions; see, e. g. Chapter 26 by Granger in this volume for further details. In particular, most of the attention focused on the implications of dealing with integrated variables which are a specific class of nonstationary variables with important economic and statistical properties. These are derived from the presence of unit roots which give rise to stochastic trends, as opposed to pure deterministic trends, with innovations to an integrated process being permanent rather than transitory.

The presence of, at least, a unit root in economic time series is implied in many economic models. Among them, there are those based on the rational use of available information or the existence of very high adjustment costs in some

markets. Interesting examples include future contracts, stock prices, yield curves, exchange rates, money velocity, hysteresis theories of unemployment, and, perhaps the most popular, the implications of the permanent income hypothesis for real consumption under rational expectations.

Statisticians, in turn, following the influential approach by Box and Jenkins (1970), had advocated transforming integrated time series into stationary ones by successive differencing of the series before modelization. Therefore, from their viewpoint, removing unit roots through differencing ought to be a prerequisite for regression analysis. However, some authors, notably Sargan (1964), Hendry and Mizon (1978) and Davidson et al. (1978), inter alia, started to criticize on a number of grounds the specification of dynamic models in terms of differenced variables only, especially because of the difficulties in inferring the long-run equilibrium from the estimated model. After all, if deviations from that equilibrium relationship affect future changes in a set of variables, omitting the former, i. e. estimating a differenced model, should entail a misspecification error. However, for some time it remained to be well understood how both variables in differences and levels could coexist in regression models.

Granger (1981), resting upon the previous ideas, solved the puzzle by pointing out that a vector of variables, all of which achieve stationarity after differencing, could have linear combinations which are stationary in levels. Later, Granger (1986) and Engle and Granger (1987) were the first to formalize the idea of integrated variables sharing an equilibrium relation which turned out to be either stationary or have a lower degree of integration than the original series. They denoted this property by cointegration, signifying comovements among trending variables which could be exploited to test for the existence of equilibrium relationships within a fully dynamic specification framework. Notice that the notion of "equilibrium" used here is that of a state to which a dynamic system tends to converge over time after any of the variables in the system is perturbed by a shock. In economics, the strength of attraction to such a state depends on the actions of a market or on government intervention. In this sense, the basic concept of cointegration applies in a variety of economic models including the relationships between capital and output, real wages and labor productivity, nominal exchange rates and relative prices, consumption and disposable income, long – and short-term interest rates, money velocity and interest rates, price of shares and dividends, production and sales, etc. In particular, Campbell and Shiller (1987) have pointed out that a pair of integrated variables that are related through a present value model, as it is often the case in macroeconomics and finance, must be cointegrated.

In view of the strength of these ideas, a burgeoning literature on cointegration has developed over the last decade. In this chapter we will explore the basic conceptual issues and discuss related econometric techniques, with the aim of offering an introductory coverage of the main developments in this new field of research. Section 2 provides some preliminaries on the implications of cointegration and the basic estimation and testing procedures in a single equation framework, when variables have a single unit root. In Section 3, we extend the previous techniques to more general multivariate setups, introducing those system-based approaches to cointegration which are now in common use. Section 4, in turn, presents some interesting developments on which the recent research on cointegration has been focusing. Finally, Section 5 draws some concluding remarks.

Nowadays, the interested reader, who wants to deepen beyond the introductory level offered here, could find a number of textbooks (e. g. Banerjee et al., 1993; Johansen, 1995; Hatanaka, 1996; Maddala and Kim, 1998) and surveys (e. g. Engle and Granger, 1991; Watson, 1994) on cointegration where more general treatments of the relevant issues covered in this chapter are presented. Likewise, there are now many software packages that support the techniques discussed here (e. g. Gauss-COINT, E-VIEWS and PC-FIML).

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