Cointegration Research Paper

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Time-series data consist of multiple observations of firms, households, persons, or other entities over several time periods. Many economic time series have empirical distributions that are nonconstant over time, with changing means and variances, making these series nonstationary. Stochastic trends occur when there are persistent longterm movements in time series data; and such trends represent a major source of nonstationarity, causing variables to drift over time. Series with stochastic trends are thus called “integrated” or “unit-root” processes.

In 1982, Charles Nelson and Charles Plosser showed that, empirically, many macroeconomic time series appear to be integrated of order one [denoted I(1)]. Growth rates of these series do not tend to drift, which is consistent with the growth rates being nonintegrated, I.E., integrated of order zero [denoted I(0)]. Moreover, efficientmarket theories in economics and finance suggest that asset and commodity prices follow a random walk, which is the simplest I(1) process.

Cointegration occurs when a relationship ties together nonstationary economic time series such that a combination of those time series is I(0). The concepts of integration and cointegration are intrinsically statistical in nature. Cointegration formalizes, in statistical terms, the property of a long-run relation between integrated economic variables.

Some Economic Implications of Cointegration

While cointegration is a statistical concept, it has economic implications. For example, it plays important roles in five aspects of economics: (1) long-run relations, (2) agent optimization, (3) the problem of nonsense regressions, (4) equilibrium correction (or error correction) models (ECMs), and (5) economic forecasting.

First, cointegration embeds the economic notion of a long-run relationship between economic variables in a statistical model of those variables. If a long-run relation exists, then the variables are cointegrated.

Second, market forces or optimizing behavior often provide an economic rationale for cointegration. For instance, consumers’ expenditure and income may be cointegrated because of economic agents’ budget constraints or because of intertemporal optimization plans for lifetime saving.

Third, the statistical theory of unit-root processes aids inference about the empirical existence of cointegration. Econometric theory historically relied on the assumption of stationary data even though many observed economic time series were trending and nonstationary. Cointegration explicitly allows for nonstationarity, thus providing a sounder basis for empirical inference. Cointegration also clarifies the problem of nonsense regressions, in which intrinsically unrelated nonstationary time series are highly correlated with each other.

Fourth, cointegration implies, and is implied by, the existence of an equilibrium correction representation of the relevant variables. Cointegration thus solidifies the statistical and economic bases for the empirically successful class of equilibrium correction models, in which past disequilibria in levels have an effect on current changes in the variables. Through ECMs, cointegration provides a systematic framework for jointly analyzing short-run (e.g., cyclical) and long-run properties. This framework also resolves the debate on whether to model data in levels or in differences, with classical econometric models and George Box and Gwilym Jenkins’s time-series models both being special cases of ECMs.

Fifth, optimal forecasts of cointegrated variables are themselves cointegrated. Hence, the existence of cointegration may improve the long-term forecasting of economic time series.

History

The history of cointegration was examined by David Hendry in “The Nobel Memorial Prize for Clive W. J. Granger” (2004). Hendry and Mary Morgan, in The Foundations of Econometric Analysis (1995), highlight the following events in that history. In 1901, R. H. Hooker illustrated and analyzed the difficulties attendant to nonstationarity, which he viewed as “common trends.” In 1926, G. Udny Yule showed that I(1) and I(2) observations would generate “nonsense correlations”; for example, high correlations lacking causal explanation, such as between church marriages and mortality rates. In 1974, Clive Granger and Paul Newbold reemphasized the dangers of nonsense correlations, and Peter Phillips presented a formal analysis in 1986, which he updated in 1998. Klein’s great ratios of economics (e.g., of consumption to income) suggested that variables’ levels can be closely related. J. Denis Sargan established the link between static-equilibrium economic theory and ECMs in 1964. In the 1980s, Granger and Robert Engle developed cointegration analysis as such.

Model Specification

In 1987, Engle and Granger established an isomorphism between cointegration and ECMs. Cointegration entails, and is entailed by, an ECM, which explicitly embeds a steady-state solution for its variables, while also allowing them to deviate from that steady state in the short run. In a nonstochastic steady state, an equilibrium relation would typically be motivated by economic theory. Hence, economic hypotheses are testable in a cointegration framework. In empirical work, conditional ECMs have been popular and may be interpretable as agents’ contingent plans. Applications include wages and prices, consumers’ expenditure, and money demand.

Testing for Integration and Cointegration

Cointegration makes the economic concept of equilibrium operational; that is, data allow tests of whether a long-run relation holds. With suitable tests, asymptotically correct inferences can be obtained. In addition, spurious regressions can be detected and avoided, as can unbalanced regressions involving variables of different orders of integration.

Economic theory rarely specifies orders of integration for variables, so a practitioner must analyze the data for both integration and cointegration. While the presence of unit roots complicates inference because some associated limiting distributions are nonstandard, critical values have been tabulated for many common cases. David Dickey and Wayne Fuller calculated critical values of tests for unit roots in univariate processes, and many robust unit-root tests have subsequently been developed.

Numerous cointegration tests have also been designed. In 1987, Engle and Granger proposed a singleequation approach that is intuitive and easy to implement, though it includes nuisance parameters in inference and may lack power (see Hendry 1986). A test of cointegration is also feasible in the corresponding ECM (see Neil Ericsson and James MacKinnon 2002).

Søren Johansen has provided a system-based approach in which cointegration relations are estimated via maximum likelihood in a vector autoregression (VAR). Johansen’s test statistics for cointegration generalize the Dickey-Fuller statistic to the multivariate context. Several authors have tabulated critical values, which are also embodied in software such as Cats for Rats and PcGive. In Johansen’s framework, hypotheses about cointegration properties are also testable. For instance, testing the longrun homogeneity of money with respect to prices is equivalent to testing whether the logs of money and prices are cointegrated with a unit coefficient. Other hypotheses, such as weak exogeneity, can be tested in Johansen’s framework as well. Weak exogeneity is satisfied if the cointegrating vector entering the conditional model does not appear in the marginal model of the conditioning variables. Under weak exogeneity, inference on those parameters from the conditional model alone is without loss of information relative to inference in the complete system.

In summary, cointegration and equilibrium correction help us understand short-run and long-run properties of economic data, and they provide a framework for testing economic hypotheses about growth and fluctuations. At the outset of an empirical investigation, economic time series should be analyzed for integration and cointegration, and tests are readily available to do so. Such analyses can aid in the interpretation of subsequent results and may suggest possible modeling strategies and specifications that are consistent with the data, while also reducing the risk of spurious regressions.

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  18. Johansen, Søren. 1988. Statistical Analysis of Cointegration Vectors. Journal of Economic Dynamics and Control 12 (2/3): 231–254.
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  20. Klein, Lawrence R. 1953. A Textbook of Econometrics. Evanston, IL: Row, Peterson.
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