Co-integration: a review

Abstract

Many nonstationary univariate time series can be made stationary by appropriate differencing before ARMA models are fitted to the differenced series. However, when it comes to nonstationary vector time series, the situation is more complex. Since the dynamic of a multivariate time series is multidimensional, even if we can make each component stationary by appropriate differencing, the vector process of the differenced components may be still nonstationary. However, it is possible that the projections of a nonstationary vector time series in some directions may result in a stationary process. Engle and Granger(1987) formally demonstrated that it is possible for some linear combinations of the components of nonstationary vector time series to be stationary. They called this phenomenon Co-Integration. This concept of cointegration turned out to be extremely important in the modeling and analysis of non-stationary time series in economics. Although economic variables individually may exhibit disequilibrium behaviors, often time, due to economic forces, these disequilibrium economic variables corporately form a dynamic equilibrium relationship. Specifically, certain linear combinations of nonstationary time series may appear to be stationary. Engle and Granger developed statistical method for detecting and estimating this equilibrium relationship. They also proposed the so called error correction model to model Co-Integrated vector time series. In this report, I give a detail review on the concept of cointegration, the 2-step estimation procedure for the error correction models, and the 7 types of tests for testing cointegration. Since the test statistics for testing cointegration do not follow any known distribution, critical values were obtained based on two models by Engle and Granger. Augmented Dickey-Fuller and Dickey-Fuller tests were recommended as it is believed that their distributions are independent of the under lying process model. The critical values table presented in their paper is widely used in testing cointegration. In this report, we'll construct tables of critical values based on different models and compare them with those obtained by Engle and Granger. Also, to demonstrate the practical usage of cointegration, applications to currency exchange rates and US stock and Asian stock indexes are presented as illustrative examples.