Introduction

In the previous chapter we discussed methods of estimation in cointegration systems. In this chapter we shall discuss tests for cointegration. Corresponding to the single equation and multiple equation methods of estimation considered in the previous chapter, we have tests for cointegration in single equation and system frameworks. Also, corresponding to the tests for unit roots discussed in chapter 4 where we considered the case of unit root as null and stationarity (no unit root) as null, we have tests for cointegration with no cointegration as null and cointegration as null.

There is one additional factor to be considered with regard to tests for cointegration in systems of multiple equations. Here we are interested in finding how many cointegrating relationships there are among the variables. In other words, we are interested in testing hypotheses about the rank of the cointegration space. In the following sections we shall start with tests in a single equation framework and then consider system methods.

Single equation methods: residual-based tests

The residual-based tests were the earlier tests for cointegration and were discussed in Engle and Granger (1987). Consider the set of (k + 1) variables yt which are I(1). If there exists a vector θ such that θ′yt is I(0), then θ is the cointegrating vector. Since θ is determined only up to a multiplicative constant, we shall normalize the first variable in yt to have coefficient 1.