COINTEGRATION AND VECTOR ERROR CORRECTION MODELS: A BRIEF REVIEW

Once a unit root has been confIrmed for a data series, the question arises as to whether there exists some long-run equilibrium relationship among the variables. This is referred to as co integration. The concept is based on the idea that, even though economic time­ series reveal non-stationary behaviour, an appropriate linear combination between variables could remove the common trend component. The resulting linear combination of the time-series variables will thus be stationary, which means the relevant time-series variables are cointegrated. Park and Phillips ( 1 988) and Sims et al. ( 1 990) argued

traditional VAR is inappropriate when variables are cointegrated, because estimating a

multivariate time-series model using differences of the time-series alone could result in serious misspecification, since important level terms will have been omitted (Engle and Granger, 1 987).

To overcome this misspecifIcation, analysis could be carried out by the cointegration technique. That is, once the variables included in the V AR model have been found to be cointegrated, the next step is to use an error correction model ( VECM). 5 The theoretical justifIcation for co integration can be found in EngIe and Granger ( 1 987). They show that a V AR with cointegrated variables need to incorporate an error correction term

5 The cointegrating model of V AR is a restricted version of traditional V A R and, as with bivariate cointegration, an error correction component is required in a Y AR containing cointegrated variables.

(ECT) and thus can be written as a VECM.6 This allows the estimated model to reflect long-run equilibrium constraints and permits flexibility in the short-run dynamics. The research on co integration tests has been developed in two main directions. Firstly, tests based on the residuals from a cointegration regression suggested by Engle and Granger ( 1 987).7 Secondly, the Engle and Granger single equation approach of co integration is subject to a shortcoming; which is overcome by the 10hansen ( 1 988, 1 992), 10hansen and l uselius ( 1 990) tests based on the system of equations utilising

V AR models. This approach provides a multivariate framework and allows for more

than one cointegration vector in the estimated model and thereby prevents any loss of efficiency. It is thus, considered superior to the Engle and Granger approach.

The 10hansen ( 1 988, 1 992) and 10hansen and luselius ( 1 990) approach commonly uses two tests to detennine the number of cointegration vectors: the 'trace' test and the 'maximum-eigenvalue' test. In the trace test, the null hypothesis is that the number of co integrating vectors is less than or equal to r, where r = 0, 1 , 2, 3 ... . . n., and in each

case, the null hypothesis is tested against the relevant alternative. In the maximum eigenvalue test, the alternative for r=0 is that r= 1 ; r= 1 is tested against the alternative of r=2 and so on. If there is any divergence of results between these two tests, reliance should be placed on the evidence produced by the maximum eigenvalue test, since the results of this test are more reliable in small samples (BaneIjee et aI., 1 986). In addition, a recent attempt by Haug ( 1 997) using the Monte Carlo Method for ten alternative tests for cointegration has also found that the 10hansen and luselius ( 1 990) maximum eigenvalue test has the overall least size distortions compared with the trace test.

6 See Zivot (2000) for econometric explanation of YARs transferred into VECM.

7 The Engle and Granger based cointegarion tests have been used widely in the literature. It has, however several shortcomings such as: if there are more than two variables in the model, there can be more than one coinetgrating vectors. That is, the variables in a model may feature as a part of several equilibrium relationships governing the joint evaluation of the variables. It is possible for up to (n-l ) linearly independent conintegration vectors to exist in a system with n variables. Assuming only one cointegrating vector, when there is more than one, leads to inefficiency in the sense that only a complex linear combination of all possible vectors can be obtained. When estimating a single-equation model, even i f there i s only 0 ne c ointegrating vector, e stimating a single e quation i s potentially i nefficient

because of the loss of information that results from inability of the model to treat all variables as potentially endogenous. Given that the number of cointegration vectors in a model is unknown, and given the need to allow all variables to be potentially endogenous, the Engle and Granger single equation approach to testing cointegration can give rise to misleading results.

If the maximum eigenvalue tests indicate the existence of more than one co integrating vector, then discrimination among these multiple long-run relationships gives rise to the issue of identification of cointegrating relationships. Imposing restrictions on the co integrating vectors motivated by economic theory does help in identifying the co integrating vectors. A likelihood ratio test can then be performed to check the validity of these identifying restrictions.

Once cointegaring relationships between relevant variables are identified, the next concern is how these variables adjust in response to a random shock. The short-run disequilibrium dynamics are an issue. The short-run dynamics of the model are studied by analysing how each variable in a cointegrated system responds or corrects itself to the residual or error from the cointegraing vector. This justifies the use of the EeT which picks up the speed of the adjustment of each variable in response to a deviation from the steady state equilibrium. A variable with a zero speed of adjustment is Granger non-causal i n determining t he s hort-run d ynamics 0 f t he other v ariables. T he p recise direction of Granger-causality can thus be detected by undertaking a likelihood ratio test to determine how significantly the ECT for each variable differs from zero.