# Nearly cointegrated systems

Even when a vector of time series is I(1), the size of the unit root in each of the series could be very different. For example, in terms of the common trend representation of a bivariate system discussed above, it could well be the case that y1t = ф1 yC + +1t and y2t = ф2yC + +2t are such that ф1 is close to zero and that ф2 is large. Then y1t will not be different from +1t which is an I(0) series while y2t will be clearly I(1). The two series are cointegrated, since they share a common trend. However, if we regress y1t on y2t, i. e. we normalize the cointegrating vector on the coefficient of y1t, the regression will be nearly unbalanced, namely, the regressand is almost I(0) whilst the regressor is I(1). In this case, the estimated coefficient on y2t will converge quickly to zero and the residuals will resemble the properties of y1t, i. e. they will look stationary. Thus, according to the Engle and Granger testing approach, we will often reject the null of no cointegration. By contrast, if we regress y2t on y1t, now the residuals will resemble the I(1) properties of the regressand and we will often reject cointegration. Therefore, normalization plays a crucial role in least squares estimation of cointegrating vectors in nearly cointegrated systems. Consequently, if one uses the static regression approach to estimate the cointegrating vector, it follows from the previous discussion that it is better to use the "less integrated" variable as the regressand. Ng and Perron

(1997) have shown that these problems remain when the equations are estimated using more efficient methods like FM-OLS and DOLS, while the Johansen’s methodology provides a better estimation approach, since normalization is only imposed on the length of the eigenvectors.

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