Intertemporal Cointegration Model: A New Approach to the Lead–Lag Relationship Between Cointegrated Time Series

We generalize the single regression-type cointegration model of Engle and Granger (Econometrica 55(2):251–276, 1987) from the lead–lag relationship perspective. A leading series is introduced into the model as an independent variable to express the long-run relationship between the leading and lagging variables if the disturbance term is stationary. The theoretical analysis shows that the estimators of the coefficients with true lead–lag intervals have stochastic characteristics equivalent to those of Engle and Granger (1987). Monte Carlo simulations suggest that inappropriate interval selection leads to seriously biased estimators and that the identification of the lead–lag intervals is successful when using the adjusted coefficient of determination. This accuracy is caused by the difference in order between the true interval and candidates, which increases the variance of disturbance proportionally. The farther the candidates are from the true interval, the more autocorrelation increases; further, it cannot be absorbed by the unit root test, which considers the autocorrelations of disturbance. This causes a severe deterioration in the power of cointegration testing. Therefore, cointegration analysis without considering the lead–lag interval may lead economists to overlook the important long-run relationship between the pair of variables.