Optimal estimation of cointegrated systems with irrelevant instruments

It has been known since Phillips and Hansen (1990) that cointegrated systems can be consistently estimated using stochastic trend instruments that are independent of the system variables. A similar phenomenon occurs with deterministically trending instruments. The present work shows that such “irrelevant” deterministic trend instruments may be systematically used to produce asymptotically efficient estimates of a cointegrated system. The approach is convenient in practice, involves only linear instrumental variables estimation, and is a straightforward one step procedure with no loss of degrees of freedom in estimation. Simulations reveal that the procedure works well in practice both in terms of point and interval estimation, having little finite sample bias and less finite sample dispersion than other popular cointegrating regression procedures such as reduced rank VAR regression, fully modified least squares, and dynamic OLS. The procedure is a form of maximum likelihood estimation where the likelihood is constructed for data projected onto the trending instruments. This “trend likelihood” is related to the notion of the local Whittle likelihood but avoids frequency domain issues.