Optimal Estimation of Cointegrated Systems with Irrelevant Instruments
It has been know 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, 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 shown to be 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 altogether. Correspondingly, the approach developed here has many potential applications beyond conventional cointegrating regression, such as the estimation of long memory and fractional cointegrating relationships.
Phillips, Peter C.B., "Optimal Estimation of Cointegrated Systems with Irrelevant Instruments" (2006). Cowles Foundation Discussion Papers. 1835.