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Nearly Efficient Estimation of Time Series Models with Predetermined, but not Exogenous, Instruments

Fumio Hayashi
Christopher Sims

Abstract

Particularly under the assumption of rational expectations, a model may have serially correlated errors and those errors may be uncorrelated with contemporaneous and lagged values of a predetermined instrument, yet the instruments may not be strictly exogenous. This paper proposes a method for transforming such a model to one without serial correlation, while keeping the instrument predetermined. Standard theory of instrumental variables estimation then applies. Furthermore, it turns out that for transformations of the class proposed, asymptotic distribution theory is the same whether the serial correlation properties of the errors are known a priori or estimated. As the number of lagged values of the predetermined variables used as instruments increases, the asymptotic variance of the standard instrumental variables estimator applied to the transformed model approaches that of the optimal estimator proposed by Hansen and Sargent [8].