Quantifying the Advantages of Forward Orthogonal Deviations for Long Time Series

Abstract

Under suitable conditions, generalized method of moments (GMM) estimates can be computed using a comparatively fast computational technique: filtered two-stage least squares (2SLS). This fact is illustrated with a special case of filtered 2SLS—specifically, the forward orthogonal deviations (FOD) transformation. If a restriction on the instruments is satisfied, GMM based on the FOD transformation (FOD-GMM) is identical to GMM based on the more popular first-difference (FD) transformation (FD-GMM). However, the FOD transformation provides significant reductions in computing time when the length of the time series (T) is not small. If the instruments condition is not met, the FD and FOD transformations lead to different GMM estimators. In this case, the computational advantage of the FOD transformation over the FD transformation is not as dramatic. On the other hand, in this case, Monte Carlo evidence provided in the paper indicates that FOD-GMM has better sampling properties—smaller absolute bias and standard deviations. Moreover, if T is not small, the FOD-GMM estimator has better sampling properties than the FD-GMM estimator even when the latter estimator is based on the optimal weighting matrix. Hence, when T is not small, FOD-GMM dominates FD-GMM in terms of both computational efficiency and sampling performance.