Quantifying the Computational Advantage of Forward Orthogonal Deviations

TL;DR

This paper demonstrates that the forward orthogonal deviations (FOD) transformation offers a significant computational advantage over the first-difference (FD) transformation in one-step GMM estimations, especially as the number of time periods increases.

Contribution

It quantifies the computational complexity difference between FOD and FD transformations, showing FOD's superior efficiency for larger T in GMM estimations.

Findings

FOD transformation requires less computational work than FD when T is large.

FOD's computational complexity increases with T at rate T^4, while FD's increases at T^6.

Simulations show FOD-based calculations are orders of magnitude faster.

Abstract

Under suitable conditions, one-step generalized method of moments (GMM) based on the first-difference (FD) transformation is numerically equal to one-step GMM based on the forward orthogonal deviations (FOD) transformation. However, when the number of time periods ($T$) is not small, the FOD transformation requires less computational work. This paper shows that the computational complexity of the FD and FOD transformations increases with the number of individuals ($N$) linearly, but the computational complexity of the FOD transformation increases with $T$ at the rate $T^{4}$ increases, while the computational complexity of the FD transformation increases at the rate $T^{6}$ increases. Simulations illustrate that calculations exploiting the FOD transformation are performed orders of magnitude faster than those using the FD transformation. The results in the paper indicate that, when one-step GMM based on the FD and FOD transformations are the same, Monte Carlo experiments can be conducted much faster if the FOD version of the estimator is used.