# Quantifying the Advantages of Forward Orthogonal Deviations for Long Time Series

- 8 Downloads

## 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.

## Keywords

Forward orthogonal demeaning Forward orthogonal deviations First differencing Computational complexity## Notes

## References

- Alvarez, J., & Arellano, M. (2003). The time series and cross-section asymptotics of dynamic panel data estimators.
*Econometrica*,*71*(4), 1121–1159.CrossRefGoogle Scholar - Arellano, M. (2003).
*Panel data econometrics*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Arellano, M., & Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations.
*The Review of Economic Studies*,*58*(2), 277–297.CrossRefGoogle Scholar - Arellano, M., & Bover, O. (1995). Another look at the instrumental variable estimation of error-components models.
*Journal of Econometrics*,*68*(1), 29–51.CrossRefGoogle Scholar - Hayakawa, K. (2009). First difference or forward orthogonal deviation- which transformation should be used in dynamic panel data models?: A simulation study.
*Economics Bulletin*,*29*, 2008–2017.Google Scholar - Hayakawa, K., & Nagata, S. (2016). On the behaviour of the GMM estimator in persistent dynamic panel data models with unrestricted initial conditions.
*Computational Statistics and Data Analysis*,*100*, 265–303.CrossRefGoogle Scholar - Hunger, R. (2007). Floating point operations in matrix-vector calculus (version 1.3). Munich: Technical University of Munich, Associate Institute for Signal Processing. https://mediatum.ub.tum.de/doc/625604. Accessed 5 Aug 2016.
- Keane, M. P., & Runkle, D. E. (1992). On the estimation of panel-data models with serial correlation when instruments are not strictly exogenous.
*Journal of Business & Economic Statistics*,*10*, 1–9.Google Scholar - Phillips, R. F. (2019). A numerical equivalence result for generalized method of moments.
*Economics Letters*,*179*, 13–15.CrossRefGoogle Scholar - Schmidt, P., Ahn, S. C., & Wyhowski, D. (1992). Comment.
*Journal of Business & Economic Statistics*,*10*, 10–14.Google Scholar - Strang, G. (2003).
*Introduction to linear algebra*. Wellesley: Wellesley-Cambridge Press.Google Scholar