References—681

Assuming that the eit are not autocorrelated, the optimal GMM weighting matrix for the differenced specification is given by,

and where Zi contains a mixture of strictly exogenous and predetermined instruments. Note that this weighting matrix is the one used in the one-step Arellano-Bond estimator.

Given estimates of the residuals from the one-step estimator, we may replace the Hd weighting matrix with one estimated using computational forms familiar from White period covariance estimation:

This weighting matrix is the one used in the Arellano-Bond two-step estimator.

Lastly, we note that an alternative method of transforming the original equation to eliminate the individual effect involves computing orthogonal deviations (Arellano and Bover, 1995). We will not reproduce the details on here but do note that residuals transformed using orthogonal deviations have the property that the optimal first-stage weighting matrix for the transformed specification is simply the 2SLS weighting matrix:

References

Arellano, M. (1987). “Computing Robust Standard Errors for Within-groups Estimators,” Oxford Bulletin of Economics and Statistics, 49, 431-434.

Arellano, M., and S. R. Bond (1991). “Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations,” Review of Economic Studies, 58, 277–297.

Arellano, M., and O. Bover (1995). “Another Look at the Instrumental Variables Estimation of Error-com- ponents Models,” Journal of Econometrics, 68, 29–51.