Arellano–Bond estimator
In econometrics, the Arellano–Bond estimator is a generalized method of moments estimator used to estimate dynamic panel data models. It was first proposed by Manuel Arellano and Stephen Bond in 1991.
Unlike static panel data models, dynamic panel data models include lagged levels of the dependent variable as regressors. Including a lagged dependent variable as a regressor violates strict exogeneity, because the lagged depdendent variable is necessarily correlated with the idiosyncratic error.
When the strict exogeneity assumption is violated, commonly used static panel data techniques such as fixed effects estimators are inconsistent, because these estimators require strict exogeneity.
Anderson and Hsiao (1981) first proposed a solution by utilising instrumental variables (IV) estimation. However, the Anderson–Hsiao estimator is asymptotically inefficient, as its asymptotic variance is higher than the Arellano–Bond estimator, which uses a similar set of instruments, but uses generalized method of moments estimation rather than instrumental variables estimation.
In the Arellano-Bond method, first difference of the regression equation are taken to eliminate the fixed effects. Then, deeper lags of the dependent variable are used as instruments for differenced lags of the dependent variable (which are endogenous).
In traditional panel data techniques, adding deeper lags of the dependent variable reduces the number of observations available. For example, if observations are available at T time periods, then after first differencing, only T-1 lags are useable. Then, if K lags of the dependent variable are used as instruments, only T-K-1 observations are useable in the regression. This creates a trade-off: adding more lags provides more instruments, but reduces the sample size. The Arellano-Bond method circumvents this problem.
Consider the static linear unobserved effects model for observations and time periods:
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