It was proposed in 1991 by Manuel Arellano and Stephen Bond,[1] based on the earlier work by Alok Bhargava and John Denis Sargan in 1983, for addressing certain endogeneity problems.
[2] The GMM-SYS estimator is a system that contains both the levels and the first difference equations.
It provides an alternative to the standard first difference GMM estimator.
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 dependent variable is likely to be correlated with the random effects and/or the general errors.
[2] The Bhargava-Sargan article developed optimal linear combinations of predetermined variables from different time periods, provided sufficient conditions for identification of model parameters using restrictions across time periods, and developed tests for exogeneity for a subset of the variables.
When the exogeneity assumptions are violated and correlation pattern between time varying variables and errors may be complicated, commonly used static panel data techniques such as fixed effects estimators are likely to produce inconsistent estimators because they require certain strict exogeneity assumptions.
Anderson and Hsiao (1981) first proposed a solution by utilising instrumental variables (IV) estimation.
In the Arellano–Bond method, first difference of the regression equation are taken to eliminate the individual effects.
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 usable.
Then, if K lags of the dependent variable are used as instruments, only T-K-1 observations are usable in the regression.
This creates a trade-off: adding more lags provides more instruments, but reduces the sample size.
are innate ability for individuals or historical and institutional factors for countries.
Unlike a static panel data model, a dynamic panel model also contains lags of the dependent variable as regressors, accounting for concepts such as momentum and inertia.
Taking the first difference of this equation to eliminate the individual effect, Note that if
had a time varying coefficient, then differencing the equation will not remove the individual effect.
This equation can be re-written as, Applying the formula for the Efficient Generalized Method of Moments Estimator, which is, where
The original Anderson and Hsiao (1981) IV estimator uses the following moment conditions: Using the single instrument
, these moment conditions form the basis for the instrument matrix
: Note: The first possible observation is t = 2 due to the first difference transformation The instrument
The Arellano-bond estimator addresses this trade-off by using time-specific instruments.
now becomes: Note that the number of moments is increasing in the time period: this is how the efficiency - sample size tradeoff is avoided.
Time periods further in the future have more lags available to use as instruments.
If serial correlation is present, then the Arellano–Bond estimator can still be used under some circumstances, but deeper lags will be required.
is close to being a random walk, then the Arellano–Bond estimator may perform very poorly in finite samples.
This is because the lagged dependent variables will be weak instruments in these circumstances.
[4] These additional moment conditions can be used to improve the small sample performance of the Arellano–Bond estimator.
In this case, the full set of moment conditions can be written:
Note that the consistency and efficiency of the estimator depends on validity of the assumption that the errors can be decomposed as in equation (1).