Difference GMM (Arellano-Bond Estimator)
First-Differenced Generalized Method of Moments Estimator · Also known as: Arellano-Bond estimator, AB-GMM, first-difference GMM, difference GMM estimator
Difference GMM, introduced by Arellano and Bond (1991), estimates dynamic panel data models by first-differencing the equation to remove fixed effects, then using lagged levels of the endogenous variables as GMM instruments. It is the standard approach when a lagged dependent variable or other endogenous regressors are present in a panel with many units and few time periods.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
+6 more
When to use it
Difference GMM is appropriate when the panel has a large number of cross-sectional units (N) and a short time dimension (T), the model contains a lagged dependent variable or other endogenous regressors, and there are no suitable external instruments. It is particularly valuable in micro-econometric settings such as firm-level, household-level, or country-level panels where individual heterogeneity is a concern. Do not use it when T is large relative to N — the instrument count can explode and instrument proliferation weakens the Sargan/Hansen test — or when variables are close to a unit root, which weakens the lagged-levels instruments (System GMM is then preferred). It is also not suitable for purely cross-sectional data or when the error term displays second-order serial correlation.
Strengths & limitations
- Eliminates time-invariant individual fixed effects without requiring the within-group transformation.
- Handles endogenous regressors and lagged dependent variables using internal (lagged-level) instruments, so no external instruments are required.
- Consistent under large N, small T asymptotics — the setting typical of micro-panel datasets.
- Provides formal post-estimation diagnostics (AR(2) test, Sargan/Hansen test) to assess model validity.
- Widely implemented in Stata (xtabond, xtabond2), R (plm, gmm), and Python, making it accessible.
- Instrument proliferation: using all available lags creates many moment conditions, which can overfit and weaken the Sargan/Hansen test — instruments should be collapsed or restricted.
- Biased in finite samples when the autoregressive parameter is large or when T grows, because lagged levels become weak instruments near a unit root.
- Less efficient than System GMM when the series are persistent, because lagged levels carry little information about current differences.
- First-differencing removes all time-invariant regressors, so their effects cannot be estimated.
Frequently asked
What is the difference between Difference GMM and System GMM?
Difference GMM uses only the first-differenced equation with lagged levels as instruments. System GMM additionally stacks the level equation with lagged differences as instruments, which improves efficiency when the series are persistent. System GMM is generally preferred for near-unit-root variables; Difference GMM suffices when persistence is moderate.
How many lags should I use as instruments?
Use as few lags as necessary to keep the instrument count well below the number of cross-sectional units N. A common guideline is to restrict instruments to a fixed lag window (e.g., lags 2 and 3 only) or to use the 'collapse' option in xtabond2. Check that the Hansen test p-value is neither too small (invalid instruments) nor suspiciously close to 1 (instrument proliferation).
What does the AR(2) test tell me?
The Arellano-Bond AR(2) test checks for second-order serial correlation in the first-differenced residuals. If AR(2) is significant it indicates that the original errors are serially correlated at lag 1, which invalidates the moment conditions built on lags 2 and earlier. The AR(1) test is expected to be significant (differencing induces first-order correlation) and is uninformative about instrument validity.
Can I use Difference GMM with a balanced panel?
Yes, but unbalanced panels are also handled. The estimator naturally accommodates missing observations because instruments are constructed observation by observation. However, severe gaps in the time dimension reduce the number of valid moment conditions and can weaken identification.
Why can I not recover the coefficients on time-invariant regressors?
First-differencing wipes out any variable that does not change over time within a unit, including time-invariant regressors. If those effects are of interest, System GMM (which preserves the level equation) or Hausman-Taylor IV estimation should be considered instead.
Sources
- Arellano, M., & Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies, 58(2), 277–297. DOI: 10.2307/2297968 ↗
- Roodman, D. (2009). How to do xtabond2: An introduction to difference and system GMM in Stata. Stata Journal, 9(1), 86–136. DOI: 10.1177/1536867X0900900106 ↗
How to cite this page
ScholarGate. (2026, June 3). First-Differenced Generalized Method of Moments Estimator. ScholarGate. https://scholargate.app/en/econometrics/difference-gmm
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Arellano-Bond GMM estimatorEconometrics↔ compare
- Dynamic Panel Data ModelEconometrics↔ compare
- Fixed Effects ModelEconometrics↔ compare
- Panel Data AnalysisEconometrics↔ compare
- Panel System GMMEconometrics↔ compare