GMM for dynamic panel data

GMM (Generalized Method of Moments) for dynamic panels handles the case where a lagged dependent variable appears on the right-hand side ($Y_{t-1}$) and/or there are endogenous regressors. In that case FEM/REM are biased (the Nickell bias for small T); GMM uses internal instruments (lags of the variables themselves) to obtain consistent estimates. Two common variants: Difference GMM (Arellano–Bond, 1991) and System GMM (Arellano–Bover/Blundell–Bond, 1998).

In EcoLab, GMM belongs to the Panel Data group and generates reproducible Stata/R/Python code. See FEM and REM and Estimation & Modeling.

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When should you use dynamic GMM?

• The model is dynamic: it contains $Y_{t-1}$ (or several lags) among the regressors.
• The panel has large N, small T (many units, few periods) — typical of firm/country data.
• There are endogenous regressors correlated with the error, but no external instruments are available.
• You need to control for unobserved unit effects in a dynamic context.

System GMM is preferred when the series is close to a random walk (persistent) — then lags are weak instruments for Difference GMM.

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Model specification

$$Y_{it} = \alpha \, Y_{i,t-1} + \beta \, X_{it} + \mu_i + \varepsilon_{it}$$

• $Y_{i,t-1}$: the lagged dependent variable (source of dynamics and endogeneity with $\mu_i$).
• Difference GMM: take first differences to remove $\mu_i$, using level lags as instruments.
• System GMM: combine the difference and level equations, adding lagged differences as instruments for the level equation.

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Required assumptions and tests

1. AR(2) — Arellano–Bond: tests for second-order autocorrelation of the differenced residuals; must not be rejected (p > 0.05) for the instruments to be valid.
2. Hansen/Sargan: tests the validity of the instrument set (overidentifying restrictions); a p-value that is too high (≈1.00) signals instrument proliferation.
3. Number of instruments ≤ number of groups (N): keep the instrument count small (collapse/limit lags) to avoid weakening Hansen.
4. Distinguish exogenous / predetermined / endogenous variables when declaring them.

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Running in EcoLab

1. Data Collection module: prepare the dynamic panel (entity + time columns), ensuring enough lags.
2. Modeling module → Panel Data group → GMM (Arellano–Bond / Blundell–Bond).
3. Declare $Y$, the lag terms, classify variables (exogenous/predetermined/endogenous), and choose Difference or System GMM.
4. Run and read the Diagnostics tab: AR(1)/AR(2), Hansen, instrument count. Get the code from the Replication Code tab.

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Input / output example

Input (illustrative): a panel of 30 countries × 15 years; growth is the dependent variable; growth_lag, invest, open are regressors.

Output (format, illustrative figures — not real results):

	Coefficient	p-value
growth_lag	0.34***	0.000
invest	0.21**	0.018
AR(2) p-value	0.41	(not rejected — valid)
Hansen p-value	0.28	(instruments valid)
Instruments / groups	18 / 30	(safe)

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Replication code

Stata

* ===== System GMM — Arellano-Bover / Blundell-Bond =====
* Panel setting
xtset country_id year

* System GMM: two-step with Windmeijer-corrected robust SE
* gmm() declares the endogenous variable and its instrument lag range
* iv() declares the strictly exogenous variable
xtabond2 growth L.growth invest open, ///
    gmm(L.growth, lag(2 4)) iv(open) ///
    twostep robust

* --- Diagnostics ---
* AR(2) p-value > 0.05 → no second-order autocorrelation (valid)
* Hansen p-value: not too low (<0.05) nor too high (≈1.00)
* Number of instruments ≤ number of groups

* Difference GMM (Arellano-Bond) for comparison
xtabond2 growth L.growth invest open, ///
    gmm(L.growth, lag(2 4)) iv(open) ///
    twostep robust noleveleq

R

# ===== System GMM — Arellano-Bover / Blundell-Bond =====
library(plm)

# Prepare panel data frame
pdata <- pdata.frame(df, index = c("country_id", "year"))

# System GMM: two-step estimator
# Dependent: growth; lagged dep. var as instrument (lags 2–4)
gmm_sys <- pgmm(
  growth ~ lag(growth, 1) + invest | lag(growth, 2:4),
  data   = pdata,
  effect = "twoways",
  model  = "twosteps"
)

# Robust summary (Windmeijer-corrected SE)
summary(gmm_sys, robust = TRUE)

# --- Diagnostics ---
# AR(2) test and Sargan/Hansen test printed in summary
# Ensure: AR(2) p > 0.05, Hansen p not too low or ≈1.00

Python

# ===== System GMM — using pydynpd =====
# Install: pip install pydynpd
import pandas as pd
from pydynpd import regression

# df must contain: country_id, year, growth, invest, open
# pydynpd uses Arellano-Bond / Blundell-Bond internally

# System GMM command string (similar to Stata syntax)
command = (
    "growth L1.growth invest open | "
    "gmm(growth, 2 4) | "
    "iv(open) | "
    "twostep nolevel"  # remove 'nolevel' for System GMM
)

# For System GMM (both level + difference equations):
command_sys = (
    "growth L1.growth invest open | "
    "gmm(growth, 2 4) | "
    "iv(open) | "
    "twostep"
)

result = regression.abond(command_sys, df, ["country_id", "year"])

# result prints: coefficients, AR(1)/AR(2), Hansen test
# Verify: AR(2) p > 0.05, Hansen p in a reasonable range

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Limitations and notes

• Instrument proliferation invalidates Hansen; always report the instrument count and use collapse/limited lags.
• Difference GMM is weak with highly persistent series → consider System GMM.
• Requires a large enough N; with small N, standard errors are unreliable (use the Windmeijer correction).
• Not appropriate when T is large (consider other dynamic-panel estimators).

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Video tutorial

Video Tutorial: Guide to running GMM (System/Difference) in EcoLab

See also

• FEM and REM · ARDL Model
• Worked example: Public debt and growth (panel data)
• Estimation & Econometric Modeling