Two-Step System GMM (Generalized Method of Moments)

 The Two-Step System GMM is a dynamic panel data estimation technique designed to address:

• Endogeneity of explanatory variables

• Unobserved heterogeneity (fixed effects)

• Autocorrelation and heteroskedasticity

It improves upon simpler estimators (like OLS or FE) and is particularly useful when the model includes lagged dependent variables.

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⚙️ System GMM: The Basics

Developed by Arellano & Bover (1995) and Blundell & Bond (1998), System GMM combines:

1. Difference GMM (Arellano & Bond, 1991):
   First-differences the model to remove fixed effects and uses lagged levels as instruments.

2. Level GMM (Blundell & Bond):
   Uses lagged differences as instruments for the level equation to improve efficiency.

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🧮 The Dynamic Panel Model

A typical dynamic panel data model is:

$$y_{it} = \alpha y_{it-1} + \beta X_{it} + \mu_i + \varepsilon_{it}$$

Where:

• $y_{it}$ is the dependent variable

• $y_{it-1}$ is the lagged dependent variable

• $X_{it}$ are explanatory variables

• $\mu_i$ = unobserved fixed effect

• $\varepsilon_{it}$ = idiosyncratic error

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🔁 Why “Two-Step”?

✅ First Step:

• Uses an initial weighting matrix assuming homoskedastic errors.

• Generates consistent but inefficient estimates of parameters and residuals.

✅ Second Step:

• Uses residuals from Step 1 to build a robust weighting matrix accounting for heteroskedasticity and autocorrelation.

• Produces efficient estimates and valid standard errors.

🔺 However, standard errors in Two-Step GMM tend to be downward biased, so Windmeijer-corrected standard errors are commonly used.

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🧰 Key Features and Assumptions

Feature	Description
Instruments	Lagged levels and lagged differences of variables
Endogeneity	Can instrument endogenous variables
Fixed Effects	Removed via differencing
Autocorrelation	Assumes no second-order serial correlation in errors
Heteroskedasticity	Robust in the second step

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📊 When to Use System GMM

• Panel data with small time dimension (T) and large cross-section (N)

• Presence of endogenous regressors

• Need to include lagged dependent variables

• Data with individual effects and potential measurement error

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🧾 Output Diagnostics

Test	Purpose
Hansen Test / Sargan Test	Validity of instruments (overidentifying restrictions)
Arellano-Bond AR(1), AR(2) Test	Checks for serial correlation in differenced residuals

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🔍 Comparison with Other Estimators

Estimator	Handles Endogeneity?	Lagged DV?	Efficiency
Pooled OLS	❌	❌	Low
Fixed Effects	❌	❌	Medium
Difference GMM	✅	✅	Medium
System GMM	✅✅	✅	✅✅ (especially in Two-Step)

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📌 Final Notes

• Overfitting with too many instruments can weaken GMM results. Rule of thumb: number of instruments < number of groups.

• Best suited for macro panels like bank performance, firm-level profitability, or investment behavior over time.