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Overview of Difference and System GMM

Aug 24, 2024

EV's Presentation on Difference GMM and System GMM

Overview

  • Goal: Estimate Difference GMM, choose between Difference GMM and System GMM, and interpret results.
  • Key Features of Estimators:
    • Large N and small T (number of groups > time periods)
    • Linear, autoregressive dependent variables
    • Endogenous regressors
    • Group-specific fixed effects (heterogeneity)
    • Heteroscedasticity (error variance) and serial correlation within groups

Difference GMM

  • Origin: Arellano and Bond (1991)
  • Method: Uses first differences to eliminate fixed effects and correct endogeneity using instrumental variables.
  • Weaknesses:
    • Differencing removes previous observations and time-invariant variables.
    • Example: Differencing a time-invariant variable results in zero.

System GMM

  • Origin: Arellano and Bover (1995), with improvements by Blundell and Bond (1998).
  • Method:
    • Introduces additional instruments to improve model efficiency.
    • Transforms instruments with orthogonal deviations instead of differencing.
    • Example: Uses average of future observations subtracted from current value.
  • Advantages:
    • More robust for unbalanced panels and missing observations.

Instrumental Variables

  • Internal Instruments: Lags of endogenous regressors.
  • Variable Types:
    • Exogenous: Uncorrelated with error term.
    • Predetermined: Correlated with past error term (e.g., lagged dependent variable).
    • Endogenous: Correlated with error term.

Model Estimation

  • Difference GMM:
    • Differencing removes fixed effects but leaves lagged error terms that correlate with lagged dependent variable.
    • Issues with biased estimates when instruments are weak and the model is persistent.
  • System GMM:
    • Uses a two-equation approach with more instruments for better parameter estimates.

Choosing Between Difference and System GMM

  • Rule of Thumb (Bond, 2001):
    • Estimate original model with OLS for upper bound coefficient.
    • Use fixed effects model for lower bound coefficient.
    • Compare Difference GMM result with fixed effect coefficient:
      • If greater, choose Difference GMM; it's correctly instrumented.
      • If less, choose System GMM due to downward bias in Difference GMM.

Example Estimation

  • Data: 247 groups, 6 years of data.
  • Variables:
    • Y: Dependent variable
    • X1, X2: Independent variables
  • Steps:
    1. Estimate with pooled OLS.
    2. Estimate with fixed effects.
    3. Estimate Difference GMM and check coefficients.

Conclusions

  • Differ GMM Results:
    • Coefficient greater than fixed effect indicates correct instrumentation.
    • Ariano-Bond test confirms no second-order serial correlation.
    • J-statistic supports validity of the model.
    • Persistence captured by lagged dependent variable with moderate degree.
    • X1 and X2 significant, positively impacting Y.

Next step: Example of System GMM where the coefficient of the lagged dependent variable with Difference GMM is less than with fixed effects estimation.

Full transcript