Dynamic Panel GMM

Arellano–Bond and Blundell–Bond estimation for dynamic panels: a lagged dependent variable with short T, where the within estimator is biased downward by order 1/T (Nickell bias) and lagged levels or differences serve as instruments instead.

Specification

\[y_{it} = \gamma y_{i,t-1} + X_{it} \beta + \mu_i + \varepsilon_{it}\]

Variants

Variant	Instruments	Notes
`DifferenceGMM()`	Lagged levels for the differenced equation	Weak when `gamma` approaches 1
`SystemGMM()`	Difference equation stacked with the levels equation	More efficient; needs the extra initial-conditions assumption

Estimation

# Y_mat: N×T matrix, X_ten: N×T×K array
m = fit_dynamic_gmm(SystemGMM(), Y_mat, X_ten;
    max_lag = 4, collapsed = false, varnames = ["x1"])

m_diff = fit_dynamic_gmm(DifferenceGMM(), Y_mat, X_ten; max_lag = 3)

max_lag caps how deep the instrument matrix reaches. Left uncapped, the instrument count grows with T² and the Hansen test loses power against an over-fitted model; collapsed = true collapses the instrument set, trading a little efficiency for a far smaller instrument count.

Reading the diagnostics

m.hansen_j, m.hansen_df   # over-identification: a *small* p-value is bad
m.ar1_z, m.ar2_z          # serial correlation in the differenced residuals

AR(1) in differences is expected and not a problem. AR(2) is the one to watch: it indicates the lag structure is misspecified and that the instruments are not valid. m.vcov_windmeijer is the covariance matrix to report for the two-step estimator; the uncorrected two-step standard errors are biased downward in short panels.

See Dynamic Spatial Panels for the case that also carries a spatial lag.