Anderson-Hsiao Instrumental Variables Estimator

First-differencing eliminates the unobserved individual fixed effect, which is correlated with all regressors including the lagged dependent variable. Without removing the fixed effect, OLS is inconsistent because the fixed effect induces correlation between the regressor and the composite error. Differencing solves this but creates a new endogeneity problem that IV then addresses.

Both are valid under no serial correlation, but the level instrument y_{i,t-2} is generally preferred when the series is stationary, as it tends to be more correlated with the endogenous differenced regressor. The differenced instrument Delta y_{i,t-2} may be preferred when levels are close to a unit root, though weak-instrument concerns arise in both cases near non-stationarity.

Anderson-Hsiao uses only one or two instruments (the second lag in levels or differences), whereas Arellano-Bond GMM stacks all available lags as instruments, exploiting additional moment conditions for efficiency gains. Anderson-Hsiao is consistent but inefficient relative to Arellano-Bond; it is simpler to implement and less prone to the instrument proliferation problem that can inflate GMM standard errors in small samples.