Nonlinear Dynamic Panel Data Model

When we observe a panel from some arbitrary starting date, the initial outcome y_i0 is correlated with the unobserved individual effect c_i because both reflect the unit's history before the sample begins. Treating y_i0 as exogenous ignores this correlation and biases the estimated state-dependence coefficient, usually upward. Wooldridge's (2005) solution conditions the distribution of c_i on y_i0 and time-averaged covariates to absorb this correlation.

Arellano-Bond GMM eliminates fixed effects by first-differencing and instruments lagged differences with deeper lags. This is valid for linear models. When the outcome is binary or count-valued, first-differencing destroys the nonlinear structure, and the transformed model no longer has a meaningful probability interpretation. The nonlinear model instead integrates out heterogeneity through a distributional assumption.

For logit, Chamberlain's conditional MLE eliminates the individual fixed effect exactly but discards units with no within-variation. For probit and most other nonlinear models there is no analogous sufficient statistic, and the incidental-parameters problem causes substantial finite-T bias. Correlated random-effects approaches (Mundlak-Chamberlain device with Wooldridge's initial-conditions extension) are generally preferred for small T.

At minimum T = 3 is required to separately identify state dependence, heterogeneity, and the initial-conditions correction. Longer panels (T >= 5) provide more precise estimates and allow richer dynamics such as longer lags or time-varying coefficients.

The raw coefficient on y_{i,t-1} in the latent index is not directly interpretable as a probability or count change. Compute the average partial effect of moving from y_{i,t-1}=0 to y_{i,t-1}=1, averaging over the sample distribution of covariates and estimated individual effects. This gives the average increase in the probability of the outcome attributable to true state dependence.