Weighted Least Squares in Integrated Galton--Watson Processes: Intercept Inference and Optimal Weights

In integrated Galton--Watson processes with immigration, Wei and Winnicki (1990) fitted weighted least squares (WLS) with weights $(1+X_{t-1})^{-1}$ and left open the asymptotic distribution of the resulting intercept estimator in the recurrent case. Lu (2026) bypasses this difficulty by proposing time-weighted WLS with weights $1/t$. While this estimator yields one Gaussian procedure valid across all regimes, its rate is only $\sqrt{\log n}$. We extend Wei--Winnicki's state-weighted WLS estimator to weights $(a+X_{t-1})^{-1}$ for any fixed positive $a$. We solve this distributional problem and show that the convergence rate is polynomial in $n$ under strict recurrence and $\log n$ at the boundary, where the limiting distribution is nonnormal. We also justify a common estimated-offset procedure across all three regimes. Simulations illustrate the finite-sample performance of the resulting inference and show that imposing the unit root substantially improves coverage, especially near the boundary. An application to Canadian flood-disaster counts shows that state-weighted drift estimates are substantially less sensitive to the sample's starting year than time-weighted estimates.