Estimation of the intercept parameter in integrated Galton-Watson processes

TL;DR

This paper introduces a new weighted least squares estimator for the intercept in integrated Galton-Watson processes, which is consistent across different process regimes and converges at a rate of sqrt(ln n).

Contribution

It proposes a novel WLS estimator with a $1/t$ weight function that is consistent regardless of process recurrence, improving estimation robustness.

Findings

The new estimator is consistent for all process types.

Convergence rate of the estimator is √ln n.

The method outperforms existing estimators in null recurrent cases.

Abstract

We study estimation of the intercept parameter in an integrated Galton-Watson process, a basic building-block for many count-valued time series models. In this unit root setting, the ordinary least squares estimator is inconsistent, whereas an existing weighted least squares (WLS) estimator is consistent only in the case where the process is transient, a condition that depends on the unknown intercept parameter . We propose an alternative WLS estimator based on the new weight function of $1/t$, and show that it is consistent regardless of whether the process is transient or null recurrent, with a convergence rate of $\sqrt{\ln n}$.