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Estimation of the intercept parameter in integrated Galton-Watson processes

Yang Lu

TL;DR

This paper tackles the estimation of the intercept parameter $\mu$ in integrated Galton-Watson count processes with a unit root. It introduces a novel $1/t$-weighted WLS estimator, proving its consistency across transient and null-recurrent regimes and establishing a $\sqrt{\ln n}$ convergence rate for $\tilde{\mu}$ via a CIR diffusion limit for the rescaled process. The work also analyzes its relationship to OLS and the prior $1/(1+X_{t-1})$-weighted WLS, presents simulation evidence of improved finite-sample performance, and discusses uniform inference and a practical KPSS-based procedure to distinguish stationarity from unit roots. The results enable robust inference for nonstationary count-valued time series and suggest potential extensions to related models such as INARCH and INAR processes.

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 $\sqrt{\ln n}$.

Estimation of the intercept parameter in integrated Galton-Watson processes

TL;DR

This paper tackles the estimation of the intercept parameter in integrated Galton-Watson count processes with a unit root. It introduces a novel -weighted WLS estimator, proving its consistency across transient and null-recurrent regimes and establishing a convergence rate for via a CIR diffusion limit for the rescaled process. The work also analyzes its relationship to OLS and the prior -weighted WLS, presents simulation evidence of improved finite-sample performance, and discusses uniform inference and a practical KPSS-based procedure to distinguish stationarity from unit roots. The results enable robust inference for nonstationary count-valued time series and suggest potential extensions to related models such as INARCH and INAR processes.

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 , and show that it is consistent regardless of whether the process is transient or null recurrent, with a convergence rate of .
Paper Structure (11 sections, 4 theorems, 39 equations, 1 figure, 1 table)

This paper contains 11 sections, 4 theorems, 39 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The estimator
  3. Discussions
  4. Comparison with the OLS estimator
  5. A simulation experiment
  6. Uniform inference?
  7. Stationarity and unit root tests
  8. Conclusion
  9. Proof of Theorem 1
  10. Proof of Theorem 2
  11. Proof of Theorem 3

Key Result

Theorem 1

As $n$ increases to infinity,

Figures (1)

  • Figure 1: Comparison of the sample distributions of the OLS estimator $\hat{\mu}$ (left panel) and the $1/t$-WLS estimator (right panel)

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4