[Paper Review] p-order rounded integer-valued autoregressive (RINAR(p)) process

Maher Kachour|ArXiv.org|Feb 10, 2009

Fault Detection and Control Systems12 references4 citations

TL;DR

This paper introduces the p-order rounded integer-valued autoregressive (RINAR(p)) process, a novel discrete-time model for integer-valued time series that uses rounding of linear combinations of past values plus noise. Unlike traditional INAR(p) models, RINAR(p) allows for arbitrary real-valued autoregressive coefficients, negative values in the series, and negative autocorrelations, while ensuring integer-valued forecasts via rounding. The key contribution is the proof of consistency for the least squares estimator under a suitable identifiability condition, validated through simulation and real data analysis.

ABSTRACT

An extension of the RINAR(1) process for modelling discrete-time dependent counting processes is considered. The model RINAR(p) investigated here is a direct and natural extension of the real AR(p) model. Compared to classical INAR(p) models based on the thinning operator, the new models have several advantages: simple innovation structure ; autoregressive coefficients with arbitrary signs ; possible negative values for time series ; possible negative values for the autocorrelation function. The conditions for the stationarity and ergodicity, of the RINAR(p) model, are given. For parameter estimation, we consider the least squares estimator and we prove its consistency under suitable identifiability condition. Simulation experiments as well as analysis of real data sets are carried out to assess the performance of the model.

Motivation & Objective

• To develop a more flexible integer-valued autoregressive model than existing INAR(p) models.
• To address limitations of INAR(p) models, including restricted coefficient ranges, complex innovation structures, and inability to model negative values or negative autocorrelations.
• To propose a model with a simple innovation structure driven solely by i.i.d. integer noise.
• To establish theoretical conditions for stationarity, ergodicity, and consistency of the least squares estimator.

Proposed method

• The RINAR(p) model is defined as Xt = ⟨∑j=1 to p αjXt−j + λ⟩ + εt, where ⟨·⟩ denotes rounding to the nearest integer.
• The innovation process εt is i.i.d. and centered, with integer support, ensuring a simple and interpretable error structure.
• The one-step-ahead predictor is inherently integer-valued due to the rounding operator.
• Least squares estimation is used for parameter inference, with a specialized algorithm developed to handle the discontinuity of the rounding operator.
• Identifiability is established by proving that the true parameter vector is the unique minimizer of the expected squared error under a specific condition on the fractional parts of the linear combination.
• Theoretical analysis includes proofs of stationarity, ergodicity, and consistency of the least squares estimator under regularity conditions.

Experimental results

Research questions

• RQ1 Can a rounding-based integer-valued autoregressive model achieve greater flexibility than classical INAR(p) models in terms of coefficient signs and autocorrelation structure?
• RQ2 Does the RINAR(p) model support negative time series values and negative autocorrelations, which classical INAR models cannot?
• RQ3 Is the least squares estimator consistent for the RINAR(p) model despite the discontinuity introduced by the rounding operator?
• RQ4 What conditions ensure the stationarity and ergodicity of the RINAR(p) process?
• RQ5 How does the model perform in finite samples, and can it be reliably estimated on real-world count data?

Key findings

• The RINAR(p) model allows autoregressive coefficients αj to take any real values, not just [0,1], enabling modeling of negative dependencies.
• The model supports time series with negative values and can produce negative autocorrelation functions, unlike classical INAR(p) models.
• The least squares estimator for the RINAR(p) model is consistent under a suitable identifiability condition, proven via continuity and compactness arguments.
• Simulation experiments with 500 replications of length 500 show that the parameter estimates ˆαj and ˆλ converge to their true values, with actual parameters θ0 = (3/25, 3/8, 1/5, -1/4, 5/2).
• The model fits real F¨urth data well, with forecast errors for the last 105 observations showing no systematic pattern, indicating good predictive performance.
• The rounding operator is shown to be a natural and practical choice for discrete time series, as it preserves integer-valued forecasts without requiring post-processing.