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A log-linear model for non-stationary time series of counts

Anne Leucht, Michael H. Neumann

Abstract

We propose a new model for nonstationary integer-valued time series which is particularly suitable for data with a strong trend. In contrast to popular Poisson-INGARCH models, but in line with classical GARCH models, we propose to pick the conditional distributions from nearly scale invariant families where the mean absolute value and the standard deviation are of the same order of magnitude. As an important prerequisite for applications in statistics, we prove absolute regularity of the count process with exponentially decaying coefficients.

A log-linear model for non-stationary time series of counts

Abstract

We propose a new model for nonstationary integer-valued time series which is particularly suitable for data with a strong trend. In contrast to popular Poisson-INGARCH models, but in line with classical GARCH models, we propose to pick the conditional distributions from nearly scale invariant families where the mean absolute value and the standard deviation are of the same order of magnitude. As an important prerequisite for applications in statistics, we prove absolute regularity of the count process with exponentially decaying coefficients.
Paper Structure (7 sections, 11 theorems, 89 equations, 3 figures, 1 table)

This paper contains 7 sections, 11 theorems, 89 equations, 3 figures, 1 table.

Table of Contents

  1. Motivation and introduction of the model
  2. Absolute regularity of the count process
  3. An application in statistics
  4. Proof of Theorem \ref{['T1']}
  5. Proof of Proposition \ref{['P3.1']}
  6. Proof of Proposition \ref{['P3.2']}
  7. A few auxiliary results

Key Result

Theorem 2.1

Suppose that 1.1a, 1.1c, 1.3a, and 1.3b are fulfilled, and let $E|\ln(\sigma_0)|+E\ln^+(Y)<\infty$. We assume that the density $p\colon [0,\infty)\rightarrow [0,\infty)$ of $Y$ is In the former case we set $\gamma=\Gamma=1$ and in the latter $\gamma=\int_0^\infty \sup\{p(y)\colon\, y\geq x\}\, dx$ and $\Gamma=( 1+\int_0^\infty x\,|p'(x)|\,dx )/2$. Then the count process $(X_t)_{t\in{\mathbb N}_0}

Figures (3)

  • Figure 1: left: Monthly immigration numbers for the Netherlands with increasing trend and strongly increasing seasonality; right: daily COVID-19 infection numbers from Italy with explosive trend.
  • Figure 2: COVID-19 infection numbers with estimated trend curve.
  • Figure 3: Boxplots for sample sizes $n=200$ and $n=500$: left: $Y\sim$ half-normal with $EY=1$, right: $Y\sim Exp(1)$.

Theorems & Definitions (26)

  • Theorem 2.1
  • Remark 1
  • Remark 2
  • Proposition 3.1
  • Remark 3
  • Remark 4
  • Proposition 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 16 more