Pretesting Count Time Series With Application to Epidemiology: A Fast-Double Bootstrap Approach

Abstract

INTRODUCTION
Integer Autoregressive Models (INAR) have demonstrated their versatility in applications involving epidemiological count data. Among the possible model specifications, the equidispersed first-order Poisson INAR (1) represents a straightforward and parsimonious solution for practitioners due to its simplicity of use and availability of bootstrap-based inference. However, two preliminary checks are relevant: testing for the presence of an INAR effect and assessing whether the arrival process can reasonably be modeled as equidispersed Poisson, which simplifies both estimation and forecasting.
OBJECTIVES
This work proposes a novel bootstrap procedure for testing the presence of an INAR(1) effect in count time series. The procedure adapts the fast-double bootstrap algorithm proposed in Davidson, et al by introducing an additional loop within each bootstrap replication. In addition, it provides diagnostic information on the equidispersion of the arrival process by extending the approach of Palazzo et al.
METHODS
To test the presence of the INAR effect, both the semi-parametric and the parametric bootstrap-based score statistics can be redesigned for the fast-double bootstrap algorithm with and without specifying a given distribution for the disturbances. Then, by exploiting the different behavior of the parametric and semi-parametric bootstrap distribution in the specific case of the Poisson score statistic under model misspecification, a procedure is implemented to diagnose the non-equidispersion of the disturbances. To show the applicability of the procedure, a small-scale Monte Carlo simulation is implemented by employing four data-generating processes for the parametric arrivals to capture under-/over-dispersion and zero-inflation. Then, freely available surveillance low-count data regarding infectious diseases have been considered for an empirical assessment of the procedure. One of the data sources regards the weekly number of syphilis cases (dataset: syph) in the United States (51 states, T=209).
RESULTS
Simulation analysis confirms that the fast-double bootstrap algorithm can provide non-negligible improvements in terms of empirical size (under the null hypothesis of no INAR effect) and power (presence of INAR effect) due to the pivotality of both the bootstrapped and the asymptotic score statistics in many cases, especially when the time series are moderately small (T=50, 75). Moreover, the procedure fails when the parametric bootstrapped Poisson score statistic is used under model misspecification (presence of under-/over-dispersion or zero-inflation) due to the non-pivotality issue. In this sense, a further comparison between bootstrapped parametric and semi-parametric score statistics can represent a diagnostic tool to empirically verify equi-dispersion in the data. In terms of empirical applications, the syph data do not always show evidence of an INAR effect, whereas the diagnostic tool suggests avoiding the use of Poisson INAR for the majority of the series investigated. After the pretests, a summary of the applications can be provided through ex-post bootstrap-based confidence intervals.
CONCLUSIONS
The proposed fast-double bootstrap procedure represents a computationally feasible method to pretest count data for the presence of a short-term mechanism. The diagnostic tool should be formalized in further research and also compared with the existing proposals in terms of performance achieved under different simulation scenarios.