Checking Model Adequacy for Count Time Series by Using Pearson Residuals

Abstract

After having fitted a model to a given count time series, one has to check the adequacy of this model fit. The (standardized) Pearson residuals, being easy to compute and interpret, are a popular diagnostic approach for this purpose. But which types of model inadequacy might be uncovered by which statistics based on the Pearson residuals? In view of being able to apply such statistics in practice, it is also crucial to ask for the properties of these statistics under model adequacy. We look for answers to these questions by means of a comprehensive simulation study, which considers diverse types of count time series models and inadequacy scenarios. We illustrate our findings with two real-data examples about strikes in the U.S., and about corporate insolvencies in the districts of Rhineland–Palatinate. We conclude with a theoretical discussion of Pearson residuals.

Testing the dispersion structure of count time series using Pearson residuals

“…In addition, one has to be aware that the Pearson residuals based on ML estimators will behave differently (not necessarily worse) than those based on moment estimators in some situations. This was observed by Weiß et al (2019); see the last paragraph of their Sect. 3, where deviations in the dispersion structure also affected the mean and autocorrelation of the ML-based residuals.…”

“…But other types of estimators might be used as well for computing the Pearson residuals, e.g., maximum likelihood (ML) estimators as this was done in the simulation study by Weiß et al (2019). Then, however, it is not possible anymore to find closedform formulae for the asymptotics of (2), because both the ML estimators themselves as well as their asymptotic distribution can only be computed numerically.…”

“…This conclusion, however, has to be done with some caution. In a recent work, Weiß et al (2019) did a comprehensive simulation study and showed that statistics based on Pearson residuals offer a good potential for detecting certain misspecifications, but a decision for or against the actual candidate model is difficult because of considerable deviations from the above target values even under model adequacy. Comparing the distributions of statistics based on either R t (θ ) or R t (θ), they also showed that main parts of these deviations are caused by estimation uncertainty.…”

Control charts for health surveillance based on residuals of negative binomial regression

“…This research is oriented to the control of count variables in Health‐Care and Public‐Health Surveillance based on the residuals of a GLM where the response variable follows a negative binomial distribution. Another type of residual that has been less applied in the design of control charts is the Pearson residual, which is easier to calculate and interpret, they have a sample mean about 0, a sample variance about 1 and are not usually correlated over time when are obtained of well‐fitted models 17 . In this work, we will compare EWMA control charts for Pearson and deviance residuals and consider their studentized versions.…”

“…Another type of residual that has been less applied in the design of control charts is the Pearson residual, which is easier to calculate and interpret, they have a sample mean about 0, a sample variance about 1 and are not usually correlated over time when are obtained of well-fitted models. 17 In this work, we will compare EWMA control charts for Pearson and deviance residuals and consider their studentized versions.…”