Model Diagnostics for Poisson INARMA Processes Using Bivariate Dispersion Indexes

Abstract

Also in a time series context, the simple Poisson distribution is very popular for modeling the marginal distribution of the generated counts. If, in contrast, a more complex count distribution is appropriate, then it is important to have tests available to detect this deviation from Poissonity. A possible approach is to use Fisher’s index of dispersion as a test statistic. For several types of Poisson INARMA process, the generated counts $$X_t$$ do not only exhibit a marginal Poisson distribution, the lagged pairs $$(X_t,X_{t-k})$$ are even bivariately Poisson distributed. So to uncover violations of the Poisson null hypothesis within these INARMA processes, the application of a bivariate dispersion index appears to be a reasonable alternative to the simple univariate Fisher index. We survey several proposals for bivariate dispersion indexes and adapt them to a time series context. For the resulting modified indexes, we derive the asymptotic distribution under the null such that they can be used as test statistics. With simulations, we investigate the finite-sample performance of the asymptotic tests, and we analyze the question if the novel tests are advantageous compared to the simple univariate dispersion index test.