This article considers testing for independence in a time series of small counts within an Integer Autoregressive (INAR) model, taking a semiparametric approach that avoids any distributional assumption on the arrivals process of the model. The nature of the testing problem is shown to differ depending on whether or not the support of the arrivals distribution is the full set of natural numbers (as would be the case for Poisson or Negative Binomial distributions for example) or some strict subset of the natural numbers (such as for a Binomial or Uniform distribution). The theory for these two cases is studied separately.
For the case where the arrivals have support on the natural numbers, a new asymptotically efficient semiparametric test, the effective score (Neyman-Rao) test, is derived. The semiparametric Likelihood-Ratio, Wald and score tests are shown to be asymptotically equivalent to the effective score test, and hence also asymptotically efficient. Asymptotic relative efficiency calculations demonstrate that the semiparametric effective score test can provide substantial power advantages over the first order autocorrelation coefficient, which is most commonly applied in practice.
For the case where the arrivals have support that is a strict subset of the natural numbers, the theory is considerably altered because the support of the observations becomes different under the null and alternative hypotheses. The semiparametric Likelihood-Ratio, Wald and score tests become asymptotically degenerate in this case, while the effective score test remains valid. Remarkably, in this case the effective score test is also found to have power against local alternatives that shrink to the null at the rate T−1. In rare cases where the arrival support is partly or totally known, additional tests exploiting this information are considered.
Finite sample properties of the tests in these various cases demonstrate the semiparametric effective score test can provide substantial power advantages over the first order autocorrelation test implied by a parametric Poisson specification. The simulations also reveal situations in which the first order autocorrelation is preferable in finite samples, so a hybrid of the effective score and autocorrelation tests is proposed to capture most of the benefits of each test.