Semi-parametric independence testing in the INAR model

Abstract: 

This paper considers testing the null hypothesis of temporal independence in a first order integer autoregressive (INAR) model for count data in which the distribution of the disturbances (the arrivals process) is not parameterised. The effective score test is derived, along with the classical score, Wald and likelihood ratio tests. The properties of these tests are shown to depend on the structure of the support of the arrivals process. If this support consists of the set of non-negative integers then the semi-parametric likelihood is shown to have standard LAN properties, the asymptotic null distributions of the tests can be derived from normal distributions in the usual way, and the tests can be shown to be asymptotically efficient against local alternatives that approach the null at the usual rate T^{-1/2}, where T is the sample size. That is, it is possible to adapt to the unspecified arrivals distribution in this case. However, if the support of the arrivals distribution is finite and/or has "gaps", then the testing problem is shown to become non-standard, since the support of the observations becomes dependent on the parameter under test. In this case the effective score test is shown to retain its asymptotically normal null distribution and to have non-trivial power against local alternatives that approach the null at the rate T^{-1}. The classical score, Wald, and likelihood ratio tests are shown to be asymptotically degenerate in this non-standard case. Simulation evidence is provided to illustrate the power gains, relative to possibly misspecified parametric tests, available in using the semi-parametric effective score test.