Two independent pivotal statistics that test location and misspecification and add-up to the Anderson-Rubin statistic

We extend the novel pivotal statistics for testing the parameters in the instrumental variables regression model. We show that these statistics result from a decomposition of the Anderson-Rubin statistic into two independent pivotal statistics. The first statistic is a score statistic that tests location and the second statistic tests misspecification. We obtain the conditional distribution of the likelihood ratio statistic that tests location in case of multiple parameters of interest. This conditional distribution is a weighted average of the distributions of the location and misspecification statistics. The weights depend on a statistic that tests the rank of a matrix. We construct a quasi likelihood ratio statistic that bounds the likelihood ratio statistic and that can be used in case of a non-Kronecker covariance matrix. When there is a single parameter of interest, the quasi likelihood ratio statistic is identical to the likelihood ratio statistic. Alongside we provide expressions for idenfication statistics that result when we evaluate the limit behavior of the different statistics when the value of the parameter of interest converges to infinity. All exact distribution results straightforwardly extend to limiting distributions, that do not depend on nuisance parameters, under mild conditions.