Higher order asymptotic theory is targeted on the development of an asymptotic expansion for the distribution function of a statistic of interest. The asymptotic inference procedures are commonly based on simple characteristics of the density function at or near a data point of interest. In particular, exponential models are useful to provide accurate approximations to general statistical models. Typically, to the third order the exponential approximation has three primary parameters, two corresponding to pure model type and one for the departure from an exponential model (termed a non-exponentiality term). Andrews, Fraser and Wong (2005) discovered that to the third order, the observed significance function does not depend on the non-exponential term for univariate models. This finding has remarkable statistical implications for inference concerning univariate models. However, it is not clear whether this property holds for multivariate models. In this paper we address this question, and explore the intrinsic discrepancy between univariate and multivariate models.