Convergence to Stochastic Integrals with Non-linear Integrands

In this paper we present a general result concerning the convergence to stochastic integrals with non-linear integrands. The key finding represents a generalization of Chan and Wei`s (1988) Theorem 2.4. and that of Ibragimov and Phillips` (2004) Theorem 8.2. This result is necessary for analysing the asymptotic properties of mis-specification tests, when applied to a unit root process, for which Wooldridge (1999) mentioned that the exiting results in the literature were not sufficient.