Testing for ARCH in ARCH-in-Mean Model

An issue that arises in applications involving the ARCH-in-Mean (ARCH-M) model is whether or not the error variance is constant over time. A proper statistical formulation of this as a test of hypothesis presents two difficulties. First, the model does not satisfy the standard regularity conditions because a nuisance parameter becomes unidentified under the null hypothesis and consequently the usual tests, such as the Lagrange Multiplier test, and their distribution theory require modification.