Deterministic exponential heteroskedasticity, a weakly stationary unit-root process and a useful diagnostic test

A specific form of deterministic exponential heteroskedasticity is examined. A non-trivial unit root process which has exponentially heteroskedastic innovation and as a consequence, a variance that vanishes asymptotically is detailed. Such a unit root stochastic process, with exponential heteroskedasticity, may be perceived as weakly stationary by the usual unit root tests. In view of the importance of deterministic exponential heteroskedasticity, a new general diagnostic test for detecting the presence of deterministic exponential heteroskedasticity is developed.