This paper presents results on the issue of weak convergence of a
TVP-GQARCH-M(1,1) process. These suggest that the weak limit of this
endogenous volatility model is an exogenous (stochastic) volatility
continuous time process. Under the appropriate assumptions, we derive
the invariant distributions at which the process converges in various
cases. They reveal an approximate distributional relation between the
GARCH or the asymmetric GQARCH variance processes and the continuous
time Ornstein–Uhlenbeck models with respect to appropriate
nonnegative Levy processes.I am grateful
to Ritsa Panagiotou, Phyllis Alexander, Antonis Demos, Bruce Hansen,
Nicholas Magginas, Nour Meddahi, Enrique Sentana, Paolo Zaffaroni, two
anonymous referees, and the seminar participants at the Department of
International and European Economic Studies for their valuable
comments.