Z-valued smooth transition GARCH models: Specification and testing

This paper introduces a new class of nonlinear models known as the -valued smooth transition
GARCH model, designed to accommodate -valued time series that display asymmetric,
nonlinear and highly persistent volatility. The paper outlines the maximum likelihood estimation
procedure and establishes its consistency and asymptotic normality of the estimated parameters.
Three types of tests are studied, including sup-type linearity test, score-based goodness-of-fit
test, and residual-based mixed portmanteau diagnostic checking test. The asymptotic properties
of these three test statistics are established. To address the computationally complex problems of
estimation, the parametrization of the smooth transition function and the optimization algorithm
for the estimation procedure in numerical simulations are discussed. The effectiveness of the
tests is demonstrated through numerical simulations, and crime and exchange rate data sets are
analyzed to showcase the superior performance of the proposed model.
Keywords: Diagnostic checking test; Discrete-valued model; GARCH