Martingale decomposition and approximations for nonlinearly dependent processes

This paper proposes a new martingale (MG) decomposition (Gordin, 1969; Hall and Heyde, 1980) for a dependent time series under a predictive dependence measure based on Wu (2005). The decomposition produces a generalized version of the Beveridge–Nelson (BN) lemma (Phillips and Solo, 1992) that accommodates many nonlinear time series, such as GARCH models and threshold autoregressive processes, thereby extending the empirical ambit of the original lemma designed for the linear process. Under this extended framework, MG approximations can be constructed for weighted sums of the nonlinear dependent processes and these approximations lead directly to a new central limit theorem whose range of application includes many practical time series models.