This paper provides approaches to estimating and testing a class of nonlinear autoregressive integrated (NLARI) models. The relative restoring coefficient controls the dynamic patterns and the stability of NLARI’s deterministic system. The critical points based on the relative restoring coefficient and the interval restrictions on the coefficients are used to identify: (i) if a time series variable is a linear or nonlinear unit root process and (ii) if deterministic structure of the variable has a stable fixed point or a stable or an unstable period cycle for the mean-removed data (variability data). The ordinary least squares (OLSs) estimators of the model parameters are asymptotically consistent under some assumptions. The asymptotic distributions of test statistics are derived and critical values in finite samples are presented. Monte Carlo experiments are carried out under small disturbance variance. Simulation results show that the test statistics have a good finite sample performance. This supports theoretical results and suggests that the assumptions to ensure asymptotic consistency of OLS estimators are met for small disturbance variance.

Brownian local time, hypothesis testing, linear regression in parameters, nonlinear univariate unit root process.