Norming Rates and Limit Theory for Some Time-Varying Coefficient Autoregressions

A time-varying autoregression is considered with a similarity-based coefficient and possible drift. It is shown that the random walk model has a natural interpretation as the leading term in a small-sigma expansion of a similarity model with an exponential similarity function as its autoregressive coefficient. Consistency of the quasi-maximum likelihood estimator of the parameters in this model is established, the behaviors of the score and Hessian functions are analyzed and test statistics are suggested. A complete list is provided of the normalization rates required for the consistency proof and for the score and Hessian functions standardization. A large family of unit root models with stationary and explosive alternatives are characterized within the similarity class through the asymptotic negligibility of a certain quadratic form that appears in the score function. A variant of the stochastic unit root model within the class is studied and a large sample limit theory provided which leads to a new nonlinear diffusion process limit showing the form of the drift and conditional volatility induced by this model. Some simulations and a brief empirical application to data on an Australian Exchange Traded Fund are included.