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 AR coefficient. Consistency of the quasi-maximum likelihood estimator of the parameters in this model is established, the behaviours of the score and Hessian functions are analysed 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 function standardization. A large family of unit root models with stationary and explosive alternatives is 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 sustained stochastic departures from unity. The findings provide a composite case for time-varying coefficient dynamic modelling. Some simulations and a brief empirical application to data on international Exchange Traded Funds are included. Copyright (c) 2014 Wiley Publishing Ltd
Autoregression, Consistency, Nonlinear diffusion, Non-stationarity, Similarity, Small-sigma approximation, Stochastic unit root, Time-varying coefficients
Journal of Time Series Analysis
Wiley: 12 months
LIEBERMAN, Offer and Peter C. B. PHILLIPS.
Norming rates and limit theory for some time-varying coefficient autoregressions. (2014). Journal of Time Series Analysis. 35, (6), 592-623. Research Collection School Of Economics.
Available at: http://ink.library.smu.edu.sg/soe_research/1835
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