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We propose a two-step local polynomial and a weighted bootstrapped estimator for the parameter functions of a time varying GARCH( p , q ) model. We also suggest a test statistic for testing the constancy of parameter functions of the model. Asymptotic distributions of the estimators and a test statistic are derived. The validity of the bootstrapped estimator and the test is established with the help of a simulation study.

Even though integer-valued time series are common in practice, the methods for their analysis have been developed only in recent past. Several models for stationary processes with discrete marginal distributions have been proposed in the literature. Such processes assume the parameters of the model to remain constant throughout the time period. However, this need not be true in practice. In this paper, we introduce non-stationary integer-valued autoregressive (INAR) models with structural breaks to model a situation, where the parameters of the INAR process do not remain constant over time. Such models are useful while modelling count data time series with structural breaks. The Bayesian and Markov Chain Monte Carlo (MCMC) procedures for the estimation of the parameters and break points of such models are discussed. We illustrate the model and estimation procedure with the help of a simulation study. The proposed model is applied to the two real biometrical data sets.

In this paper, a non-stationary time-varying GARCH (tvGARCH) model has been introduced by allowing the parameters of a stationary GARCH model to vary as functions of time. It is shown that the tvGARCH process is locally stationary in the sense that it can be locally approximated by stationary GARCH processes at fixed time points. We develop a two-step local polynomial procedure for the estimation of the parameter functions of the proposed model. Several asymptotic properties of the estimators have been established, including the asymptotic optimality. It is found that the tvGARCH model performs better than many of the standard GARCH models for various real data sets.

Summary This paper develops a new approach for order selection in autoregressive moving average models using the focused information criterion. This criterion minimizes the asymptotic mean squared error of the estimator of a parameter of interest. Simulation studies indicate that the suggested criterion is quite effective and comparable to the Akaike information criterion, the corrected Akaike information criterion and the Bayesian information criterion in autoregressive moving average order selection. The use of the focused information criterion for the simultaneous selection of regression variables and order of the error process in a linear regression model with autoregressive moving average errors is also considered.

In this paper, we consider a general family of asymmetric volatility models with stationary and ergodic coefficients. This family can nest several non-linear asymmetric GARCH models with stochastic parameters into its ambit. It also generalizes Markov-switching GARCH and GJR models. The geometric ergodicity of the proposed process is established. Sufficient conditions for stationarity and existence of moments have also been investigated. Geometric ergodicity of various volatility models with stochastic parameters has been discussed as special cases.