Time Series Model Of Likelihood Method

Data gathered sequentially in time are called a time series, which demon-strate the development patterns and trends of research process. The developing rule is helpful to our understanding of the development and can provide a theo-retical basis for the prediction, thus the time series models are studied more and more, and have already formed some perfect methods.The idea of penalization is very useful in statistical modeling, particularly in variable selection, which is fundamental to the field. Penalized likelihood is a useful variable selection method which will be applied to the time series analysis in this paper. We have established good property and the asymptotic normality of penalized likelihood estimator. In this paper, we discusses two kinds of time series model:AR(p) model and AR.MA(p, q) model, and will give the consistency of penalized likelihood estimator.We use the adaptive Lasso method for AR(p). The adaptive Lasso is a tech-nique for simultaneous estimation and variable selection and enjoys the oracle properties. The adaptive Lasso is that adaptive weights are used for penalizing different coefficients. The weights are data-dependent:big weights for small co-efficients and small weights for big coefficients. Some small coefficients will be set to0. We show that, adaptive Lasso method can determine order automati-cally; meanwhile nonzero parameters are identified and estimated.For ARMA(p, q) model, We have two steps to do:firstly, we convert it to AR(∞) model and estimate coefficients of AR(∞) model with SCAD penalty; secondly, determine order of ARMA(p,q) model according to coeffi-cients estimation of AR(oo) model, and then gain ARMA(p, q) model coefficients estimation by calculation.