A Mixture Model Based On Fractional Differencing Time Series

One of the most distinguished features of financial data is its property of long memory, which means that the data sequence is significantly autocorrelated even the time interval is large enough. We choose models possessing the property of long memory to describe this phenomenon. However, it is gradually unreasonable to merely modelling the aspect of long memory as the financial data develop. In the actual situation, even an economical process generally shows the property of long memory for the long term, it still possess the feature of short memory when we consider it in a short term sense, this is why we can no longer count on the long memory model. With this in mind, we come up with this mixture model which possesses the property of long memory as well as the quality of short memory.Here we say a time series possesses the property of long memory if its autocorrelation function decays hyperbolically. And a time series is called a short-memory model when having the geometric decaying ACFs. In this context, we use ARFIMA model for possess of long memory and AR model to describe short memory property, and we put forward this mixture model-ARFIMA model.We get the conditional distribution function of the new mixture model by summing the weighted conditional distribution functions of the AR model and the ARFIMA model.Then we discuss the stationarity of the new mixture model and estimate the parameters.The definition of the new mixture model is as follows, cumulative distribution function of the standard normal distribution and α1+α2=1,α1,α2>0.The polynomial (?)(B)=1-02iB,|021|<1,B is the backward operator,Byt=yt-1,∈k,tThe first-order stationarity conditions for the mixture model are given in theorem1.And the second-order stationarity conditions for the model is given in theorem2.Theorem1. A condition for the process{yt} following the mixture model to be station-Theorem2. Suppose that the process{yt} following the mixture model is first order stationary.A condition for the process to be second order stationary isAnd in the derivation of the two theorems we also got the mean and the variance of the mixture model.Estimating the parameters of the mixture model is a two-step process:firstly,replace the ARFIMA(1,d,0) model with the AR(1) model,then the mixture AR-ARFIMA model simpli-fies to the MAR model,the parameters of which will be easily estimated; secondly,estimate the fractional differencing parameter d.We estimate the parameter d by Hurst index for they have a relationship.In this contex-t,to estimate H we use the method of aggregation of variance.In the last part,we estimate the parameters of the mixture model in R.And the result shows that the estimation method is feasible.