Covariate Driven Binomial Autoregressive Processes

In recent years,the analysis and application of integer-valued time series data has become a popular research field.Among them,the integer-valued time series data defined on a finite range has attracted a lot of attention because of its practical background,for example,the number of seventeen countries in European Union which are experiencing inflation(see Wei? and Kim(2015)[1]).Modelling count data without upper limit has been studied intensively by researchers and many useful models have been proposed.For the count data with upper limit which has many practical application backgrounds,they have also been deeply studied.Among them,the binomial autoregressive(BAR(1))model,because of its simplicity and practicality,has a significant advantage in fitting this kind of data,and has become a more successful model for describing the count data with upper limit.It is because of its superiority,domestic and foreign scholars on the study of the model have achieved fruitful results.However,the model still has some limitations in explaining some practical phenomena,for example,the autoregressive coefficient in the model is non random,and it is a little inadequate to use a simple parameter to describe it Therefore,some scholars considered the influence of observations on the thinning probability into the model,and proposed a dependent BAR(1)model.This kind of model takes into account the influence of the internal factors of the system on the thinning probability,which greatly enhances the ability of BAR(1)model to explain many practical problems.However,the influence of external factors on the thinning probability is very common in practice,and the modeling of this kind of problem has not been studied.In this paper,we do the following research:take the influence of covariates on autoregressive coefficient into account,because the value of autoregressive coefficient is in the interval(0,1),we introduce Logistic regression into the model,so as to better describe the influence of covariates on autoregressive coefficient.Based on the above discussion,a covariates driven binomial autoregressive model is proposed and given by:(?) where n is a known positive integer,?t,?t?(0,1),Zt is a p-dimensional explanatory vari-able which can be observed,Zt and Xt-1 are independent,? and ? are two p-dimensional unknown parameter vectors.In this paper,the statistical properties of the model are studied,and the model parameters are estimated by using conditional least square estimation and conditional maximum likelihood estimation.Since the Logistic function is hard to handle,it is dif-ficult for us to find out the explicit expression for the estimators.Based on this,in the process of parameter estimation,the estimation equation is directly optimized to obtain the parameter estimation value At the same time,using numerical simulation method,the estimation effect of the two methods is compared by mean absolute deviation error and mean square error.Finally,we apply the new model and the classical BAR(1)model to a set of stock data,and the fitting results verify the effectiveness of the new model.