Randomness Test Of Coefficients For Several Autoregressive Models

In recent years,time series data of non-linear structure has attracted great atten-tion from statisticians.This kind of non-linear structure exists not only in classical time series data with continuous values,but also in count data with only non-negative integer values.For example,Canadian plutonium data,material corrosion data in o-ceanography,and the closing index of a stock exchange on each trading day throughout the year are typical real-value time series with a non-linear structure.On the other hand,global volcanoes The number of eruptions,the number of skin cancers in a hospi-tal,and the number of unemployed in a certain period of time are typical integer time series with a non-linear structure.In production and scientific research,linear models are used to fit such data.It is relatively rough and often cannot effectively reflect the objective laws of observation dataThis paper studies the test of coefficient randomness of several types of autore-gressive models.The main content is divided into three parts.In the first part,we consider the RCAR(1)and RCAR(p)model randomness test problems.First,using the local maximum power method,we construct the test statistics of RCAR(1)model parameters.Under the null hypothesis,it is proved that the limit distribution of the statistics is normal.In the process of numerical simulation,it is assumed that the error terms obey the normal distribution,the T distribution,and the mixed normal distribu-tion.The local maximum power method is compared by random simulation.The pros and cons of the empirical likelihood test under the above three distribution conditions The simulation results show that the local maximum power method has a better test result.The proposed test method is used to fit a set of economic data.Second,westudy the hypothesis testing problem of RCAR(p)model parameters,construct the test statistics of the model parameters,and give the limit properties of the statistics.The effect of the test is discussed through numerical simulations and applied to the empirical analysis of a set of economic data in Italy.In the second part,in order to better characterize the two sets of continuous value time series data with correlation,we model the model from The univariate random coefficient autoregressive model is extended to the binary case.The binary random coefficient autoregressive BRCAR(1)model randomness test problem is studied.The test statistics of the model parameters are constructed and obtained under the null hypothesis.The limit distribution of the test statistics is also shown.We also use a large number of numerical simulations to show that our proposed test method is feasible.Combining the characteristics of actual data,not all non-linear data are time series data with continuous values.Integer non-linear time series data is also ubiquitous in real life.Therefore,in the third part,we considered the randomness test of the coefficients of the binary integer autoregressive model,based on the local maximum power method,constructed the test statistics of the model parameters and proved the limit properties of the statistics.The main research results of this article are detailed below.1.Test of coefficient randomness in one-variable autoregressive model1.1 Randomness test of coefficient for RCAR(1)modelIn order to overcome the limitation of the AR(1)model autoregressive coefficient being constant,Conlisk(1974)proposed a first-order random coefficient autoregressive model,namely RCAR(1)model,its definition as follows:Definition 1 Th.e process {Xt} is called RCAR(1)process if it satisfies the following regression equationXt=?txt-1+Zt,t>1,(1)where(?){?t} is an i.i.d.random variables with distribution function F?;E(?t)=??,Var(?t)=??2,??2+??2<1;(?){Zt} is an i.i.d.random variables with density function fz;and E(Zt)=0,Var(Zt)=?z2 are all finite values.For a fixed time t,Assume Xo,{?t} and {Zt}are two mutually independent sequences.Note that when ??2=0.The random coefficient autoregressive model is reduced to the standard autoregressive model.In the following,we consider the testing problem of RCAR(1)model.In practice,whether at is a constant is an important research topic;This is equiv-alent to testing the variance ??2 of coefficient ?t is zero or not.So,we consider the following hypothesis testing problem:H0:??2=0(?)H1:??2>0.We assume that the process of {Xt} is a strictly stable traversal.{Xt}t=1n is a set of observations from model(1).Let ?=(?,?z2)T denotes the unknown parameter.?0=(?0,?z02)T is the true value.We assume that {Xt}t=1n is a strictly stable traversal set from model(1).According to Aitchison and Silvey(1958),the following LMP test statistics can be obtained:(?)where Rt(?)=1/2f"z??(xt-??xt-1)/fz(xt-??xt-1),f"z??(xt-??xt-1)=(?)2/(?)??2fz(xt-??xt-1).In order to study the limit properties of the LMP test statistics,we give the following assumptions:Suppose U?0 is the field of ?0 and a real function is N(x)make(C.1)??2???2<1;(C.2)E|Xt|4<?,E|?t|3<?;(C.3)For (?)t?1.The first derivative of fz about unknown parameters ? in U?0 is continuous and starts with N(x)is the bound;(C.4)For V t>1.The third-order derivative and mixed partial derivative of the log fz with respect to unknown parameters ? are continuous and bounded by N(x).Where is a ? domain generated by {R0,R1,...,Rt} and ? represent the maximum likelihood estimates of the parameters.The following theorem shows that the test statistic Tn(?)converges to normal distribution.Theorem 1 Assuming these conditions(C.1)-(C.4)are true,under H0 when n??,we have that(?)where(?)(?)In the following simulation process,we assume that the error terms Zt obey the normal distribution,T distribution,and mixed normal distribution,respectively.At the same time,the local maximum power test method and empirical likelihood test method are compared in the above three distributions.The simulation results show that the test effect of the local maximum power method is better than empirical likelihood.Finally,the LMP test method is used to analyze a set of Australian economic data,and the random coefficient autoregressive model is used to analyze the actual data.1.2 Randomness test of Coefficients for RCAR(p)modelThis section generalizes the correlation of the sequence from the first order to the p order,considering the RCAR(p)model randomness test problem.This article is in-terested in testing whether the coefficients ?t=(?1t,?2t,…,?pt)' are random.That is,check whether the covariance matrix of the random coefficient is zero.Because?i,t,i=1,2 are independent.So it is equivalent to testing whether the variance is zero.For the sake of narrative convenience,let ?=?i=12 ??i2.Therefore,we consider the following inspection questions:H0:?=0??H1:?>0.Similar to the verification steps of RCAR(1)in the model to obtain the test statistics,test statistics of the RCAR(p)model coefficients are T(?)=(Tn1(?),Tn2(?),…,Tnp(?))'.The following theorem gives the limit distribution of the test statisticTheorem 2 Under the assumptions(C.1)-(C.5)and the null hypothesis H0 conditions,when n??,we have(?),b??.where (?).The pecific definition of VR(?),J(?),I(?)are presented in the section 2.2.The effect of the test was discussed through numerical simulations and applied to an empirical analysis of a set of economic data in Italy.2.Coefficient Randomness Test of bivariate Autoregressive ModelIn order to better characterize the two sets of continuous-valued time series data with correlation,Nicholls and Quinn(1982)proposed a binary random coefficient autoregressive model,namely the BRCAR(1)model,which is defined as follows:Definition 2 The bivariate process {Xt}tEZ is called BRCAR(1)process if it satisfies the following regression equation(?)(2)where(?){?1,t} and {?2,t} are two independent random variable sequences of i.i.d.whose distribution function is F?i,E(?i,t)=??i,Var(?i,t)=??2,i=1,2;(?)(?) is a random matrix,statisfied E(At)=A,For all t ?Z,when ?i,t(i=1,2)is given,?1,tX1,t-1 and ?2,tX2,t-1 are mutually independent seque'nces;(?)The error term {Zt} is a sequence of bivariate dependent random variables whose joint density function is fz(x,y)>0.E(Zi,t)=0,V ar(Zi,t)=?zi2,Cov(Z1,t,Z2,t)=??z1(1,2,where i=1,2;(?)For fixed moments t and V s<t,Zt,AtXt-1 and X s are mutually independent sequences.In the following,we discuss the BRCAR(1)model coefficient test problem.For the model(2),we are interested in testing whether the coefficient matrix is a constant matrix,that is H0:At=A??H1:At is a random matrix;That is to test whether the covariance matrix of At is zero,which At is a diagonal matrix and ?i,t,i=1,2,independent of each other,which is equivalent to testing whether the variance??i2,i=1,2 of the coefficients ?i,t,i=1,2 is zero.For the convenience of description,let?=?i=12??i2.Therefore,we consider the following inspection questions:H0:?=0?? H1:?>0.The variance of at least one of the parameters under the alternative hypothesis is zero.Let {Xt}t?Z is a set of observations from model(2),The unknown parameterof the model is ?=(??1,??2,?z12,?zee,?)T,Let ?0 represents the truth value of the unknown parameter.According to Aitchison and Silvey(1958),the following LMP test statistics can be obtained:where Tni(?)=?t=1n 1/2f"z?i(z1,t,z2,t)/fz(z1,t,z2,t)=?t=1n Sti(?),i=1,2,t(?)=(Tn1(?),Tn2(?))',S(?)=(St1(?),St2(?))',f"z?i(z1,t,z2,t)=(?)2/(?)?i2fz(z1,t,z2,t)i=1,2.Suppose U?0 is the field of ?0 and a real function is N(x)make(?),i=1,2;(C.2)E|Xi,t|4<?,i=1,2;(C.3)E|?it|3<?,i=1,2;(C.4)For (?) t?1.The first derivative of fz about unknown parameters ? in U?0 is continuous and starts with N(x)is the bound;(C.5)For V t>1.The third-order derivative and mixed partial derivative of the log fz with respect to unknown parameters ? are continuous in U?0.and bounded by N(x).Where ? represent the maximum likelihood estimates of the parameters.The following theorem shows that the test statistic Tn(?)converges to normal distribution.Theorem 3 Under the assumptions(C.1)-(C.5)and the null hypothesis H0 conditions,when n??,we have where(?);(?);(?).Based on the Theorem 3 we get that the quadratic of Tn1(?),Tn2(?)form converges according to the distribution.Theorem 4 Let {Xt}t?z be a sequence of random variables rigorously stationary.Under the assumptions(C.1)-(C.5)and the null hypothesis,we have Where ?2(2)represents a chi-square distribution with degrees of freedom 2.3.Coefficient Randomness Test of Bivariate Integer-valued Autoregressive ModelInteger value time series data is also often encountered in production and scien-tific research.In order to characterize two sets of integer value time series data with correlation,Yu et al.(2019)proposed a bivariate first order random coefficient integer value autoregression.The process,which is the BRCINZR(1)process,is defined as follows:Definition 3 The bivariate process {Xt}t?Z is called BRCINAR(1)process if it satisfies the following regression equation where(?){?1,t} and {?2,t} are two independent random variable sequences on[0,1)of i.i.d.whose cumulative distribution function(CDF)is P?i(ui),E(?i,t)=?i,Var(?i,t)=??i2,??i+??i2<1,i=1,2;(?)(?) is a random matrix,statisfied E(?t)=?,For all t?Z,given(i=1,2),?1,t(?) X1,t-1 and ?2,t(?) X2,t-1 are mutually independent sequences;(?){Zt} is a sequence of i.i.d.bivariate non-negative integer-valued random vec-tors and follow some bivariate distribution with joint probability mass function fz(x,y)>0.And E(Zi,t)=?i,Var(Zi,t)=?zi2,Cov(Z1,t,Z2,t)=?,i=1,2.(?)For the given moment t,assume Zt and Xt-l,?it and ?it(?)Xi,t-l,i=1.2,l?1.are all mutually independent sequences.Consider the following inspection questions:where ?=?i=12??i2,i=1,2.That is,the variance of the coefficients has at least one non-zero under alternative assumptions.Let {Xt}t?Z is a set of observations from model(3),The unknown parameter of the model is ?=(??1,??2,?1,?2,?z12,?z22,?)'=(?1,?2,?3,?4,?5,?6,?7)',Let ?0 repre-sents the truth value of the unknown parameter.According to Aitchison and Silvey(1958),the following LMP test statistics can be obtained:T(?)=(Tn1(?),Tn2(?))'.where Tni(?)=?t=1n 1/2f"z?i(zi,t,z2,t)/fz(zi,t,z2,t)=?t=1n Qti(?),i=1,2.Theorem 5 Under the assumptions(C.1)-(C.5)and the null hypothesis H0 conditions,when n??,we have((?))-1(Tn1(?),Tn2(?))'(?)N(0,?),where?=V(?)-J(?)'I-1(?)J(?);J(?)=E[(?)Qti(?)/(?)?],i=1,2,I(?)=E[(?)logfz(?)/(?)?·(?)logfz(?)/(?)?']V(?)=(vij)2×2,vii=Var(Qti(?)),i=1,2,v12=v21=Cov(Qt1(?),Qt2(?)).Based on Theorem 5 we get Quadratic forms of Tn1(?),Tn2(?)converge by dis-tribution Theorem 6 Let {Xt}t?Z be a sequence of random variables rigorously stationary.Under the assumptions(C.1)-(C.5)and the null hypothesis,we have n-1T(?)?-1T(?)(?)?2(2),n??,Where ?2(2)represents a chi-square distribution with degrees of freedom 2.