Parameter Estimation Of Some Autoregressive Models And Its Application

In practice,integer value time series is commonly encountered,It can be divided into integer-value linear time series and integer-value nonlinear time series.For the integer-value linear time series,such as the number of unemployed population,the number of crime,the number of claims in a certain insurance policy,etc.In this paper,we consider quantile regression for the INAR(1)model based on binomial thinning operators.For the integer-value nonlinear time series,such as the number of patients in a certain epidemic,the number of natural disasters in a certain country or region,etc.In this paper,we propose a new threshold autoregressive model(BNBTINAR(1))based on binomial thinning and negative binomial thinning operator,and makes some statistical analysis of the model.Random coefficient autoregressive(RCAR)model is very useful to model contin-uous data in applications.It is commonly observed that the random autoregressive coefficient is assumed to be an independent identically distributed(i.i.d.)random vari-able sequence.However,in practical problems,the random autoregressive coefficient may be affected by some known or observable variables,the sequence of independent identically distributed random variables cannot give a dynamic description of the ran-dom autoregressive coefficient.In order to make the RCAR model more suitable,a new RCAR model driven by explanatory variable and observations is considered in this paper.Based on the above discussion,we introduce the main results of this thesis.1.Quantile Regression(QR)for the INAR(1)processTo make the quantile regression method feasible,we first smoothed the INAR(1)model.Definition 1(Smoothing INAR(1)process).Let U be a random variable with uniform distribution on interval[0,1),which is independent of {Xt}.Then {Xt+U}is called a smoothing INAR(1)process ifXt=Xt+U=?(?)Xt-1+Zt,Zt=zt+U.Considering the existence of binomial thinning operators,the conditional quantiles of the smoothing INAR(1)model are given by the following theorem.Theorem 1 Consider the random variables Xt and Zt from smoothing INAR(1)process(2.1.1),and let QXt|Xt-1(?)denote the conditional quantile of Xt conditional on Xt-1 fixed at xt-1.Then the following equation holds:QXt|Xt-1(?)(?)?xt-1+QZt(?),where QZt(?)denotes the 100?th quantile of Zt.Now we are ready to develop the quantile regression for process(2.1.1).Let{Xt,U}t=1n be a random sample generated from(2.1.1).Then,the QR estimator(?,QZt)can be obtained by minimizing the following QR objective function Ln{?,QZt(?))=1/n(?)??(Xt+U-?Xt-1-QZt(?)),To derive the asymptotic distribution of the estimator,we give the following reg-ularity Conditions.(Cl)E|Xt-1|3<?.(C2)When t is an integer,the derivative of the probability of Zt(fz)is not derivable,we defined it by f'Z{Zt)=(f'Z-(Zt)+f'Z+(Zt))/2,where f'Z-(Zt)and f'Z+(Zt)denote the left and right limits of fZ.Theorem 2 Under the Conditions(C1)-(C2).the QR estimators(?,QZt)are consistent and asymptotically normal,i.e.,where and QZt*(?)denote the true values of a and QZt(?).2.Parameter estimation of a new threshold autoregressive model(BNBTIAN-R(1)).In order to characterize the different process characteristics of the integer threshold autoregressive model before and after the threshold,we propose a new threshold au-toregressive model based on the binomial thinning operator and the negative binomial thinning operator.Definition 2 if {Xt}t?Z satisfied:Xt=(?1(?)Xt-1+Z1,t)I1,t+(?2*Xt-1+Z2,t)I2,t,t ?Z.we call it BNBTINAR(1)process.where(?)I1,t=I{Xt-1?r},I2,t=1-I1,t=I{Xt-1>r};(?)the binomial thinning operator“(?)”,introduced by Steutel and Van Harn(1979),is defined as?1(?)X=?i=1 X Bi,where ?1?(0:1),{Bi} is a sequence of i.i.d.Befrnoulli random variables satisfying P(Bi=1)=1-P(Bi=0)=?1;(iii)the negative binomial thinning operator“*”,introduced by Ristic et al.(2009),is defined as ?2*X=?i=1 XWi,where ?2?(0,1),{Wi} is a sequence of i.i.d.geo'metric random variables with parameter ?2/1+?2;(?){Z1,t} is a sequence of i.i.d.Poisson distributed random variables with mean X,{Z2,t} is a sequence of i.i.d.geometric distributed random variables with pa-rameter ?/1+? and mean ?;(?)For fixed t,Z1,t is assumed to be independent of ?1(?)Xt-1 and Xt-1 for all l?1,Z2,t is assumed to be independent of ?2*Xt-1 and Xt-1 for all 1>1.Now we are ready to state the strict stationarity and ergodicity of BNBTINAR(1)process(3.1.1).These properties will be useful to derive the asymptotic properties of the parameter estimates.Proposition 1 Let {Xt}t?Z be the process defined in(3.1.1).Then {Xt}t?Z is an irreducible,aperiodic,and positive recurrent(and hence ergodic)Markov chain.Thus,there exists a strictly stationary process satisfying(3.1.1).The following proposition ensures the first three moments exist.To show this,we denote ?max=max{?1,?2}.Proposition 2 Let {Xt} be the process defined by(3.1.1).Then E(Xtk)<?,for k=1,2,3.Since the moments and conditional moments will be useful in obtaining the ap-propriate estimating equations for parameter estimation.We give some moments and conditional moments of BNBTINAR(1)process in the following proposition.For conve-nience,we denote E(I1,t)=q,E(I2,t)=1-q,u1:=E(Xt|Xt?r),u2:=E(Xt|Xt>r),?12:=Var(Xt|Xt?r),?22:=Var(Xt|Xt>r).Proposition 3 Let {Xt}t?Z be the process defined by(3.1.1).Then(?)E(Xt|Xt-1)=?1Xt-1I1,t+?2Xt-1I2,t+?;(?)E(Xt)=q?1u1+(1-q)?2u22+?;(?)Var(Xt)=q(?12?12+?1(1-?1+q(1-q)?12?12+q?+(1-q)(?22?22+?2(1+?2)?2)+q(1-q)?22?22+(1-q)?(1+?)-2q(1-q)(?1?1+?)(?2?2+?).The conditional least squares estimator(CLS)of the model is given below.The CLS-estimators ?CLS:=(?1,CLS,?2,CLS,?CLS)T of ? are obtained by mini-mizing the expressionwhereUt(0)=Xt-?1Xt-1I1,t-?2Xt-1I2,t-?.From the partial derivatives of first order,we obtain the systemThe following theorem states the strong consistency and asymptotic normality of the CLS-estimators.Theorem 3 Let {Xt}t?z be a BNBTINAR(1)process.Then the CLS-cstimators?CLS:=(?1,CLS,?2,CLS,?CLS)T are strongly consistent and asymptotically normal,where V and W are square matrices of order 3,with elements respectively,with ?1=?1,?2=?2 and ?3=?.The conditional maximum likelihood estimation(CML)of the model BNBTI-NAR(1)is given below.The CML-estimators ?CML:=(?1,CML,?2,CML,?CML)T of ? are obtained by max-imizing the conditional log-likelihood function l(?)=(?)log(A1(xt-1,xt,?1,?2,?)),whereA1(x_t-1,xt,?1,?2,?)=p1(x_t-1,x_t,?₁,?)I_1,t+p2(x_t-1,x_t,?₂,?)I₂,t,Thus the maximum likelihood estimators can be obtained by solving the score equations below,whereA2(x_t-1,x_t,?₁,?)=(p1(x_t-1,x_t-1,?₁,?)-p1(x_t-1,xt,?)1,?))I_1,t,The following results establish the consistency and the asymptotic normality of the CML-estimators.Theorem 4 Let {Xt}tez be a BNBTINAR(1)process satisfying(C1)-(C6).Then there exists a consistent solution ?CML=(?1,CML,?2,CML,?CML)Of(3.2.1)which is a local maximum of l(0)with probability tending to one.Moreover,any other consistent solution of(3.2.1)coincides with ?CML with probability tending to one as n??.Theorem 5 Under the assumptions of Theorem 4,the CML-estimators ?CML is asymptotically normal,i.e.,where m is the Fisher information matrix.We present a new threshold variable r estimation algorithm(SIS).Step 1.Choose some appropriate positive integer L,let ?(j)=?+j(?-?)/L,j=0,1,…,L.Step 2.For each j ? {0,1,…,L},calculate r?(j)by(3.2.3)and(3.2.4).Step 3.The final r is calculated by r=arg min0?j?L Sn(r?(j),?(j)).3.Parameter estimation of a new RCAR(1)model based on explanatory variables.To make the RCAR model more practical,this paper considers a new RCAR model driven by explanatory variable and observations.Definition 3 A process {Xt}t ?Z is said to be a RCAR(1)process,if it satisfied where {Zt} is a stationary sequence of random variables which can be observed and be independent of Xt-1,{?t}.?1,?2 are two parameters.{?t} is a sequence of i.i.d.random variables and is independent of {Xt-1},with mean 0 and variance ??2.The following propositions give the statistical properties of the RCAR(1)model.Proposition 4 Let {?1,?2} satisfy sup(z,x)?R2 |?1Z+?2X|<?,then the process{Xt}t?Z defined by the model(4-1-1)is an ergodic Markov chain.Proposition 5 Let {X_t}t?Z be the process defined by(4.1.1).Then(?)(?)(?)Var(X_t|X_t-1)=??2.Let {X_t}t=1n be a random sample generated from the model(4.1.1).Then the CLS estimates(?1,CLS,?2,CLS)T of(?1,?2)T are obtained by minimizing the expressionThe weak consistency and asymptotic normality for(?1,CLS,?2,CLS)T are given in Theorem 4.2.1.Theorem 5 Let ?0:=(?10,?20)T,under condition(A1)-(A3).for the CLS esti-mates(?1,CLS,?2,CLS)T,we havewhere ?10 and ?20 denote the true values of ?1 and ?2,whereConditional maximum likelihood estimation based on Model ? and Model ?.Model ?:Let {X_t} be the process defined by(4.1.1),where {?t}t=1n i.i.d.?N(0,1),Zt satisfied Zt=0.5Zt-1+?t,Z0=0,?t?N(0,1).The maximum likelihood estimates can be obtained by solving the score equations below,where Gt(?1,?2)=exp{?1zt+?2xt-1},Gt(?1,?2)=Gt(?1,?2)/Gt(?1,?2)+1.The following results establish the consistency and the asymptotic normality of the CML estimates.Theorem 6 Let {X_t} be a RCAR process satisfying(A2)-(A3),?0:=(?10,?20)T.Then there exists a consistent solution ?1,CML=(?1,CML,?2,CML)T of(4.2.2)which is a local maximum of L1(?)with probability tending to one.Moreover,any other consistent solution of(4.2.2)coincides with ?1,CML with probability tending to one as n??.Theorem 7 Under the assumptions of Theorem 6,the CML estimates ?1,CML are asymptotically normal,i.e.,where I₁(?⁰)=(?_ij)2×2 is the Fisher information matrix,i.e.,?₁₁=E((?/??₁)log f1(X₀,X₁,Z₁)(?)log f1(X₀,X₁,Z₁)).?₁₂=?21=E((?)log f1(X₀,X₁,Z₁)(?)log f1(X₀,X₁,Z₁)).Model ?:Let {Xt} be the process defined by(4.1.1),where {?t}t=1n i.i.d.?t(3),Zt satisfied Zt=0.5Zt-1+?t,Z0=0,?t?N(0,1).The maximum likelihood estimates can be obtained by solving the score equations,where Gt(?1,?2)=exp{?1zt+?2xt-1},Gt(?1,?2)=Gt(?1,?2)/Gt(?1,?2)+1.The following results establish the consistency and the asymptotic normality of the CML estimates.Theorem 8 Let {X_t} be a RCAR process satisfying(A2)-(A3),?0:=(?10,?20)T.Then there exists a consistent solutio ?2,CML=(?1,CML,?2,CML)T of(4.2.3)which is a local maximum of L2(0)with probability tending to one.Moreover,any other consistent solution of(4.2.3)coincides with ?2,CML with probability tending to one as n??.Theorem 9 Under the assumptions of Theorem 8,the CML estimates ?2,CML are asymptotically normal,i.e.,where I₂(?⁰)=(?ij)2×2 is the Fisher information matrix,i.e.,?11=E((?/??₁)log f2(X₀,X₁,Z₁))²,?22=E((?/??₁)logf2(X₀,X₁,Z₁))²,The QR estimates(?1,QR,?2,QR)T can be obtained by minimizing the following functionTheorem 10 Under the assumptions(A2)-(A6),the QR estimates(?1,QR,?2,QR)T are consistent and asymptotically normal,where?1=E(v_1tX_t-1)²,?₁=E(f_?(Q_?t(?))(v_1tX_t-1)²),?2=E(v_2tX_t-1)²,?₂=E(f_?(Q_?t(?))(v_2tX_t-1)²).