Studies On Inference For Some Classes Of Threshold Time Series Models

Nonlinear time series data are widespread in our life. This nonlinear structure data exists not only in the classical continuous time series data, but also in the integer-valued time series data. For example, the financial return series, the sunspot data in astronomy (Tong and Lim,1980) and the southern oscillation index (SOI) in meteorology are all typically continuous time series data with nonlinear structure. On the other hand, the lynx data of Canada (Elton,1942; Tong,1990), the number of some special crime data and the number of major earthquake in the world are all typically integer-valued time series data with nonlinear structure. It is very routh using a linear model to describe this kind of data, often cannot reflect and explain the law of the observed data effectivelyIn this paper, we studied the issues about modeling and statistical inference of some nonlinear time series data. The main content is divided into three parts. In the first part, we focused on the volatility and the variance asymmetry of financial return series, proposed a class of threshold stochastic volatility models with explanatory vari-ables, gave the algorithm of Bayesian estimation via MCMC method. Also, we studied the accuracy of the algorithm, the sensitivity to the assumptions of the model and ro-bustness to the priors based on the simulations, and then fitted a set of real data using the proposed model. In the second part, we focused on the count data with thresh-old features and so-called piecewise phenomenon, proposed a new first-order threshold integer-valued autoregressive process, proved the strict stability and the ergodicity of the process, studied some basic probability statistics properties and the estimation methods. The strong consistency, asymptotic normality properties of the estimators are obtained. An algorithm was given to search the root for the integer-valued pa-rameter. Finally, we use the model to fit a set of real data, and compare the fitting results with other integer-valued time series models. In the third part, we weaken the conditions of the original SETINAR(2,1) model, redefined the SETINAR(2,1) model, considered the quasi-likelihood inference for the new SETINAR(2,1) process, gave the quasi maximum likelihood estimators for the parameters of the model, discussed the asymptotic distribution of the estimators and the confidence regions of the parameters. We also studied the performance of the estimators via simulations.In what follows, we introduce the main results of this thesis.1. Modeling and statistical inference of the threshold stochastic volatility model (THSV-X) with explanatory variablesTo capture the asymmetric reaction of financial return series, investigated the impact of exogenous factors on the volatility, we introduce the univariate threshold stochastic volatility model with exogenous variables, which is named THSV-X model and defined as follows:Definition 2 The THSV-X model is a sequence of random variables defined by the following equations, where(ⅰ) {εt, t ≥1} is a white noise process with unit variance, generated independently of {ηt,t≥1},{ηt,t≥1} is also a white noise process with mean zero and variance σ2;(ⅱ) {yt, t≥1} is standardized financial time series of interest, {X1,t,X2,t,…,Xk,t} denotes a set of designed explanatory variables, ht denotes the log-volatility of yt;(ⅲ) {st, t≥11} is a sequence of Bernoulli random variables defined bywhere r is the threshold value, which is assumed to be known. In practice, the value of r can be determined by the decision makers.Assume that Y = (y1, y2,…, yn)T be the sample generated from the above THSV-X model based on h0, we can write the likelihood function as follows, where ω= (ωθ,ωh),ωθ= (θ1T,θ2T,σ2) are the parameters of interest, and ωh= (h1,…, hn) are the latent variables.We studied the parameter estimation problem via MCMC method.We choose the priors for the parameters as follows. We take θ1 and θ2 to be independent N(μi,Vi), where μi are ρ×1 vectors and Vi are square matrixs of order p, p = k + 2, i = 1, 2. We take σ2 to be IGa(v,λ), where A is scale parameter and v is shape parameter. The priors of latent variables ht (t = 1, …,n) can be derived from the volatility equation in the following form:Using standard Bayesian techniques, we obtain the following full conditional pos-terior distributions: ⅰ. The full conditional posterior distribution of θi (i = 1,2) is independent of θj for i≠j, whereπi be the timeindex of the ith smallest value of {X1,0,X1,1,…,X1,n-1}, s =∑t=1n(1-st), i.e., s satisfies X1,π1≤X1,π2≤…≤X1,πs<X1,πs+1≤…≤X1,πn. ⅱ. The full conditional posterior distribution of σ2 is where ⅲ. The full conditional posterior distribution of ht is where C and D take the formsA summary of our MCMC sampling algorithm is given below:Algorithm 1 (MCMC algorithm)① Choose the starting points ω(0) and set i=l.② Draw ht(i) (t = 1,…,n) with a random-walk Metropolis step (2.2.6).③ Draw σ2(i) from π(σ2|θ1(i-1),θ2(i-1),ωh(i),Y).④ Draw θj(i) (j = 1,2.) from π(θj|σ2(i),ωh(i),Y)⑤ Set i=i+1 and go to ② until convergence is achieved.We also studied the accuracy, sensitivity and robustness of the MCMC algorithm via some simulations. Finally, we use the proposed model to fit the data set of S&P500 index, which achieves a very good fitting effect. As can be seen from the fitting results that there are asymmetrical features in the data set of S&P500 index.2. Modeling and statistical inference of the first order integer-valued threshold autoregressive (NBTINAR(1)) process.Many integer-valued data often exhibit the threshold features and the so-called piecewise phenomenon. To capture these features, we propose a first-order integer-valued threshold autoregressive process, namely the NBTINAR(1) process, which is defined as follows:Definition 2 The NBTINAR(1) process is a sequence of random variables {Xt}t∈Z, defined by the following recursive equation Xt =αst+1*Xt-1+Zst+1,t, t∈Z, (1) where(ⅰ) {st} is a sequence of Bernoulli random variables defined aswhere r denotes the threshold variable;(ⅱ) The negative-binomial thinning operator "*" is defined as{Wk(i)} represents a sequence of i.i.d. geometric random variables with parameter αi/1+αi; (ⅲ) {Zi,t} is a sequence of i.i.d. NB(v,αi/1+αi) random variables with probability massfunction(ⅳ) For fixed t andi (i = 1,2), Zi,t is assumed to be independent of αi*Xt-l and Xt-l for all l≥1;(ⅴ) {Z1,t} and {Z2,t} are mutual independent.We first introduce some basic properties of NBTINAR(l) process. Proposition 1 states the strict stationarity and ergodicity of NBTINAR(l) process.Proposition 1 Let {Xt}t∈Zi be the process defined by (1). Then(i) {Xt}t∈Z is an irreducible, aperiodic and positive recurrent Markov chain, hence {Xt}t∈Z ergodic.(ii) There exists a strictly stationary process satisfying (1).The following result shows the first three moments of NBTINAR(l) process exist.Proposition 2 Let {Xt}t∈Z, be the process defined by the NBTINAR(1) process. Then E(Xtk) <∞ for k = 1,2,3.The expectation and variance of the NBTINAR(1) process are given by the fol-lowing proposition.Proposition 3 Let {Xt}t∈Z, be the process defined by the NBTINAR(1) process.ThenNext, we discussed the parameter estimation problem of the NBTINAR(1) pro-cess. To begin with, we make some assumptions about the underlying process and the parameter space.Assumptions:(A1) The observed sequence {Xt}t=1n is generated from the NBTINAR(1) process, with true parameter θ0∈D×N+, D= (0,1)×(0,1) is a compact subset of R2.(A2) The model (1) is identifiable, i.e., pθ≠pθ0, if θ≠θ0, where pθ denotes the marginal distribution law of Xt with parameter θ.The following result state the strong consistency of θcls.Theorem 1 Under the assumptions (A1)-(A2), the CLS-estimators are strongly consistent, i.e.,Since v is integer-valued, the consistency of Vcls implies that vcls =v0 eventually. Therefore, the efficiency of the other estimates with v being estimated together is asymptotically the same as that when v is known.Therefore, we remove v from the parameter vector θ and only consider a central limit theorem for the least squares estimator with a known v.Theorem 2 Under the assumptions (A1)-(A2) except that the v is known, the conditional least squares estimator αCLS= (α1,CLS,α2,CLS)T is asymptotically normal, i.e., where V and W are diagonal matrix of order two with elements Vii = pi[σi2+ (ui+v)2] andThe following two theorems give the strong consistency and the asymptotically normality of the conditional maximum Likelihood estimators.Theorem 3 Under the assumptions (A1)-(A2), the CML-estimators are strongly consistent, i.e.,Theorem 4 Under the assumptions (A1)-(A2) except that the v is known, the conditional maximum likelihood estimator αCML= (α1,cml, α2,CML)T is asymptotically normal, i.e., Furthermore, the matrix G can be estimated consistently byWe also consider the case of r is unknown. Using the NeSS algorithm proposed by Li and Tong (2015), we give the estimation of r. For the integer parameter v of the model, we give an optimizing algorithm, namely: "Min-Min" algorithm. Now, we summary the "Min-Min" algorithm of CLS and CML methods as follows:Algorithm 2 (CLS)① Choose the starting point v(0) (e.g., v(0)= 1), and set i = 0.② Determine whether r is known, if known, go to step ?.③ Choose an upper bound f and a lower bound r, set Dr = {k∈G Z+|l≤k < r-[r]}; where [a] is the largest integral part of a.④ For each k k∈Dr, calculate Jn(rk) by (3.2.18) with v = v(0).⑤ Calculate r by r = argmaxk Jn(rk), set r = r.⑥ Calculate αj(i+1) (j = 1,2), by (3.2.20).⑦ Calculate v(i+1) by (3.2.21), and set v(i+1) = [v(i+1)+ 0.5].⑧ Set i = i + 1 and go to step ⑥ until convergence is achieved.Algorithm 3 (CML)① Choose the starting points α1(0),α2(2) and v(0) and set i= 0.② Determine whether r is known, if unknown, estimate r by using step ③ to step ⑤ of Algorithm 3.1, set r = r③ Choose an upper bound v(i) and a lower bound v(i) such that④ Calculate v(i+1) by (3.2.22), set v(i+1) =|v(i+1)+0.5].⑤ For j = 1,2, Choose an upper bound αj(i) and a lower bound αj(i) such that⑥ Calculate αj(i+1) (j= 1,2) by (3.2.23).⑦ Set i = i + 1 and go to step ③ until convergence is achieved.We compared the effect of CLS estimation and CML estimation via simulations. The simulation results show that the CML method is better than the CLS method. Finally, we use the proposed model to fit a set of global earthquake data, and compare the fitting result with other integer-valued models.The results showed that the our NBTINAR(1) model is the best fitting model and may well explain the data.3. Quasi-likelihood inference for SETINAR(2,1) processThe SETINAR(2,1) process, proposed by Monteiro et al. (2012), also known as first order integer-valued self-exciting threshold autoregressive process, is defined as follows:Definition 3 The SETINAR(2,1) process is a sequence of random variables {Xt}t∈Z defined by the following recursive equation where(ⅰ) I1,t= I{Xt-1≤r}, I2,t=1-I1,t =I{Xt-1 > r}, r is the known threshold variable;(ⅱ) The binomial thinning operator "o" is defined bywhere {Bk(i)} is a sequence of i.i.d. Bernoulli random variables with parameter αi, i=1,2;(ⅲ) {Zt} is a sequence of i.i.d. random variables with E(Zt) =λ, Var(Zt) =σz2 <∞(iv) For fixed t and i (i=1,2), Zt is assumed to be independent of αi o Xt-l and Xt-for all l≥1.The following proposition states the ergodicity of{Xt}. The ergodicity is useful in deriving the asymptotic properties of the estimators.Proposition 4 The process{Xt,t≥1} defined in (2) is an ergodic Markov chain.By definition 2, it is easy to obtain the conditional expectation and conditional variance of SETINAR(2,1) process as follows:We make some assumptions:(Cl){Xt} is a strictly stationary ergodic solution of SETINAR(2,1) process.(C2) E|X0|4<∞.Let{Xt}t=1 n be the sample generated form the SETINAR(2,1) process, β= (α1,α2,λ)T is the unknown parameters. Let θ= (θ1,θ2,σz2)T, where θi=αi(1-αi), i= 1,2. We write the conditional variance in the following form: Using the proposed quasi-likelihood method (Wedderburn,1974), we establish the fol-lowing standard estimation equations: The root of the above equations, if exist, gives the quasi-maximum likelihood estimator for β, i.e. whereThe following theorem gives the asymptotically normality of the quasi-maximum likelihood estimator βMQL.Theorem 5 Under the assumptions (C1)-(C2), the quasi-maximum likelihood estimator βMQL is asymptotically normal, i. e whereNote that the consistency of βMQL follows readily from the above result.In what follows we give a consistent estimator of θ based on Lemma 2.2 in Monteiro et al. (2012) and the CLS-estimators (α1,CLS,α2,CLS,λCLS).Theorem 6 Under the assumptions (C1)-(C2), the following estimators are con-sistent: and where αi= αi,CLS (i=1,2),Theorem 6 is essentially a moment based estimate, its expression is too compli-cated. The following theorem shows that there are more simple consistency estimators of σz2.Theorem 7 Under the assumptions (C1)-(C2), the following estimator is con-sistent: whereBy the above theorem, it is easy to give a confidence region of β in the following form:Theorem 8 Under the assumptions (C1)-(C2). For 0<δ<1, the 100(1-δ)% confidence region of β is given by: where T(θ) is the consistent estimator of T(θ), X32(δ) denotes the δ-upper quantile of X2 distribution with degrees of freedom 3.We also do some simulations to study the performance of the quasi-maximum likelihood estimator βMQL.We conclude from the simulations that, the performance of the quasi-maximum likelihood estimator βMQL is better than the conditional least square estimator under certain conditions, and except for some extreme cases, the performance of the quasi-maximum likelihood estimator βMQL is not inferior to the conditions maximum likelihood estimator. Furthermore, the confidence region based on quasi-likelihood method is more robust than the other methods.