| Conditional heteroscedastic time series models as one kind of nonlinear timeseries models, including new Laplace autoregressive (NLAR) models and autoregressiveconditional heteroscedastic (ARCH) models, have received considerable attentionin recent years.The NLAR model is one of the representative non-Gaussian time series modelswhich were developed in the 1980s, whose marginal distribution is Laplace. Thismodel can be used for modeling large kurtosis and long tailed data. Because of thecomplexity of the model structure, there is very little literature on this model. Inthis thesis we estimate parameters in the model using three different approaches.In the last few years ARCH models were very popular for their extensive applicationsto economy and finance. Although the ARCH model characterizes thefact that variances of financial data are time-varying, it doesn't characterize other"stylized facts" (such as asymmetry, heavy tails, long memory and leverage effect),so there are various generalizations or modifications of the ARCH model.First, volatility functions of the ARCH model are parameterized and are assumedto be of specific form. The drawback of this set-up has been stressed by many authors in the econometric and statistical literature. A good alternative isto assume that volatility functions in a flexible nonparametric way because non-parametric methods free the traditional parametric estimators of volatility from theconstraints related to their specific models. Regarding approaches to estimatingnonparametric volatility functions, for example, see Hardle and Tsybakov (1997),Ruppert et al. (1997), Fan and Yao (1998) and Ziegelmann (2002). In this thesiswe propose the local log-linear approach and give asymptotic properties of the localconstant and the local log-linear estimators. Our simulation study and an appli-cation to finance lead to superior performance of our approach compared with theconventional treatments in the literature.Second, the ARCH model can not deal with integer-valued time series, butdata of this class are fairly common in practice, for instance, counts arise in marketmicrostructure as soon as one starts looking at tick-by-tick data; the price processfor a stock can be viewed as a sum of discrete price changes and the daily numberof these price changes constitutes an integer-valued time series whose properties areof interest. An integer-valued analogue of the ARCH model with Poisson deviateswas proposed by Ferland et al. (2006) to model the number of cases of campylobac-terosis infections from January 1990 to the end of October 2000 in the north of theProvince of Québec. We study the problem of estimating parameters in the model,i.e. maximum likelihood estimation, weighted least squares estimation and maxi-mum empirical likelihood estimation. We also consider the problem of hypothesistest for the model. We develop an asymptotic theory for quadratic forms of sam-ple autocorrelations of conditional residuals from the model, based on this result,and we propose a statistic to check the adequacy of a fitted integer-valued ARCHspecification. Testing the presence of ARCH effects and testing parameters under asimple ordered restriction are also noted.In the following we introduce the main results of this thesis.1. Estimation of Parameters in the NLAR(p) Model Consider the NLAR(p) model defined aswhereα0,α1,…,αp are non-negative and their sum is one, and the distribution ofthe i.i.d. innovation sequence {εt} is chosen so that the marginal distribution ofthe stationary {Xt} is a standard Laplace. Assume that observations of {Xt} areavailable for t=1,2,…,n. Define ai =αiβi,σii=αiβi2(1-αi), i=1,2,…,p.Let a =(a1…ap)?,σ=(σ11,…,σpp)?, and (?)t-1 be theσ-field generated by X1,X2,…,Xt-1,Then conditional least squares estimators of a andσareWeighted conditional least squares estimators of a andσhave similar expressionswith (?) and (?), respectively. Maximum quasi-likelihood estimators can be obtainedby solving the estimating equationswhere 2. Estimating nonparametric volatility functions in AR modelsConsider the following modelwhereεt is a sequence of i.i.d. random variables with mean 0 and variance 1, andX0 is a random variable independent of {εt : t≥1}.We transform model (1) into the following regression modelwhere Yt=log(Xt -φXt-1)2, g{Xt-1)=β+log f(Xt-1)=β+ log(σt2),β=E(log(εt2)), Ut=log(εt2)-β. For a fixed fitting point x∈R, the local constantquasi log-likelihood of Yt can be written aswith Kh(u)=(1/h)K(u/h) andwhere K is a. kernel function, h = hn is a bandwidth. We replaceφby its n1/2consistent estimator (?). Minimizing (2) with respect to g(x) leads to the local constantestimator (?)(x) (and consequently the estimator (?)(x)).Let c0 = E(εt4), v =∫K2(u)du, the density of Xt is p(x). Assume that n1/3h→k as n→∞, and the following limit exists:Theorem 1 (Local constant estimation) For every fitting point x, the localconstant estimator (?)(x) for the function f(x) is given by Moreover, under necessary assumptions, (?)(x) is consistent, and nh1/2((?)(x)-f(x)) isasymptotically normal with mean k3/2∫f*(x,u)K(u)du and variance (c0 -1)vf2(x)/p(x).Define residuals (?)t=Xt-(?)Xt-1, we minimizesto obtain our local log-linear estimatorTheorem 2 (Local log-linear estimation) Under necessary assumptions, forevery fitting point x, the local log-linear estimator (?) (x) is consistent,and nh1/2((?)(x)-f(x)+op(1)) is asymptotically normal with mean 0 and variance (c0 -1)vf2(x)/p(x).A possible extension of the estimation procedure is also described.3. Statistical inference for integer-valued ARCH modelsConsider the integer-valued ARCH(p) modelwhereα0 > 0,αi≥0,i=1,…p, (?)t-1 is theσ-field generated by {Xt-1,Xt-2,…}.Letα=(α0,α1,…,αp)?. Suppose that X1-p,…,X0,X1,…,Xn are generated bymodel (3) with the true parameter valueα0.First, we give the maximum likelihood estimation and the weighted least squaresestimation ofα0. The conditional log-likelihood function can be written asThe maximizer (?) of Ln(α) is the maximum likelihood estimator ofα0. Theorem 3 (Maximum likelihood estimation) Under necessary assumptions,(?)→α0 a.s., and n1/2((?)-α0)(?)N(0,I-1(α0)),where the Fisher informationmatrix is given bywith Zt = (1,Xt-1,…,Xt-p)?.We can minimize∑t=1n(Xt-λt)2=∑t=1n(Xt-Zt?α)2 with respect toαtoobtain a preliminary least squares estimator ofα0 asBased on (?), an improved estimator (?) ofα0 is given aswhere (?)t=Zt?(?).Theorem 4 (Weighted least squares estimation) Under necessary assumptions.n1/2((?) -α0)(?) N(0,I-1(α0)).Next, we consider empirical likelihood inference for model (3). Let Dt(α) =(?)lt(α)/(?)α,Pt(α)=(?)2lt(α)/(?)α(?)α?. Then the log empirical likelihood function canbe defined aswhere b(α) is a continuous differentiable function of a and satisfiesThe minimizer (?)E of LE(α) is called the maximum empirical likelihood estimator.and the corresponding Lagrange multiplier denoted by (?) = b((?)E). Theorem 5 (Maximum empirical likelihood estimation) Under necessaryassumptions, P((?)E is the inner point of B(α0,n-1/3)→1 as n→∞. andwhen (?)E is an inner point, it satisfieswhereFurthermore, under necessary assumptions, n1/2((?)E-α0)(?)N(0,I-1(α0)).Last, we develop an asymptotic theory for quadratic forms of the autocorrela-tions of the conditional residuals from model (3). Definewhere the conditional residuals (?)t= Xt/(?)t with (?)t= (?)0C+∑i=1p(?)iCXt-i, Here (?)Cis a n1/2-consistent estimator ofα0.Theorem 6 (Asymptotic distribution of the test statistic) Under necessaryassumptions, for any 1≤i1<…K with 1≤K < n.The results in Theorem 6 lead to and provide a rigorous justification for Portmanteaugoodness-of-fit tests of integer-valued ARCH specification, whose usefulnessare showed by a simulation study. Testing the presence of ARCH effects and testingparameters under a simple ordered restriction are also noted.
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