| During the past two decades, there has been much interest in both theoretical and empirical analysis of nonparametric time series models, as traditional parametric methodsare inadequate in capturing possible hidden relationship between the response variablesand their associated covariates in many practical applications. The main advantageof nonparametric methods is that the data may be allowed to speak for themselves in the sense of determining the form of mathematical relationships between time series variables. In fact, the applications of nonparametric time series analysis can be traced back to the 1940s at least. Recently, modern computers and the information age have brought us opportunities with challenges. For example, technological inventions have led to the explosion in data collection such as the data of stock market trading. Nonparametricmethods can provide useful exploratory tools for this venture and many authors have studied the asymptotic properties of nonparametric methods. For recent developments, we refer to Fan & Gijbels (1996), Fan & Yao (2003), Li & Racine (2006) and the references therein.However, in the multivariate setting with more than two covariates, the underlying regression function can not be estimated with reasonable accuracy due to the so-called "curse of dimensionality" of Bellman (1961). How to circumvent the cure of dimensionalityis an important topic in nonparametric statistical inference and there are many powerful approaches to avoid this problem (cf. Hastie & Tibshirani 1990, Hastie & Tibshirani1993, Fan & Yao 2003). Among them, the semiparametric partially linear modellingis one of the most commonly used methods. An advantage of the partially linear approach is that any existing information concerning possible linearity of some of the components can be taken into account in such models. Engle, Granger, Rice & Weiss (1986) were among the first to study the partially linear models. Since then, the partiallylinear approach has been studied extensively in both econometrics and statistics literature. The concentration has been mainly on estimation and testing of both the parametricand nonparametric components in partially linear models, see Robinson (1988), Hardle, Liang & Gao (2000) and Gao (2007) for reference.The books and papers mentioned above mainly study nonparametric and semiparametric methods for the case where the observed time series satisfy a type of stationarity.However, as pointed out in literature, the stationarity condition may be too restrictivein practice. In fact, when tackling economic and financial issues from time perspective,we often deal with nonstationary components. For example, neither prices nor consumption, nor exchange rates, GDP nor macroeconomic variables follow an invariantstationary law over time. Hence, practitioners might feel more comfortable avoiding restrictions like stationarity for processes involved in time series models. There is much literature on parametric linear and nonlinear models for nonstationary time series, but very few work has been done in nonparametric and semiparametric nonlinear cases. In fact, the statistical inference for nonstationary time series is quite different from that for stationary observations. In traditional stationary time series, the independence or some sort of mixing condition is assumed for the underlying processes to obtain the asymptoticproperties of the proposed estimators. It is well known that a minimal condition for undertaking the asymptotic analysis of nonparametric or semiparametric estimators at some point x0 is that, as the number of observations tends to infinity, there should be infinitely many observations in any neighborhood of x0, which means that the underlyingprocesses must return to a neighborhood of x0 infinitely often. Hence, the class of nonstationary processes we are dealing with in this dissertation is not the general one. Instead, we need to impose some restrictions on it. We considerφ-irreducible Harris recurrentMarkov chains which include many interesting nonstationary cases such as the random walk and unit root process. We also study trending time-varying coefficient semiparametric models.On the other hand, research on nonparametric estimation is mainly concentratedon time series case. Studies on nonparametric statistical inference for spatial processes (or random fields) are relatively few. However, spatial data modelling has received increasingattention in the last two decades. Applications of spatial statistical models are quite popular in many fields, such as econometrics, epidemiology, environmental scienceand image analysis. Many authors (cf. Ripley 1981, Cressie 1991) have studied parametric methods for statistical inference in spatial models. Recently, nonparametric statistical modelling for spatial processes has become a hot topic. There have been some developments in nonparametric estimation for spatial processes. Among them, Tran (1990), Carbon, Tran & Wu (1997) and Hallin, Lu & Tran (2001, 2004a) discussed kinds of asymptotic properties of density estimation for spatial processes. Hallin, Lu & Tran (2004b) and Gao, Lu & Tj(?)stheim (2006) studied spatial nonparametric and semiparametricregression estimation, respectively. We will consider local linear M-estimation for dependent spatial processes in Chapter 5. In addition, an additive model is consideredto avoid the curse of dimensionality and a method based on a combination of the local linear M-smoothing and the marginal integration technique is applied to deal with the estimation problem in the spatial additive model.In this dissertation, various nonparametric and semiparametric statistical inference techniques such as the estimation methods, test statistics and variable selection, are developed for nonstationary time series and spatial processes. The main contributions of this dissertation are summarized as follows.At first, we study a nonlinear model Yk = m(Xk) +εk, where m(·) is an unknownregression function, {Yk,Xk} are the observations and {εk} is a stationary sequence oferrors. When {Yk, Xk} is a stationary process such as i.i.d. or stationaryα-mixing, manyauthors have studied various statistical properties of robust nonparametric estimatorsof m(·) and its derivative m'(·). For example, when the observations are i.i.d., Fan &Jiang (2000) established the weak consistency and asymptotic normality of local linearM-estimators of m(·) and m'(·). They also proposed a one-step iterative procedureto reduce the computational burden in the numerical implementation of the local linearM-technique. Jiang & Mack (2001) obtained consistency and asymptotic normality of the local polynomial M-estimators under stationarity and weak dependence. Hong (2003) studied Bahadur representation of nonparametric M-estimators. However, so far as we know, little work has been done in studying the behavior of robust nonparametric estimators for nonstationary processes. So in Chapter 2, we will discuss the statistical properties of local linear M-estimator of m(·) for nonstationary time series. In fact, thederivation of the large sample properties of local linear M-estimators for nonstationarytime series is more complicated and difficult than that for stationary case. Many powerful devices in the limit theorems of stationary time series, such as the ergodic theorem and the Lindeberg-Feller central limit theorem, are no longer valid for nonstationarynull recurrent processes. Hence, we should turn to other methods and tools in the asymptotic analysis of statistical inference for nonstationary time series. In Chapter2, we use an independence decomposition method based on the stopping time of Markov chains (cf. Nummelin 1984). With the help of this method, we can decompose the partial sums of nonstationary random variables into i.i.d. parts and asymptotically negligible parts, which makes the research on asymptotic properties possible. We first obtain the weak convergence rate and asymptotic normality of the estimators under mild conditions. We find that the weak convergence rate and the normalization rate of the asymptotic distribution in nonstationary case are slower than those in stationarycase. We also study strong Bahadur representation of the estimators and establish their strong consistency. From our asymptotic results, we can directly obtain the asymptoticproperties of some well-known estimators for nonstationary case, such as the local linear estimators and the least absolute distance estimators. Furthermore, we apply a one-step iterative procedure to reduce the computational burden in the Monte-Carlo simulation. If the initial value satisfies certain conditions, we show that the one-step local linear M-estimators are still asymptotic normal. Furthermore, some numerical examplesare provided to show that the proposed methods perform well even when the observations are contaminated or the errors are heavy-tailed.In Chapter 3, we consider a partially linear model of the form Yk = Xkτα+g(Vk)+εk,k = 1,…, n, where {Vk} is aβ-null recurrent Markov chain, {Xk} is a sequence of eitherstrictly stationary or nonstationary regressors and {εk} is a stationary sequence. For the case where {Vk} is a sequence of either fixed designs or strictly stationary regressors but there is some type of unit-root structure in {Xk}, existing studies, such as Juhl &Xiao (2005), have already discussed some estimation and testing problems. However, to the best of our knowledge, the case where either {Vk} is a sequence of nonstationary regressors or both {Xk} and {Vk} are nonstationary, has not been discussed in literature. So in Chapter 3, we consider the following two cases: (i) {Xk} is a sequence of strictly stationary regressors and {Vk} is a sequence of nonstationary regressors; and (ii) both {Xk} and {Vk} are nonstationary. When {Xk, Vk} is a sequence of stationary random variables or fix-design points, many authors usually apply the weighted leastsquaremethod (cf.§1.3) to estimate the parameter a and the function g(·). In the case where {Vk} is a sequence of stationary random variables (or fixed-designs) with compactsupport, the uniform consistency of the nonparametric estimator guarantees that the weighted least-square estimator behaves well in both large sample theory and applications,see Hardle, Liang & Gao (2000). However,βnull recurrent processes do not have any compact support, which causes many difficulties in our discussion. In addition,there are nonstationary random variables in the denominator of the least-square estimator in the nonstationary case, so we can not use the same method as that in the stationarycase to study the asymptotic properties of the estimator. To avoid the compact support restriction, we then apply a truncated weighted least-square method, which was proposed by Robinson (1988), to estimate the regression parameterαand the functiong(·). In the proofs of our main results, the independence decomposition methodmentioned above and many asymptotic properties of nonparametric estimates for null recurrent time series (cf. Karlsen & Tj(?)stheim 2001, Karlsen, Mykelbust & Tj(?)stheim 2007) are used. It is interesting to find that the normalized estimator ofαis asymptoticallynormal whenever {Xk} is stationary or nonstationary and the convergence rate is the same as that in the case of stationary time series. This is because the asymptotic distribution and the asymptotic variance of the estimator ofαare mainly determined by {εk} and {Uk = Xk- E(Xk|Vk)}, and both {εk} and {Uk = Xk- E(Xk|Vk)} are assumedto be stationary in our dissertation. On the other hand, the asymptotic distribution of the nonparametric estimator of g(·) is the same as Theorem 3.1 in Karlsen, Mykelbust& Tj(?)stheim (2007) and its convergence rate is slower than that in stationary case. Furthermore,we establish the strong uniform consistency of the nonparametric regression estimator and the kernel density estimator. In the proofs of the strong uniform consistencyresults, the independence decomposition method, the truncation method and a Bernstein-type inequality are used. We not only weaken the bandwidth condition inKarlsen & Tj(?)stheim (2001), but also extend their result (point-wise strong consistency) to uniform strong consistency. From the Monte-Carlo simulation results, we can find that our methods work well when {Vk} is a random walk.In Chapter 4, we consider the specification testing theory in the above partiallylinear model when {Vk} is a random walk. In the case of stationary time series, many authorshave discussed hypothesis testing problems in partially linear models, see GonzalezManteiga& Aneiros-Perez (2003), Fan & Huang (2005) and Gao (2007). In order tostudy the asymptotic properties of the test statistics, we often need to deal with the leading terms-quadratic form. However, there is few literature concerning the limit theory of quadratic forms for nonstationary time series. This causes many difficulties in the proofs of large sample properties of semiparametric test statistics in the nonstationary case. In Chapter 4, we first consider a parametric testing problem in the partially linear model and apply the Wald statistic. With the help of the asymptotic normality of the truncated least-square estimator of the parameterα, we can show that the asymptotic distribution of the parametric test statistic (after normalization) is the same as that for the stationary case and the Wilks phenomenon holds. Then, we considera nonparametric testing problem in the partially liner model. A test statistic of the quadratic form is constructed and its asymptotic distribution is obtained, although the convergence rate is different from that in stationary time series. We not only extend corresponding result in Gao, King, Lu & Tj(?)stheim (2007), who considered a nonlinear regression model, but also remove the Gaussian assumption in their paper. In the proofs of asymptotic properties of the quadratic form, we apply the martingale approximationmethod, that is, a sequence of martingale differences is constructed to approximate the quadratic form. However, the sequence of martingale differences is nonstationary, which causes many difficulties in our proofs. Furthermore, how to choose the critical value of the test statistic is important in applications. In the case of finite samples, a bootstrap scheme is developed to choose the critical value. We also study the asymptoticproperties for the bootstrap scheme, which extend corresponding results in Li & Wang (1998). Some Monte-Carlo simulated examples are provided to show that the proposed methods work well in practice.During the last two decades, the trending time-varying models have become an importanttopic, since they have many important applications in econometrics and finance. For example, a market model in finance relates the return of an individual stock to the return of a market index or another individual stock and the coefficient is usually called a beta-coefficient in a capital assets pricing model. Some recent studies such as Wang (2003) show that the beta-coefficient might vary over time. In order to establish the asymptotic properties of time-varying models, we need to deal with weighted partial sums or fixed-design quadratic forms, which is not an easy task. There is now a large literature on parametric and nonparametric models with time-varying coefficients, see Robinson (1989), Phillips (2001), Cai (2007) and the references therein. However, so far as we know, little work has been done in semiparametric time-varying models. In Chapter 5, we will introduce a partially time-varying coefficient model (PTVCM) to characterize the nonlinearity, nonstationarity and trending phenomenon. Our model covers many interesting time series models such as the fixed-design partially linear model and the nonlinear time-varying model. Since the coefficient function varies over time, the response variables in PTVCM are nonstationary with trend. To estimate the regression coefficients and the coefficient function, the profile least-square (PLS) techniqueis applied and the asymptotic distributions of the proposed estimators are establishedunder mild conditions, from which, Theorem 2.1.1 of Hardle, Liang & Gao (2000) can be obtained as an immediate corollary. Meanwhile, we consider the parametric and nonparametric testing problems. A generalized likelihood ratio test statistic, introducedby Fan, Zhang & Zhang (2001), is applied and the proposed test statistic is shown to follow an asymptoticχ2-distribution under the null hypothesis. On the other hand, variable selection is fundamental in statistical modelling. For example, in the fields of biostatistics and econometrics, a number of variables are available in the initial analysis, but some of them might not be significant and should be excluded. By variable selection,we can simplify the model and reduce the computational burden in estimation and testing problems. Hence, there has been a lot of attention in variable selection during the past several decades. In Chapter 5, the penalized least-square method is applied to select significant variables in our model and the convergence rate together with the sparsity and oracle property of the resulting estimator is established. We also discuss some extensions of our model such as PTVCM with heteroscedastic errors and generalizedPTVCM. Some Monte Carlo simulation studies are given to show that our theory and method work well in practice.There is extensive literature concerning asymptotic properties of local linear Mestimatorsfor stationary time series. To the best of our knowledge, however, nonparametricM-type estimation has not been developed for spatial data. In Chapter 6, we will study the asymptotic properties of local linear M-estimators for dependent spatialprocesses. At first, the weak consistency as well as the asymptotic normality for the proposed estimators are established under some mild conditions, from which the asymptotic properties for some well-known estimators, such as local linear estimators and least absolute distance estimators, are obtained as immediate corollaries. Furthermore,an additive model is considered to avoid the curse of dimensionality for spatial data when the dimension of the covariates is larger than two. An estimation procedure based on a combination of the marginal integration technique and the local linear Mestimation is applied and the asymptotic distribution of the resulting estimator is established.Our Monte-Carlo simulation results show that the proposed robust estimation method works well numerically even when the spatial observations are contaminated or the errors are heavy-tailed.
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