Multifrequency-band Tests For Serial Correlation And Inference For Two Types Of Integer-valued Time Series Models

In this thesis,two types of important problems in time series analysis are studied.One is testing for serial correlation,and this thesis introduces the wavelet transform as a tool to propose innovative tests in the frequency domain.The second is the modeling of integer-valued time series,where two cores concerned are nonlinear features in integer-valued data and the analysis of categorical time series from the perspective of integer-valued time series.Therefore,the main content of this thesis is divided into three parts as follows:1.Multifrequency-Band Tests for Serial Correlation.Considering a stochastic sequence{y_t}^T_t=1with E(y_t)=0 for all t,testing for the null hypothesis that{y_t}^T_t=1is an uncorrelated process is a highly sought-after research.This thesis proposes a new family of multifrequency-band tests for serial correlation by using the maximum overlap discrete wavelet packet transform(MODWPT).The MODWPT allows the variance of a process to be decomposed into the variance of its components on different equal-length frequency sub-bands,and the multifrequency-band tests then measure the distance between the MODWPT-based variance ratio and its theoretical null value jointly over several frequency sub-bands.At each scale,the multifrequency-band test has the chi-square asymptotic null distribution under mild conditions that allow the data to be heteroskedastic,and then it is shown to possess the non-trivial power to detect a large family of local alternatives.Moreover,to over-come the drawback that the multifrequency-band test is sensitive to the choice of the decomposition scale,an automatic multifrequency-band test is further proposed by using a data-driven method to select the scale,and its asymptotic null distribution is chi-square with one degree of freedom.Both multifrequency-band and automatic multifrequency-band tests are shown to have the desirable size and power performance by simulation studies.The practicality of tests proposed is further illustrated by ap-plying them to the S&P500 log-return data to test the efficient market hypothesis.Further,the idea of multifrequency-band tests are extended to check the adequacy of linear time series regression models,and similar tests for unobservable residuals are constructed by correcting the covariance matrix.2.Hysteretic Negative Binomial Autoregression.In the study of integer-valued time series,the introduction of the threshold au-toregressive model allows the nonlinear features in the data to be reflected.As an extension of the classical two-regime threshold process,the hysteretic autoregression enjoys a more flexible regime-switching mechanism and is therefore considered as an important tool to characterize the nonlinearity of the data.In addition,the negative binomial distribution is often used to characterize over-dispersion in data because of its objective property that the variance is larger than the expectation.Hence,to account for the nonlinearity and over-dispersion in integer-valued time series,this thesis studies an observation-driven model for time series of counts,in which the observations are supposed to follow a negative binomial distribution conditioned on past information with the form of the hysteretic autoregression.Stability properties of the model are established by the e-chain and Lyapunov's method.The estimation is obtained by the quasi-likelihood with Poisson-based score function,and the corresponding asymptotic properties are established.Moreover,unlike the classical two-regime threshold process,there are two thresholds to be estimated in the hysteretic autoregression.A reasonable method for selecting search ranges for thresholds is proposed to alleviate the compu-tational pressure.As an application,we bring attention to some features of the daily number of trades of Siparex Croissance which have been overlooked in previous studies.3.Modeling Normalcy-dominant Categorical Time Series.Inspired by the study of air quality level data,this thesis proposes a new model for the normalcy-dominant categorical time series,aiming to provide a new idea for the research of categorical time series from the perspective of integer-value time series.There is a common feature in the air quality level data of major cities in China:two categories,‘excellent'and‘good',dominate,and the percent of other four categories decreases with their air quality level(from‘slightly polluted'to‘severely polluted').In other words,‘excellent'and‘good'occur with relatively larger probabilities than other categories,and we thus treat them as normalcy categories.The proposed model is based on a new zero-one-inflated bounded Poisson distribution with an autoregressive feedback mechanism in intensity.Compared to the classical Poisson distribution and zero-inflated Poisson distribution,the zero-one-inflated bounded Poisson distribution has finite states,{0,1,···,M},and it is thus more suitable to fit the air quality level data(M=5),which have six states corresponding to six air quality levels,respective-ly.Under certain conditions,the stationarity and maximum likelihood estimation are established.Moreover,a Lagrange multiplier test is constructed to detect the inflation phenomenon in the model.Applications find that the model can adequately capture the air quality level data in 30 major cities in China.Most importantly,the fitted models are used to make the overall and dynamic air quality rankings for these cities,and both rankings are rational.