Modelling Discrete-Valued Time Series And Applied Research

In economic and financial activities, discrete time series is encontered frequently. For such data, the process follows a discrete distribution, such as the number of large claims of a accedents in the insurance market, Special Treatment companies processed particularly of listed company, and daily large transactions and so on. Therefore, it is essential in economic or financial analysis to build relevant time series models that enables us to study the statistical properties of the discrete stochstic processes. Data evidence extracted from statistical models will deepen our understanding of economic activities and their operation rules, as well as to improve our ability in solving the economic problem’s. However, there is a little study in current financial time series literature, especially concerning model selection, unit roots test, and forecast, all these aspects need to be studied thoroughly.This thesis focuses on five topics:(1)In both INAR(p) and INMA(q) models, I compared the performances of maximum likelihood estimator, conditional least squares estimator and Yule-Walker estimator from various scenarios; (2) I discussesed the rationale through the means of traditional model selection criteria and cross validation (such as AICc, BIC, EIC and K-CV method) for the purpose determining the order p in the Poisson INAR(p) model; (3) I discussed the theory of discrete-valued random walk process, and in particular I proved the limiting distribution for auto-regressive coefficients of the unit root process; (4) I introduced random coefficient integer-valued moving average model,and established the existence and ergodic properties, as well as some moment estimator in some especial cases; (5)Applying the INAR(p) model, I studied the trading volume of personal shares in Chinese Stock markets, including point prediction and probability based prediction for trading volume in the future.The main contributions of this thesis are detailed as fllows:1. I present a review regarding severals major established methods available for INAR(p) model, including maximum likelihood estimator(MLE), conditional least squares estimator(CLSE), and Yule-Walker estimator(YWE). In the Meanwhile, I discuss both advantages and disadvantages of different estimatin methods through Monte Carlo simulation studies from 4 perspectives; They are different lags, different model coefficients, and different error distributions, different sample size. For the cases of INAR(1) model, the simulaton results allow us to draw following conclusions:(1) When the alpha and lambda are relatively small, CLSE performs the best;(2) When the alpha and lambda are of moderate size, MLE performs the best. For INAR(p) model, it is in general difficult to determine the dominant established method.I reviewed major estimation methods in INMA(q) models including conditional least squares estimator(CLSE), Yule-Walker estimator(YWE), and GMM estimator (GMME). I concluded the comparisons concerning their performance in different settings between YWE and CLSE with different order of lags, different error distributions and different sample sizes. In the cases of large sample size, I achieved the following conclusions:(ⅰ) Based on the Bias, YWE performs better; Based on the MSE, CLSE perfoms better;(ⅱ) Based on the Bias or MSE of error distributions parameters, CLSE perfoms better.2.I present the framework of model selectoins, with detail concerning AICc、BIC, EIC and cross-validation. I explored the possibility of adophing these selection methods into the INAR(p) models. Using Monte Carlo simulation studies, I attempted to vadidated the selection of oreder p in possions INAR(p) models in which MLE is proposed by Rujui BU.McCabe and Hadri(2008). We found that when the order of lags is low, the performances of all three method are satisfactory. But when the order of lags is large, the positive selection rate of these three method can reach more than 50%. In contrast, using the cross-validation method, the selection results is unstable, regarding of small or large lags.3. I generalized the concept of integral-valued random walk process. I have shown that the proposed six forms of random walk processes all converge to constants respective, and their convergence rates are faster than the forms of corresponding continuous random walk processes. Moreover, I proved that in the unit root process with no intercept or time trend, the limit distribution of model coefficients CLSE dose not take a functional form of the Wiener process, instead converges in probability to the true values one. Monte Carlo simulation study was also used to confirm the theoretical findings.4. I enriched the clan of INARMA models, and extended Poisson INMA(q) models such that the coefficient obey Beta distributions. I theoretical proved the existence of proposed processes. Also, I proved that the proposed process is a stable process, and its mean and variance have ergodicity. Varing a andβparameters in the moment estimator’s performance of the model parameter through Monte Carlo Simulation Method.The results show that the proposed estimator works well.5. There is no publications regarding studies of the number of personal share’s transaction in China’s Stock markets by using INAR(p) models as this thesis does. This thesis studies the features of trading volume of Lutianhua’s personal shares in a given day morning, and predicts probability of the trading volume. The probability of predictions provides more meaningful information to investors.