<p>In recent years, two estimators have been proposed to correct the bias exhibited by the leastsquares (LS) estimator of the lagged dependent variable (LDV) coefficient in dynamic regression models when the sample is finite. They have been termed as &lsquo / mean-unbiased&rsquo / and &lsquo / medianunbiased&rsquo / estimators. Relative to other similar procedures in the literature, the two locationbased estimators have the advantage that they offer an exact and uniform methodology for LS estimation of the LDV coefficient in a first order autoregressive model with or without exogenous regressors i.e. ARX(1).</p> <p><br /> However, no attempt has been made to accurately establish and/or compare the statistical properties among these estimators, or relative to those of the LS estimator when the LDV coefficient is restricted to realistic values. Neither has there been an attempt to&nbsp / compare their performance in terms of their mean squared error (MSE) when various forms of the exogenous regressors are considered. Furthermore, only implicit confidence intervals have been given for the &lsquo / medianunbiased&rsquo / estimator. Explicit confidence bounds that are directly usable for inference are not available for either estimator. In this study a new estimator of the LDV coefficient is proposed / the &lsquo / most-probably-unbiased&rsquo / estimator. Its performance properties vis-a-vis the existing estimators are determined and compared when the parameter space of the LDV coefficient is restricted. In addition, the following new results are established: (1) an explicit computable form for the density of the LS estimator is derived for the first time and an efficient method for its numerical evaluation is proposed / (2) the exact bias, mean, median and mode of the distribution of the LS estimator are determined in three specifications of the ARX(1) model / (3) the exact variance and MSE of LS estimator is determined / (4) the standard error associated with the determination of same quantities when simulation rather than numerical integration method is used are established and the methods are compared in terms of computational time and effort / (5) an exact method of evaluating the density of the three estimators is described / (6) their exact bias, mean, variance and MSE are determined and analysed / and finally, (7) a method of obtaining the explicit exact confidence intervals from the distribution functions of the estimators is proposed.</p> <p><br /> The discussion and results show that the estimators are still biased in the usual sense: &lsquo / in expectation&rsquo / . However the bias is substantially reduced compared to that of the LS estimator. The findings are important in the specification of time-series regression models, point and interval estimation, decision theory, and simulation.</p>

Magister Scientiae - MSc / In recent years, two estimators have been proposed to correct the bias exhibited by the leastsquares (LS) estimator of the lagged dependent variable (LDV) coefficient in dynamic regression models when the sample is finite. They have been termed as &lsquo;mean-unbiased&rsquo; and &lsquo;medianunbiased&rsquo; estimators. Relative to other similar procedures in the literature, the two locationbased estimators have the advantage that they offer an exact and uniform methodology for LS estimation of the LDV coefficient in a first order autoregressive model with or without exogenous regressors i.e. ARX(1). However, no attempt has been made to accurately establish and/or compare the statistical properties among these estimators, or relative to those of the LS estimator when the LDV coefficient is restricted to realistic values. Neither has there been an attempt to&nbsp; compare their performance in terms of their mean squared error (MSE) when various forms of the exogenous regressors are considered. Furthermore, only implicit confidence intervals have been given for the &lsquo;medianunbiased&rsquo; estimator. Explicit confidence bounds that are directly usable for inference are not available for either estimator. In this study a new estimator of the LDV coefficient is proposed; the &lsquo;most-probably-unbiased&rsquo; estimator. Its performance properties vis-a-vis the existing estimators are determined and compared when the parameter space of the LDV coefficient is restricted. In addition, the following new results are established: (1) an explicit computable form for the density of the LS estimator is derived for the first time and an efficient method for its numerical evaluation is proposed; (2) the exact bias, mean, median and mode of the distribution of the LS estimator are determined in three specifications of the ARX(1) model; (3) the exact variance and MSE of LS estimator is determined; (4) the standard error associated with the determination of same quantities when simulation rather than numerical integration method is used are established and the methods are compared in terms of computational time and effort; (5) an exact method of evaluating the density of the three estimators is described; (6) their exact bias, mean, variance and MSE are determined and analysed; and finally, (7) a method of obtaining the explicit exact confidence intervals from the distribution functions of the estimators is proposed. The discussion and results show that the estimators are still biased in the usual sense: &lsquo;in expectation&rsquo;. However the bias is substantially reduced compared to that of the LS estimator. The findings are important in the specification of time-series regression models, point and interval estimation, decision theory, and simulation. / South Africa

過去實證研究多以時間序列模型搭配 GARCH 模型針對台灣股價指數進行分析。然而，Gourieroux and Zakoian(2017) 提出，當一時間序列具有泡沫現象時，noncausal Cauchy AR(1) process 是可能的優選模型。此外，Sarno and Taylor(1999) 的研究認為，台灣股價指數具有泡沫現象，故我們以 noncausal Cauchy AR(1) with Gaussian component 分析台灣股價指數，進而判斷其泡沫效果係來自 noncausal linear process 之 local explosive，並根據 noncausal Cauchy AR(1) 與 Gaussian component 之係數變動，捕捉泡沫效果之形成與來源。 / Most of the previous studies focused on analyzing Taiwan Stock Price Index using time series models with GARCH effects. However, Gourieroux and Zakoian (2017) have demonstrated that noncausal Cauchy AR(1) process may be a possible model in which the bubbles are observed. Besides, according to the studies of Sarno and Taylor (1991), some bubbles exactly existed in Taiwan Stock Price Index before 1990. Accordingly, this study aims at investigating the possible bubbles in Taiwan Stock Price Index from 2005 to 2015 by employing noncausal Cauchy AR(1) with Gaussian component method. As a result, we find out he bubbles which modeled by the noncausal linear process are local explosive. And based on the changes of the coefficients from noncausal Cauchy AR(1) and Gaussian component, this study successfully captures the form of bubbles.

In this thesis, we focus on a forecasting tool, Martingale Model of Forecast Evolution (MMFE), to model the evolution of forecasts in a production-distribution system. Additive form is performed to represent the evolution process. Variance-Covariance (VCV) matrix is defined to express the forecast updates. The selected demand pattern is stationary and it is normally distributed. It follows an Autoregressive Order-1 (AR(1)) model. Two forecasting procedures are selected to compare the MMFE with. These are MA (Moving average) and ES (Exponential smoothing) methods. A production-distribution model is constructed to represent a two-stage supply chain environment. The performance measures considered in the analyses are the total costs, fill rates and forecast accuracy observed in the operation of the production-distribution system. The goal is to demonstrate the importance of good forecasting in supply chain environments.

自從90年代以來，許多機構因為金融商品的操縱不當或是金融風暴的衝擊數度造成全球金融市場的動盪，使得風險管理的重要性與日俱增，而量化風險模型的準確性也益受重視，基於財務資料的相關性質如異質變異、厚尾現象等，本文主要結合AR(1)-GARCH(1,1)模型、極值理論、copula函數三種模型應用在風險值的估算，且將報酬分配的假設區分為三類，一是無母數模型的歷史模擬法，二是基於常態分配假設下考量隨機波動度的有母數模型，三是利用歷史資料配適尾端分配的極值理論法來對聯電、鴻海、國泰金、中鋼四檔個股和台幣兌美元、日圓兌美元、英鎊兌美元三種外匯資料作一日風險值、十日風險值、組合風險值的測試。 實證結果發現，在一日風險值方面，95%信賴水準下以動態風險值方法表現相對較好，99%信賴水準下動態極值理論法和動態歷史模擬法皆有不錯的估計效果；就十日風險值而言，因為未來十日資產的報酬可能受到特定事件影響，所以估計上較為困難，整體看來在99%信賴水準下以條件GPD＋蒙地卡羅模擬的表現相對較理想；以組合風險值來說， copula、Clayton copula+GPD marginals模擬股票或外匯組合的聯合分配不論在95%或99%信賴水準下對其風險值的估計都獲得最好的結果；雖然台灣個股股價受到上下漲跌幅7%的限制，台幣兌美元的匯率也受到央行的干涉，但以極值理論來描述資產尾端的分配情形相較於假設其他兩種分配仍有較好的估計效果。

The hidden state Markov (HSM) model is introduced as a new conceptual framework for modelling long-term persistence in hydrological time series. Unlike the stochastic models currently used, the conceptual basis of the HSM model can be related to the physical processes that influence long-term hydrological time series in the Australian climatic regime. A Bayesian approach was used for model calibration. This enabled rigourous evaluation of parameter uncertainty, which proved crucial for the interpretation of the results. Applying the single site HSM model to rainfall data from selected Australian capital cities provided some revealing insights. In eastern Australia, where there is a significant influence from the tropical Pacific weather systems, the results showed a weak wet and medium dry state persistence was likely to exist. In southern Australia the results were inconclusive. However, they suggested a weak wet and strong dry persistence structure may exist, possibly due to the infrequent incursion of tropical weather systems in southern Australia. This led to the postulate that the tropical weather systems are the primary cause of two-state long-term persistence. The single and multi-site HSM model results for the Warragamba catchment rainfall data supported this hypothesis. A strong two-state persistence structure was likely to exist in the rainfall regime of this important water supply catchment. In contrast, the single and multi-site results for the Williams River catchment rainfall data were inconsistent. This illustrates further work is required to understand the application of the HSM model. Comparisons with the lag-one autoregressive [AR(1)] model showed that it was not able to reproduce the same long-term persistence as the HSM model. However, with record lengths typical of real data the difference between the two approaches was not statistically significant. Nevertheless, it was concluded that the HSM model provides a conceptually richer framework than the AR(1) model. / PhD Doctorate

長期性資料(longitudinal data)，是指對同一個觀察個體(subject)或實驗單位(experiment unit)，在不同時間點上重複觀察或測量一個或多個變數。雖然觀察個體之間互相獨立，但就同一個個體而言，不同時間的觀察或測量常常是有相關性的。且觀察的個體之間可能由於一些無法測量的環境因素造成個體之間有差異，因此在傳統橫斷面分析中，假設其有相同迴歸係數的邊際模型可能不合理。隨機效果模型可以解決長期資料分析的相關，並假設每個個體的迴歸係數不同；此模型不但可以說明橫斷面資料的cohort效果，也可直接解釋長期資料的age效果；更可以區分個體之間與個體之內的變異。 本研究以1995年至2000年台灣11個產業中的100家公司之每股盈餘與各財務比率，作為實證分析的資料；分別配適每股盈餘與時間、產業別、時間產業別交互作用及財務比率及排除每股盈餘有異常值後之邊際效果模型(一般迴歸分析)及隨機效果模型，並比較其參數估計之異同。實證結果顯示，一般迴歸分析與假設誤差不相關且等變異下的隨機效果模型參數估計相似，但後者能區分變異為個體之間(between-subjects)與個體之內(within-subject)的變異。而假設誤差不相關且不等變異與假設誤差服從AR(1)且不等變異下的隨機效果模型估計相近。實證結果並顯示，在排除異常值後的模型參數估計，一般迴歸分析不論是估計值及顯著性大多沒有很大差別；而隨機效果模型的估計在排除異常值前後較有差別。特別是現金流量比率(CFR)原本為不顯著變數，在排除異常值後的模型配適全部變顯著性變數。 / The defining characteristic of a longitudinal study is that individuals are measured repeatedly through time. Although it is independent between subjects, the set of observations on one subject tends to be inter-correlated. Because there is some natural heterogeneity due to unmeasured factors between subjects, it is not corrected to assume they have the same regression coefficients. A random effect model is a reasonable description about the different regression coefficients, and it can resolve the inter-correlation of the observations on one subject. The major advantages of the random effect model are its capacity to separate what in the context of population studies are called cohort and age effects, and it can distinguish the variations between subjects and within subjects. This study describes the marginal model and random effect model, and shows their difference by real data analysis. We apply these models to the earnings per share (EPS) and other financial ratios of one hundred companies in Taiwan, which are distributed in eleven industries. The results show that the parameter estimates of the marginal model and random effect model are similar when error structure is independent and of equal variance. Furthermore, the latter can distinguish the variations between subjects and within subjects. However, the residual analysis reveals that the error structure may not be constant. Therefore, we consider heteroscedasticity error in random effect model. We also assume that error follows an autoregressive process (e.g. AR(1) model), which leads to the optimum among our results in terms of residual analysis. There are some observations that appear to be outlying from the majority of data. The results show little difference in the marginal models no matter whether those outliers are included. However, we obtain different results in the random effect models. Especially, the variable of “cash flow ratio” becomes significant once those potential outliers have been excluded, while it is not significant when all cases are fitted in the model.

本論文主要在探討以移動平均法則為基礎的簡單技術分析指標，以及時間序列模型在台灣股票市場是否具有獲利能力，研究期間為1987/01/01-2006/12/31共20年的樣本期間。我們發現只有使用(1,50,0)和(1,50,0.01) 這兩個移動平均交易法則時才有顯著的報酬；並以AR(1)-GARCH(1,1)-M作為時間序列的預測模型。研究發現在股價上漲的時候，技術分析指標的確有較好的預測能力；而在股價下跌時，利用時間序列模型有較佳的獲利能力。因為技術分析指標與時間序列模型分別捕捉到不同的資訊，將兩預測工具結合在一起應該可以得到一個更好的組合預測指標。本文的實證研究發現此一組合預測指標，不管是在多頭或空頭期間時，都可以比使用單一分析工具獲得更高報酬。

In time series models, autoregressive processes are one of the most popular stochastic processes, which are stationary under certain conditions. In this study we consider nonstationary autoregressive models of order one, which have iid random errors. One of the important nonstationary time series models is the unit root process in AR (1), which simply implies that a shock to the system has permanent effect through time. Therefore, testing unit root is a very important problem. However, under nonstationarity, any estimator of the autoregressive coefficient does not have a known exact distribution and the usual t &ndash / statistic is not accurate even if the sample size is very large. Hence,Wiener process is invoked to obtain the asymptotic distribution of the LSE under normality. The first four moments of under normality have been worked out for large n. In 1998, Tiku and Wong proposed the new test statistics and whose type I error and power values are calculated by using three &ndash / moment chi &ndash / square or four &ndash / moment F approximations. The test statistics are based on the modified maximum likelihood estimators and the least square estimators, respectively. They evaluated the type I errors and the power of these tests for a family of symmetric distributions (scaled Student&rsquo / s t). In this thesis, we have extended this work to skewed distributions, namely, gamma and generalized logistic.