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We discuss the Normal inverse Gaussian (NIG) distribution in modeling volatility in the financial markets. Refining the work of Barndorff-Nielsen (1997) and Andersson (2001), we introduce a new parameterization of the NIG distribution to build the GARCH(p,q)-NIG model. This new parameterization allows the model to be a strong GARCH in the sense of Drost and Nijman (1993). It also allows us to standardized the observed returns to be i.i.d., so that we can use standard inference methods when we evaluate the fit of the model. We use the realized volatility (RV), calculated from intraday data, to standardize the returns of the ECU/USD foreign exchange rate. We show that normality cannot be rejected for the RV-standardized returns, i.e., the Mixture-of-Distributions Hypothesis (MDH) of Clark (1973) holds. {We build a link between the conditional RV and the conditional variance. This link allows us to use the conditional RV as a proxy for the conditional variance. We give an empirical justification of the GARCH-NIG model using this approximation. In addition, we introduce a new General GARCH(p,q)-NIG model. This model has as special cases the Threshold-GARCH(p,q)-NIG model to model the leverage effect, the Absolute Value GARCH(p,q)-NIG model, to model conditional standard deviation, and the Threshold Absolute Value GARCH(p,q)-NIG model to model asymmetry in the conditional standard deviation. The properties of the maximum likelihood estimates of the parameters of the models are investigated in a simulation study.

In 1988, Owen introduced empirical likelihood as a nonparametric method for constructing confidence intervals and regions. Since then, empirical likelihood has been studied extensively in the literature due to its generality and effectiveness. It is well known that empirical likelihood has several attractive advantages comparing to its competitors such as bootstrap: determining the shape of confidence regions automatically using only the data; straightforwardly incorporating side information expressed through constraints; being Bartlett correctable. The main part of this thesis extends the empirical likelihood method to several interesting and important statistical inference situations. This thesis has four components. The first component (Chapter II) proposes a smoothed jackknife empirical likelihood method to construct confidence intervals for the receiver operating characteristic (ROC) curve in order to overcome the computational difficulty when we have nonlinear constrains in the maximization problem. The second component (Chapter III and IV) proposes smoothed empirical likelihood methods to obtain interval estimation for the conditional Value-at-Risk with the volatility model being an ARCH/GARCH model and a nonparametric regression respectively, which have applications in financial risk management. The third component(Chapter V) derives the empirical likelihood for the intermediate quantiles, which plays an important role in the statistics of extremes. Finally, the fourth component (Chapter VI and VII) presents two additional results: in Chapter VI, we present an interesting result by showing that, when the third moment is infinity, we may prefer the Student's t-statistic to the sample mean standardized by the true standard deviation; in Chapter VII, we present a method for testing a subset of parameters for a given parametric model of stationary processes.

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。

My dissertation consists of three essays in three different areas: international trade; agricultural markets; and nonparametric econometrics. The first and third essays are theoretical papers, while the second essay is empirical. In the first essay, I developed a political economy model of trade agreements where the set of policy instruments are endogenously determined, providing a rationale for countervailing duties (CVDs). Trade-related policy intervention is assumed to be largely shaped in response to rent seeking demand as is often shown empirically. Consequently, the uncertain circumstance during the lifetime of a trade agreement involves both economic and rent seeking conditions. The latter approximates the actual trade policy decisions more closely than the externality hypothesis and thus provides scope for empirical testing. The second essay tests whether normal mixture (NM) generalized autoregressive conditional heteroscedasticity (GARCH) models adequately capture the relevant properties of agricultural commodity prices. Volatility series were constructed for ten agricultural commodity weekly cash prices. NM-GARCH models allow for heterogeneous volatility dynamics among different market regimes. Both in-sample fit and out-of-sample forecasting tests confirm that the two-state NM-GARCH approach performs significantly better than the traditional normal GARCH model. For each commodity, it is found that an expected negative price change corresponds to a higher volatility persistence, while an expected positive price change arises in conjunction with a greater responsiveness of volatility. In the third essay, I propose an estimator for a nonparametric additive unconditional quantile regression model. Unconditional quantile regression is able to assess the possible different impacts of covariates on different unconditional quantiles of a response variable. The proposed estimator does not require d-dimensional nonparametric regression and therefore has no curse of dimensionality. In addition, the estimator has an oracle property in the sense that the asymptotic distribution of each additive component is the same as the case when all other components are known. Both numerical simulations and an empirical application suggest that the new estimator performs much better than alternatives. / the Canadian Agricultural Trade Policy and Competitiveness Research Network, the Structure and Performance of Agriculture and Agri-products Industry Network, and the Institute for the Advanced Study of Food and Agricultural Policy.

This thesis consists of four research papers. The main focus is on building the multivariate Conditional Correlation (CC-) GARCH models. In particular, emphasis lies on considering an extension of CC-GARCH models that allow for interactions or causality in conditional variances. In the first three chapters, misspecification testing and parameter restrictions in these models are discussed. In the final chapter, a computer package for building major variants of the CC-GARCH models is presented. The first chapter contains a brief introduction to the CC-GARCH models as well as a summary of each research paper. The second chapter proposes a misspecification test for modelling of the conditional variance part of the Extended Constant CC-GARCH model. The test is designed for testing the hypothesis of no interactions in the conditional variances. If the null hypothesis is true, then the conditional variances may be described by the standard CCC-GARCH model. The test is constructed on the Lagrange Multiplier (LM) principle that only requires the estimation of the null model. Although the test is derived under the assumption of the constant conditional correlation, the simulation experiments suggest that the test is also applicable to building CC-GARCH models with changing conditional correlations. There is no asymptotic theory available for these models, which is why simulation of the test statistic in this situation has been necessary. The third chapter provides yet another misspecification test for modelling of the conditional variance component of the CC-GARCH models, whose parameters are often estimated in two steps. The estimator obtained through these two steps is a two-stage quasi-maximum likelihood estimator (2SQMLE). Taking advantage of the asymptotic results for 2SQMLE, the test considered in this chapter is formulated using the LM principle, which requires only the estimation of univariate GARCH models. It is also shown that the test statistic may be computed by using an auxiliary regression. A robust version of the new test is available through another auxiliary regression. All of this amounts to a substantial simplification in computations compared with the test proposed in the second chapter. The simulation experiments show that, under both under both Gaussian and leptokurtic innovations, as well as under changing conditional correlations, the new test has reasonable size and power properties. When modelling the conditional variance, it is necessary to keep the sequence of conditional covariance matrices positive definite almost surely for any time horizon. In the fourth chapter it is demonstrated that under certain conditions some of the parameters of the model can take negative values while the conditional covariance matrix remains positive definite almost surely. It is also shown that even in the simplest first-order vector GARCH representation, the relevant parameter space can contain negative values for some parameters, which is not possible in the univariate model. This finding makes it possible to incorporate negative volatility spillovers into the CC-GARCH framework. Many new GARCH models and misspecification testing procedures have been recently proposed in the literature. When it comes to applying these models or tests, however, there do not seem to exist many options for the users to choose from other than creating their own computer programmes. This is especially the case when one wants to apply a multivariate GARCH model. The last chapter of the thesis offers a remedy to this situation by providing a workable environment for building CC-GARCH models. The package is open source, freely available on the Internet, and designed for use in the open source statistical environment R. With this package can estimate major variants of CC-GARCH models as well as simulate data from the CC-GARCH data generating processes with multivariate normal or Student's t innovations. In addition, the package is equipped with the necessary functions for conducting diagnostic tests such as those discussed in the third chapter of this thesis. / <p>Diss. Stockholm : Handelshögskolan, 2010. Sammanfattning jämte 4 uppsatser.</p>

Forecasting volatility with precision in financial market is very important. This paper examines the use of various forms of GARCH models for forecasting volatility. Three financial data sets from Japan (NIKKEI 225 index), the United States (Standard & Poor 500) and Germany (DAX index) are considered. A number of GARCH models, such as EGARCH, IGARCH, TGARCH, PGARCH and QGARCH models with normal distribution and student’s t distribution are used to fit the data sets and to forecast volatility. The Maximum Likelihood method and the Bayesian approach are used to estimate the parameters in the family of the GARCH models. The results show that the QGARCH model under student’s t distribution is the precise model for the NIKKEI 225 index in terms of fitting the data and forecasting volatility. The TGARCH under the student’s t distribution fits the S&P 500 index data better while the traditional GARCH model under the same distribution performs better in forecasting volatility. The PGARCH with student’s t distribution is the precise model for the DAX index in terms of fitting the data and forecasting volatility. / Department of Mathematical Sciences

本篇論文的目的在驗證台股期貨報酬率與其波動度之間的相對應關係是由槓桿效果或是波動度回饋效果之因果關係所驅動，並且分別以日資料以及高頻率日內資料進行實證。實證結果發現在高頻率日內資料的應用下，能夠比日資料揭露出更詳細的波動度資訊，將報酬率與波動度間的對應關係描繪得更加明瞭。且在大多數資料期間內，同期下，台股期貨報酬率與其波動度之間會呈現負相關性，而負相關的程度會隨著報酬率遞延期數越長而逐漸遞減，因此可以發現報酬率與其波動度間呈現一個經由報酬率進而影響波動度的對應關係，與槓桿效果的因果關係雷同。最後，本文亦採用了常見的波動度預測模型，歷史模擬法、GARCH(1,1)模型、EGARCH(1,1)模型以及GJR-GARCH(1,1)模型，觀察這些波動度模型所預測出之波動度是否含有上述驗證的資訊意涵，並比較各波動度模型的預測能力，結果發現GJR-GARCH模型於樣本外期間所預測之波動度，其與報酬率之間不但具有槓桿效果的因果關係，且預測能力亦於四個波動度模型中表現最佳。

台灣風險基礎資本額制度實施至今已將近七年，但風險係數卻從未調整，本研究主要針對股票指數與匯率之風險係數探討其是否有更新之必要，藉由1986年12月至2009年12月之資料，利用GARCH模型及EGARCH模型進行風險係數之估計，除了和風險基礎資本額制度相同，以風險值為考量外，另外加入條件尾端期望值，並比較其與風險值之差別。 實證結果發現，僅部分財務時間序列有顯著之槓桿效果，因此使用GARCH模型估計風險係數較為合適；所估計之風險係數，無論是股價指數或是匯率，其估計結果皆比現行標準高出許多。 / In Taiwan, Risk-based capital (RBC) is set up in 2003. From 2003 until now, no matter how the economical environment has changed, the risk factors have remained all the same.This research mainly focuses on the risk factors of stock index and foreign exchange and wants to know if the risk factors need to be changed. The data this research encompasses is from December 1986 to December 2009.The risk factors are estimated by GARCH model and EGARCH model, utilizing not only the VaR but also the conditional tail expectation (CTE). From the result, only a few financial time series have shown leverage effect, therefore it is indeed more appropriate to apply GARCH model in risk factors estimation. Moreover, the risk factors from the result of this research, whether it is stock index or foreign exchange rate, are significantly higher than the risk factors standard applicable in Taiwan at the present.

Hamilton (1989)發展出馬可夫轉換模型(Markov-switching Model)，由於該模型允許母體參數在不同時期，具有間斷性跳動性質，且跳動次數並不限定為一，並利用馬可夫鏈(Markov chain)的機制來掌控狀態間切換，解決混合分配模型狀態跳動毫無規則的問題，將可適可掌握金融與經濟變數所面臨的結構改變，以及解決在計測風險值(valued at risk)過程中，所存在報酬分配的高峰厚尾問題。 本文非僅是嘗試另一種方法，而是我們在探討股市報酬波動與景氣循環變數行為後，推判它較能夠捕捉實際的報酬波動與景氣循環行為。我們除作過去文獻較未顧及的，系統性地分析各種潛在風險值計測方法所適用與不適用報酬率變異情境，並嘗試使用允許參數來自不同波動狀態，對傳統ARCH模型加以修正之SWARCH模型，希對股市報酬波動提供更佳的分析。在景氣循環探討，針對馬可夫轉換模型加以修正，掌握台灣與南韓經濟結構與美國及日本等國迥異的問題。此外，我們也回溯、討論各種處理財經變數結構問題之實證模型差異，分析馬可夫轉換模型相對優、劣點。