<p>This work consists of four articles concerning Gaussian probability laws with stochastic means and variances. The first paper introduces a new way of approximating the probability distribution of a function of random variables. This is done with a Gaussian probability law with stochastic mean and variance. In the second paper an extension of the Generalized Hyperbolic class of probability distributions is presented. The third paper introduces, using a Gaussian probability law with stochastic mean and variance, a GARCH type stochastic process with skewed innovations. </p><p>In the fourth paper a Lévy process with second order stochastic volatility is presented, option pricing under such a process is also considered.</p>

This work consists of four articles concerning Gaussian probability laws with stochastic means and variances. The first paper introduces a new way of approximating the probability distribution of a function of random variables. This is done with a Gaussian probability law with stochastic mean and variance. In the second paper an extension of the Generalized Hyperbolic class of probability distributions is presented. The third paper introduces, using a Gaussian probability law with stochastic mean and variance, a GARCH type stochastic process with skewed innovations. In the fourth paper a Lévy process with second order stochastic volatility is presented, option pricing under such a process is also considered.

此論文研究有什麼因子會影響台灣股票加權指數報酬率之偏態係數。過去的文獻顯示，交易量和報酬率為可能的因子。實證的結果確實發現，交易量和報酬率顯著地影響偏態係數。 / This study examines the determinants for conditional skewness of the return distribution of the Taiwan Stock Exchange Value-Weighted Index. Important driving factors that affect conditional skewness, based on the theory literature, include trading volumes and returns. To capture the skewness in the data, the family of time series model we consider focuses on the specifications of higher-order moments than mean and volatility that conventional models look at. With the specifications, we are able to test whether the factors, volumes and returns, can influence conditional skewnees of the return distribution. Our results suggest the significance of the factors using data from the Taiwan Stock Exchange Value-Weighted Index.

ABSTRACT It may be of no surprise that water quality data is right-skewed, but what appears to be overlooked by some is that the arithmetic mean and standard deviation most often fail as measures of central tendency in skewed data. When using the arithmetic mean and arithmetic standard deviation with nutrient data, one standard deviation about the arithmetic mean can capture nearly all of the data and extend into negative values. Representing nutrient data this way can be misleading to viewers who are using the statistics, and making assumptions, to understand the characteristics of those waters. Through an in-depth statistical analysis of Florida's nitrogen and phosphorus data, I have found the geometric mean and multiplicative standard deviation capture a better representation of the central region of skewed data. Including the geometric mean and multiplicative standard deviation in the descriptive statistics of nutrient data is relatively simple with today's tools and helps to better describe the data. Adding these statistics can contribute to more effective understanding of nutrient concentrations, better application of data, and the development of better data-derived policy. While the suggestions of this paper are by no means original, it is with added evidence provided by the study of the skewness, distributions, and central regions of 53 nutrient data sets that I intend to help reiterate the argument that a few additional descriptive statistics can greatly empower the communication of data, and because of the ease with which they can now be calculated, there is no excuse to ignore them.

Σκοπός της παρούσας είναι ο έλεγχος της ύπαρξης ημερολογιακών ανωμαλιών στη λειτουργία των διεθνών αγορών, όπως αυτή αποτυπώνεται στην πορεία των χρηματιστηριακών δεικτών κατά την περίοδο 01.01.1988 έως 31.03.2008. Όταν μια ημερολογιακή ανωμαλία γίνεται γνωστή στην αγορά, είναι αναμενόμενο η αντίδραση των επενδυτών στην αναμονή της να βαίνει φθίνουσα με το χρόνο. Για την απόρριψη ή την επιβεβαίωση της υπόθεσης αυτής, στην παρούσα μελέτη, ελέγχεται η ύπαρξη ημερολογιακών ανωμαλιών στις αποδόσεις 15 διεθνών χρηματιστηριακών αγορών τα τελευταία 20 έτη. Κυρίαρχος σκοπός της συγκεκριμένης μελέτης είναι ο έλεγχος της μεταβολής των τάσεων στις προτιμήσεις και τις προσδοκίες των επενδυτών τόσο κατά το πέρασμα των χρόνων όσο και με βάση τη θέση κάθε αγοράς στον παγκόσμιο χάρτη ανάπτυξης. Η εξεταζόμενη περίοδος χωρίζεται σε δύο επιμέρους υποπεριόδους με γνώμονα την διεθνή κρίση που ξεκίνησε στα τέλη του 1999 και επηρέασε καθοριστικά την πορεία της παγκόσμιας οικονομίας. Προσπαθούμε, επομένως, να μελετήσουμε την προσαρμογή των αγορών στις συνθήκες και τα δεδομένα που δημιουργούνται έπειτα από ένα σημαντικό γεγονός. Η μεγάλη βάση του δείγματος - ξεπερνά τα είκοσι έτη, το πλήθος των εξεταζόμενων δεικτών και η προσπάθεια μελέτης διαφορετικών τάσεων με βάση ένα κομβικό σημείο της πορείας της ιστορίας αλλά και με βάση την κατηγοριοποίηση των αγορών ανάλογα με το βαθμό ανάπτυξής τους, διαφοροποιούν την παρούσα μελέτη, και πιστεύουμε ότι αποτελούν ένα ισχυρό κίνητρο για έναν μελετητή ή επενδυτή να αφιερώσει χρόνο στην παρούσα μελέτη. Εξάλλου, τόσο η αξιοσημείωτη διαφοροποίηση των αποτελεσμάτων στις δύο εξεταζόμενες υποπεριόδους, όσο και η χαλαρή τάση που εμφανίζεται σε κάθε κατηγορία αγορών, θεωρούμε ότι δικαιώνουν το εν λόγω εγχείρημα. / When a calendar anomaly becomes acquaintance in the market, the reaction of investors is expected to go declining with the time. For the reject or the confirmation of this affair, in the present study, we check the existence of calendar anomalies in the output of 15 International Stock Exchange markets the last 20 years. The examined period (1988-2008) is separated in two subperiods taking into consideration the international crisis that began in the dues of 1999 in order to be checked the stability of results. The empirical results show important differentiation of results in the two examined subperiods. In any case, the segregation of indicators according to their places in the world market does not appear to attribute substantially conclusions. Specifically in the case of emerging markets, these show that they mark an autonomous movement and to be influenced more by other (internal, mainly) factors.

This thesis includes three chapters regarding influence functions, higher moments, and futures hedging. In Chapter 2, the objective is to use an influence function to better understand semi-kurtosis for use in analyzing peakedness and tail heaviness on one side of a distribution. Also, it is shown that both the right side semi-kurtosis and left side semi-kurtosis summed together, equal kurtosis, so the ratio of semi-kurtosis to kurtosis can be used to analyze asymmetry, as an alternative to skewness. In Chapter 3, the objective is to analyze higher moments of daily, weekly, and monthly stock market returns using large stocks, technology stocks, and small cap stocks. Kurtosis is found to be positive (greater than 3) and statistically significant for all of the daily and weekly stock market returns, indicating peakedness and fat tails. Similar to kurtosis, the left side semi-fourth moment (semi-kurtosis) is also found to be positive (greater than 1.5) for all of daily and weekly returns, indicating peakedness and fat tails on the left sides of the distributions. Skewness is found to be both positive and negative in the daily stock returns data, indicating asymmetry but with no consistent patterns. The fifth moment is also used to analyze asymmetry, as an alternative to skewness. The fifth moment and skewness (third moment) sometimes indicate opposite asymmetry results, as evidenced by different signs for the two moments. This is because the exponent of five for the fifth moment amplifies observations further from the mean, more so than the exponent of three for skewness. In Chapter 4, the objective is to analyze research on futures hedging and to identify the major factors affecting the use of futures hedging by commodity producers. A multifactor conceptual model is developed that explains the factors and subfactors that are likely to affect the commodity producers’ hedging decisions. Factors include industry characteristics, business operation characteristics, management characteristics, futures hedging costs, and substitute risk management instruments. This model provides a more complete understanding of the factors and subfactors affecting futures hedging, and should be of interest to academics and practitioners working with hedging models.

This thesis includes three chapters regarding influence functions, higher moments, and futures hedging. In Chapter 2, the objective is to use an influence function to better understand semi-kurtosis for use in analyzing peakedness and tail heaviness on one side of a distribution. Also, it is shown that both the right side semi-kurtosis and left side semi-kurtosis summed together, equal kurtosis, so the ratio of semi-kurtosis to kurtosis can be used to analyze asymmetry, as an alternative to skewness. In Chapter 3, the objective is to analyze higher moments of daily, weekly, and monthly stock market returns using large stocks, technology stocks, and small cap stocks. Kurtosis is found to be positive (greater than 3) and statistically significant for all of the daily and weekly stock market returns, indicating peakedness and fat tails. Similar to kurtosis, the left side semi-fourth moment (semi-kurtosis) is also found to be positive (greater than 1.5) for all of daily and weekly returns, indicating peakedness and fat tails on the left sides of the distributions. Skewness is found to be both positive and negative in the daily stock returns data, indicating asymmetry but with no consistent patterns. The fifth moment is also used to analyze asymmetry, as an alternative to skewness. The fifth moment and skewness (third moment) sometimes indicate opposite asymmetry results, as evidenced by different signs for the two moments. This is because the exponent of five for the fifth moment amplifies observations further from the mean, more so than the exponent of three for skewness. In Chapter 4, the objective is to analyze research on futures hedging and to identify the major factors affecting the use of futures hedging by commodity producers. A multifactor conceptual model is developed that explains the factors and subfactors that are likely to affect the commodity producers’ hedging decisions. Factors include industry characteristics, business operation characteristics, management characteristics, futures hedging costs, and substitute risk management instruments. This model provides a more complete understanding of the factors and subfactors affecting futures hedging, and should be of interest to academics and practitioners working with hedging models.

This thesis concerns the problem of statistical hypothesis testing for mean vector as well as testing for non-normality in a high-dimensional setting which is called the Kolmogorov condition. Since we consider mainly the first and the second moment in testing for mean vector and we utilize the third and the fourth moment in testing for non-normality, this thesis concerns a more general moment testing problem. The research question is related to a data matrix with $p$ rows, which is the number of parameters and $n$ columns which is the sample size, where $p$ can exceed $n$, assuming that the ratio $\frac{p}{n}$ converges when both the number of parameters and the sample size increase.  The first paper reviews the Dempster's non-exact test for mean vector, with a focus on one-sample case. We investigated its size and power properties compared to Hotelling's $\mathit{T}^2$ test as well as Srivastava's test using Monte Carlo simulation.  The second paper concerns the problem of testing for multivariate non-normality in high-dimensional data. We proposed three test statistics which are based on marginal skewness and kurtosis. Simulation studies are carried out for examining the size and power properties of the three test statistics. / Avhandlingen undersöker hypotesprövning i höga dimensioner, under förutsättning att det så kallad Kolmogorovvillkoret (Kolmogorov condition) är uppfyllt. Villkoret innerbär att antalet parametrar ökar tillsammans med storleken på stickprovet med en konstant hastighet. Till kategorin multivariat analys räknas de statistiska metoder som analyserar stickprov från flerdimensionella fördelningar, särskilt multivariat normalfördelning. För högdimensionella data fungerar klassiska skattningar av kovariansmatris inte tillfredställande eftersom komplexiteten med att skatta den inversa kovariansmatrisen ökar när dimensionen ökar. I den första uppsatsen utförs en genomgång av Dempsters (non-exact) test där skattning av den inversa kovariansmatrisen inte behövs. Istället används spåret (trace) av en kovariansmatris. I den andra uppsatsen testas antagandet om normalfördelning med hjälp av tredje och fjärde ordningens moment. Tre olika testvariabler har föreslagits där sumuleringar också presenteras för att jämföra hur väl en icke-normalfördelning identifieras av testet.

Submitted by Cristina Tessari (tinatessari@gmail.com) on 2016-06-09T13:51:42Z No. of bitstreams: 1 DissertationEPGE_CristinaTessari2016.pdf: 1264081 bytes, checksum: 14e65157457bfe8deea5353bb192a0af (MD5) / Approved for entry into archive by Marcia Bacha (marcia.bacha@fgv.br) on 2016-06-29T14:03:25Z (GMT) No. of bitstreams: 1 DissertationEPGE_CristinaTessari2016.pdf: 1264081 bytes, checksum: 14e65157457bfe8deea5353bb192a0af (MD5) / Approved for entry into archive by Marcia Bacha (marcia.bacha@fgv.br) on 2016-06-29T14:06:59Z (GMT) No. of bitstreams: 1 DissertationEPGE_CristinaTessari2016.pdf: 1264081 bytes, checksum: 14e65157457bfe8deea5353bb192a0af (MD5) / Made available in DSpace on 2016-06-29T14:07:23Z (GMT). No. of bitstreams: 1 DissertationEPGE_CristinaTessari2016.pdf: 1264081 bytes, checksum: 14e65157457bfe8deea5353bb192a0af (MD5) Previous issue date: 2016-03-22 / We use Brazilian data to compute monthly idiosyncratic moments (expected skewness, realized skewness, and realized volatility) for equity returns and assess whether they are informative for the cross-section of future stock returns. Since there is evidence that lagged skewness alone does not adequately forecast skewness, we estimate a cross-sectional model of expected skewness that uses additional predictive variables. Then, we sort stocks each month according to their idiosyncratic moments, forming quintile portfolios. We find a negative relationship between higher idiosyncratic moments and next-month stock returns. The trading strategy that sells stocks in the top quintile of expected skewness and buys stocks in the bottom quintile generates a significant monthly return of about 120 basis points. Our results are robust across sample periods, portfolio weightings, and to Fama and French (1993)’s risk adjustment factors. Finally, we identify a return reversal of stocks with high idiosyncratic skewness. Specifically, stocks with high idiosyncratic skewness have high contemporaneous returns. That tends to reverse, resulting in negative abnormal returns in the following month. / In the first chapter, we test some stochastic volatility models using options on the S&P 500 index. First, we demonstrate the presence of a short time-scale, on the order of days, and a long time-scale, on the order of months, in the S&P 500 volatility process using the empirical structure function, or variogram. This result is consistent with findings of previous studies. The main contribution of our paper is to estimate the two time-scales in the volatility process simultaneously by using nonlinear weighted least-squares technique. To test the statistical significance of the rates of mean-reversion, we bootstrap pairs of residuals using the circular block bootstrap of Politis and Romano (1992). We choose the block-length according to the automatic procedure of Politis and White (2004). After that, we calculate a first-order correction to the Black-Scholes prices using three different first-order corrections: (i) a fast time scale correction; (ii) a slow time scale correction; and (iii) a multiscale (fast and slow) correction. To test the ability of our model to price options, we simulate options prices using five different specifications for the rates or mean-reversion. We did not find any evidence that these asymptotic models perform better, in terms of RMSE, than the Black-Scholes model. In the second chapter, we use Brazilian data to compute monthly idiosyncratic moments (expected skewness, realized skewness, and realized volatility) for equity returns and assess whether they are informative for the cross-section of future stock returns. Since there is evidence that lagged skewness alone does not adequately forecast skewness, we estimate a cross-sectional model of expected skewness that uses additional predictive variables. Then, we sort stocks each month according to their idiosyncratic moments, forming quintile portfolios. We find a negative relationship between higher idiosyncratic moments and next-month stock returns. The trading strategy that sells stocks in the top quintile of expected skewness and buys stocks in the bottom quintile generates a significant monthly return of about 120 basis points. Our results are robust across sample periods, portfolio weightings, and to Fama and French (1993)’s risk adjustment factors. Finally, we identify a return reversal of stocks with high idiosyncratic skewness. Specifically, stocks with high idiosyncratic skewness have high contemporaneous returns. That tends to reverse, resulting in negative abnormal returns in the following month.

Defining ambiguity as investor's uncertainty about the precision of the observed information, Chapter One constructs an empirical measure of ambiguity based on analysts' earnings forecast information, and finds that the market tends to react more negatively to highly ambiguous bad news, while it tends to be less responsive to highly ambiguous good news. This result supports the theoretical argument of Epstein and Schneider (2003, 2008) that ambiguity-averse investors take a worst-case assessment of the information precision, when they are uncertain about the information precision. In addition, Chapter One shows that returns on stocks exposed to highly ambiguous and intangible information are more negatively skewed. Chapter Two finds that certain traders are informed about either the forthcoming analysts' forecasts or long-term value of the stock, and informed traders prefer to use medium-size trades to exploit their private information advantage. Specifically, medium-size trade imbalance prior to the forecast announcements is positively correlated with the nature of forecast revisions, while in the days immediately after the forecasts medium-size trade imbalance is positively correlated with future stock returns for up to four months. Small-size trade imbalance is also positively correlated with future returns but only following downward revisions. In contrast, it is also shown that large trades placed right after the forecasts are unprofitable and generate slightly negative profits in the long run. Overall, our results are consistent with the "stealth trading hypothesis" proposed by Barclay and Warner (1993).