Contributions published at Empirical Finance (Marc Paolella)

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The saddlepoint method of approximation is attributed to Daniels (1954), and can be described in basic terms as yielding an accurate and usually fast and very numerically reliable approximation to the mass or density function (hereafter pdf), and the cumulative distribution function (cdf), of a random variable, say X, based on knowledge of its moment generating function (mgf). Denote the latter by $M_{X}(s)$, where s is the real argument of the function, such that s is contained in the convergence strip of $M_{X}(s)$, to be defined below. Several surveys and monographs are available; the best starting point is the currently definitive exposition in Butler (2007), along with the first textbook dedicated to the subject, Jensen (1995). Our goal is to outline the basics of the methodology in the easiest way possible, and then to illustrate a small subset of its many applications.

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It has been nearly 50 years since the appearance of the pioneering paper of Mandelbrot (1963) on the non-Gaussianity of financial asset returns, and their highly fat-tailed nature is now one of the most prominent and accepted stylized facts. The recent book by Jondeau et al. (2007) is dedicated to the topic, while other chapters and books discussing the variety of non-Gaussian distributions of use in empirical finance include McDonald (1997), Knight and Satchell (2001), and Paolella (2007).

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Standard statistical methods in the empirical economics and finance literature are mostly applicable to data that is aggregated on equally spaced time points. However, a key characteristic of many economic and financial variables is that they occur randomly and are observed irregularly in time. Since the pathbreaking work of Robert Engle in the last years of the twentieth century, there are new approaches that do not require aggregated data but are able to account for their irregular timing nature. These developments were mainly supported by the increasing availability of high frequency transaction data due to the implementation of electronic order recording systems at stock exchanges all over the world. Typically, financial markets data are irregularly observed along the time axis. As pointed out by some authors, time series analysis of fixed time interval data annihilates the natural timing dependence of transaction data and possibly neglects relevant information. Further, the selection of inappropriate equidistant aggregation schemes and the exclusion of data points might lead to misspecifications. Hence, the inclusion of all events in an empirical analysis provides additional information about the timing relation of transaction variables and allows to revisit old and to analyze new questions delivered by financial markets theory. The statistical modeling framework to account for characteristics of irregularly spaced event data is provided by the theory of point processes. A point process statistically describes the history of events that occur consecutively in time. A process consisting of points at which we simultaneously observe variables that mark the points is conceived as marked point process. This thesis' aim is to present new univariate and multivariate empirical point processes applied in the field of financial and monetary econometrics. In particular, this thesis analyzes the following topics. In the second chapter, we suggest a univariate discrete marked point process model for the federal funds rate target and investigate its point and probability forecast performance. Chapter three presents a model for daily return variation. Total daily return variation is disentangled into a continuous and jump variation component. While daily continuous variation is modeled by an autoregressive conditional time series model, irregularly occurring jumps are conceived as a univariate marked point process. Finally, the fourth chapter introduces a new information share that measures the home and foreign market share in price discovery. For this purpose, a multivariate point process based on high frequency transaction data is used.

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Two computable expressions for the exact density of a ratio of quadratic forms in Gaussian random vectors are derived, one of which is restricted to special cases of the problem. Ratios of this type are ubiquitous in econometrics, but their density, unlike the corresponding cumulative distribution function, has not received much attention to date. The new algorithms complement those available for the latter. The included performance study demonstrates the accuracy of the two algorithms, both absolute and relative to each other, and allows general recommendations on their use to be made.

An asymmetric multivariate generalization of the recently proposed class of normal mixture GARCH models is developed. Issues of parametrization and estimation are discussed. Conditions for covariance stationarity and the existence of the fourth moment are derived, and expressions for the dynamic correlation structure of the process are provided. In an application to stock market returns, it is shown that the disaggregation of the conditional (co)variance process generated by the model provides substantial intuition. Moreover, the model exhibits a strong performance in calculating out–of–sample Value–at–Risk measures.

This paper shows how independent component analysis can be used to estimate the generalized orthogonal GARCH model in a fraction of the time otherwise required. The proposed method is a two-step procedure, separating the estimation of the correlation structure from that of the univariate dynamics, thus facilitating the incorporation of non-Gaussian innovations distributions in a straightforward manner. The generalized hyperbolic distribution provides an excellent parametric description of financial returns data and is used for the univariate fits, but its convolutions, necessary for portfolio risk calculations, are intractable. This restriction is overcome by saddlepoint approximations for the Value at Risk and expected shortfall, which are computationally cheap and retain excellent accuracy far into the tails. It is further shown that the mean-expected shortfall portfolio optimization problem can be solved efficiently in the context of the model. A simulation study and an application to stock returns demonstrate the validity of the procedure.

The development of unit root tests continues unabated, with many recent contributions using techniques such as generalized least squares (GLS) detrending and recursive detrending to improve the power of the test. In this article, the relation between the seemingly disparate tests is demonstrated by algebraically nesting all of them as ratios of quadratic forms in normal variables. By doing so, and using the exact sampling distribution of the ratio, it is straightforward to compute, examine, and compare the test' critical values and power functions. It is shown that use of GLS detrending parameters other than those recommended in the literature can lead to substantial power improvements. The open and important question regarding the nature of the first observation is addressed. Tests with high power are proposed irrespective of the distribution of the initial observation, which should be of great use in practical applications.

Knowledge of the statistical distribution of the prices of emission allowances, and their forecastability, are crucial in constructing, among other things, purchasing and risk management strategies in the emissions-constrained markets. This paper analyzes the two emission permits markets, CO2 in Europe, and SO2 in the US, and investigates a model for dealing with the unique stylized facts of this type of data. Its effectiveness in terms of model fit and out-of-sample value-at-risk-forecasting, as compared to models commonly used in risk-forecasting contexts, is demonstrated.

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Ratios of quadratic forms in correlated normal variables which introduce noncentrality into the quadratic forms are considered. The denominator is assumed to be positive (with probability 1). Various serial correlation estimates such as least-squares, Yule–Walker and Burg, as well as Durbin–Watson statistics, provide important examples of such ratios. The cumulative distribution function (c.d.f.) and density for such ratios admit saddlepoint approximations. These approximations are shown to preserve uniformity of relative error over the entire range of support. Furthermore, explicit values for the limiting relative errors at the extreme edges of support are derived.