Contributions published at Empirical Finance (Marc Paolella)

A large proportion of the most viable time series models used in empirical finance for density and value-at-risk forecasting are estimated with maximum likelihood methods. By way of its definition, the likelihood implicitly places equal weight on each of the observations in the sample, but this need not be optimal, depending on the extent to which the model and the true data generating process deviate. For example, in the context of modeling financial asset returns, schemes that place relatively more weight on observations in the recent past result in considerable improvement of out-of-sample density forecasts, compared with the default of equal weights. If instead of accurate forecasting of the, entire density, interest is restricted to just downside risk and risk model validation, then it would seem wise to (also) place more weight on the negative observations in the sample. In this paper, such weighted likelihood schemes are proposed and demonstrated to yield considerable improvements in forecast accuracy using a variety of data sets and different GARCH models. Further improvement is realized by combining the two weighting schemes, giving rise to a doubly weighted asymmetric risk forecasting method or, in short, a DWARF-like method.

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A new point estimator for the AR(1) coefficient in the linear regression model with arbitrary exogenous regressors and stationary AR(1) disturbances is developed. Its construction parallels that of the median--unbiased estimator, but uses the mode as a measure of central tendency. The mean--adjusted estimator is also considered, and saddlepoint approximations are used to lower the computational burden of all the estimators. Large--scale simulation studies for assessing their small--sample properties are conducted. Their relative performance depends almost exclusively on the value of the autoregressive parameter, with the new estimator dominating over a large part of the parameter space.

Closed-form approximations for the density and cumulative distribution function of the doubly noncentral t distribution are developed based on saddlepoint methods. They exhibit remarkable accuracy throughout the entire support of the distribution and are vastly superior to existing approximations. An application in finance is considered which capitalizes on the enormous increase in computational speed.

Intermediate Probability is the natural extension of the author's Fundamental Probability. It details several highly important topics, from standard ones such as order statistics, multivariate normal, and convergence concepts, to more advanced ones which are usually not addressed at this mathematical level, or have never previously appeared in textbook form. The author adopts a computational approach throughout, allowing the reader to directly implement the methods, thus greatly enhancing the learning experience and clearly illustrating the applicability, strengths, and weaknesses of the theory.

Probability is a vital measure in numerous disciplines, from bioinformatics and econometrics to finance/insurance and computer science. Developed from a successful course, Fundamental Probability provides an engaging and hands-on introduction to this important topic. Whilst the theory is explored in detail, this book also emphasises practical applications, with the presentation of a large variety of examples and exercises, along with generous use of computational tools.

A resampling method based on the bootstrap and a bias-correction step is developed for improving the Value-at-Risk (VaR) forecasting ability of the normal-GARCH model. Compared to the use of more sophisticated GARCH models, the new method is fast, easy to implement, numerically reliable, and, except for having to choose a window length L for the bias-correction step, fully data driven. The results for several different financial asset returns over a long out-of-sample forecasting period, as well as use of simulated data, strongly support use of the new method, and the performance is not sensitive to the choice of L.

The use of Markov-switching models to capture the volatility dynamics of financial time series has grown considerably during past years, in part because they give rise to a plausible interpretation of nonlinearities. Nevertheless, GARCH-type models remain ubiquitous in order to allow for nonlinearities associated with time-varying volatility. Existing methods of combining the two approaches are unsatisfactory, as they either suffer from severe estimation difficulties or else their dynamic properties are not well understood. In this article we present a new Markov-switching GARCH model that overcomes both of these problems. Dynamic properties are derived and their implications for the volatility process discussed. We argue that the disaggregation of the variance process offered by the new model is more plausible than in the existing variants. The approach is illustrated with several exchange rate return series. The results suggest that a promising volatility model is an independent switching GARCH process with a possibly skewed conditional mixture density

The accuracy and speed of numerical methods for computing the moments of a ratio of quadratic forms in normal variables is examined, with particular application to the sample autocorrelation function. Methods based on a saddlepoint approximation are demonstrated to be not only superior to existing approximations, but are numerically reliable and virtually as accurate as the method suitable for exact computations, while taking only a fraction of the time to compute. The new method also maintains its accuracy for time series models near the nonstationary border, which is of significant interest for unit-root inference and also a case for which first-order mean and variance approximations break down. As a wide variety of test statistics and their power functions arising in econometric models are expressible in the general form considered, the method should prove very useful for data analysis and model building.