Forecasting volatility: does continuous time do better than discrete time?

In this paper we compare the forecast performance of continuous and discrete-time volatility models. In discrete time, we consider more than ten GARCH-type models and an asymmetric autoregressive stochastic volatility model. In continuous-time, a stochastic volatility model with mean reversion, volatility feedback and leverage. We estimate each model by maximum likelihood and evaluate their ability to forecast the two scales realized volatility, a nonparametric estimate of volatility based on highfrequency data that minimizes the biases present in realized volatility caused by microstructure errors. We find that volatility forecasts based on continuous-time models may outperform those of GARCH-type discrete-time models so that, besides other merits of continuous-time models, they may be used as a tool for generating reasonable volatility forecasts. However, within the stochastic volatility family, we do not find such evidence. We show that volatility feedback may have serious drawbacks in terms of forecasting and that an asymmetric disturbance distribution (possibly with heavy tails) might improve forecasting.Asymmetry, Continuous and discrete-time stochastic volatility models, GARCH-type models, Maximum likelihood via iterated filtering, Particle filter, Volatility forecasting

Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems

We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-) equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Lévy-driven SDEs. We construct multivariate infinitesimally over dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.Continuous time, Counting Markov process, Birth-death process, Environmental stochasticity, Infinitesimal over-dispersion, Simultaneous events

Forecasting volatility: does continuous time do better than discrete time?

In this paper we compare the forecast performance of continuous and discrete-time volatility models. In discrete time, we consider more than ten GARCH-type models and an asymmetric autoregressive stochastic volatility model. In continuous-time, a stochastic volatility model with mean reversion, volatility feedback and leverage. We estimate each model by maximum likelihood and evaluate their ability to forecast the two scales realized volatility, a nonparametric estimate of volatility based on highfrequency data that minimizes the biases present in realized volatility caused by microstructure errors. We find that volatility forecasts based on continuous-time models may outperform those of GARCH-type discrete-time models so that, besides other merits of continuous-time models, they may be used as a tool for generating reasonable volatility forecasts. However, within the stochastic volatility family, we do not find such evidence. We show that volatility feedback may have serious drawbacks in terms of forecasting and that an asymmetric disturbance distribution (possibly with heavy tails) might improve forecasting

Trajectory composition of Poisson time changes and Markov counting systems

Changing time of simple continuous-time Markov counting processes by independent unit-rate Poisson processes results in Markov counting processes for which we provide closed-form transition rates via composition of trajectories and with which we construct novel, simpler infinitesimally over-dispersed processes

On idiosyncratic stochasticity of financial leverage effects

We model leverage as stochastic but independent of return shocks and of volatility and perform likelihood-based inference via the recently developed iterated filtering algorithm using S&P500 data, contributing new evidence to the still slim empirical support for random leverage variation.This work was supported by Spanish Government Project ECO2012-32401 and Spanish Program Juan de la Cierva (JCI-2010-06898)

An entropy-based machine learning algorithm for combining macroeconomic forecasts

This paper applies a Machine Learning approach with the aim of providing a single aggregated prediction from a set of individual predictions. Departing from the well-known maximum-entropy inference methodology, a new factor capturing the distance between the true and the estimated aggregated predictions presents a new problem. Algorithms such as ridge, lasso or elastic net help in finding a new methodology to tackle this issue. We carry out a simulation study to evaluate the performance of such a procedure and apply it in order to forecast and measure predictive ability using a dataset of predictions on Spanish gross domestic product

Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems

We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-) equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Levy-driven SDEs. We construct multivariate infinitesimally over-dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.Comment: 26 page

Statistical Inference for Nonlinear Dynamical Systems

To my family, for their love and support ii ACKNOWLEDGEMENTS This dissertation is the end result of my stay at the University of Michigan during which I have benefited from contact and varied relationships with many members of its community. I thank my advisor Professor Edward Ionides for his patient advice, generosity with his time and for the encouragement to join the statistics research community as a co-author of some of his work. In addition, Professor Ionides recruited me for the cholera project, which has served as a motivation for the statistical results in this dissertation as well as an important funding source throughout the program. I also thank the rest of the people involved in the cholera project, Mercedes Pascual in particular for her role in its organization. Access to Aaron King’s cluster of computers has been crucial for reasonable computation times, which have made the whole process substantially more enjoyable. I also thank him for organizing and inviting me to the NCEAS group on inference for dynamical systems. I thank all three for very interesting and stimulating discussions. This dissertation has benefited enormously from the resources that the university provides students with. The faculty and libraries have been a bottomless well of knowledge and the staff at the statistics department kind and helpful. I also thank my fellow graduate students, many of which offered their friendship, help and company in the long hours spent in the shared space. Finally, I would like to add that without the love and support of my family, especially my wife Maria, this thesis would not have been possible. iii Additional acknowledgements are required regarding collaborations. The second chapter has been published, with some minor changes, in the National Proceeding