arXiv subjects

Gael M. Martin

Publications and source records attributed to Gael M. Martin.

At least 19 recordsLinked to original sources

arXiv2026

Probabilistic Predictions of Option Prices with Modular Approximate Bayesian Inference

A new approximate Bayesian inferential framework is proposed that exploits multiple information sources -- daily spot returns, high-frequency spot data and option prices -- and enables fast calculation of probabilistic predictions of future option prices. This approach operates directly from the theoretical option pricing model, and does not require an explicit statistical model, or likelihood, for the observed option prices. We demonstrate that our approach produces accurate probabilistic option-price predictions in realistic scenarios and, despite not explicitly modelling option-pricing errors via a statistical model, the method is shown to be robust to the presence of such errors. Predictive accuracy based on the Heston option pricing model is illustrated empirically for short-maturity options, with the rapidity of real-time updates of the predictive distributions highlighted.

q-fin.ST↗

arXiv2023

ABC-based Forecasting in State Space Models

Approximate Bayesian Computation (ABC) has gained popularity as a method for conducting inference and forecasting in complex models, most notably those which are intractable in some sense. In this paper we use ABC to produce probabilistic forecasts in state space models (SSMs). Whilst ABC-based forecasting in correctly-specified SSMs has been studied, the misspecified case has not been investigated, and it is that case which we emphasize. We invoke recent principles of 'focused' Bayesian prediction, whereby Bayesian updates are driven by a scoring rule that rewards predictive accuracy; the aim being to produce predictives that perform well in that rule, despite misspecification. Two methods are investigated for producing the focused predictions. In a simulation setting, 'coherent' predictions are in evidence for both methods: the predictive constructed via the use of a particular scoring rule predicts best according to that rule. Importantly, both focused methods typically produce more accurate forecasts than an exact, but misspecified, predictive. An empirical application to a truly intractable SSM completes the paper.

stat.ME↗

arXiv2023

Solving the Forecast Combination Puzzle

We demonstrate that the forecasting combination puzzle is a consequence of the methodology commonly used to produce forecast combinations. By the combination puzzle, we refer to the empirical finding that predictions formed by combining multiple forecasts in ways that seek to optimize forecast performance often do not out-perform more naive, e.g. equally-weighted, approaches. In particular, we demonstrate that, due to the manner in which such forecasts are typically produced, tests that aim to discriminate between the predictive accuracy of competing combination strategies can have low power, and can lack size control, leading to an outcome that favours the naive approach. We show that this poor performance is due to the behavior of the corresponding test statistic, which has a non-standard asymptotic distribution under the null hypothesis of no inferior predictive accuracy, rather than the {standard normal distribution that is} {typically adopted}. In addition, we demonstrate that the low power of such predictive accuracy tests in the forecast combination setting can be completely avoided if more efficient estimation strategies are used in the production of the combinations, when feasible. We illustrate these findings both in the context of forecasting a functional of interest and in terms of predictive densities. A short empirical example {using daily financial returns} exemplifies how researchers can avoid the puzzle in practical settings.

econ.EM↗

arXiv2023

Bayesian Forecasting in Economics and Finance: A Modern Review

The Bayesian statistical paradigm provides a principled and coherent approach to probabilistic forecasting. Uncertainty about all unknowns that characterize any forecasting problem -- model, parameters, latent states -- is able to be quantified explicitly, and factored into the forecast distribution via the process of integration or averaging. Allied with the elegance of the method, Bayesian forecasting is now underpinned by the burgeoning field of Bayesian computation, which enables Bayesian forecasts to be produced for virtually any problem, no matter how large, or complex. The current state of play in Bayesian forecasting in economics and finance is the subject of this review. The aim is to provide the reader with an overview of modern approaches to the field, set in some historical context; and with sufficient computational detail given to assist the reader with implementation.

econ.EM↗

arXiv2023

Optimal probabilistic forecasts for risk management

This paper explores the implications of producing forecast distributions that are optimized according to scoring rules that are relevant to financial risk management. We assess the predictive performance of optimal forecasts from potentially misspecified models for i) value-at-risk and expected shortfall predictions; and ii) prediction of the VIX volatility index for use in hedging strategies involving VIX futures. Our empirical results show that calibrating the predictive distribution using a score that rewards the accurate prediction of extreme returns improves the VaR and ES predictions. Tail-focused predictive distributions are also shown to yield better outcomes in hedging strategies using VIX futures.

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arXiv2022

ABC of the Future

Approximate Bayesian computation (ABC) has advanced in two decades from a seminal idea to a practically applicable inference tool for simulator-based statistical models, which are becoming increasingly popular in many research domains. The computational feasibility of ABC for practical applications has been recently boosted by adopting techniques from machine learning to build surrogate models for the approximate likelihood or posterior and by the introduction of a general-purpose software platform with several advanced features, including automated parallelization. Here we demonstrate the strengths of the advances in ABC by going beyond the typical benchmark examples and considering real applications in astronomy, infectious disease epidemiology, personalised cancer therapy and financial prediction. We anticipate that the emerging success of ABC in producing actual added value and quantitative insights in the real world will continue to inspire a plethora of further applications across different fields of science, social science and technology.

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arXiv2022

Computing Bayes: From Then 'Til Now'

This paper takes the reader on a journey through the history of Bayesian computation, from the 18th century to the present day. Beginning with the one-dimensional integral first confronted by Bayes in 1763, we highlight the key contributions of: Laplace, Metropolis (and, importantly, his co-authors!), Hammersley and Handscomb, and Hastings, all of which set the foundations for the computational revolution in the late 20th century -- led, primarily, by Markov chain Monte Carlo (MCMC) algorithms. A very short outline of 21st century computational methods -- including pseudo-marginal MCMC, Hamiltonian Monte Carlo, sequential Monte Carlo, and the various `approximate' methods -- completes the paper.

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arXiv2022

The Impact of Sampling Variability on Estimated Combinations of Distributional Forecasts

We investigate the performance and sampling variability of estimated forecast combinations, with particular attention given to the combination of forecast distributions. Unknown parameters in the forecast combination are optimized according to criterion functions based on proper scoring rules, which are chosen to reward the form of forecast accuracy that matters for the problem at hand, and forecast performance is measured using the out-of-sample expectation of said scoring rule. Our results provide novel insights into the behavior of estimated forecast combinations. Firstly, we show that, asymptotically, the sampling variability in the performance of standard forecast combinations is determined solely by estimation of the constituent models, with estimation of the combination weights contributing no sampling variability whatsoever, at first order. Secondly, we show that, if computationally feasible, forecast combinations produced in a single step -- in which the constituent model and combination function parameters are estimated jointly -- have superior predictive accuracy and lower sampling variability than standard forecast combinations -- where constituent model and combination function parameters are estimated in two steps. These theoretical insights are demonstrated numerically, both in simulation settings and in an extensive empirical illustration using a time series of S&P500 returns.

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arXiv2022

Loss-Based Variational Bayes Prediction

We propose a new approach to Bayesian prediction that caters for models with a large number of parameters and is robust to model misspecification. Given a class of high-dimensional (but parametric) predictive models, this new approach constructs a posterior predictive using a variational approximation to a generalized posterior that is directly focused on predictive accuracy. The theoretical behavior of the new prediction approach is analyzed and a form of optimality demonstrated. Applications to both simulated and empirical data using high-dimensional Bayesian neural network and autoregressive mixture models demonstrate that the approach provides more accurate results than various alternatives, including misspecified likelihood-based predictions.

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arXiv2022

Variational Bayes in State Space Models: Inferential and Predictive Accuracy

Using theoretical and numerical results, we document the accuracy of commonly applied variational Bayes methods across a range of state space models. The results demonstrate that, in terms of accuracy on fixed parameters, there is a clear hierarchy in terms of the methods, with approaches that do not approximate the states yielding superior accuracy over methods that do. We also document numerically that the inferential discrepancies between the various methods often yield only small discrepancies in predictive accuracy over small out-of-sample evaluation periods. Nevertheless, in certain settings, these predictive discrepancies can become meaningful over a longer out-of-sample period. This finding indicates that the invariance of predictive results to inferential inaccuracy, which has been an oft-touted point made by practitioners seeking to justify the use of variational inference, is not ubiquitous and must be assessed on a case-by-case basis.

stat.ME↗

arXiv2022

Forecasting: theory and practice

Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases.

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arXiv2021

Approximating Bayes in the 21st Century

The 21st century has seen an enormous growth in the development and use of approximate Bayesian methods. Such methods produce computational solutions to certain intractable statistical problems that challenge exact methods like Markov chain Monte Carlo: for instance, models with unavailable likelihoods, high-dimensional models, and models featuring large data sets. These approximate methods are the subject of this review. The aim is to help new researchers in particular -- and more generally those interested in adopting a Bayesian approach to empirical work -- distinguish between different approximate techniques; understand the sense in which they are approximate; appreciate when and why particular methods are useful; and see the ways in which they can can be combined.

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arXiv2020

Computing Bayes: Bayesian Computation from 1763 to the 21st Century

The Bayesian statistical paradigm uses the language of probability to express uncertainty about the phenomena that generate observed data. Probability distributions thus characterize Bayesian analysis, with the rules of probability used to transform prior probability distributions for all unknowns - parameters, latent variables, models - into posterior distributions, subsequent to the observation of data. Conducting Bayesian analysis requires the evaluation of integrals in which these probability distributions appear. Bayesian computation is all about evaluating such integrals in the typical case where no analytical solution exists. This paper takes the reader on a chronological tour of Bayesian computation over the past two and a half centuries. Beginning with the one-dimensional integral first confronted by Bayes in 1763, through to recent problems in which the unknowns number in the millions, we place all computational problems into a common framework, and describe all computational methods using a common notation. The aim is to help new researchers in particular - and more generally those interested in adopting a Bayesian approach to empirical work - make sense of the plethora of computational techniques that are now on offer; understand when and why different methods are useful; and see the links that do exist, between them all.

stat.CO↗

arXiv2020

Optimal probabilistic forecasts: When do they work?

Proper scoring rules are used to assess the out-of-sample accuracy of probabilistic forecasts, with different scoring rules rewarding distinct aspects of forecast performance. Herein, we re-investigate the practice of using proper scoring rules to produce probabilistic forecasts that are `optimal' according to a given score, and assess when their out-of-sample accuracy is superior to alternative forecasts, according to that score. Particular attention is paid to relative predictive performance under misspecification of the predictive model. Using numerical illustrations, we document several novel findings within this paradigm that highlight the important interplay between the true data generating process, the assumed predictive model and the scoring rule. Notably, we show that only when a predictive model is sufficiently compatible with the true process to allow a particular score criterion to reward what it is designed to reward, will this approach to forecasting reap benefits. Subject to this compatibility however, the superiority of the optimal forecast will be greater, the greater is the degree of misspecification. We explore these issues under a range of different scenarios, and using both artificially simulated and empirical data.

econ.EM↗

arXiv2020

Focused Bayesian Prediction

We propose a new method for conducting Bayesian prediction that delivers accurate predictions without correctly specifying the unknown true data generating process. A prior is defined over a class of plausible predictive models. After observing data, we update the prior to a posterior over these models, via a criterion that captures a user-specified measure of predictive accuracy. Under regularity, this update yields posterior concentration onto the element of the predictive class that maximizes the expectation of the accuracy measure. In a series of simulation experiments and empirical examples we find notable gains in predictive accuracy relative to conventional likelihood-based prediction.

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arXiv2020

High-Frequency Jump Tests: Which Test Should We Use?

We conduct an extensive evaluation of price jump tests based on high-frequency financial data. After providing a concise review of multiple alternative tests, we document the size and power of all tests in a range of empirically relevant scenarios. Particular focus is given to the robustness of test performance to the presence of jumps in volatility and microstructure noise, and to the impact of sampling frequency. The paper concludes by providing guidelines for empirical researchers about which test to choose in any given setting.

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arXiv2019

Forecasting observables with particle filters: Any filter will do!

We investigate the impact of filter choice on forecast accuracy in state space models. The filters are used both to estimate the posterior distribution of the parameters, via a particle marginal Metropolis-Hastings (PMMH) algorithm, and to produce draws from the filtered distribution of the final state. Multiple filters are entertained, including two new data-driven methods. Simulation exercises are used to document the performance of each PMMH algorithm, in terms of computation time and the efficiency of the chain. We then produce the forecast distributions for the one-step-ahead value of the observed variable, using a fixed number of particles and Markov chain draws. Despite distinct differences in efficiency, the filters yield virtually identical forecasting accuracy, with this result holding under both correct and incorrect specification of the model. This invariance of forecast performance to the specification of the filter also characterizes an empirical analysis of S&P500 daily returns.

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arXiv2018

Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models

A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach - including a computationally efficient version based on the auxiliary score - achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the paper.

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