Special Issue in Honour of Stephen J. Taylor: Guest Editors' Introduction

Stephen John Taylor was born in England in 1954 and dedicated his career to research in Financial Econometrics. He obtained an MA in Mathematics from Trinity College, University of Cambridge, UK, an MA and a PhD in Operational Research from Lancaster University, UK, in the 1970s. From 1977 until his emeritation in 2020, as Professor of Finance in the Accounting and Finance Department at Lancaster University, he held positions as Lecturer in Operational Research (1977–88), Lecturer in Finance (1988-89), Reader in Finance (1989–93) and Professor of Finance (1993–2020) at Lancaster. Stephen is a key authority in the area of Time Series Econometrics, especially regarding Stochastic Volatility and Option Pricing modelling. He has published more than 60 papers in the broader areas of Finance and Econometrics including in top journals such as the Journal of Econometrics, Journal of Financial & Quantitative Analysis and Journal of Financial Econometrics. Stephen has been cited extensively with more than 14,000 google-scholar citations as of 2025, and he has contributed to the careers of over 20 PhD students and numerous co-authors. Stephen was one of the very first contributors to the European Finance Association and a founding member of the Society of Financial Econometrics. His work has inspired generations of scholars in the area, and he is referenced in the Engle and Granger 2003 Nobel Prize review. His Taylor (1982) paper, introducing stochastic volatility models, is arguably his most prominent work, and it has been re-printed three times. Stephen's influential books Modelling Financial Time Series (1986) and Asset Pric Dynamics, Volatility and Prediction (2011) have been adopted globally over the last few decades as key readings for courses in Time Series Analysis and Financial Econometrics at top universities and thereby shaped the field by teaching generations of students. The Centre for Financial Econometrics, Asset Markets and Macroeconomic Policy at Lancaster University, UK, hosted a Financial Econometrics Conference to mark Stephen Taylor's Retirement in 2023 with over 100 international participants coming together to celebrate Stephen's career and contributions. We are honoured to serve as guest editors for this special issue in the Journal of Time Series Analysis dedicated to Stephen Taylor with papers solicited from the conference submissions and then undergoing the journal's rigorous review process. This special issue contains eight papers on the latest topics in Time Series Econometrics building on and reflecting on Stephen's earlier work in the area and one paper by Stephen himself on his latest work concerning market microstructure noise components. Stephen's paper focuses on the differential impact of discreteness versus other (residual) components of microstructure noise (MN) and how to draw inference about their size and statistical properties. The overriding point is that the literature dealing with MN is dominated by simplifying i.i.d. and continuous distributional assumptions, for example, Gaussian noise. Stephen demonstrates that such assumptions are counterfactual and impede the analysis of the key MN components and their dynamic features. Specifically, he distinguishes the discrete pricing effects arising from the tick size and bid-ask spread from other sources of noise such as asymmetric information, order flow, price pressure, inventory control and block trades. To derive identifying constraints on the two components, Stephen exploits the (arbitrage-induced) cointegration relation between the spot and futures price along with a set of basic assumptions on their respective components, including similarity of their (non-discrete) regular noise terms, to determine the second order moments of the joint spot and futures return distribution. The setting generates novel expressions for the return variances, covariances, auto-covariances and cross-covariances. These relations along with direct observations on differential tick sizes and spreads for the spot and forward returns restrict the size and dynamics of the individual terms, rendering estimation of the average size and persistence of the various noise components feasible. The approach is illustrated through an empirical analysis for the spot exchange traded fund for the S&P 500 index (ticker SPY) and the e-mini S&P 500 futures contract. The contribution from Ahsan, Dufour and Rodriguez-Rondon provides inference techniques for a multivariate extension of the classical stochastic volatility model of Taylor (1982), labelled SVL( p $$ p $$ ). The model includes higher order autoregressive components ( p > 1 $$ p>1 $$ ) as well as an asymmetric return-volatility relation, also known as a leverage effect. The emphasis is on simple and efficient inference, which can be extremely expensive for higher order systems for other existing approaches. The proposed moment-based closed-form estimators exploit ARMA representations for SVL models along with winsorization techniques to stabilize estimation. Moreover, hypothesis tests are available through simulation or bootstrap procedures. In sum, their results provide additional flexibility in implementing estimation and inference procedures for the latent stochastic volatility model embodying empirically relevant extensions. The entry from Jonathan Wright studies the information content of options, a topic also explored in a variety of papers by Stephen Taylor, for example, Shackleton et al. (2010), Taylor et al. (2010), Taylor et al. (2014, 2018). Specifically, Wright relies on the recent introduction of very short maturity options on Treasury futures to measure the incremental risk-neutral interest rate uncertainty implied by options expiring just after versus just before an FOMC or employment report. Furthermore, these measures are then compared to the corresponding physical measures to imply an average estimate of the associated risk premiums. Finally, option-implied densities are constructed on the eve of FOMC and employment report days. The paper by Blasques, Koopman and Moussa relates to Taylor (1982), introducing stochastic volatility models and focuses on a class of stochastic volatility models with asymmetric stable errors. It proposes an indirect inference approach for parameter estimation and extracting latent volatility using extremum Monte Carlo methods, thereby improving shortcomings of alternative approaches considered in the existing literature. Hurn, Martin, Tian and Xu investigate the properties of multi-horizon forecasts within a setup that facilitates investigation of horizon-specific induced biases within a rational forecasts framework. A generalised method of moments approach for parameter estimation is used, using a combination of fixed-target, differential-information forecasts and fixed-information, differential-target forecast moments, to achieve identification. The work is related to Stephen's work in the area of forecasting and builds most notably on his Shackleton et al. (2010) paper. The paper by Shi, Yu and Zhang focuses on fractional Brownian motions providing a computationally feasible expression for the spectral density of fractional Brownian motions which then enables the authors to assess the accuracy of a range of existing approximation methods including a truncation method, Paxson's approximation and a Taylor expansion for the estimation of the fractional integration parameter. The paper by Nielsen, Pedersen, Rahbek and Thorsen proposes a fixed shrinkage bootstrap for a class of GARCH models with explanatory variables for testing whether one or more of the covariates can be excluded from the model. This leads to a non-standard testing problem, where the limiting distribution further depends on whether nuisance parameters are on the boundary or in the interior of the parameter space. The fixed shrinkage bootstrap takes the presence of nuisance parameters into account, and the authors show its asymptotic validity. An empirical illustration shows that the presence of nuisance parameters (especially whether or not on the boundary) are vital for interpreting the dynamics of conditional volatility in financial stock market indices. This relates both to Stephen's book from 1986, and several of his works on modelling and forecasting volatility, such as Taylor and Xu (1997) and Areal and Taylor (2002). Gregoir and Meddahi introduce an approach based on the characteristic function or Laplace transform of the observed process and show that, for a large class of state-space models (with finite second-order moments and non-zero higher order cumulants), it is possible to recover the cumulants of the structural shocks and the measurement errors from the cumulants and cross-cumulants of the observed process and the first-order parameters. This allows them to design specification tests related to the properties of the structural shocks or measurement errors, separately or jointly, or of the data generating process of the observed time series. The size and power properties of this test are applied to a simple stochastic volatility model. The paper has close links to the burgeoning literature on stochastic volatility, including Stephen's 1982 paper, his 1986 book, and his subsequent work in this area. Li, Nolte, Nolte and Yu extend the price-duration based volatility estimation framework of Hong et al. (2021) with an adaptive price change threshold that allows the authors then to disentangle daily and intraday volatility dynamics from price durations, which greatly simplifies the parametric modelling of price durations and hence generates more accurate volatility estimators. Simulation results demonstrate superior finite-sample performance of these duration-based estimators for both spot and integrated volatility compared to some established methods and an empirical application based on intraday data for the SPDR S&P 500 ETF shows improved forecasting accuracy. Torben G. Andersen thanks the Kenney Fund at Kellogg for research support.

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