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Keynes Fund


Summary of Project Results

The aim of the project was to study the statistical properties of new methods for modelling volatility and heavy-tailed distributions. Implementation of the models required writing new computer software.

High volatility can damage economic growth in that it creates uncertainty. It can also be an indication of deep-seated problems in the way markets operate. Understanding and modelling volatility is therefore important for managing and regulating financial markets. As such it is relevant for the Keynes Fund in that it has implications for the interface between ‘best private sector practice and public policy’.

Extreme observations, which can be seen as arising from heavy-tailed distributions, are also a feature of uncertainty. Much of our work has been concerned with the interactions between heavy tails and volatility in financial markets. The performance of standard time series models can often be adversely affected by outliers, whereas our models are, by construction, much more robust. Although much of the work was motivated by financial issues and applied to financial time series, the methodology developed has applications in all areas where nonlinear time series models are used. For example one of our recent working papers (Harvey and Ito) applies the methods to time series on rainfall.


Research Output

Published Papers


Circuit Breakers on the London Stock Exchange: Do they improve subsequent market quality?, James Brugler and Oliver Linton

Abstract: This paper uses proprietary data to evaluate the efficacy of single-stock circuit breakers on the London Stock Exchange during July and August 2011. We exploit exogenous variation in the length of the uncrossing periods that follow a trading suspension to estimate the effect of auction length on market quality, measured by volume of trades, frequency of trading and the change in realised variance of returns. We also estimate the effect of a trading suspension in one FTSE-100 stock on the volume of trades, trading frequency and the change in realised variance of returns for other FTSE-100 stocks in the same industrial sector as the event stock and in other sectors. We find that auction length has a significant detrimental effect on market quality for the suspended security when returns are negative but no discernible effect when returns are positive. We also find that trading suspensions help to ameliorate the spread of market microstructure noise and price inefficiency across securities during falling markets but the reverse is true during rising markets. Although trading suspensions may not improve the trading process within a particular security, they do play an important role preventing the spread of poor market quality across securities in falling markets and therefore can be effective tools for promoting market-wide stability.


EGARCH models with fat tails, skewness and leverage, Harvey, A.C. and G. Sucarrat (2014) Computational Statistics and Data Analysis, Vol 26, pp. 320-338 

Abstract: An EGARCH model in which the conditional distribution is heavy-tailed and skewed is proposed. The properties of the model, including unconditional moments, autocorrelations and the asymptotic distribution of the maximum likelihood estimator, are set out. Evidence for skewness in a conditional t-distribution is found for a range of returns series, and the model is shown to give a better fit than comparable skewed-t GARCH models in nearly all cases. A two-component model gives further gains in goodness of fit and is able to mimic the long memory pattern displayed in the autocorrelations of the absolute values.

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Filtering with Heavy Tails, Harvey, A.C. and A. Luati. (2014) Journal of the American Statistical Association, Vol 109, pp. 1112-1122.

Abstract: An unobserved components model in which the signal is buried in noise that is non-Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation-driven model, based on a conditional Student’s t-distribution, which is tractable and retains some of the desirable features of the linear Gaussian model. Letting the dynamics be driven by the score of the conditional distribution leads to a specification that is not only easy to implement, but which also facilitates the development of a comprehensive and relatively straightforward theory for the asymptotic distribution of the maximum likelihood estimator. The methods are illustrated with an application to rail travel in the United Kingdom. The final part of the article shows how the model may be extended to include explanatory variables.

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Time series models with an EGB2 conditional distribution, M. Caivano and Andrew Harvey (2014), Journal of Time Series Analysis, Vol 35(6), pp. 558-571

Abstract: A time-series model in which the signal is buried in noise that is non-Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation-driven model, based on an exponential generalized beta distribution of the second kind (EGB2), in which the signal is a linear function of past values of the score of the conditional distribution. This specification produces a model that is not only easy to implement but which also facilitates the development of a comprehensive and relatively straightforward theory for the asymptotic distribution of the maximum-likelihood (ML) estimator. Score-driven models of this kind can also be based on conditional t distributions, but whereas these models carry out what, in the robustness literature, is called a soft form of trimming, the EGB2 distribution leads to a soft form of Winsorizing. An exponential general autoregressive conditional heteroscedastic (EGARCH) model based on the EGB2 distribution is also developed. This model complements the score-driven EGARCH model with a conditional t distribution. Finally, dynamic location and scale models are combined and applied to data on the UK rate of inflation.

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Computation of Maximum Likelihood Estimates for Score Driven Models for Positive Valued Observations, Philipp Andres, (2014), Computational Statistics and Data Analysis, Vol 76, pp. 34-42.

Abstract: Recently, the Dynamic Conditional Score (DCS) or Generalized Autoregressive Score (GAS) time series models have attracted considerable attention. This motivates the need for a software package to estimate and evaluate these new models. A straightforward to operate program called the Dynamic Score (DySco) package is introduced for estimating models for positive variables, in which the location/scale evolves over time. Its capabilities are demonstrated using a financial application.

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Testing against changing correlation, Harvey, A.C. and S. Thiele (2016), Journal of Empirical Finance, Vol 38, pp. 575-89.  

Abstract: A test for time-varying correlation is developed within the framework of a dynamic conditional score (DCS) model for both Gaussian and Student t-distributions. The test may be interpreted as a Lagrange multiplier test and modified to allow for the estimation of models for time-varying volatility in the individual series. Unlike standard moment-based tests, the score-based test statistic includes information on the level of correlation under the null hypothesis and local power arguments indicate the benefits of doing so. A simulation study shows that the performance of the score-based test is strong relative to existing tests across a range of data generating processes. An application to the Hong Kong and South Korean equity markets shows that the new test reveals changes in correlation that are not detected by the standard moment-based test.

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Robust time series models with trend and seasonal components, Michele Caivano, Andrew Harvey and Alessandra Luati (2016), Journal of the Spanish Economic Association, Vol 7(1), pp. 99-120

Abstract: We describe observation driven time series models for Student-t and EGB2 conditional distributions in which the signal is a linear function of past values of the score of the conditional distribution. These specifications produce models that are easy to implement and deal with outliers by what amounts to a soft form of trimming in the case of t and a soft form of Winsorizing in the case of EGB2. We show how a model with trend and seasonal components can be used as the basis for a seasonal adjustment procedure. The methods are illustrated with US and Spanish data.

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Volatility Modelling with a Generalized t-distribution, Harvey, A.C. and R-J. Lange (2017), Journal of Time Series Analysis (forthcoming).

Abstract: Exponential generalized autoregressive conditional heteroscedasticity models in which the dynamics of the logarithm of scale are driven by the conditional score are known to exhibit attractive theoretical properties for the t distribution and general error distribution. A model based on the generalized t includes both as special cases. We derive the information matrix for the generalized t and show that, when parameterized with the inverse of the tail index, it remains positive definite in the limit as the distribution goes to a general error distribution. We generalize further by allowing the distribution of the observations to be skewed and asymmetric. Our method for introducing asymmetry ensures that the information matrix reverts to the usual case under symmetry. We are able to derive analytic expressions for the conditional moments of our exponential generalized autoregressive conditional heteroscedasticity model as well as the information matrix of the dynamic parameters. The practical value of the model is illustrated with commodity and stock return data. Overall, the approach offers a unified, flexible, robust, and effective treatment of volatility.

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Working Papers


Modeling the Interactions between Volatility and Returns, Harvey, A.C. and R-J. Lange (2015) CWPE 1518

Abstract: Volatility of a stock may incur a risk premium, leading to a pos- itive correlation between volatility and returns. On the other hand the leverage effect, whereby negative returns increase volatility, acts in the opposite direction. We propose a reformulation and extension of the ARCH in Mean model, in which the logarithm of scale is driven by the score of the conditional distribution. This EGARCH-M model is shown to be theoretically tractable as well as practically useful. By employing a two component extension we are able to distinguish between the long and short run effects of returns on volatility. The EGARCH formulation allows more flexibility in the asymmetry of the response (leverage) and this enables us to find that the short-term response is, in some cases, close to being anti-asymmetric. The long and short run volatility components are shown to have very different effects on returns, with the long-run component yielding the risk pre- mium. A model in which the returns have a skewed t distribution is shown to fit well to daily and weekly data on some of the major stock market indices.

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Modeling Time Series with Zero Observations, Harvey, A.C. and R. Ito (2017), Nuffield College Economics Working Paper 2017-W01, Oxford University

Abstract: We consider situations in which a significant proportion of obser- vations in a time series are zero, but the remaining observations are positive and measured on a continuous scale. We propose a new dy- namic model in which the conditional distribution of the observations is constructed by shifting a distribution for non-zero observations to the left and censoring negative values. The key to generalizing the censoring approach to the dynamic case is to have (the logarithm of) the location/scale parameter driven by a filter that depends on the score of the conditional distribution.  An exponential link function means that seasonal effects can be incorporated into the model and this is done by means of a cubic spline (which can potentially be time- varying). The model is fitted to daily rainfall in northern Australia and compared with a dynamic zero-augmented model.

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Prof. Andrew Harvey


Professor Andrew Harvey is Emeritus Professor of Econometrics at the Faculty of Economics, University of Cambridge and Fellow of the Econometric Society and a Fellow of the British Academy (FBA). His research expertise is in Time Series, Financial Econometrics, State Space Models, Signal Extraction, Volatility.




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