Andrew C. Harvey

University of Cambridge | Cam · Faculty of Economics

The score‐driven approach to time series modelling is able to handle circular data and switching regimes with intra‐regime dynamics. Furthermore it enables a dynamic model to be fitted to a linear and a circular variable when their joint distribution is a cylinder. The viability of the new method is illustrated by estimating models for hourly data...

The construction of score-driven filters for nonlinear time series models is described, and they are shown to apply over a wide range of disciplines. Their theoretical and practical advantages over other methods are highlighted. Topics covered include robust time series modeling, conditional heteroscedasticity, count data, dynamic correlation and a...

The time-dependent reproduction number, R t , is a key metric used by epidemiologists to assess the current state of an outbreak of an infectious disease. This quantity is usually estimated using time-series observations on new infections combined with assumptions about the distribution of the serial interval of transmissions. Bayesian methods are...

Score-driven models provide a solution to the problem of modeling time series when the observations are subject to censoring and location and/or scale may change over time. The method applies to generalized t and EGB2 distributions, as well as to the normal distribution. Explanatory variables can be included, making static Tobit models a special ca...

This article shows how new time series models can be used to track the progress of an epidemic, forecast key variables and evaluate the effects of policies. The univariate framework of Harvey and Kattuman (2020, Harvard Data Science Review , Special Issue 1—COVID-19, https://hdsr.mitpress.mit.edu/pub/ozgjx0yn ) is extended to model the relationship...

A new class of time series models is used to track the progress of the COVID-19 epidemic in the UK in early 2021. Models are fitted to England and the regions, as well as to the UK as a whole. The growth rate of the daily number of cases and the instantaneous reproduction number are computed regularly and compared with those produced by SAGE. The r...

Control groups can provide counterfactual evidence for assessing the impact of an event or policy change on a target variable. We argue that fitting a multivariate time series model offers potential gains over a direct comparison between the target and a weighted average of controls. More importantly, it highlights the assumptions underlying method...

Sometimes a significant proportion of observations in a time series are zero, but the remaining observations are positive and measured on a continuous scale. We propose a new dynamic 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...

Circular observations pose special problems for time series modeling. This article shows how the score-driven approach, developed primarily in econometrics, provides a natural solution to the difficulties and leads to a coherent and unified methodology for estimation, model selection and testing. The new methods are illustrated with hourly data on...

This paper sets up a statistical framework for modeling realised volatility (RV) using a Dynamic Conditional Score (DCS) model. It first shows how a preliminary analysis of RV, based on fitting a linear Gaussian model to its logarithm, confirms the presence of long memory effects and suggests a two component dynamic specification. It also indicates...

Score-driven models provide a solution to the problem of modelling time series when the observations are subject to censoring and location and/or scale may change over time. The method applies to generalized-t and EGB2 distributions, as well as to the normal distribution. A set of Monte Carlo experiments show that the score-driven model provides go...

An EGARCH‐M model, in which the logarithm of scale is driven by the score of the conditional distribution, is shown to be theoretically tractable as well as practically useful. A two‐component extension makes it possible to distinguish between the short‐ and long‐run effects of returns on volatility, and the resulting short‐ and long‐run volatility...

Electronic supplement

James Durbin joined the LSE Statistics Department in 1950 and remained there until he retired in 1988. He was a key figure in the development of econometrics and statistics. An early contribution, which came to be known as the Durbin–Watson test, was the first ‘diagnostic’ to be routinely applied in the analysis of regression models and it exerted...

The state space form opens the way to the statistical treatment of a wide range of dynamic models in a unified framework. For models formulated in unobserved components it offers algorithms for filtering, signal extraction and prediction. Data irregularities can be handled and recent work on computational methods has extended the range of nonlinear...

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 th...

The question of defining a trend is one which has exercised the minds of statisticians for many years. In much of the statistical literature, a trend is conceived of as that part of a series that changes relatively slowly over time. Viewed in terms of prediction, the estimated trend is that part of the series that when extrapolated gives the cleare...

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...

This book is a tribute to Professor Andrew Harvey, who has been an active researcher for four decades, writing on many aspects of time series modeling with a particular focus on economic and more recently financial applications. As well as generating path breaking research articles, he has been unusually influential in writing research monographs a...

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-base...

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...

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...

We compare two EGARCH models, which belong to a new class of models in which the dynamics are driven by the score of the conditional distribution of the observations. Models of this kind are called dynamic conditional score (DCS) models and their form facilitates the development of a comprehensive and relatively straightforward theory for the asymp...

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, a...

In dynamic conditional score models, the innovation term of the dynamic specification is the score of the conditional distribution. These models are investigated for non-negative variables, using distributions from the generalized beta and generalized gamma families. The log-normal distribution is also considered. Applications to the daily range of...

A time-varying probability density function, or the corresponding cumulative distribution function, may be estimated nonparametrically by using a kernel and weighting the observations using schemes derived from time series modelling. The parameters, including the bandwidth, may be estimated by maximum likelihood or cross-validation. Diagnostic chec...

The relationship between inflation and the output gap can be modelled simply and effectively by including an unobserved random walk component in the model. The dynamic properties match the stylized facts and the random walk component satisfies the properties normally required for core inflation. The model may be generalized so as to include a term...

The volatility of financial returns changes over time and, for the last thirty years, Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models have provided the principal means of analyzing, modeling and monitoring such changes. Taking into account that financial returns typically exhibit heavy tails – that is, extreme values can oc...

A copula defines the probability that observations from two time series lie below given quantiles. It is proposed that stationarity tests constructed from indicator variables be used to test against the hypothesis that the copula is changing over time. Tests associated with different quantiles may point to changes in different parts of the copula....

Quantiles provide a comprehensive description of the properties of a variable, and tracking changes in quantiles over time using signal extraction methods can be informative. It is shown here how departures from strict stationarity can be detected using stationarity tests based on weighted quantile indicators. Corresponding tests based on expectile...

A copula models the relationships between variables independently of their marginal distributions. When the variables are time series, the copula may change over time. Recursive procedures based on indicator variables are proposed for tracking these changes over time. Estimation of the unknown parameters is by maximum likelihood. When the marginal...

The local quadratic trend model provides a flexible response to underlying movements in a macroeconomic time series in its estimates of level and change. If the underlying movements are thought of as a trend plus cycle, an estimate of the cycle may be obtained from the quadratic term. Estimating the cycle in this way may offer a useful alternative...

State-space methods permit a flexible treatment of unobserved components models. Furthermore, data irregularities such as missing observations are easily handled. For example, irregularly spaced observations can be dealt with since, as discussed in [3, Chap. 3], unobserved components models can be set up in continuous time, and the implied discrete...

State space models is a rather loose term given to time series models, usually formulated in terms of unobserved components, that make use of the state space form for their statistical treatment.

The asymptotic distribution of maximum likelihood estimators is derived for a class of exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models. The result carries over to models for duration and realised volatility that use an exponential link function. A key feature of the model formulation is that the dynamics are dr...

A time-varying quantile can be fitted by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modified state space signal extraction algorithm. It is shown that such quantiles satisfy the defining property of fixed quantiles in having the appropriate number of observations above and below. Li...

Algorithms are presented for computing mean square errors in a misspecified unobserved components model when the true model is known. It is assumed that both the true and misspecified models can be put in linear state space form. The algorithm for filtering is based on the Kalman filter while that for smoothing modifies the fixed-point smoother. Il...

A copula models the relationships between variables independently of their marginal distributions. When the variables are time series, the copula may change over time. A statistical framework is suggested for tracking these changes over time. When the marginal distribu- tions change, pre-filtering is necessary before constructing the indicator vari...

The GARCH-t model is widely used to predict volatilty. However, modeling the conditional variance as a linear combination of past squared observations may not be the best approach if the standardized observations are non-Gaussian. A simple modi.cation lets the conditional variance, or its logarithm, depend on past values of the score of a t-distrib...

A copula defines the probability that observations from two time series lie below given quantiles. It is proposed that stationarity tests constructed from indicator variables be used to test against the hypothesis that the copula is changing over time. Tests associated with different quantiles may point to changes in different parts of the copula,...

If the process generating a time series contains a deterministic component the differencing operations carried out to achieve stationarity may lead to an ARMA model which is strictly noninvertible. This is known as overdifferencing but it is shown here that overdifferencing need not have serious implications for prediction provided that a finite sa...

The paper examines various tests for assessing whether a time series model requires a slope component. We first consider the simple t-test on the mean of first differences and show that it achieves high power against the alternative hypothesis of a stochastic nonstationary slope and also against a purely deterministic slope. The test may be modifie...

The article analyses the relationship between unobserved component trend-cycle models and the Hodrick-Prescott filter. Consideration is given to the consequences of using an inappropriate smoothing constant and the effect of changing the observation interval.

It is well established that while financial variables such as stock returns are serially uncorrelated over time, their squares are not. The most common way of modeling this serial correlation in volatility is by means of the generalized autoregressive conditional heteroscedasticity (GARCH) class. This chapter provides a long memory stochastic volat...

Trends and cyclical components in economic time series are modeled in a Bayesian framework. This enables prior notions about the duration of cycles to be used, while the generalized class of stochastic cycles employed allows the possibility of relatively smooth cycles being extracted. The posterior distributions of such underlying cycles can be ver...

Quantiles provide a comprehensive description of the properties of a variable and tracking changes in quantiles over time using signal extraction methods can be informative. It is shown here how sta-tionarity tests can be generalized to test the null hypothesis that a particular quantile is constant over time by using weighted indica-tors. Correspo...

Quantiles provide a comprehensive description of the properties of a variable and tracking changes in quantiles over time using signal extraction methods can be informative. It is shown here how stationarity tests can be generalized to test the null hypothesis that a particular quantile is constant over time by using weighted indicators. Correspond...

This article compares and contrasts structural time series models and the common features methodology. The way in which trends are handled is highlighted by describing a recent structural time series model that allows convergence to a common growth path. Postsample data are used to test its forecasting performance for income per head in U.S. region...

We study the convergence properties of inflation rates among the countries of the European Monetary Union over the period 1980-2004. Given the Maastricht agreements and the adoption of the single currency, the sample can be naturally split into two parts, before and after the birth of the euro. We study convergence in the first subsample by means o...

Structural time series models are formulated in terms of components, such as trends, seasonals and cycles, that have a direct interpretation. As well as providing a framework for time series decomposition by signal extraction, they can be used for forecasting and for ‘nowcasting’. The structural interpretation allows extensions to classes of models...

We consider how unit-root and stationarity tests can be used to study the convergence of prices and rates of inflation. We show how the joint use of these tests in levels and first differences allows the researcher to distinguish between series that are converging and series that have already converged, and we set out a strategy to establish whethe...

A time-varying quantile can be fitted to a sequence of observations by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modified state space signal extraction algorithm. Quantiles estimated in this way provide information on various aspects of a time series, including dispersion, asymmetr...

This book presents a collection of readings which give the reader an idea of the nature and scope of unobserved components (UC) models and the methods used to deal with them. It contains four parts, three of which concern recent theoretical developments in classical and Bayesian estimation of linear, nonlinear, and non Gaussian UC models, signal ex...

Multivariate unobserved components (structural) time series models are fitted to annual post-war observations on real income per capita in countries in the Euro-zone. The aim is to establish stylized facts about convergence as it relates both to long-run and short-run movements. A new model, in which convergence components are combined with a commo...

Changes in variance, or volatility, over time can be modelled using the approach based on autoregressive conditional heteroscedasticity (ARCH). However, the generalizations to multivariate series can be difficult to estimate and interpret. Another approach is to model variance as an unobserved stochastic process. Although it is not easy to obtain t...

This article reports the results of fitting unobserved components (structural) time series models to data on real income per capita in eight regions of the United States. The aim is to establish stylised facts about cycles and convergence. A new model is developed in which convergence components are combined with a common trend and cycles. These co...

Cyclical components in economic time series are analysed in a Bayesian framework, thereby allowing prior notions about periodicity to be used. The method is based on a general class of unobserved component models that allow relatively smooth cycles to be extracted. Posterior densities of parameters and smoothed cycles are obtained using Markov chai...

The implied signal extraction filters in unobserved components models depend on key signal-noise ratios. This paper examines how these ratios change with the observation interval. The analysis is based on continuous time models and is carried out for both stocks and flows. As a by-product, a connection is established between continuous time flow mo...

The aim of this article is the development of models for converging economies. After discussing models of balanced growth, univariate models of the gap between per capital income in two economies are examined. The preferred models combine unobserved components with an error correction mechanism and allow a decomposition into trend, cycle and conver...

The paper presents various tests for assessing whether a time series is subject to drift. We first consider departures from the null hypothesis of no drift against the alternative of a deterministic and/or a non-stationary stochastic drift with initial value zero. We show that the standard t-test on the mean of first differences achieves high power...

This article presents a model-based approach to the investigation of cyclical properties of a time series. It is shown how models for stochastic cycles, both stationary and nonstationary, may be set up and how deterministic cycles emerge as a special case. The Lagrange multiplier principle is used to formulate a test of the null of a deterministic...

We present algorithms for computing the weights implicitly assigned to observations when estimating unobserved components, by filtering or smoothing, using a model in state space form. The algorithms are based on recursions derived from the Kalman filter and associated smoother. Since the method applies to any model with a linear state space form,...

A class of model-based filters for extracting trends and cycles in economic time series is presented. These lowpass and bandpass filters are derived in a mutually consistent manner as the joint solution to a signal extraction problem in an unobserved-components model. The resulting trends and cycles are computed in finite samples using the Kalman f...

This article modifies and extends the test against nonstationary stochastic seasonality proposed by Canova and Hansen. A simplified form of the test statistic in which the nonparametric correction for serial correlation is based on estimates of the spectrum at the seasonal frequencies is considered and shown to have the same asymptotic distribution...

We examine the properties of a multivariate Dickey-Fuller t-statistic designed to test for a unit root in a panel while taking account of cross-sectional dependence. The asymptotic distribution is presented and critical values provided. When intercepts are present, a modification along the lines of Elliot, Rothenberg and Stock (1996) can be impleme...

Cyclical components in economic time series are analysed in a Bayesian framework, thereby allowing prior notions about periodicity to be used. The method is based on a general class of unobserved component models that encompasses a range of dynamics in the stochastic cycle. This allows for instance relatively smooth cycles to be extracted from time...

The paper presents various tests for assessing whether a time series is subject to drift. We first consider departures from the null hypothesis of no drift against the alternative of a deterministic and/or a non-stationary stochastic drift with initial value zero. We show that the standard t-test on the mean of first differences achieves high power...

Cyclical components in economic time series are analysed in a Bayesian framework, thereby allowing prior notions about periodicity to be used. The method is based on a general class of unobserved component models that allow relatively smooth cycles to be extracted.

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated...

This article first discusses ways of decomposing a time series into trend and cyclical components, paying particular attention to a new class of model for cycles. It is shown how using an auxiliary series can help to achieve a more satisfactory decomposition. A discussion of balanced growth then leads on to the construction of new models for conver...

Cyclical components in economic time series are analysed in a Bayesian framework, thereby allowing prior notions about periodicity to be used. The method is based on a general class of unobserved component models that allow relatively smooth cycles to be extracted. Posterior densities of parameters and smoothed cycles are obtained using Markov chai...

A trend estimated from an unobserved components model tends to be smoother when it is modelled as an integrated random walk rather than a random walk with drift. This article derives a test of the null hypothesis that the trend is deterministic against the alternative that it is an integrated random walk. It is assumed that the other component in t...

We consider tests for the presence of a random walk component in a stationary or trend stationary time series and extend them to series that contain structural breaks. The locally best invariant (LBI) test is derived and the asymptotic distribution is obtained. Then a modified test statistic is proposed. The advantage of this statistic is that its...

This article reviews recent work on testing for the presence of non-stationary unobserved components and presents it in a unified way. Tests against random walk components and seasonal components are given and it is shown how the procedures may be extended to multivariate models and models with structural breaks. Many of the test statistics have an...

Lagrange multiplier tests against nonstationary unobserved components such as stochastic trends and seasonals are based on statistics which, under the null hypothesis, have asymptotic distributions belonging to the class of generalized Cramér-von Mises distributions. Conversely, unit root tests can be formulated, again using the Lagrange multiplier...

We present algorithms for computing the weights implicitly assigned to observations when estimating unobserved components using a model in state space form. The algorithms are for both filtering and signal extraction. In linear time-invariant models such weights can sometimes be obtained analytically from the Wiener-Kolmogorov formulae. Our method...

This note defines a Beveridge–Nelson smoother, that is a two-sided signal extraction filter for trends. The smoother is shown to be the optimal estimator of the trend when the ARIMA model can be decomposed into an uncorrelated random walk trend and stationary cycle components. The conditions under which such a decomposition is possible are discusse...

This paper is concerned with tests in multivariate time series models made up of random walk (with drift) and stationary components. When the stationary component is white noise, a Lagrange multiplier test of the hypothesis that the covariance matrix of the disturbances driving the multivariate random walk is null is shown to be locally best invari...

By setting up a suitable time series model in state space form, the latest estimate of the underlying current change in a series may be computed by the Kalman filter. This may be done even if the observations are only available in a time-aggregated form subject to survey sampling error. A related series, possibly observed more frequently, may be us...

This paper calculates indices of central bank autonomy (CBA) for 163 central banks as of end-2003, and comparable indices for a subgroup of 68 central banks as of the end of the 1980s. The results confirm strong improvements in both economic and political CBA over the past couple of decades, although more progress is needed to boost political auton...

This paper looks at unobserved components models and examines the implied weighting patterns for signal extraction. There are three main themes. The first is the implications of correlated disturbances driving the components, especially those cases in which the correlation is perfect. The second is how setting up models with t-distributed disturban...

The paper considers tests for the presence of a random walk component in a stationary or trend stationary time series and extends them to series which contain structural breaks. The locally best invariant (LBI) test is derived and the asymptotic distribution obtained. Then a modified test statistic is proposed. The advantage of this statistic is th...

An indeterministic cycle can be represented as an infinite moving-average process and has a smooth peak in its spectrum. A deterministic cycle can be formulated in such a way that it is still stationary but its cyclical behaviour is characterized by a sudden jump in the spectral distribution function. This paper shows how both processes can be nest...

A test for the presence of a stationary first-order autoregressive process embedded in white noise is constructed so as to be relatively powerful when the autoregressive parameter is close to one. The test statistic is shown to have a Cramér–von Mises distribution in large samples. A comparison is made with some standard tests for serial correlatio...

Many series are subject to data irregularities such as missing values, outliers, structural breaks, and irregular spacing. Data can also be messy, and hence difficult to handle by standard procedures, when they are intrinsically non-Gaussian or contain complicated periodic patterns because they are observed on an hourly or weekly basis. This paper...