Circuit Breakers and the Tail Index of Equity Returns

Using the tail index of returns on US equities as a summary measure of extreme behaviour, we examine changes in the equity markets surrounding the development of program trading for portfolio insurance, the crash of 1987, and the subsequent introduction of circuit breakers and other changes in market architecture. Recently-developed tests for the null of constancy of the tail index, versus the alternative of a change at an unknown date, permit inference on changes in extreme behaviour over a long time period while allowing for second-moment dependence in the return data. We find strong evidence of a decrease in the tail index (increase in the probability of extreme events) around the beginning of large scale program trading, and weaker, but still substantial, evidence of further significant change in the tail index following the introduction of circuit breakers. Point estimates of the tail index suggest that the tail index has nonetheless not regained pre-program-trading levels. Utilisant l'indice de queue de distribution (index tail) des rendements financiers sur actions dans les marchés américains comme mesure sommaire des comportements extrêmes, nous examinons les changements dans le marché des actions entourant le développement de programmes automatiques de transaction (Trading Program) pour l'assurance de portefeuille, le krach de 1987, l'introduction des coupe-circuits et autres changements dans les systèmes financiers. De nouveaux tests, récemment développés, permettent l'inférence statistique sur le changement des comportements extrêmes sur une longue période ; tests qui sont valides dans le cas d'hétéroscédasticité conditionnelle. L'hypothèse nulle est que l'indice de queue de distribution est constant alors que l'hypothèse alternative est le changement de cet indice à une date inconnue. Nous avons trouvé de manière très significative que d'une part, l'indice de queue de distribution a diminué (la probabilité d'évènements extrêmes a augmenté) au début de la période des programmes de transactions. D'autre part, l'introduction de coupe-circuits a augmenté cet indice mais est resté plus faible que sa valeur avant l'introduction des programmes de transactions. Les estimateurs de l'indice de queue de distribution suggèrent qu'il n'a pas retrouvé sa valeur initiale.Circuit breaker, structural change tests, tail index, Coupe-circuits, tests sur le changement structurel, indice de queue de distribution

Information Content of Volatility Forecasts at Medium-term Horizons

Using realized volatility to estimate daily conditional volatility of financial returns, we compare forecasts of daily volatility from standard QML-estimated GARCH models, and from projections on past realized volatilities obtained from high-frequency data. We consider horizons extending to thirty trading days. The forecasts are compared with the unconditional sample variance of daily returns treated as a daily volatility forecast, allowing us to estimate the maximum horizon at which the model-based forecasts provide forecasting power, measured by MSE reduction. Using data from a Toronto Stock Exchange equity index and foreign exchange returns (DM/$US and Yen/$US), we find evidence of forecasting power at horizons of up to thirty trading days, on each of the three financial returns series. We also find some evidence that the result of (e.g.) Bollerslev and Wright (2001), that projections on past realized volatility provide better 1-step forecasts than the QML-GARCH forecasts, appears to extend to longer horizons up to around ten to fifteen trading days. At longer horizons, there appears to be little to distinguish the forecast methods. En utilisant la volatilité réalisée pour estimer la volatilité conditionnelle quotidienne des rendements financiers, nous comparons les prévisions de volatilité quotidienne effectuées à partir de modèles GARCH-QVM standard et à partir de projections directes sur les volatilités réalisées. Nous considérons un horizon maximal de trente jours de transaction. Les prévisions sont comparées à la variance non conditionnelle des rendements quotidiens, ce qui nous permet d'estimer l'horizon maximal pour lequel les modèles détiennent un pouvoir de prévision. Nous utilisons des données de l'indice TSE 35 et des taux de change DM/US$et Yen/US$, et nos résultats montrent qu'il y a un pouvoir de prédiction jusqu'à un horizon de trente jours, et ce, pour chacune des trois séries. Nous montrons aussi que le résultat de Bollerslev et Wright (2001), résultat indiquant que les projections sont supérieures sur l'horizon d'un jour, reste valide dans un horizon s'étendant jusqu'à dix ou quinze jours. Pour des horizons plus longs, les deux types de méthodes de prévision ne se différencient guère.GARCH, high-frequency data, integrated volatility, realized volatility, GARCH, données à haute fréquence, volatilité intégrée, volatilité réalisée

A GENERALIZED ASYMMETRIC STUDENT-T DISTRIBUTION WITH APPLICATION TO FINANCIAL ECONOMETRICS

This paper proposes a new class of asymmetric Student-t (AST) distributions, and investigates its properties, gives procedures for estimation, and indicates applications in financial econometrics. We derive analytical expressions for the cdf, quantile function, moments, and quantities useful in financial econometric applications such as the expected shortfall. A stochastic representation of the distribution is also given. Although the AST density does not satisfy the usual regularity conditions for maximum likelihood estimation, we establish consistency, asymptotic normality and efficiency of ML estimators and derive an explicit analytical expression for the asymptotic covariance matrix. A Monte Carlo study indicates generally good finite-sample conformity with these asymptotic properties.

How Far Can Forecasting Models Forecast? Forecast Content Horizons for Some Important Macroeconomic Variables

For stationary transformations of variables, there exists a maximum horizon beyond which forecasts can provide no more information about the variable than is present in the unconditional mean. Meteorological forecasts, typically excepting only experimental or exploratory situations, are not reported beyond this horizon; by contrast, little generally accepted information about such maximum horizons is available for economic variables. The authors estimate such content horizons for a variety of economic variables, and compare these with the maximum horizons that they observe reported in a large sample of empirical economic forecasting studies. The authors find that many published studies provide forecasts exceeding, often by substantial margins, their estimates of the content horizon for the particular variable and frequency. The authors suggest some simple reporting practices for forecasts that could potentially bring greater transparency to the process of making and interpreting economic forecasts.Econometric and statistical methods, Business fluctuations and cycles

Electronic Transactions as High-Frequency Indicators of Economic Activity

Since the advent of standard national accounts data over 60 years ago, economists have traditionally relied on monthly or quarterly data supplied by central statistical agencies for macroeconomic modelling and forecasting. However, technological advances of the past several years have resulted in new high-frequency data sources that could potentially provide more accurate and timely information on the current level of economic activity. In this paper we explore the usefulness of electronic transactions as real-time indicators of economic activity, using Canadian debit card data as an example. These data have the advantages of daily availability and the high market penetration of debit cards. We find that (i) household transactions vary greatly according to the day of the week, peaking every Friday and falling every Sunday; (ii) debit card data can help lower consensus forecast errors for GDP and consumption (especially non-durable) growth; (iii) debit card transactions are correlated with Statistics Canada’s revisions to GDP; (iv) high-frequency analyses of transactions around extreme events are possible, and in particular we are able to analyze expenditure patterns around the September 11 terrorist attacks and the August 2003 electrical blackout.Business fluctuations and cycles

Properties of Estimates of Daily GARCH Parameters Basaed on Intra-Day Observations

We consider estimates of the parameters of GARCH models of daily financial returns, obtained using intra-day (high-frequency) returns data to estimate the daily conditional volatility.Two potential bases for estimation are considered. One uses aggregation of high-frequency Quasi- ML estimates, using aggregation results of Drost and Nijman (1993). The other uses the integrated volatility of Andersen and Bollerslev (1998), and obtains coefficients from a model estimated by LAD or OLS, in the former case providing consistency and asymptotic normality in some cases where moments of the volatility estimation error may not exist. In particular, we consider estimation in this way of an ARCH approximation, and obtain GARCH parameters by a method related to that of Galbraith and Zinde-Walsh (1997) for ARMA processes. We offer some simulation evidence on small-sample performance, and characterize the gains relative to standard quasi-ML estimates based on daily data alone. Nous considérons les estimés des paramètres des modèles GARCH pour les rendements financiers journaliers, qui sont obtenus à l'aide des données intra-jour (haute fréquence) pour estimer la volatilité journalière. Deux bases potentielles sont evaluées. La première est fondée sur l'aggrégation des estimés quasi-vraisemblance-maximale, en profitant des résultats de Drost et Nijman (1993). L'autre utilise la volatilité integrée de Andersen et Bollerslev (1998), et obtient les coefficients d'un modèle estimé par LAD ou MCO; la première méthode résiste mieux à la possibilité de non-existence des moments de l'erreur en estimation de volatilité. En particulier, nous considérons l'estimation par approximation ARCH, et nous obtenons les paramètres par une méthode liée à celle de Galbraith et Zinde-Walsh (1997) pour les processus ARMA. Nous offrons des résultats provenant des simulations sur la performance des méthodes en échantillons finis, et nous décrivons les atouts relatifs à l'estimation standard de quasi-VM basée uniquement sur les données journalières.GARCH, high frequency data, integrated volatility, LAD, GARCH, données haute fréquence, volatilité intégrée, LAD

FORECAST CONTENT AND CONTENT HORIZONS FOR SOME IMPORTANT MACROECONOMIC TIME SERIES

For quantities that are approximately stationary, the information content of statistical forecasts tends to decline as the forecast horizon increases, and there exists a maximum horizon beyond which forecasts cannot provide discernibly more information about the variable than is present in the unconditional mean (the content horizon). The pattern of decay of forecast content (or skill) with increasing horizon is well known for many types of meteorological forecasts; by contrast, little generally-accepted information about these patterns or content horizons is available for economic variables. In this paper we attempt to develop more information of this type by estimating content horizons for variety of macroeconomic quantities; more generally, we characterize the pattern of decay of forecast content as we project farther into the future. We find wide variety of results for the different macroeconomic quantities, with models for some quantities providing useful content several years into the future, for other quantities providing negligible content beyond one or two months or quarters.

HOW FAR CAN WE FORECAST? FORECAST CONTENT HORIZONS FOR SOME IMPORTANT MACROECONOMIC TIME SERIES

For stationary transformations of variables, there exists a maximum horizon beyond which forecasts can provide no more information about the variable than is present in the unconditional mean. Meteorological forecasts, typically excepting only experimental or exploratory situations, are not reported beyond this horizon; by contrast, little generally-accepted information about such maximum horizons is available for economic variables. In this paper we estimate such content horizons for a variety of economic variables, and compare these with the maximum horizons which we observe reported in a large sample of empirical economic forecasting studies. We find that there are many instances of published studies which provide forecasts exceeding, often by substantial margins, our estimates of the content horizon for the particular variable and frequency. We suggest some simple reporting practices for forecasts that could potentially bring greater transparency to the process of making the interpreting economic forecasts.

A Generalized Asymmetric Student-t Distribution with Application to Financial Econometrics

This paper proposes a new class of asymmetric Student-t (AST) distributions, and investigates its properties, gives procedures for estimation, and indicates applications in financial econometrics. We derive analytical expressions for the cdf, quantile function, moments, and quantities useful in financial econometric applications such as the expected shortfall. A stochastic representation of the distribution is also given. Although the AST density does not satisfy the usual regularity conditions for maximum likelihood estimation, we establish consistency, asymptotic normality and efficiency of ML estimators and derive an explicit analytical expression for the asymptotic covariance matrix. A Monte Carlo study indicates generally good finite-sample conformity with these asymptotic properties. Le présent document propose une nouvelle catégorie de distributions asymétriques suivant la loi t de Student (Asymmetric Student-t Distribution - AST). Il en examine les propriétés, suggère des procédures d’estimation et propose des applications dans le domaine de l’économétrie financière. Nous établissons des expressions analytiques pour la fonction de distribution cumulative, la fonction quantile, les moments et les quantités, ces aspects étant utiles dans certaines applications liées à l’économétrie financière, par exemple l’estimation du manque à gagner prévu. Nous mettons aussi de l’avant une représentation stochastique de la distribution. Même si la densité suivant la loi t de Student ne répond pas aux conditions habituelles de régularité pour l’estimation du maximum de vraisemblance, nous établissons néanmoins la consistance, la normalité asymptotique et l’efficacité des estimateurs du maximum de vraisemblance et arrivons à une expression analytique explicite en ce qui concerne la matrice de covariance asymptotique. Une étude selon la méthode Monte Carlo indique généralement une bonne conformité des échantillons finis avec ces propriétés asymptotiques.asymmetric distribution, expected shortfall, maximum likelihood estimation, distribution asymétrique, manque à gagner prévu, estimation du maximum de vraisemblance