There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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 with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

This paper provides a general framework for integration of high-frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency volatility and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on restrictive and complicated parametric multivariate ARCH or stochastic volatility models, which often perform poorly at intraday frequencies. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time series procedures for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we formally develop the links between the conditional covariance matrix and the concept of realized volatility. Next, using continuously recorded observations for the Deutschemark / Dollar and Yen / Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably compared to popular daily ARCH and related models. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution implied by the theoretically and empirically grounded assumption of normally distributed standardized returns, gives rise to well-calibrated density forecasts of future returns, and correspondingly accurate quantile estimates. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation and financial risk management applications.

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Modeling and forecasting realized volatility

Summary. The availability of intraday data on the prices of speculative assets means that we can use quadratic variation-like measures of activity in financial markets, called realized volatility, to study the stochastic properties of returns. Here, under the assumption of a rather general stochastic volatility model, we derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility). These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods.

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Econometric analysis of realized volatility and its use in estimating stochastic volatility models

Matrix Differential Calculus with Applications in Statistics and Econometrics

The paper evaluates the performance of several recently proposed tests for structural breaks in the conditional variance dynamics of asset returns. The tests apply to the class of ARCH and SV type processes as well as data-driven volatility estimators using high-frequency data. In addition to testing for the presence of breaks, the statistics identify the number and location of multiple breaks. We study the size and power of the new tests for detecting breaks in the conditional variance under various realistic univariate heteroscedastic models, change-point hypotheses and sampling schemes. The paper concludes with an empirical analysis using data from the stock and FX markets for which we find multiple breaks associated with the Asian and Russian financial crises. These events resulted in changes in the dynamics of volatility of asset returns in the samples prior and post the breaks. Copyright © 2002 John Wiley & Sons, Ltd.

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Detecting multiple breaks in financial market volatility dynamics

Regression models with mixed sampling frequencies

We introduce easy-to-implement, regression-based methods for predicting quarterly real economic activity that use daily financial data and rely on forecast combinations of mixed data sampling (MIDAS) regressions. We also extract a novel small set of daily financial factors from a large panel of about 1000 daily financial assets. Our analysis is designed to elucidate the value of daily financial information and provide real-time forecast updates of the current (nowcasting) and future quarters of real GDP growth.

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Should Macroeconomic Forecasters Use Daily Financial Data and How

We propose extensions of the continuous record asymptotic analysis for rolling sample variance estimators developed for estimating the quadratic variation of asset returns, referred to as integrated or realized volatility. We treat integrated volatility as a continuous time stochastic process sampled at high frequencies and suggest rolling sample estimators which share many features with spot volatility estimators. We discuss asymptotically efficient window lengths and weighting schemes for estimators of the quadratic variation and establish links between various spot and integrated volatility estimators. Theoretical results are complemented with extensive Monte Carlo simulations and an empirical investigation.

https://www.ucy.ac.cy/econ/documents/working_papers/0108.pdf

Rolling-Sample Volatility Estimators: Some New Theoretical, Simulation, and Empirical Results

/pdf/forecasting-with-mixed-frequency-data-rq1yi6tp22.pdf

Elena Andreou

Papers

Detecting multiple breaks in financial market volatility dynamics

Regression models with mixed sampling frequencies

Should Macroeconomic Forecasters Use Daily Financial Data and How

Rolling-Sample Volatility Estimators: Some New Theoretical, Simulation, and Empirical Results

Forecasting with Mixed-Frequency Data