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Book part
Publication date: 24 April 2023

Peter C. B. Phillips

The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency…

Abstract

The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

Book part
Publication date: 24 April 2023

Xiaohu Wang, Weilin Xiao and Jun Yu

This chapter derives asymptotic properties of the least squares (LS) estimator of the autoregressive (AR) parameter in local to unity processes with errors being fractional…

Abstract

This chapter derives asymptotic properties of the least squares (LS) estimator of the autoregressive (AR) parameter in local to unity processes with errors being fractional Gaussian noise (FGN) with the Hurst parameter H(0,1). It is shown that the estimator is consistent for all values of H(0,1). Moreover, the rate of convergence is n1 when H[0.5,1). The rate of convergence is n2H when H(0,0.5). Furthermore, the limiting distribution of the centered LS estimator depends on H. When H=0.5, the limiting distribution is the same as that obtained in Phillips (1987a) for the local to unity model with errors for which the standard functional central limit theorem is applicable. When H > 0.5 or when H < 0.5, the limiting distributions are new to the literature. The asymptotic properties of the LS estimator with fitted intercept are also derived. Simulation studies are performed to check the reliability of the asymptotic approximation for different values of sample size.

Book part
Publication date: 24 April 2023

Han-Ying Liang, Yu Shen and Qiying Wang

Joon Y. Park is one of the pioneers in developing nonlinear cointegrating regression. Since his initial work with Phillips (Park & Phillips, 2001) in the area, the past two…

Abstract

Joon Y. Park is one of the pioneers in developing nonlinear cointegrating regression. Since his initial work with Phillips (Park & Phillips, 2001) in the area, the past two decades have witnessed a surge of interest in modeling nonlinear nonstationarity in macroeconomic and financial time series, including parametric, nonparametric and semiparametric specifications of such models. These developments have provided a framework of econometric estimation and inference for a wide class of nonlinear, nonstationary relationships. In honor of Joon Y. Park, this chapter contributes to this area by exploring nonparametric estimation of functional-coefficient cointegrating regression models where the structural equation errors are serially dependent and the regressor is endogenous. The self-normalized local kernel and local linear estimators are shown to be asymptotic normal and to be pivotal upon an estimation of co-variances. Our new results improve those of Cai et al. (2009) and open up inference by conventional nonparametric method to a wide class of potentially nonlinear cointegrated relations.

Book part
Publication date: 24 March 2006

Eric Hillebrand

Apart from the well-known, high persistence of daily financial volatility data, there is also a short correlation structure that reverts to the mean in less than a month. We find…

Abstract

Apart from the well-known, high persistence of daily financial volatility data, there is also a short correlation structure that reverts to the mean in less than a month. We find this short correlation time scale in six different daily financial time series and use it to improve the short-term forecasts from generalized auto-regressive conditional heteroskedasticity (GARCH) models. We study different generalizations of GARCH that allow for several time scales. On our holding sample, none of the considered models can fully exploit the information contained in the short scale. Wavelet analysis shows a correlation between fluctuations on long and on short scales. Models accounting for this correlation as well as long-memory models for absolute returns appear to be promising.

Details

Econometric Analysis of Financial and Economic Time Series
Type: Book
ISBN: 978-1-84950-388-4

Content available
Book part
Publication date: 19 March 2018

Abstract

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Global Tensions in Financial Markets
Type: Book
ISBN: 978-1-78714-839-0

Abstract

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Fundamentals of Transportation and Traffic Operations
Type: Book
ISBN: 978-0-08-042785-0

Book part
Publication date: 24 March 2006

Torben G. Andersen, Tim Bollerslev, Francis X. Diebold and Ginger Wu

A large literature over several decades reveals both extensive concern with the question of time-varying betas and an emerging consensus that betas are in fact time-varying…

Abstract

A large literature over several decades reveals both extensive concern with the question of time-varying betas and an emerging consensus that betas are in fact time-varying, leading to the prominence of the conditional CAPM. Set against that background, we assess the dynamics in realized betas, vis-à-vis the dynamics in the underlying realized market variance and individual equity covariances with the market. Working in the recently popularized framework of realized volatility, we are led to a framework of nonlinear fractional cointegration: although realized variances and covariances are very highly persistent and well approximated as fractionally integrated, realized betas, which are simple nonlinear functions of those realized variances and covariances, are less persistent and arguably best modeled as stationary I(0) processes. We conclude by drawing implications for asset pricing and portfolio management.

Details

Econometric Analysis of Financial and Economic Time Series
Type: Book
ISBN: 978-1-84950-388-4

Book part
Publication date: 29 March 2006

Borus Jungbacker and Siem Jan Koopman

In this chapter, we aim to measure the actual volatility within a model-based framework using high-frequency data. In the empirical finance literature, it is widely discussed that…

Abstract

In this chapter, we aim to measure the actual volatility within a model-based framework using high-frequency data. In the empirical finance literature, it is widely discussed that tick-by-tick prices are subject to market micro-structure effects such as bid-ask bounces and trade information. These market micro-structure effects become more and more apparent as prices or returns are sampled at smaller and smaller time intervals. An increasingly popular measure for the variability of spot prices on a particular day is realised volatility that is typically defined as the sum of squared intra-daily log-returns. Recent theoretical results have shown that realised volatility is a consistent estimator of actual volatility, but when it is subject to micro-structure noise and the sampling frequency increases, the estimator diverges. Parametric and nonparametric methods can be adopted to account for the micro-structure bias. Here, we measure actual volatility using a model that takes account of micro-structure noise together with intra-daily volatility patterns and stochastic volatility. The coefficients of this model are estimated by maximum likelihood methods that are based on importance sampling techniques. It is shown that such Monte Carlo techniques can be employed successfully for our purposes in a feasible way. As far as we know, this is a first attempt to model the basic components of the mean and variance of high-frequency prices simultaneously. An illustration is given for three months of tick-by-tick transaction prices of the IBM stock traded at the New York Stock Exchange.

Details

Econometric Analysis of Financial and Economic Time Series
Type: Book
ISBN: 978-0-76231-274-0

Book part
Publication date: 28 October 2019

Angelo Corelli

Abstract

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Understanding Financial Risk Management, Second Edition
Type: Book
ISBN: 978-1-78973-794-3

Book part
Publication date: 1 January 2005

Jose A. Lopez

Foreign exchange rates are examined using cointegration tests over various time periods linked to regime shifts in central bank behavior. The number of cointegrating vectors…

Abstract

Foreign exchange rates are examined using cointegration tests over various time periods linked to regime shifts in central bank behavior. The number of cointegrating vectors appears to vary across these regime changes within the foreign exchange market. For example, cointegration is not generally found prior to the Plaza Agreement of September 22, 1985, but it is present after that date. The significance of these changes is evaluated using a likelihood ratio procedure proposed by Quintos (1994). The changing nature of the cointegrating relationships indicate that certain aspects of central bank activity do have long-term effects on exchange rates.

Details

Research in Finance
Type: Book
ISBN: 978-0-76231-277-1

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