Using high-frequency data on deutschemark and yen returns against the dollar, we construct model-free estimates of daily exchange rate volatility and correlation that cover an entire decade. Our estimates, termed realized volatilities and correlations, are not only model-free, but also approximately free of measurement error under general conditions, which we discuss in detail. Hence, for practical purposes, we may treat the exchange rate volatilities and correlations as observed rather than latent. We do so, and we characterize their joint distribution, both unconditionally and conditionally. Noteworthy results include a simple normality-inducing volatility transformation, high contemporaneous correlation across volatilities, high correlation between correlation and volatilities, pronounced and persistent dynamics in volatilities and correlations, evidence of long-memory dynamics in volatilities and correlations, and remarkably precise scaling laws under temporal aggregation.

Abstract not found

This paper discusses the use of fractional exponential models (Robinson (1990), Beran (1994)) to model the spectral density f(x) of a covariance stationary process when f(x) may be decomposed as f(x) = x \Gamma2d f (x), where f (x) is bounded and bounded away from zero. A form of log-periodogram regression technique is presented both in the parametric context (i.e. f (x) is a finite order exponential model in the sense of Bloomfield (1973)) and the semi-parametric context (f (x) is regarded as a nuisance parameter). Assuming gaussianity and additional conditions on the regularity of f (x) which seem mild, asymptotic normality of the parameter estimates in the parametric and the semi-parametric context is established. As a by-product, some improvements over the results presented by Robinson (1994) have been obtained for the large sample distribution of log-periodogram ordinates for Gaussian processes.

This paper reviews several periodogram-based methods for estimating the long-memory parameter H in time series and suggests a way to robustify them. The high frequencies tend to bias the estimates. Using only low frequencies eliminates the bias but increases the variance. We hence suggest plotting the estimates of H as a function of a parameter which balances bias versus variance and, if the plot flattens in a central region, to use the flat part for estimating H. We apply this technique to the periodogram regression method, the Whittle approximation to maximum likelihood and to the local Whittle method. We investigate its effectiveness on several simulated fractional ARIMA series and also apply it to estimate the long-memory parameter H in computer network traffic. 1 Introduction Time series with long memory have been considered in many fields including hydrology, biology and computer networks. Unfortunately, estimating the long memory (long-range dependence) parameter H in a given d...

The huge theoretical and empirical econometric literatures on long memory and on structural change have evolved largely independently, as the phenomena appear distinct. We argue, in contrast, that they are intimately related. In particular, we show analytically that stochastic regime switching is easily confused with long memory, so long as only a "small" amount of regime switching occurs (in a sense that we make precise). A Monte Carlo analysis supports the relevance of the asymptotic theory in finite samples and produces additional insights.

Moving from univariate to bivariate jointly dependent long memory time series introduces a phase parameter (), at the frequency of principal interest, zero; for short memory series = 0 automatically. The latter case has also been stressed under long memory, along with the "fractional di¤erencing " case = ( 2 1) =2; where 1; 2 are the memory parameters of the two series. We develop time domain conditions under which these are and are not relevant, and relate the consequent properties of cross-autocovariances to ones of the (possibly bilateral) moving average representation which, with martingale di¤erence innovations of arbitrary dimension, is used in asymptotic theory for local Whittle parameter estimates depending on a single smoothing number. Incorporating also a regression parameter ( ) which, when non-zero, indicates cointegration, the consistency proof of these implicitly-de…ned estimates is nonstandard due to the estimate converging faster than the others. We also establish joint asymptotic normality of the estimates, and indicate how this outcome can apply in statistical inference on several questions of interest. Issues of implemention are discussed, along with implications of knowing and of correct or incorrect speci…cation of; and possible extensions to higher-dimensional systems and nonstationary series.

This thesis describes methods of analysis and synthesis of long memory processes. Long memory processes are those which exhibit correlations between events separated by a long period of time. This phenomenon is characterized in the frequency domain by a sharp peak in the spectral density function as the frequency approaches zero. This characteristic is observed in many physical time series, including those in the fields of geophysics, astronomy and finance. A class of models that captures such long memory behaviour are fractionally differenced processes, the simplest of these processes is obtained by differencing white noise a fractional number of times. We employ two methods of analyzing such processes: Multitaper spectral estimation and Wavelet analysis. Multitaper spectral analysis uses the average of several direct spectral estimators evaluated using orthogonal tapers. We look at two sets of tapers: the discrete prolate spheroidal sequences and sinusoidal tapers. This method of s...

We discuss how prior regression on seasonal dummies leads to singularities in periodogram regression procedures for the detection of long memory. We suggest a modified procedure. We illustrate the problems using monthly inflation data from Hassler and Wolters (1995). Keywords Long Memory, Seasonal Adjustment 1 A Numerical Problem In this section we propose a numerical problem which came up when we redid a regression analysis of simple fractionally integrated models for inflation, where two packages came up with different answers. In the next sections we propose solutions. Periodogram regression is by now a standard procedure to start the examination of long memory in a time series. A popular model with long memory is the ARFIMA(p; d; q) model: \Phi(L)(1 \Gamma L) d y t = \Theta(L)AE t , t = 1; 2; : : : with \Phi(L) and \Theta (L) polynomials of orders p and q in the lag operator L : L k y t = y t\Gammak and (1 \Gamma L) d = 1 \Gamma dL \Gamma d(1\Gammad) 2 L 2 \Gamma...

This paper investigates several strategies for consistently estimating the so-called Hurst parameter H responsible for the long-memory correlations in a linear class of ARCH time series, known as LARCH(1) models, as well as in the continuous-time Gaussian stochastic process known as fractional Brownian motion (fBm). A LARCH model’s parameter is estimated using a conditional maximum likelihood method, which is proved to have good stability properties and to perform well numerically. A local Whittle estimator is also discussed. The article further proposes a specially designed conditional maximum likelihood method for estimating the H which is closer in spirit to one based on discrete observtions of fBm. In keeping with the popular …nancial interpretation of ARCH models, all estimators are based only on observation of the “returns ” of the model, not on their “volatilities”. 1