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.

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COWLES FOUNDATION DISCUSSION PAPER NO. 1263

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...

This paper studies fractional processes that may be perturbed by weakly dependent time series. The model for a perturbed fractional process has a components framework in which there may be components of both long and short memory. All commonly used estimates of the long memory parameter (such as log periodogram (LP) regression) may be used in a components model where the data are affected by weakly dependent perturbations, but these estimates can suffer from serious downward bias. To circumvent this problem, the present paper proposes a new procedure that allows for the possible presence of additive perturbations in the data. The new estimator resembles the LP regression estimator but involves an additional (nonlinear) term in the regression that takes account of possible perturbation effects in the data. Under some smoothness assumptions at the origin, the bias of the new estimator is shown to disappear at a faster rate than that of the LP estimator, while its asymptotic variance is inflated only by a multiplicative constant. In consequence, the optimal rate of convergence to zero of the asymptotic MSE of the new estimator is faster than that of the LP estimator. Some simulation results demonstrate the viability and the bias-reducing feature of the new estimator relative to the LP estimator in finite samples. A

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The long range dependence paradigm appears to be a suitable description of the data generating process for many observed economic time series. This is mainly due to the fact that it naturally characterizes time series displaying a high degree of persistence, in the form of a long lasting effect of unanticipated shocks, yet exhibiting mean reversion. Whereas linear long range dependent time series models have been extensively used in macroeconomics, empirical evidence from financial time series prompted the development of nonlinear long range dependent time series models, in particular models of changing volatility.We discuss empirical evidence of long range dependence as well as the theoretical issues, both for economics and econometrics, such evidence has stimulated.

Recently, there has been an upsurge of interest on the possibility of confusing long memory and structural changes in level. Many studies have shown that when a stationary short memory process is contaminated by level shifts the estimate of the fractional differencing parameter is biased away from zero and the autocovariance function exhibits a slow rate of decay, akin to a long memory process. We analyze the properties of the log periodogram estimate of the memory parameter when the jump component is specified by a simple mixture model. Our theoretical results explain many findings reported and uncover new features. Simulations are presented to highlight the properties of the distributions and to assess the adequacy of our approximations. We also show the usefulness of our results to distinguish between long memory and level shifts via an application to the volatility of daily returns for wheat commodity futures. JEL Classification Number: C22.

Many empirical studies find a negative correlation between the returns on the nominal spot exchange rate and the lagged forward discount. This forward discount anomaly implies that the current forward rate is a biased predictor of the future spot rate. A large number of studies in the existing literature try to explain this anomaly, and recent work has tried to explain the anomaly as a statistical artifact based on (1) the long memory behavior of the forward discount; or (2) the existence of structural breaks in the forward discount. In this paper, we evaluate the evidence for long memory and structural change in the forward discount. Our approach is as follows. First, we nonparametrically estimate the long memory parameter for a number of forward discount series without allowing for structural breaks. Second, we test for and estimate a multiple mean break model and then adjust for the structural breaks in the forward discount. Finally, we re-estimate the long memory parameter on the mean-break adjusted data. To check the robustness of our results, we use Monte Carlo simulations to evaluate our procedure to estimate structural breaks in the presence of long-memory. We show that allowing for structural breaks drastically reduces the persistence of the forward discount. However, after removing the breaks, we still find evidence of stationary long memory in all of the forward discount series. Our results have important implications for understanding the statistical properties of the forward discount, because we confirm not only the presence of long memory behavior in the forward discount but also the importance of structural breaks.