Using theory on (conditionally) Gaussian processes with stationary increments developed in Barndorff-Nielsen et al. (2009, 2011), this paper presents a general semiparametric approach to conducting inference on the fractal index , \alpha, of a time series. Our setup encompasses a large class of Gaussian processes and we show how to extend it to a large class of non-Gaussian models as well. It is proved that the asymptotic distribution of the estimator of \alpha does not depend on the specifics of the data generating process for the observations, but only on the value of \alpha and a "heteroskedasticity" factor. Using this, we propose a simulation-based approach to inference, which is easily implemented and is valid more generally than asymptotic analysis. We detail how the methods can be applied to test whether a stochastic process is a non-semimartingale . Finally, the methods are illustrated in two empirical applications motivated from finance. We study time seriesof log-prices and log-volatility from 29 individual US stocks; no evidence of non-semimartingality in asset prices is found, but we do find evidence of non-semimartingality in volatility. This confirms a recently proposed conjecture that stochastic volatility processes of financial assets are rough (Gatheral et al., 2014) .
We study a well-known estimator of the fractal index of a stochastic process. Our framework is very general and encompasses many models of interest; we show how to extend the theory of the estimator to a large class of non-Gaussian processes. Particular focus is on clarity and ease of implementation of the estimator and the associated asymptotic results, making it easy for practitioners to apply the methods. We further develop a new estimator which is robust to measurement noise in the observations . Finally, the methods are illustrated on two time series; one of turbulent velocity flows and one of financial prices .
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Using theory on (conditionally) Gaussian processes with stationary increments developed in Barndorff-Nielsen et al. (2009, 2011), this paper presents a general semiparametric approach to conducting inference on
We study a well-known estimator of
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, \alpha, of a time series. Our setup encompasses a large class of Gaussian processes and
of a stochastic process. Our framework is very general and encompasses many models of interest;
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the theory of the estimator
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models as well. It is proved that the asymptotic distribution of the estimator of \alpha does not depend on the specifics of the data generating process for the observations, but only on the value of \alpha and a "heteroskedasticity" factor. Using this, we propose a simulation-based approach to inference, which is easily implemented and is valid more generally than asymptotic analysis. We detail how the methods can be applied to test whether a stochastic process is a non-semimartingale
processes. Particular focus is on clarity and ease of implementation of the estimator and the associated asymptotic results, making it easy for practitioners to apply the methods. We further develop a new estimator which is robust to measurement noise in the observations
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in two empirical applications motivated from finance. We study time seriesof log-prices and log-volatility from 29 individual US stocks; no evidence of non-semimartingality in asset prices is found, but we do find evidence of non-semimartingality in volatility. This confirms a recently proposed conjecture that stochastic volatility processes of financial assets are rough (Gatheral et al., 2014)
on two time series; one of turbulent velocity flows and one of financial prices