The curve time series framework provides a convenient vehicle to accommodate some nonstationary features into a stationary setup. We propose a new method to identify the dimensionality of curve time series based on the dynamical dependence across different curves. The practical implementation of our method boils down to an eigenanalysis of a finite-dimensional matrix. Furthermore, the determination of the dimensionality is equivalent to the identification of the nonzero eigenvalues of the matrix, which we carry out in terms of some bootstrap tests. Asymptotic properties of the proposed method are investigated. In particular, our estimators for zero-eigenvalues enjoy the fast convergence rate n while the estimators for nonzero eigenvalues converge at the standard $\sqrt{n}$-rate. The proposed methodology is illustrated with both simulated and real data sets.Comment: Published in at http://dx.doi.org/10.1214/10-AOS819 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Kernel smoothing techniques free the traditional parametric estimators of volatility from the constraints related to their specific models. In this paper the nonparametric local exponential estimator is applied to estimate conditional volatility functions, ensuring its nonnegativity. Its asymptotic properties are established and compared with those for the local linear estimator. It theoretically enables us to determine when the exponential is expected to be superior to the linear estimator. A very strong and novel result is achieved: the exponential estimator is asymptotically fully adaptive to unknown conditional mean functions. Also, our simulation study shows superior performance of the exponential estimator.

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In this paper we compare several multi‐period volatility forecasting models, specifically from MIDAS and HAR families. We perform our comparisons in terms of out‐of‐sample volatility forecasting accuracy. We also consider combinations of the models' forecasts. Using intra‐daily returns of the BOVESPA index, we calculate volatility measures such as realized variance, realized power variation and realized bipower variation to be used as regressors in both models. Further, we use a nonparametric procedure for separately measuring the continuous sample path variation and the discontinuous jump part of the quadratic variation process. Thus MIDAS and HAR specifications with the continuous sample path and jump variability measures as separate regressors are estimated. Our results in terms of mean squared error suggest that regressors involving volatility measures which are robust to jumps (i.e. realized bipower variation and realized power variation) are better at forecasting future volatility. However, we find that, in general, the forecasts based on these regressors are not statistically different from those based on realized variance (the benchmark regressor). Moreover, we find that, in general, the relative forecasting performances of the three approaches (i.e. MIDAS, HAR and forecast combinations) are statistically equivalent. Copyright © 2014 John Wiley & Sons, Ltd.