Computational Statistics and Data Analysis 56 (2012) 3080–3090 Contents lists available at SciVerse ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda On the online estimation of local constant volatilities Roland Fried ∗ Department of Statistics, TU Dortmund University, 44221 Dortmund, Germany article info Article history: Available online 27 February 2011 Keywords: Heteroscedasticity Structural breaks Heavy tails Outliers Tests for equality of variances abstract Time varying volatilities in financial time series are commonly modeled by GARCH or by stochastic volatility models. Models with piecewise constant volatilities have been proposed recently as nonparametric alternatives. Following the latter approach, a procedure for online approximation of the current volatility is constructed by combining one-sided localized estimation of the variability with sequential testing for a change in it. A robust nonparametric framework is assumed since many financial time series show tails heavier than the Gaussian. A two-sample test for a change in variability is proposed, which works well even in case of skewed distributions. © 2011 Elsevier B.V. All rights reserved. 1. Introduction The returns z (t ) = log(p(t )/p(t − 1)) of risky assets in a period t ∈ Z, with p(t ) being the price at the end of period t , are commonly modeled as Z (t ) = σ(t )E (t ), (1) where (σ(t ) : t ∈ Z) is a sequence of time-varying volatilities and (E (t ) : t ∈ Z) is a white noise process with unit variance, which is independent from the volatilities (σ(t ) : t ∈ Z). The GARCH(1, 1) model is popular for describing the time-varying behavior of the latter process, σ 2 (t ) = α 0 + α 1 Z 2 (t − 1) + β 1 σ 2 (t − 1). (2) As an alternative, Mercurio and Spokoiny (2004) and Granger and Stărică (2005) point out that the volatility of many financial time series can be represented adequately by piecewise constant models. A piecewise constant behavior of the volatility provides an explanation of the long memory effects found in many financial time series, since these can be artificially generated by structural breaks (Mikosch and Stărică, 2000). For fitting piecewise constant volatilities one needs to determine time intervals within which the volatility can be approximated by a constant and detect changes between subsequent intervals. Davies et al. (2012) make use of the fact that under the additional assumptions that (E (t ) : t ∈ Z) is Gaussian and (σ(t ) : t ∈ Z) a sequence of constants we have in model (1)  t ∈I Z 2 (t ) σ 2 (t ) ∼ χ 2 |I | , (3) where χ 2 m denotes the χ 2 -distribution with m degrees of freedom and I ⊂{1, 2,..., N } is a nonempty time interval of width |I |. These authors then search a piecewise constant volatility function which minimizes the number of intervals on ∗ Tel.: +49 231 755 3119; fax: +49 231 755 3454. E-mail address: fried@statistik.tu-dortmund.de. 0167-9473/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2011.02.012