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In this paper we extend the concept of graphical models for multivariate data to multivariate time series. We de ne a partial correlation graph for time series and use the partial spectral coherence between two components given the remaining components to identify the edges of the graph. As an example we consider multivariate autoregressive processes. The method is applied to air pollution data. 1 This work has been supported by a European Union Capital and Mobility Programme (ERB CHRX-CT 940693) AMS 1991 subject classi cations. Primary 62M15 secondary 62F10. Key words and phrases. Graphical models, multivariate time series, partial spectral coherence, spectral estimates, multivariate autoregressive processes, air pollution data.

A class of processes with a time varying spectral representation is established. As an example we study time varying autoregressions. Several results on the asymptotic norm behaviour and trace behaviour of covariance matrices of such processes are derived. As a consequence we prove a Kolmogorov formula for the local prediction error and calculate the asymptotic Kullback Leibler information divergence.

In this paper the class of ARCH$(\infty)$ models is generalized to the nonstationary class of ARCH$(\infty)$ models with time-varying coefficients. For fixed time points, a stationary approximation is given leading to the notation ``locally stationary ARCH$(\infty)$ process.'' The asymptotic properties of weighted quasi-likelihood estimators of time-varying ARCH$(p)$ processes ($p<\infty$) are studied, including asymptotic normality. In particular, the extra bias due to nonstationarity of the process is investigated. Moreover, a Taylor expansion of the nonstationary ARCH process in terms of stationary processes is given and it is proved that the time-varying ARCH process can be written as a time-varying Volterra series.Comment: Published at http://dx.doi.org/10.1214/009053606000000227 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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The article contains an overview over locally stationary processes. At the beginning time varying autoregressive processes are discussed in detail -both as as a deep example and an important class of locally stationary processes. In the next section a general framework for time series with time varying finite dimensional parameters is discussed with special emphasis on nonlinear locally stationary processes. Then the paper focusses on linear processes where a more general theory is possible. First a general definition for linear processes is given and time varying spectral densities are discussed in detail. Then the Gaussian likelihood theory is presented for locally stationary processes. In the next section the relevance of empirical spectral processes for locally stationary time series is discussed. Empirical spectral processes play a major role in proving theoretical results and provide a deeper understanding of many techniques. The article concludes with an overview of other results for locally stationary processes. Keywords: locally stationary process, time varying parameter parameter, local likelihood, derivative process, time varying autoregressive process, shape curve, empirical spectral process, time varying spectral density Acknowledgement: I am grateful to Suhasini Subba Rao for helpful comments on an earlier version which lead to significant improvements. 0 arXiv:1109.4174v2 [math.ST]