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For more than half a century, Manfred Deistler has been contributing to the construction of the rigorous theoretical foundations of the statistical analysis of time series and more general stochastic processes. Half a century of unremitting activity is not easily summarized in a few pages. In this short note, we chose to concentrate on a relatively little-known aspect of Manfred’s contribution that nevertheless had quite an impact on the development of one of the most powerful tools of contemporary time series and econometrics: dynamic factor models.

This article develops an information criterion for determining the number q of common shocks in the general dynamic factor model developed by Forni et al., as opposed to the restricted dynamic model considered by Bai and Ng and by Amengual and Watson. Our criterion is based on the fact that this number q is also the number of diverging eigenvalues of the spectral density matrix of the observations as the number n of series goes to infinity. We provide sufficient conditions for consistency of the criterion for large n and T (where T is the series length). We show how the method can be implemented and provide simulations and empirics illustrating its very good finite-sample performance. Application to real data adds a new empirical facet to an ongoing debate on the number of factors driving the U.S. economy.

This paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity. We establish the convergence, under the null hypothesis and contiguous alternatives, of the log ratio of the joint densities of the sample covariance eigenvalues to a Gaussian process indexed by the norm of the perturbation. When the perturbation norm is larger than the phase transition threshold studied in Baik, Ben Arous and Péché [Ann. Probab. 33 (2005) 1643—1697] the limiting process is degenerate, and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the limiting process is nondegenerate, and the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous. Using the asymptotic theory of statistical experiments, we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region. In particular, we show that the asymptotic power of the Tracy—Widom-type tests is trivial (i.e., equals the asymptotic size), whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.

Classical spectral methods are subject to two fundamental limitations: they can account only for covariance-related serial dependences, and they require second-order stationarity. Much attention has been devoted lately to quantile-based spectral methods that go beyond covariance-based serial dependence features. At the same time, covariance-based methods relaxing stationarity into much weaker local stationarity conditions have been developed for a variety of time series models. Here, we combine those two approaches by proposing quantilebased spectral methods for locally stationary processes. We therefore introduce a time varying version of the copula spectra that have been recently proposed in the literature, along with a suitable local lag window estimator. We propose a new definition of local strict stationarity that allows us to handle completely general non-linear processes without any moment assumptions, thus accommodating our quantile-based concepts and methods. We establish a central limit theorem for the new estimators and illustrate the power of the proposed methodology by means of a simulation study. Moreover, in two empirical studies (namely of the Standard & Poor's 500 series and a temperature data set recorded in Hohenpeissenberg), we demonstrate that the new approach detects important variations in serial dependence structures both across time and across quantiles. Such variations remain completely undetected and are actually undetectable, via classical covariance-based spectral methods.

We propose rank-based estimators of principal components, both in the one-sample and, under the assumption of common principal components , in the m -sample cases. Those estimators are obtained via a rank-based version of Le Cam’s one-step method, combined with an estimation of cross-information quantities . Under arbitrary elliptical distributions with, in the m -sample case, possibly heterogeneous radial densities, those R-estimators remain root- n consistent and asymptotically normal, while achieving asymptotic efficiency under correctly specified radial densities. Contrary to their traditional counterparts computed from empirical covariances, they do not require any moment conditions. When based on Gaussian score functions, in the one-sample case, they uniformly dominate their classical competitors in the Pitman sense. Their AREs with respect to other robust procedures are quite high—up to 30, in the Gaussian case, with respect to minimum covariance determinant estimators. Their finite-sample performances are investigated via a Monte Carlo study.

This paper proposes a factor model with infinite dynamics and nonorthogonal idiosyncratic components. The model, which we call the generalized dynamic-factor model, is novel to the literature and generalizes the static approximate factor model of Chamberlain and Rothschild (1983), as well as the exact factor model à la Sargent and Sims (1977). We provide identification conditions, propose an estimator of the common components, prove convergence as both time and cross-sectional size go to infinity at appropriate rates, and present simulation results. We use our model to construct a coincident index for the European Union. Such index is defined as the common component of real GDP within a model including several macroeconomic variables for each European country.

Hallin and Ley [Bernoulli 18 (2012) 747-763] investigate and fully characterize the Fisher singularity phenomenon in univariate and multivariate families of skew-symmetric distributions. This paper proposes a refined analysis of the (univariate) problem, showing that singularity can be more or less severe, inducing n1/4 (\"simple singularity\") n⅙ (\"double singularity\"), or n1/8 (\"triple singularity\") consistency rates for the skewness parameter. We show, however, that simple singularity (yielding n⅙ consistency rates), if any singularity at all, is the rule, in the sense that double and triple singularities are possible for generalized skew-normal families only. We also show that higher-order singularities, leading to worse-than-n⅛ rates, cannot occur. Depending on the degree of the singularity, our analysis also suggests a simple reparametrization that offers an alternative to the so-called centred parametrization proposed, in the particular case of skew-normal and skew-f families, by Azzalini [Scand. J. Stat. 12 (1985) 171-178], Arei lano-Val le and Azzalini [J. Multivariate Anal. 113 (2013) 73-90], and DiCiccio and Monti [Quaderni di Statistica 13 (2011) 1-21], respectively.

Skew-symmetric densities recently received much attention in the literature, giving rise to increasingly general families of univariate and multivariate skewed densities. Most of those families, however, suffer from the inferential drawback of a potentially singular Fisher information in the vicinity of symmetry. All existing results indicate that Gaussian densities (possibly after restriction to some linear subspace) play a special and somewhat intriguing role in that context. We dispel that widespread opinion by providing a full characterization, in a general multivariate context, of the information singularity phenomenon, highlighting its relation to a possible link between symmetric kernels and skewing functions - a link that can be interpreted as the mismatch of two densities.

Interconnectedness between stocks and firms plays a crucial role in the volatility contagion phenomena that characterize financial crises, and graphs are a natural tool in their analysis. We propose graphical methods for an analysis of volatility interconnections in the Standard & Poor's 100 data set during the period 2000–2013, which contains the 2007–2008 Great Financial Crisis. The challenges are twofold: first, volatilities are not directly observed and must be extracted from time series of stock returns; second, the observed series, with about 100 stocks, is high dimensional, and curse-of-dimensionality problems are to be faced. To overcome this double challenge, we propose a dynamic factor model methodology, decomposing the panel into a factor-driven and an idiosyncratic component modelled as a sparse vector auto-regressive model. The inversion of this auto-regression, along with suitable identification constraints, produces networks in which, for a given horizon h, the weight associated with edge (i, j) represents the h-step-ahead forecast error variance of variable i accounted for by variable j's innovations. Then, we show how those graphs yield an assessment of how systemic each firm is. They also demonstrate the prominent role of financial firms as sources of contagion during the 2007–2008 crisis.

We propose new concepts of statistical depth, multivariate quantiles, vector quantiles and ranks, ranks and signs, based on canonical transportation maps between a distribution of interest on ℝd and a reference distribution on the d-dimensional unit ball. The new depth concept, called Monge-Kantorovich depth, specializes to halfspace depth for d = 1 and in the case of spherical distributions, but for more general distributions, differs from the latter in the ability for its contours to account for non-convex features of the distribution of interest. We propose empirical counterparts to the population versions of those Monge-Kantorovich depth contours, quantiles, ranks, signs and vector quantiles and ranks, and show their consistency by establishing a uniform convergence property for empirical (forward and reverse) transport maps, which is the main theoretical result of this paper.