This. paper analyzes identificat.io~ conditions, and proposes an estimator, for a dy-namJc factor model.where the 1d10syncratic components are allowed to be mutually non-ort~ogonal. ThJS mode!, which we cali generalized dynamic fador model is nove! to. the hterature,. an d generalizes the stati c approximate • factor mode! of Chamberlam and Rothschild (1983), as well as the exact factor mode! à la Sargent and Sims (197~). We prove rr:ean-s~uare convergence of our estimator to the common factor as t~e t1me cross-~ect10nal d1~ensions go to infinity at appropriate rates. Simulations yJe!d encouragmg results m small samples. An empirica! example on the out t growth of US states illustrates the method. pu JEL classification nos.: C13, C33, C43.

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.

The paper uses a large data set, made up by 447 monthly macroeconomic time series concerning the main countries of the Euro area to simulate out-of-sample predictions of the Euro area industrial production and the harmonized inflation indexes and to evaluate the role of financial variables in forecasting. We considered two models which allows forecasting based on large panels of time series: Forni, Hallin, Lippi and Reichlin (2001b) and Stock and Watson (1999). Performance of both models were compared to that of a simple univariate AR model. Results show that, in general, multivariate methods outperform univariate methods and that financial variables help forecasting inflation, but do not help forecasting industrial production. We also show that Forni et al.'s method outperforms SW's. JEL subject classification : C13, C33, C43.

Key words and phrases: High -dimensional time series. Generalized dynamic factor models.Vector processes with singular spectral density. One-sided representations of dynamic factor models. Consistency and rates.

A new multivariate concept of quantile, based on a directional version of Koenker and Bassett's traditional regression quantiles, is introduced for multivariate location and multiple-output regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow for efficient depth contour computations by means of parametric linear programming, but also for transferring from the quantile to the depth universe such asymptotic results as Bahadur representations. Finally, linear programming duality opens the way to promising developments in depth-related multivariate rank-based inference. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2010, Vol. 38, No. 2, 635-669. This reprint differs from the original in pagination and typographic detail. 1 2 M. HALLIN, D. PAINDAVEINE AND M.ŠIMANideas of this definition were exposed in an unpublished master thesis by Laine [21], quoted in [16]. In this paper, we carefully revive Laine's ideas, and systematically develop and prove the main properties of the concept he introduced.A huge literature has been devoted to the problem of extending to a multivariate setting the fundamental one-dimensional concept of quantile; see, for instance, [1, 3-7, 10, 15, 19, 34] and [37] or [33] for a recent survey. An equally huge literature-see [9,22,39] and [40] for a comprehensive account-is dealing with the concept of (location) depth. The philosophies underlying those two concepts at first sight are quite different, and even, to some extent, opposite. While quantiles resort to analytical characterizations through inverse distribution functions or L 1 optimization, depth often derives from more geometric considerations such as halfspaces, simplices, ellipsoids and projections. Both carry advantages and some drawbacks. Analytical definitions usually bring in efficient algorithms and tractable asymptotics. The geometric ones enjoy attractive equivariance properties and intuitive contents, but their probabilistic study and asymptotics are generally trickier, while their implementation, as a rule, leads to heavy combinatorial algorithms; a highly elegant analytical approach to depth has been proposed in [24], but does not help much in that respect.Yet, beyond those sharp methodological differences, quantiles and depth obviously exhibit a close conceptional kinship. In the univariate case, all definitions basically agree that the depth of a point x ∈ R with respect to a probability distribution P with strictly monotone distribution function F should be min(F (x), 1 − F (x)), so that the only points with depth d are x d := F −1 (d) and x 1−d := F −1 (1 − d)-the quantiles of orders ...