On complex and real identifiability of tensors

We report about the state of the art on complex and real generic identifiability of tensors, we describe some of our recent results obtained in [6] and we present perspectives on the subject.Comment: To appear on Rivista di Matematica dell'Universit\`a di Parma, Volume 8, Number 2, 2017, pages 367-37

Higher secants of spinor varieties

Let $S_h$ be the even pure spinors variety of a complex vector space $V$ of even dimension $2h$ endowed with a non degenerate quadratic form $Q$ and let $\sigma_k(S_h)$ be the $k$-secant variety of $S_h$. We decribe a probabilistic algorithm which computes the complex dimension of $\sigma_k(S_h)$. Then, by using an inductive argument, we get our main result: $\sigma_3(S_h)$ has the expected dimension except when $h\in \{7,8\}$. Also we provide theoretical arguments which prove that $S_7$ has a defective 3-secant variety and $S_8$ has defective 3-secant and 4-secant varieties.Comment: 23 page

On the identifiability of ternary forms

We describe a new method to determine the minimality and identifiability of a Waring decomposition $A$ of a specific form (symmetric tensor) $T$ in three variables. Our method, which is based on the Hilbert function of $A$, can distinguish between forms in the span of the Veronese image of $A$, which in general contains both identifiable and not identifiable points, depending on the choice of coefficients in the decomposition. This makes our method applicable for all values of the length $r$ of the decomposition, from $2$ up to the generic rank, a range which was not achievable before. Though the method in principle can handle all cases of specific ternary forms, we introduce and describe it in details for forms of degree $8$

Econometric analyses with backdated data: unified Germany and the euro area

In this paper we compare alternative approaches for the construction of time series of macroeconomic variables for Unified Germany prior to 1991, and then use them for the construction of corresponding time series for the euro area. The resulting series for Germany and the euro area are compared with existing ones on the basis of both descriptive statistics and results of econometric analyses conducted with the alternative time series. We find that more sophisticated time series methods for backdating can yield sizeable gains. JEL Classification: C32, C43, C82Backdating, euro area, factor model, Unified Germany

Identifiability for a class of symmetric tensors

We use methods of algebraic geometry to find new, effective methods for detecting the identifiability of symmetric tensors. In particular, for ternary symmetric tensors T of degree 7, we use the analysis of the Hilbert function of a finite projective set, and the Cayley-Bacharach property, to prove that, when the Kruskal's rank of a decomposition of T are maximal (a condition which holds outside a Zariski closed set of measure 0), then the tensor T is identifiable, i.e. the decomposition is unique, even if the rank lies beyond the range of application of both the Kruskal's and the reshaped Kruskal's criteria

Estimating and forecasting the euro area monthly national accounts from a dynamic factor model

We estimate and forecast growth in euro area monthly GDP and its components from a dynamic factor model due to Doz et al. (2005), which handles unbalanced data sets in an efficient way. We extend the model to integrate interpolation and forecasting together with cross-equation accounting identities. A pseudo real-time forecasting exercise indicates that the model outperforms various benchmarks, such as quarterly time series models and bridge equations in forecasting growth in quarterly GDP and its components. JEL Classification: E37, C53Dynamic Factor Models, Interpolation, nowcasting

Waring decompositions and identifiability via Bertini and Macaulay2 software

Starting from our previous papers [AGMO] and [ABC], we prove the existence of a non-empty Euclidean open subset whose elements are polynomial vectors with 4 components, in 3 variables, degrees, respectively, 2,3,3,3 and rank 6, which are not identifiable over $\mathbb{C}$ but are identifiable over $\mathbb{R}$. This result has been obtained via computer-aided procedures suitably adapted to investigate the number of Waring decompositions for general polynomial vectors over the fields of complex and real numbers. Furthermore, by means of the Hessian criterion ([COV]), we prove identifiability over $\mathbb{C}$ for polynomial vectors in many cases of sub-generic rank.Comment: 15 page

Diffusion index-based inflation forecasts for the euro area

Diffusion indexes based on dynamic factors have recently been advocated by Stock and Watson (1998), and further used to perform forecasting tests by the same authors on US data. This technique is explored for the euro area using a multi-country data set and a broad array of variables, in order to test the inflation forecasting performance of extracted factors at the aggregate euro area level. First, a description of factors extracted from different data sets is performed using a number of different approaches. Conclusions reached are that nominal phenomena in the original variables might be well captured in-sample using the factor approach. Out-of-sample tests have more ambiguous interpretation, as factors seem to be good leading indicators of inflation, but the comparative advantage of the factors is less clear. Nevertheless, alternative indicators such as unemployment or money growth do not outperform them JEL Classification: C53, E31, E37

Interpolation and backdating with a large information set

Existing methods for data interpolation or backdating are either univariate or based on a very limited number of series, due to data and computing constraints that were binding until the recent past. Nowadays large datasets are readily available, and models with hundreds of parameters are fastly estimated. We model these large datasets with a factor model, and develop an interpolation method that exploits the estimated factors as an efficient summary of all the available information. The method is compared with existing standard approaches from a theoretical point of view, by means of Monte Carlo simulations, and also when applied to actual macroeconomic series. The results indicate that our method is more robust to model misspecification, although traditional multivariate methods also work well while univariate approaches are systematically outperformed. When interpolated series are subsequently used in econometric analyses, biases can emerge, depending on the type of interpolation but again be reduced with multivariate approaches, including factor-based ones. JEL Classification: C32, C43, C82factor model, Interpolation, Kalman filter, spline

The Italian block of the ESCB multi-country model

This paper documents the structure, estimation and simulation properties of the Italian block of the ESCB-multi-country model (MCM). The model is used regularly as an input into Eurosystem projection exercises and, to a lesser extent, in simulation analysis. The specification of the Italian model follows closely that of the Area-Wide Model (AWM) and indeed the other MCM country blocks (in terms of specification and accounting framework). The MCM is a quarterly estimated structural macroeconomic model that treats the economy in a relatively closed manner. It has a long-run classical equilibrium with a vertical Phillips curve but with some short-run frictions in price/wage setting and factor demands. Consequently, activity is demand-determined in the short-run but supply-determined in the longer run with employment having converged to a level consistent with an exogenously given level of equilibrium unemployment. The precise properties of the model are illustrated using a number of standard variant simulations. JEL Classification: C3, C5, E1, E2Italy, Macro-econometric Modelling