Maria Grazia Zoia

ON A PARTITIONED INVERSION FORMULA HAVING USEFUL APPLICATIONS IN ECONOMETRICS

In this paper a novel partitioned inversion formula is obtained in terms of the orthogonal complements of off-diagonal blocks, with the emblematic matrix of unit-root econometrics springing up as the leading diagonal block of the inverse. On the one hand, the result paves the way to a stimulating reinterpretation of restricted least-squares estimation and, on the other, to a straightforward derivation of a key-result of time-series econometrics.

Tailoring the Gaussian Law for Excess Kurtosis and Skewness by Hermite Polynomials

This article aims at reshaping the normal law to account for tail-thickness and asymmetry, of which there is plenty of evidence in financial data. The inspiration to address the issue was provided by the orthogonality of Hermite polynomials with the Gaussian density as a weight function, with the Gram–Charlier expansion as background. A solution is then devised accordingly, by embodying skewness and excess-kurtosis in a normal kernel, via third- and forth-degree polynomial tune-up. Features of the densities so obtained are established in the main theorem of this article. In addition, a glance is cast at the issue of embodying between-squares correlation, and a solution is outlined.

Orthogonal polynomials for tailoring density functions to excess kurtosis, asymmetry, and dependence

Abstract This article deals with the problem of tailoring distributions to embody evidence of moments and dependence structure deviating from those of a given parent law. First, we show that finite-moment distributions can be reshaped, to allow for extra kurtosis, asymmetry, and dependence by using orthogonal polynomials. Then, we derive a set of orthogonal polynomials for adjusting any symmetric density to given requirements in terms of moments. Conditions for positiveness of the resulting polynomially modified distribution are further established. This provides a broader approach to reshaping parent distributions by means of polynomial adjustments than that currently found in the literature.

Cooperative Innovation: In Quest of Effective Partners. Evidence from Italian Firms

In recent years, rapid technological change, shorter product life cycles and globalization have deeply transformed the current competitive environment. These changes are inducing firms to face stronger competitive pressures which push them to develop new products, improve production processes or implement new technologies. Thus, firms need to continually acquire new knowledge and innovate. At the same time, entrepreneurs have become aware that technological innovation is less and less dependent on an isolated effort of an individual firm. For small- and medium-sized enterprises (SMEs), R&D cooperation with sources of external knowledge is becoming increasingly essential for fostering innovation activities. Using firm-level data from the Community Innovation Survey for the years 2006–2008 (CIS 2008) and applying a Heckman probit model with sample selection, we analyze the determinants of cooperative innovation for the different types of partners (competitors, customers, suppliers, universities and government laboratories). Results show that internal and external R&D acquisitions, public financial support, as well as belonging to a scientific sector or to a business group are significant determinants of choice in collaborations, although with different magnitude across various types of collaborations.

Tailoring the Logistic Distribution to Fit the Empirical Distribution of Financial Asset Returns

Risk measures, including Value-at-Risk (VaR) and Conditional VaR (Expected Shortfall), turn out to be quite sensitive to the degree to which distributions are thick tailed and asymmetric. Lack of encoding information about asymmetry and leptokurtosis is a well-known drawback of the Gaussian law. This has led to a search for alternative distributions (Szego (2004)). In this paper, we will tackle the issue of accounting for asymmetry, (possibly severe) excess kurtosis and dependence by following the alternative approach of adjusting bell-shaped distributions using orthogonal polynomials as shape adapters. Our focus will be on polynomial transformations of parent symmetric probability density functions to match the empirical moments of target distributions characterized by possibly substantial heavy-tails and asymmetry. We will demonstrate a simple but powerful novel result, i.e. that the function that achieves the transformation from a given parent to a target distribution, depends on both the orthogonal polynomials associated to the former and the moment-differentials between these two distributions. We will then apply this result to the modelling of heavy-tailed and skewed distributions of financial asset returns.

Tailoring Density Functions via Orthogonal Polynomials

This paper re-examines the issue of how to tailor distributions to embody evidence of moments and dependence structure deviating from those of a given parent distribution. It is well known that the function that achieves the transformation from a given parent to a target distribution can be expressed as a linear combination of a set of orthogonal polynomials associated to parent distribution. We show that the coefficients of such linear combination are simple algebraic functions of the difference between the moments of the parent and target distributions. These results facilitate reshaping a considerably broader class of distributions than well-known approaches based on Hermite polynomials, which can be used only to reshape the normal distribution and allow to do so only to a limited extent. We provide applications to modeling distributions of financial asset returns, which are known to exhibit considerable skewness and fat tails.

An inversion formula for a matrix polynomial about a (unit) root

A solution to the problem of a closed-form representation for the inverse of a matrix polynomial about a unit root is provided by resorting to a Laurent expansion in matrix notation, whose principal-part coefficients turn out to depend on the non-null derivatives of the adjoint and the determinant of the matrix polynomial at the root. Some basic relationships between principal-part structure and rank properties of algebraic function of the matrix polynomial at the unit root as well as informative closed-form expressions for the leading coefficient matrices of the matrix-polynomial inverse are established.

A new proof of the representation theorem for I(2) processes

Abstract In this paper matrix polynomial inversion – by Laurent expansion about a second order pole – is linked to partitioned inversion of a block matrix mirroring the underlying rank assumptions. This paves the way to find closed-form solutions for the principal-part coefficient matrices of the Laurent expansion entering the representation of I(2) processes and shaping their cointegration spaces. On this basis a neat formulation and a new proof of the representation theorem of I(2) processes are provided.

Closed-form expressions for the regular-part coefficients in matrix polynomial inversion and related results

Abstract In this paper closed-form expressions for the coefficient matrices of the regular part of the Laurent expansion for a matrix-polynomial inverse about a pole are obtained. This sets the stage for deriving the dynamic multipliers of a vector autoregressive model (VAR) in a straightforward way.

Restricted Least Square revisited

Abstract In this paper new light is shed on restricted least-squares thanks to a recent result on partitioned inversion [2] which paves the way to finding a novel closed-form expression for the constrained estimator that is more general than the classical Theil’s solution.