Mario Faliva

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.

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.

Recursiveness vs. interdependence in econometric models: A comprehensive analysis for the linear case

The paper gives an explicit solution to the problem of determining the causal structure of linear multi-equation econometric models.

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.

Detecting and testing causality in linear econometric models

The ultimate objective of this paper is to arrive at an operational testing procedure enabling us to diagnose the causal or non-causal nature of an econometric relationship in a linear framework. The profferred approach to the problem relies on a close examination of causal issues in simultaneous equation models and on a sensible definition of the bidirectionality concept inherently associated with feedback mechanisms.

Trend-cycle detection as a filtering problem

This paper discusses the problem of detecting the trend-cycle of economic time series (x t), using frequency-domain arguments which lead to an analytical approach focusing on optimal filter design. After providing an unambiguous characterization of the components-i.e., trend (f t), cycle (c t), seasonality (s t) and error term (η1)-, a time-domain model is proposed as follows:

The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

\chi _t = f_t + c_t^{(e v)} + s_t^{(e v)} + \eta _t