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A mother formula for econometric estimation, the issue of parent linear models of best-fit solutions and its dual problem

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A mother formula for econometric estimation, the issue of parent linear models of best-fit solutions and its dual problem
Article
journal STATISTICA & APPLICAZIONI
issue STATISTICA & APPLICAZIONI - 2009 - 2
title A mother formula for econometric estimation, the issue of parent linear models of best-fit solutions and its dual problem
author
publisher Vita e Pensiero
format Article | Pdf
online since 02-2009
issn 18246672 (print)
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The body of econometric estimation theory in linear models must necessarily hinge, as a frame of reference, on Rao’s unified theories of linear estimation and least squares. The mathematical counterpart of the basic statistical setups turns out to be quadratic optimization problems, whose solutions yield the optimal estimators. These solutions rest on the inversion of the fundamental bordered matrix of the first-order conditions for optimality. A recently devised partitioned inversion rule leads to a mother formula for estimators within a linear framework. In addition, this paper casts further light on the link between the best fit approach to estimation and model specifications. Indeed, by taking least squares as a bridge-head and best unbiasedness as a benchmark, quite a deep insight into parameter inference is gained, whose applicative spin-offs are brought to light by a wide-ranging reappraisal of statistic and econometric estimators. Keywords: Least squares, Econometric models, Best unbiasedness, Orthogonal complements,Inversion rules. L’econometria metodologica ha nelle teorie unificate della stima lineare e dei minimi quadrati di Rao i suoi riscontri naturali. Ai sensi di tali teorie, il problema di stima si riconduce ad un problema di ottimizzazione matematica che trova nell’inversione della cosiddetta ‘‘matrice orlata fondamentale’’ la chiave per la sua soluzione. Una recente formula di inversione per parti consente di pervenire ad una formula madre per la classe degli stimatori ottimali nei modelli lineari. Da questa analisi emergono altresı`interessanti collegamenti col problema del modello di riferimento, per un dato metodo di stima di accostamento ottimale, e dualmente col problema dei minimi quadrati che conduce allo stimatore ottimale per una data specificazione lineare. L’articolo fornisce un contributo chiarificatore, con apporti innovativi, a questi temi di preminente interesse per l’econometria nelle sue interazioni con la statistica.