On Quadratic Expansions of Log-Likelihoods and a General Asymptotic Linearity Result

18 Pages Posted: 7 Sep 2013 Last revised: 14 Feb 2014

See all articles by Marc Hallin

Marc Hallin

ECARES, Universite Libre de Bruxelles

Ramon Van den Akker

Tilburg University - CentER and department of Econometrics & OR

Bas J. M. Werker

Tilburg University - Center for Economic Research (CentER)

Date Written: September 4, 2013

Abstract

Irrespective of the statistical model under study, the derivation of limits, in the Le Cam sense, of sequences of local experiments (see [7]-[10]) often follows along very similar lines, essentially involving differentiability in quadratic mean of square roots of (conditional) densities. This chapter establishes two abstract and very general results providing sufficient and nearly necessary conditions for (i) the existence of a quadratic expansion, and (ii) the asymptotic linearity of local log-likelihood ratios (asymptotic linearity is needed whenever unspecified model parameters are to be replaced, in some statistic of interest, with some preliminary estimator). Such results have been established, for locally asymptotically normal (LAN) models involving independent and identically distributed observations, by, e.g., [1], [11] and [12]. Similar results are provided here for models exhibiting serial dependencies which, so far, have been treated on a case-by-case basis (see [4] and [5] for typical examples) and, in general, under stronger regularity assumptions. Unlike their i.i.d. counterparts, our results moreover extend beyond the context of LAN experiments, so that non-stationary unit-root time series and cointegration models, for instance, also can be handled (see, e.g., [6]).

Keywords: limit experiment, differentiability in quadratic mean, asymptotic linearity

JEL Classification: C14

Suggested Citation

Hallin, Marc and Van den Akker, Ramon and Werker, Bas J.M., On Quadratic Expansions of Log-Likelihoods and a General Asymptotic Linearity Result (September 4, 2013). Available at SSRN: https://ssrn.com/abstract=2320807 or http://dx.doi.org/10.2139/ssrn.2320807

Marc Hallin (Contact Author)

ECARES, Universite Libre de Bruxelles ( email )

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Ramon Van den Akker

Tilburg University - CentER and department of Econometrics & OR ( email )

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HOME PAGE: http://sites.google.com/site/rvdnakker/

Bas J.M. Werker

Tilburg University - Center for Economic Research (CentER) ( email )

Econometrics and Finance Group
5000 LE Tilburg
Netherlands

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