One of the main advantages of panel data is that it allows one to study the dynamics of economic behaviour at an individual level. Unfortunately, when dynamic models are estimated using time series of cross sections data, the usual least squares methods (such as those presented in Chapters 3 and 4) do not lead to consistent estimates for the parameters of the two most commonly used models for panel data (i.e., fixed effects and error components models). This inconsistency results from the fact that the disturbance terms are serially correlated in these models, which causes the lagged endogenous variable to be correlated with those disturbances. As in the time series context, we do not have analytical results about the small sample properties of the various estimators of these models. The only available results come from Monte-Carlo simulation studies (see Nerlove [1967], [1971]). Hence, one must rely on the asymptotic properties of these methods and assume that the size of the sample grows to infinity. Since a panel data set has two dimensions, it is possible to increase the size of the sample in various ways. Firstly one could increase the time dimension of the sample, i.e., make T tend to infinity as is done when working with time series data. As most available panel data sets contain a large number of observations on individuals (N) over a limited number of periods (T), this is not a very relevant option. It is therefore, probably more useful to study the so-called semi-asymptotic behaviour of the various estimators, by making N --+ 00 but keeping T finite. 1 This has two notable implications when one works with a dynamic model such as

ABSTRACT We study the invariance properties of various test criteria which have been proposed for hypothesis testing in the context of incompletely specified models, such as models which are formulated in terms of estimating functions (Godambe, 1960) or moment conditions and are estimated by generalized method of moments (GMM) procedures (Hansen, 1982), and models estimated by pseudo-likelihood (Gouriéroux, Monfort, and Trognon, 1984b,c) and M-estimation methods. The invariance properties considered include invariance to (possibly nonlinear) hypothesis reformulations and reparameterizations. The test statistics examined include Wald-type, LR-type, LM-type, score-type, and C(α)−type criteria. Extending the approach used in Dagenais and Dufour (1991), we show first that all these test statistics except the Wald-type ones are invariant to equivalent hypothesis reformulations (under usual regularity conditions), but all five of them are not generally invariant to model reparameterizations, including measurement unit changes in nonlinear models. In other words, testing two equivalent hypotheses in the context of equivalent models may lead to completely different inferences. For example, this may occur after an apparently innocuous rescaling of some model variables. Then, in view of avoiding such undesirable properties, we study restrictions that can be imposed on the objective functions used for pseudo-likelihood (or M-estimation) as well as the structure of the test criteria used with estimating functions and generalized method of moments (GMM) procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant score-type and C(α)−type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by Newey and West (1987) to obtain a test statistic that is invariant to general reparameterizations. The invariance associated with linear exponential pseudo-likelihood functions is interpreted as a strong argument for using such pseudo-likelihood functions in empirical work.

We propose generalized C(alpha) tests for testing linear and nonlinear parameter restrictions in models specified by estimating functions. The proposed procedures allow for general forms of serial dependence and heteroskedasticity, and can be implemented using any root-n consistent restricted estimator. The asymptotic distribution of the proposed statistic is established under weak regularity conditions. We show that earlier C(alpha)-type statistics are included as special cases. The problem of testing hypotheses fixing a subvector of the complete parameter vector is discussed in detail as another special case. We also show that such tests provide a simple general solution to the problem of accounting for estimated parameters in the context of two-step procedures where a subvector of model parameters is estimated in a first step and then treated as fixed.

Résumé Nous abordons de manière originale les conditions de la reconnaissance de soi, constitutive de l’émergence de la conscience de soi humaine par le biais d’une discordance observée en psychothérapie chez un jeune garçon de 7 ans et 4 mois, portant sur l’intégration des représentations de soi dans la situation spéculaire. Avec une problématique d’analyse du discours produit dans l’interaction, la logique interlocutoire (Trognon, Batt, 2007 a , 2007 b ), nous recourons à la théorie des modèles et à la théorie des jeux d’investigation et de recherche développées par Hintikka (1976, 1981, 1984, 1998 b ) afin de proposer des analyses qui permettent d’expliciter formellement la clinique des phénomènes mentaux observée chez le patient. Rétrospectivement nous questionnons dans les termes de la théorie sémantique des modèles (Hintikka, 1989) le moment fécond, entre 18 et 36 mois, qui voit l’émergence d’un nouveau mode de fonctionnement de l’intentionnalité lorsque celle-ci intègre les propriétés du signe linguistique et en particulier, dans ce cadre, le statut du signifiant autoréférentiel dans la formation de la fonction du « je ».

L’économétrie des panels a connu depuis une quarantaine d’années d’existence une progression exceptionnelle tant par la variété des applications que par la constitution d’un corps de méthodes appropriées. Cet article tente un court bilan sur ce second plan de la méthodologie. Il montre qu’autour d’un cadre de modélisation linéaire, fondé sur la notion d’effets individuels destinés à la maîtrise de l’hétérogénéité des comportements individuels, le progrès général de l’économétrie des panels, lié à la connaissance pure, à la disponibilité croissante des données et à la puissance des moyens de calculs, est extraordinairement dynamique en ce début de siècle. La capacité grandissante des économètres appliqués à mettre en oeuvre des applications informatiques complexes leur permet d’intégrer de plus en plus rapidement ces nouvelles méthodes et procédures, comme l’illustre le présent numéro.

In general, when one considers a set of orthogonality conditions, the parameters can be divided into parameters of interest that the econometrician wants to estimate — the coefficient of the lagged endogenous variable in the case of an autoregressive error component (AREC) model, for instance, — and nuisance parameters — most of the second-order terms in an AREC model. We demonstrate that the elimination of such nuisance parameters using their empirical counterpart does not entail an efficiency loss when only the parameters of interest are estimated. Applications of our results to both autoregressive error component models and time-varying individual fixed effects models are discussed at length. They show the nature of the efficiency losses when some orthogonality conditions are left aside.

In many situations, assuming exogeneity for the regressors is likely to be incorrect. Indeed, it is well—known that in simultaneous equations models, some of the regressors in a given equation are the dependent variables in others and then, are correlated with the disturbances of the equation under consideration. Further examples, when exogeneity cannot be assumed for regressors, are when they are subject to measurement errors or the case of autoregressive models.

We develop a model for decomposing the covariance structure of panel data on firms into a part due to permanent heterogeneity, a part due to differential histories with unknown ages, and a part due to the evolution of economic shocks to the firm. Our model allows for the endogenous death of firms and correctly handles the problems arising from the estimation of this death process. We implement this model on an unbalanced longitudinal sample of French firms which have both known and unknown ages and histories. For firms with unknown birthdates, we find that the structural autocorrelation in employment, compensation and capital is dominated by the part due to initial heterogeneity and random growth rates. Serial correlation in the periodic shocks is less important. For these firms, profitability, value-added and indebtedness have processes in which the heterogeneity components are less important. Firms with known birthdates and histories (which are younger than the censored firms) have autocorrelation structures dominated by the heterogeneity.

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