Alain Trognon

A note on autoregressive error components models

Abstract The purpose of this paper is to shed some light on the asymptotic behavior of a wide class of estimators for a dynamic error components model when only the number of individuals tends to infinity, the number of time periods being kept fixed. In particular, it is shown that this asymptotic behavior is highly dependent on the assumption about the initial observations and that it offers very good approximations to the small sample behavior of the various estimators under consideration.

Testing nested or non-nested hypotheses

This paper presents a test procedure for nested or non-nested hypotheses. The test statistic is based on the difference between two estimators of the pseudo-true value as defined for instance by Sawa. This statistic is similar to the usual Wald statistic in the case of nested hypotheses and it can be replaced by an asymptotically equivalent one deduced from the score function.

Dynamic Linear Models

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.

A General Approach to Serial Correlation

In this paper the testing and estimation problems are discussed in the case of serial correlation. Various models are particular cases of the general framework considered: the nonlinear simultaneous equations models, the probit models, the tobit models, the disequilibrium models, the frontier models, etc. In this context, it is shown that the score test can be written explicitly and that the statistic obtained is a generalization of that of Durbin and Watson; moreover, the maximum likelihood estimation procedure is shown to be robust with respect to serial correlation.

Linear Dynamic Models

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.

Linear Models with Random Regressors

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.

Specification pre-test estimator

Abstract This paper is related to the specification test introduced by J. Hausman. A specification test is often used as a first step in an estimation procedure. In the case of the linear model the biases and the mean square errors of the classical and of the specification pre-test estimators are computed and compared. It is shown that the latter is not uniformly better than the former and conversely. In the case of a four-variables model, regions in the parameters space where the specification pre-test estimator outdoes the classical one are numerically settled.

Invariant tests based on M-estimators, estimating functions, and the generalized method of moments

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.

Generalized \(C(\alpha )\) Tests for Estimating Functions with Serial Dependence

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.