We define, in a dynamic framework, the notions of binding functions, images, reflecting sets, indirect identification, indirect information, and encompassing. We study the properties of the notion of encompassing when the true distribution does not necessarily belong to one of the two competing models of interest. In this context we propose various test procedures of the encompassing hypothesis. Some of these procedures are based on simulations, and some of them are linked with the notion of indirect estimation (in particular, the GET and simulated GET procedures). As a by-product, we get an asymptotic theory of the tests of non-nested hypotheses in the stationary dynamic case.

The aim of this paper is the study of the path solutions of a multivariate rational expectations model. We describe several procedures for solving such dynamic systems based on either the adjoint operator method or the Smith form. As a by-product, we derive the dimension of the set of solutions in terms of martingale differences and the dimension of the set of linear stationary solutions when we restrict ourselves to the linear case. These dimensions are functions of the number of equations in the system, of the maximum lead, and of the orders of some eigenvalues of the characteristic equation associated with the system.

We consider the estimation and identification of the functional structures of nonlinear econometric systems of the ARCH type. We employ nonparametric kernel estimates for the nonlinear functions characterizing the systems, and we establish strong consistency along with sharp rates of convergence under mild regularity conditions. We also prove the asymptotic normality of the estimates.

Conditional moment tests check to see whether or not population moment equalities, implied by the null model specification, hold approximately in the sample. Asymptotically valid conditional statistics can easily be calculated from the output of a so-called outer product of the gradient (OPG) artificial regression. However, several studies have now found that this OPG variant exhibits extremely poor finite sample behavior and that significant improvements can be made by employing the efficient variant. In the light of such evidence, this paper develops new artificial regressions that can be used to calculate the efficient variant of the test statistic. These artificial regressions can also serve several other purposes, including the construction of Hausmantype tests of parameter estimator consistency.

Asymptotic distributions of some test statistics in near-integrated AR processes are studied. Some exact formulas for the distribution functions are given as well as approximative results obtained by saddlepoint approximation techniques.

This paper considers unit root tests based on M estimators. The asymptotic theory for these tests is developed. It is shown how the asymptotic distributions of the tests depend on nuisance parameters and how tests can be constructed that are invariant to these parameters. It is also shown that a particular linear combination of a unit root test based on the ordinary least-squares (OLS) estimator and on an M estimator converges to a normal random variate. The interpretation of this result is discussed. A simulation experiment is described, illustrating the level and power of different unit root tests for several sample sizes and data generating processes. The tests based on M estimators turn out to be more powerful than the OLS-based tests if the innovations are fat-tailed.

This paper provides weak and strong laws of large numbers for weakly dependent heterogeneous random variables. The weak laws of large numbers presented extend known results to the case of trended random variables. The main feature of our strong law of large numbers for mixingale sequences is the less strict decay rate that is imposed on the mixingale numbers as compared to previous results.

In the solution to Problem 93.2.3 by Paolo Paruolo, γ* ≡ (β, ,Ik)′ should be replaced by γ* ≡ (βí,I*)′ throughout the text; the same change should be made for *; γ should be replaced by γ* in the second line and two lines before the references; “left-hand side” should be “right-hand side” in the second line after (4); “column” should be replaced by “row” in the last line before the references. Finally, reference \1] is Preprint 1992.2, Institute of Mathematical Statistics, University of Copenhagen; submitted to Econometric Theory.