We define, in a dynamic framework, the notions ofbinding functions, images, reflecting sets, indirectidentification, indirect information, andencompassing. We study the properties of the notionof encompassing when the true distribution does notnecessarily belong to one of the two competingmodels of interest. In this context we proposevarious test procedures of the encompassinghypothesis. Some of these procedures are based onsimulations, and some of them are linked with thenotion of indirect estimation (in particular, theGET and simulated GET procedures). As a by-product,we get an asymptotic theory of the tests ofnon-nested hypotheses in the stationary dynamiccase.

The aim of this paper is the study of the pathsolutions of a multivariate rational expectationsmodel. We describe several procedures for solvingsuch dynamic systems based on either the adjointoperator method or the Smith form. As a by-product,we derive the dimension of the set of solutions interms of martingale differences and the dimension ofthe set of linear stationary solutions when werestrict ourselves to the linear case. Thesedimensions are functions of the number of equationsin the system, of the maximum lead, and of theorders of some eigenvalues of the characteristicequation associated with the system.

We consider the estimation and identification of thefunctional structures of nonlinear econometricsystems of the ARCH type. We employ nonparametrickernel estimates for the nonlinear functionscharacterizing the systems, and we establish strongconsistency along with sharp rates of convergenceunder mild regularity conditions. We also prove theasymptotic normality of the estimates.

Conditional moment tests check to seewhether or not population moment equalities, impliedby the null model specification, hold approximatelyin the sample. Asymptotically valid conditionalstatistics can easily be calculated from the outputof a so-called outer product of thegradient (OPG) artificial regression.However, several studies have now found that thisOPG variant exhibits extremely poor finite samplebehavior and that significant improvements can bemade by employing the efficient variant. In thelight of such evidence, this paper develops newartificial regressions that can be used to calculatethe efficient variant of the test statistic. Theseartificial regressions can also serve several otherpurposes, including the construction of Hausmantypetests of parameter estimator consistency.

Asymptotic distributions of some test statistics innear-integrated AR processes are studied. Some exactformulas for the distribution functions are given aswell as approximative results obtained bysaddlepoint approximation techniques.

This paper considers unit root tests based on Mestimators. The asymptotic theory for these tests isdeveloped. It is shown how the asymptoticdistributions of the tests depend on nuisanceparameters and how tests can be constructed that areinvariant to these parameters. It is also shown thata particular linear combination of a unit root testbased on the ordinary least-squares (OLS) estimatorand on an M estimator converges to a normal randomvariate. The interpretation of this result isdiscussed. A simulation experiment is described,illustrating the level and power of different unitroot tests for several sample sizes and datagenerating processes. The tests based on Mestimators turn out to be more powerful than theOLS-based tests if the innovations arefat-tailed.

This paper provides weak and strong laws of largenumbers for weakly dependent heterogeneous randomvariables. The weak laws of large numbers presentedextend known results to the case of trended randomvariables. The main feature of our strong law oflarge numbers for mixingale sequences is the lessstrict decay rate that is imposed on the mixingalenumbers as compared to previous results.

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