This paper proposes a version of the generalized method of moments procedure that handles both the case where the number of moment conditions is finite and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its values in an interval. The objective function to minimize is then the norm of the moment conditions in a Hilbert space. The estimator is shown to be consistent and asymptotically normal. The optimal estimator is obtained by minimizing the norm of the moment conditions in the reproducing kernel Hilbert space associated with the covariance. We show an easy way to calculate this estimator. Finally, we study properties of a specification test using overidentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.

In this paper we introduce the generalized fluctuation test for monitoring structural changes and establish a result characterizing the limiting behavior of this class of tests. As applications of the generalized fluctuation test, tests based on the maximum and range of the fluctuation of moving estimates are proposed. We also derive the boundary functions for the proposed tests and tabulate simulated critical values. Our simulations indicate that these tests compare favorably with the recursive-estimates-based test considered by Chu, Stinchcombe, and White (1996, Econometrica 64, 1045–1065) when a change occurs late.

In productivity analysis, the free disposal hull (FDH) is a nonparametric estimator for the production set, the set of inputs and outputs that are technically feasible. It is defined as the smallest free disposal set containing all observations in a sample of production units. One can then derive the production frontier and efficiency scores from the FDH. In the literature the method is used as if the FDH estimator were the true feasible set. However, assuming that individuals are drawn independently from a distribution where the support is the true production set, FDH efficiency scores are random variables. This paper investigates its stochastic properties.

This paper studies asymptotic likelihood inference on cointegration parameters in systems integrated of order two. We start with so-called triangular systems and then extend the analysis to vector autoregressions. We show that even when all unit root restrictions have been imposed, the asymptotic observed information is not (locally) ancillary, which implies that the log-likelihood ratio is not locally asymptotically mixed normal. The results are applied to inference on polynomial cointegration. Some similarities and differences with I(1) systems are also discussed.

This paper develops a new estimation method for nonstationary vector autoregressions (VAR's) with unknown mixtures of I(0), I(1), and I(2) components. The method does not require prior knowledge on the exact number and location of unit roots in the system. It is, therefore, applicable for VAR's with any mixture of I(0), I(1), and I(2) variables, which may be cointegrated in any form. The limit theory for the stationary component of our estimator is still normal, thereby preserving the usual VAR limit theory. Yet, the leading term of the nonstationary component of the estimator has mixed normal limit distribution and does not involve unit root distribution. Our method is an extension of the FM-VAR procedure by Phillips (1995, Econometrica 63, 1023–1078) and yields an estimator that is optimal in the sense of Phillips (1991, Econometrica 59, 283–306). Moreover, we show for a certain class of linear restrictions that the Wald tests based on the estimator are asymptotically distributed as a weighted sum of independent chi-square variates with weights between zero and one. For such restrictions, the limit distribution of the standard Wald test is nonstandard and nuisance parameter dependent. This has a direct application for Granger-causality testing in nonstationary VAR's.

Time series data are often well modeled by using the device of an autoregressive root that is local to unity. Unfortunately, the localizing parameter (c) is not consistently estimable using existing time series econometric techniques and the lack of a consistent estimator complicates inference. This paper develops procedures for the estimation of a common localizing parameter using panel data. Pooling information across individuals in a panel aids the identification and estimation of the localizing parameter and leads to consistent estimation in simple panel models. However, in the important case of models with concomitant deterministic trends, it is shown that pooled panel estimators of the localizing parameter are asymptotically biased. Some techniques are developed to overcome this difficulty, and consistent estimators of c in the region c < 0 are developed for panel models with deterministic and stochastic trends. A limit distribution theory is also established, and test statistics are constructed for exploring interesting hypotheses, such as the equivalence of local to unity parameters across subgroups of the population. The methods are applied to the empirically important problem of the efficient extraction of deterministic trends. They are also shown to deliver consistent estimates of distancing parameters in nonstationary panel models where the initial conditions are in the distant past. In the development of the asymptotic theory this paper makes use of both sequential and joint limit approaches. An important limitation in the operation of the joint asymptotics that is sometimes needed in our development is the rate condition n/T → 0. So the results in the paper are likely to be most relevant in panels where T is large and n is moderately large.

The exact discrete model satisfied by equispaced data generated by a linear stochastic differential equations system is derived by a method that does not imply restrictions on observed discrete data per se. The method involves integrating the solution of the continuous time model in state space form and a nonstandard change in the order of three types of integration, facilitating the representation of the exact discrete model as an asymptotically time-invariant vector autoregressive moving average model. The method applying to the state space form is general and is illustrated using the prototypical higher order model for mixed stock and flow data discussed by Bergstrom (1986, Econometric Theory 2, 350–373).

We point out the close relationship between the integrated conditional moment tests in Bierens (1982, Journal of Econometrics 20, 105–134) and Bierens and Ploberger (1997, Econometrica 65, 1129–1152) with the complex-valued exponential weight function and the kernel-based tests in Härdle and Mammen (1993, Annals of Statistics 21, 1926–1947), Li and Wang (1998, Journal of Econometrics 87, 145–165), and Zheng (1996, Journal of Econometrics 75, 263–289). It is well established that the integrated conditional moment tests of Bierens (1982) and Bierens and Ploberger (1997) are more powerful than kernel-based nonparametric tests against Pitman local alternatives. In this paper we analyze the power properties of the kernel-based tests and the integrated conditional moment tests for a sequence of “singular” local alternatives, and show that the kernel-based tests can be more powerful than the integrated conditional moment tests for these “singular” local alternatives. These results suggest that integrated conditional moment tests and kernel-based tests should be viewed as complements to each other. Results from a simulation study are in agreement with the theoretical results.