Lyapunov exponents may be used to provide information on attractors of nonlinear models and, if strange, their aperiodic dynamics, in much the same way as eigenvalues of a linear model. An advantage of calculating these from an estimated model, rather than calculating the largest ones from a time series, is that economic theory is used to help distinguish between deterministic and stochastic behavior. Estimates of the Bergstrom/Wymer model of the United Kingdom, and of some other models of a similar form, were unstable in a classical sense, to some extent being caused by policy parameters. The Lyapunov exponents show that this model, and variants of it, have strange attractors in the neighborhood of the estimated parameter values and hence are stable but with aperiodic oscillations.

We consider an omnibus test for the correct specification of the dynamics of a sequence $\left\{ {x\left( t \right)} \right\}_{t \in Z^d } $ in a lattice. As it happens with causal models and d = 1, its asymptotic distribution is not pivotal and depends on the estimator of the unknown parameters of the model under the null hypothesis. One first main goal of the paper is to provide a transformation to obtain an asymptotic distribution that is free of nuisance parameters. Secondly, we propose a bootstrap analog of the transformation and show its validity. Thirdly, we discuss the results when $\left\{ {x\left( t \right)} \right\}_{t \in Z^d } $ are the errors of a parametric regression model. As a by product, we also discuss the asymptotic normality of the least squares estimator of the parameters of the regression model under very mild conditions. Finally, we present a small Monte Carlo experiment to shed some light on the finite sample behavior of our test.

The work of Professor P.C.B. Phillips, even if it is focused on the area of linear nonstationary models, is enormous. So it is hard for me to explore the whole of his work in this paper. Therefore, I have decided to take up only a few results of his work. The topics chosen here are applications of the martingale approximation and the problem of choosing between stochastic and deterministic trends, which I discuss and, hopefully, extend.

Wu introduced a new technique for proving consistency of least-squares estimators in nonlinear regression. This paper extends his results in three directions. First, we consider the minimization of arbitrary functions (M-estimators instead of least squares). Second, we use an improved type 2 inequality. Third, an extension of Kronecker's lemma yields a more powerful result.

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. The limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply in stationary and nonstationary cases. Implications for inference are discussed and a feasible pre-test inference procedure is proposed that takes unknown potential flatness into consideration and provides a practical approach to inference.

This article investigates the instrumental variable quantile regression model (Chernozhukov and Hansen, 2005, Econometrica 73, 245–261; 2013, Annual Review of Economics, 5, 57–81) with a binary endogenous treatment. It offers two identification results when the treatment status is not directly observed. The first result is that, remarkably, the reduced-form quantile regression of the outcome variable on the instrumental variable provides a lower bound on the structural quantile treatment effect under the stochastic monotonicity condition. This result is relevant, not only when the treatment variable is subject to misclassification, but also when any measurement of the treatment variable is not available. The second result is for the structural quantile function when the treatment status is measured with error; the sharp identified set is characterized by a set of moment conditions under widely used assumptions on the measurement error. Furthermore, an inference method is provided in the presence of other covariates.

We discuss the effects of temporal aggregation on the estimation of cointegrating vectors and on testing linear restrictions on this vector. We adopt a discrete time approach and demonstrate, in contrast with the findings of Chambers (2003, Econometric Theory 19, 49–77), who adopts a continuous time approach, that in some situations, when the regressand must be aggregated, systematic sampling is preferable to average sampling for estimation purposes. Like Chambers, we show that the best aggregation scheme for regressors, in terms of asymptotic estimation efficiency, is always average sampling. We also show that different types of aggregation have no influence on the relative size of tests of linear restrictions on the cointegration vector.We thank Soren Johansen, Niels Haldrup, Raquel Waters, the associate editor, and two anonymous referees for their helpful comments. Of course, any remaining error is the responsibility of the authors. The first author gratefully acknowledges the financial support of a Marie Curie Fellowship of the European Community Programme “Improving the Human Research Potential and the Socio-Economic Knowledge Base” under contract HPMF-CT-2002-01662 and the Danish Research Council. The second author gratefully acknowledges the financial support of the Spanish Ministry of Science and Technology SEC2002-01512.

This paper derives the joint moment generating function of quadratic forms occurring in seasonal autoregressive models under stationary, unit root, and explosive specifications. The results are then used to investigate the impact of the seasonal periodicity parameter on various distributional results for both the normalized ordinary least squares coefficient and t-ratio and its effects on the asymptotic bias of parameter estimates.

This paper introduces structural equations that do not satisfy the rank and/or order condition(s) for identification but still are identifiable. These equations are called seemingly unidentified structural equations. The key to the identifiability of these equations is that the right-hand-side endogenous variables undergo structural changes with respect to the exogenous and/or predetermined variables. To estimate the seemingly unidentified structural equations, this paper uses the classical minimum distance (MD) estimator and the principal components instrumental variables (PCIV) estimator. The PCIV estimator is different from conventional IV estimators for structural equations in that nonlinear functions of exogenous and/or predetermined variables are used as instruments. Simulation results comparing the estimator efficiency of the MD and PCIV estimators are reported in this paper. The results indicate that the MD and PCIV estimators are complementary to each other. The estimation methods proposed in this paper are applied to the Japanese export and GDP data to study the effect of export growth rate on that of GDP.

In this comment on Gurmu and Trivedi's “Variable Augmentation Specification Tests in the Linear Exponential Family,” I show how their generalized linear model (GLM) approach relates to other work in econometrics on specification testing in the linear exponential family. In addition to shedding light on the relationship between the statistics and econometrics literatures on testing in quasi-likelihood frameworks, this comparison reveals some important limitations of GLM as a general framework for devising specification tests.

In the AR(2) model, with a double root at unity, we consider the asymptotic distribution of the likelihood ratio with respect to a nearly nonstationary alternative. It is shown how the distribution can be represented as a Radon-Nikodym derivative of an Ito process with respect to Brownian motion. Using this result, we point out how standard contiguity arguments can be applied to obtain a representation of the asymptotic power function in nearly nonstationary alternatives.

The paper presents the asymptotic theory of the efficient method of moments when the model of interest is not correctly specified. The paper assumes a sequence of independent and identically distributed observations and a global misspecification. It is found that the limiting distribution of the estimator is still asymptotically normal, but it suffers a strong impact in the covariance matrix. A consistent estimator of this covariance matrix is provided. The large sample distribution on the estimated moment function is also obtained. These results are used to discuss the situation when the moment conditions hold but the model is misspecified. It also is shown that the overidentifying restrictions test has asymptotic power one whenever the limit moment function is different from zero. It is also proved that the bootstrap distributions converge almost surely to the previously mentioned distributions and hence they could be used as an alternative to draw inferences under misspecification. Interestingly, it is also shown that bootstrap can be reliably applied even if the number of bootstrap replications is very small.The authors express their gratitude to Professor A.R. Gallant for his support and for helpful discussions about this topic. They also thank Professor Donald Andrews for his revision and for the suggestion to read the work by Hall and Inoue (2003). They thank Hall and Inoue for sending them their technical report. Finally the authors thank two anonymous referees for their helpful comments, which led to a significant improvement of the paper. The views expressed in this paper are our sole responsibility. Víctor Aguirre-Torres acknowledges partial support from the Asociación Mexicana de la Cultura A.C. Manuel A. Domínguez acknowledges funding from CONACYT, research project J38076-D, and partial support from Asociación Mexicana de la Cultura A.C.

This paper establishes central limit theorems (CLTs) and proposes how to perform valid inference in factor models. We consider a setting where many counties/regions/assets are observed for many time periods, and when estimation of a global parameter includes aggregation of a cross-section of heterogeneous microparameters estimated separately for each entity. The CLT applies for quantities involving both cross-sectional and time series aggregation, as well as for quadratic forms in time-aggregated errors. This paper studies the conditions when one can consistently estimate the asymptotic variance, and proposes a bootstrap scheme for cases when one cannot. A small simulation study illustrates performance of the asymptotic and bootstrap procedures. The results are useful for making inferences in two-step estimation procedures related to factor models, as well as in other related contexts. Our treatment avoids structural modeling of cross-sectional dependence but imposes time-series independence.