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This paper proposes simple tests of error cross-sectional dependence which are applicable to a variety of panel data models, including stationary and unit root dynamic heterogeneous panels with short T and large N. The proposed tests are based on the average of pair-wise correlation coefficients of the OLS residuals from the individual regressions in the panel and can be used to test for cross-sectional dependence of any fixed order p, as well as the case where no a priori ordering of the cross-sectional units is assumed, referred to as CD(p) and CD tests, respectively. Asymptotic distribution of these tests is derived and their power function analyzed under different alternatives. It is shown that these tests are correctly centred for fixed N and T and are robust to single or multiple breaks in the slope coefficients and/or error variances. The small sample properties of the tests are investigated and compared to the Lagrange multiplier test of Breusch and Pagan using Monte Carlo experiments. It is shown that the tests have the correct size in very small samples and satisfactory power, and, as predicted by the theory, they are quite robust to the presence of unit roots and structural breaks. The use of the CD test is illustrated by applying it to study the degree of dependence in per capita output innovations across countries within a given region and across countries in different regions. The results show significant evidence of cross-dependence in output innovations across many countries and regions in the World.

This paper introduces a quantile regression estimator for panel data (QRPD) with nonadditive fixed effects, maintaining the nonseparable disturbance term commonly associated with quantile estimation. QRPD estimates the impact of exogenous or endogenous treatment variables on the outcome distribution using “within” variation in the instruments for identification purposes. Most quantile panel data estimators include additive fixed effects which separates the disturbance term and assumes the parameters vary based only on the time-varying components of the disturbance term. QRPD produces consistent estimates for small T. I estimate the effect of the 2008 tax rebates on the short-term household consumption distribution.

Cross-validation using randomized subsets of data—known as k-fold cross-validation—is a powerful means of testing the success rate of models used for classification. However, few if any studies have explored how values of k (number of subsets) affect validation results in models tested with data of known statistical properties. Here, we explore conditions of sample size, model structure, and variable dependence affecting validation outcomes in discrete Bayesian networks (BNs). We created 6 variants of a BN model with known properties of variance and collinearity, along with data sets of n = 50, 500, and 5000 samples, and then tested classification success and evaluated CPU computation time with seven levels of folds (k = 2, 5, 10, 20, n − 5, n − 2, and n − 1). Classification error declined with increasing n, particularly in BN models with high multivariate dependence, and declined with increasing k, generally levelling out at k = 10, although k = 5 sufficed with large samples (n = 5000). Our work supports the common use of k = 10 in the literature, although in some cases k = 5 would suffice with BN models having independent variable structures.

The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator. We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325,  2005 ). Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions.

We show that prominent centrality measures in network analysis are all based on additively separable and linear treatments of statistics that capture a node’s position in the network. This enables us to provide a taxonomy of centrality measures that distills them to varying on two dimensions: (i) which information they make use of about nodes’ positions, and (ii) how that information is weighted as a function of distance from the node in question. The three sorts of information about nodes’ positions that are usually used—which we refer to as “nodal statistics”—are the paths from a given node to other nodes, the walks from a given node to other nodes, and the geodesics between other nodes that include a given node. Using such statistics on nodes’ positions, we also characterize the types of trees such that centrality measures all agree, and we also discuss the properties that identify some path-based centrality measures.

Inference, prediction, and control of complex dynamical systems from time series is important in many areas, including financial markets, power grid management, climate and weather modeling, or molecular dynamics. The analysis of such highly nonlinear dynamical systems is facilitated by the fact that we can often find a (generally nonlinear) transformation of the system coordinates to features in which the dynamics can be excellently approximated by a linear Markovian model. Moreover, the large number of system variables often change collectively on large time- and length-scales, facilitating a low-dimensional analysis in feature space. In this paper, we introduce a variational approach for Markov processes (VAMP) that allows us to find optimal feature mappings and optimal Markovian models of the dynamics from given time series data. The key insight is that the best linear model can be obtained from the top singular components of the Koopman operator. This leads to the definition of a family of score functions called VAMP- r which can be calculated from data, and can be employed to optimize a Markovian model. In addition, based on the relationship between the variational scores and approximation errors of Koopman operators, we propose a new VAMP-E score, which can be applied to cross-validation for hyper-parameter optimization and model selection in VAMP. VAMP is valid for both reversible and nonreversible processes and for stationary and nonstationary processes or realizations.

Extended dynamic mode decomposition (EDMD) (Williams et al. in J Nonlinear Sci 25(6):1307–1346, 2015 ) is an algorithm that approximates the action of the Koopman operator on an N -dimensional subspace of the space of observables by sampling at M points in the state space. Assuming that the samples are drawn either independently or ergodically from some measure μ , it was shown in Klus et al. (J Comput Dyn 3(1):51–79, 2016 ) that, in the limit as M → ∞ , the EDMD operator K N , M converges to K N , where K N is the L 2 ( μ ) -orthogonal projection of the action of the Koopman operator on the finite-dimensional subspace of observables. We show that, as N → ∞ , the operator K N converges in the strong operator topology to the Koopman operator. This in particular implies convergence of the predictions of future values of a given observable over any finite time horizon, a fact important for practical applications such as forecasting, estimation and control. In addition, we show that accumulation points of the spectra of K N correspond to the eigenvalues of the Koopman operator with the associated eigenfunctions converging weakly to an eigenfunction of the Koopman operator, provided that the weak limit of the eigenfunctions is nonzero. As a by-product, we propose an analytic version of the EDMD algorithm which, under some assumptions, allows one to construct K N directly, without the use of sampling. Finally, under additional assumptions, we analyze convergence of K N , N (i.e., M = N ), proving convergence, along a subsequence, to weak eigenfunctions (or eigendistributions) related to the eigenmeasures of the Perron–Frobenius operator. No assumptions on the observables belonging to a finite-dimensional invariant subspace of the Koopman operator are required throughout.

We examine spectral operator-theoretic properties of linear and nonlinear dynamical systems with globally stable attractors. Using the Kato decomposition, we develop a spectral expansion for general linear autonomous dynamical systems with analytic observables and define the notion of generalized eigenfunctions of the associated Koopman operator. We interpret stable, unstable and center subspaces in terms of zero-level sets of generalized eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions and the new notion of open eigenfunctions—defined on subsets of state space—to extend these results to nonlinear dynamical systems with an equilibrium. We provide a characterization of (global) center manifolds, center-stable, and center-unstable manifolds in terms of joint zero-level sets of families of Koopman operator eigenfunctions associated with the nonlinear system. After defining a new class of Hilbert spaces, that capture the on- and off-attractor properties of dissipative dynamics, and introducing the concept of modulated Fock spaces, we develop spectral expansions for a class of dynamical systems possessing globally stable limit cycles and limit tori, with observables that are square-integrable in on-attractor variables and analytic in off-attractor variables. We discuss definitions of stable, unstable, and global center manifolds in such nonlinear systems with (quasi)-periodic attractors in terms of zero-level sets of Koopman operator eigenfunctions. We define the notion of isostables for a general class of nonlinear systems. In contrast with the systems that have discrete Koopman operator spectrum, we provide a simple example of a measure-preserving system that is not chaotic but has continuous spectrum, and discuss experimental observations of spectrum on such systems. We also provide a brief characterization of the data types corresponding to the obtained theoretical results and define the coherent principal dimension for a class of datasets based on the lattice-type principal spectrum of the associated Koopman operator.

The analysis of data from experiments in economics routinely involves testing multiple null hypotheses simultaneously. These different null hypotheses arise naturally in this setting for at least three different reasons: when there are multiple outcomes of interest and it is desired to determine on which of these outcomes a treatment has an effect; when the effect of a treatment may be heterogeneous in that it varies across subgroups defined by observed characteristics and it is desired to determine for which of these subgroups a treatment has an effect; and finally when there are multiple treatments of interest and it is desired to determine which treatments have an effect relative to either the control or relative to each of the other treatments. In this paper, we provide a bootstrap-based procedure for testing these null hypotheses simultaneously using experimental data in which simple random sampling is used to assign treatment status to units. Using the general results in Romano and Wolf (Ann Stat 38:598–633, 2010 ), we show under weak assumptions that our procedure (1) asymptotically controls the familywise error rate—the probability of one or more false rejections—and (2) is asymptotically balanced in that the marginal probability of rejecting any true null hypothesis is approximately equal in large samples. Importantly, by incorporating information about dependence ignored in classical multiple testing procedures, such as the Bonferroni and Holm corrections, our procedure has much greater ability to detect truly false null hypotheses. In the presence of multiple treatments, we additionally show how to exploit logical restrictions across null hypotheses to further improve power. We illustrate our methodology by revisiting the study by Karlan and List (Am Econ Rev 97(5):1774–1793, 2007 ) of why people give to charitable causes.

This paper develops a new method for testing for Granger non-causality in panel data models with large cross-sectional (N) and time series (T) dimensions. The method is valid in models with homogeneous or heterogeneous coefficients. The novelty of the proposed approach lies in the fact that under the null hypothesis, the Granger-causation parameters are all equal to zero, and thus they are homogeneous. Therefore, we put forward a pooled least-squares (fixed effects type) estimator for these parameters only. Pooling over cross sections guarantees that the estimator has a NT convergence rate. In order to account for the well-known “Nickell bias”, the approach makes use of the well-known Split Panel Jackknife method. Subsequently, a Wald test is proposed, which is based on the bias-corrected estimator. Finite-sample evidence shows that the resulting approach performs well in a variety of settings and outperforms existing procedures. Using a panel data set of 350 U.S. banks observed during 56 quarters, we test for Granger non-causality between banks’ profitability and cost efficiency.