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We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is

The event study model is a powerful econometric tool used for the purpose of estimating dynamic treatment effects. One of its most appealing features is that it provides a built-in graphical summary of results, which can reveal rich patterns of behavior. Another value of the picture is the estimated pre-event pseudo-\"effects\", which provide a type of placebo test. In this essay I aim to provide a framework for a shared understanding of these models. There are several (sometimes subtle) decisions and choices faced by users of these models, and I offer guidance for these decisions.

Here’s why. (a) The Hodrick-Prescott (HP) filter introduces spurious dynamic relations that have no basis in the underlying data-generating process. (b) Filtered values at the end of the sample are very different from those in the middle and are also characterized by spurious dynamics. (c) A statistical formalization of the problem typically produces values for the smoothing parameter vastly at odds with common practice. (d) There is a better alternative. A regression of the variable at date t on the four most recent values as of date t − h achieves all the objectives sought by users of the HP filter with none of its drawbacks.

We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by α, is of great importance as the resulting system is known to blow-up for large values of α. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when α is small enough.

Understanding the fluid flow and solute transport in karst conduits is significant for preventing pollutants (treated as solutes) in karst areas. This study evaluates the evolution of eddies in semi‐circular rough conduits under different hydrodynamic conditions and their effects on solute transport. As inlet flow velocity increases, the non‐Fickian coefficient (a parameter used for qualifying the tailing of BTCs) increases first then decreases afterward. A critical equilibrium concentration is found when the concentration of the main flow stream and the concentration of the eddy zone reaches the same value, signaling the moment at which the mass transfer (due to diffusion) between the main flow stream and the eddy zone drops to zero. Such a critical equilibrium concentration and its corresponding moment of occurrence are found to follow two distinctive logarithmic functions of the inlet flow velocity. These findings provide crucial technical support for groundwater pollution control in karst areas. Plain Language Summary Eddies are commonly seen when groundwater moves through highly permeable porous media, fractures, or conduit media, and understanding their effects on pollutants (treated as solutes) transport is crucial. In this study, we designed a semicircular rough conduit to reveal the mechanism of solute transport under different hydrodynamic conditions using a combined experimental and numerical simulation approach. The occurrence of the eddy zone and its influence on solute transport are revealed in great detail by analyzing the breakthrough curves (BTCs). This study provides a scientific basis for groundwater pollution control, especially in karst areas, demonstrating the relevance and applicability of our research to the field of geophysical fluid dynamics. Key Points The non‐Fickian effect increases first and then decreases gradually with the increase of inlet flow velocity The transformation mechanism process of advection and hydrodynamic diffusion is revealed The relationship between the equilibrium concentration, corresponding time and flow velocity at different stages is quantified

High-dimensional matrix-variate time series data are becoming widely available in many scientific fields, such as economics, biology, and meteorology. To achieve significant dimension reduction while preserving the intrinsic matrix structure and temporal dynamics in such data, Wang, Liu, and Chen proposed a matrix factor model, that is, shown to be able to provide effective analysis. In this article, we establish a general framework for incorporating domain and prior knowledge in the matrix factor model through linear constraints. The proposed framework is shown to be useful in achieving parsimonious parameterization, facilitating interpretation of the latent matrix factor, and identifying specific factors of interest. Fully utilizing the prior-knowledge-induced constraints results in more efficient and accurate modeling, inference, dimension reduction as well as a clear and better interpretation of the results. Constrained, multi-term, and partially constrained factor models for matrix-variate time series are developed, with efficient estimation procedures and their asymptotic properties. We show that the convergence rates of the constrained factor loading matrices are much faster than those of the conventional matrix factor analysis under many situations. Simulation studies are carried out to demonstrate finite-sample performance of the proposed method and its associated asymptotic properties. We illustrate the proposed model with three applications, where the constrained matrix-factor models outperform their unconstrained counterparts in the power of variance explanation under the out-of-sample 10-fold cross-validation setting. Supplementary materials for this article are available online.

We propose a class of observation-driven time series models referred to as generalized autoregressive score (GAS) models. The mechanism to update the parameters over time is the scaled score of the likelihood function. This new approach provides a unified and consistent framework for introducing time-varying parameters in a wide class of nonlinear models. The GAS model encompasses other well-known models such as the generalized autoregressive conditional heteroskedasticity, autoregressive conditional duration, autoregressive conditional intensity, and Poisson count models with time-varying mean. In addition, our approach can lead to new formulations of observation-driven models. We illustrate our framework by introducing new model specifications for time-varying copula functions and for multi variate point processes with time-vary ing parameters. We study the models in detail and provide simulation and empirical evidence.

An approximate factor model of high dimension has two key features. First, the idiosyncratic errors are correlated and heteroskedastic over both the cross-section and time dimensions; the correlations and heteroskedasticities are of unknown forms. Second, the number of variables is comparable or even greater than the sample size. Thus, a large number of parameters exist under a high-dimensional approximate factor model. Most widely used approaches to estimation are principal component based. This paper considers the maximum likelihood-based estimation of the model. Consistency, rate of convergence, and limiting distributions are obtained under various identification restrictions. Monte Carlo simulations show that the likelihood method is easy to implement and has good finite sample properties.

The classic papers by Newey and West (1987) and Andrews (1991) spurred a large body of work on how to improve heteroscedasticity- and autocorrelation-robust (HAR) inference in time series regression. This literature finds that using a larger-than-usual truncation parameter to estimate the long-run variance, combined with Kiefer-Vogelsang (2002, 2005) fixed-b critical values, can substantially reduce size distortions, at only a modest cost in (size-adjusted) power. Empirical practice, however, has not kept up. This article therefore draws on the post-Newey West/Andrews literature to make concrete recommendations for HAR inference. We derive truncation parameter rules that choose a point on the size-power tradeoff to minimize a loss function. If Newey-West tests are used, we recommend the truncation parameter rule S = 1.3T 1/2 and (nonstandard) fixed-b critical values. For tests of a single restriction, we find advantages to using the equal-weighted cosine (EWC) test, where the long run variance is estimated by projections onto Type II cosines, using ν = 0.4T 2/3 cosine terms; for this test, fixed-b critical values are, conveniently, t ν or F. We assess these rules using first an ARMA/GARCH Monte Carlo design, then a dynamic factor model design estimated using a 207 quarterly U.S. macroeconomic time series.

Since Quetelet's work in the nineteenth century, social science has iconified the average man, that hypothetical man without qualities who is comfortable with his head in the oven and his feet in a bucket of ice. Conventional statistical methods since Quetelet have sought to estimate the effects of policy treatments for this average man. However, such effects are often quite heterogeneous: Medical treatments may improve life expectancy but also impose serious short-term risks; reducing class sizes may improve the performance of good students but not help weaker ones, or vice versa. Quantile regression methods can help to explore these heterogeneous effects. Some recent developments in quantile regression methods are surveyed in this review.