Narrow Results

To avoid spurious inferences, researchers analyzing the dimensions of uncertainty need to determine whether it is nonstationary. The degree of persistence of uncertainty also indicates the duration of the negative impact of an uncertainty shock on the economy. We use a new panel residual augmented least squares unit root test that allows for heterogeneous structural breaks in both intercepts and slopes of a series to determine the degree of persistence of the reports-based measure of uncertainty and whether it is nonstationary for 143 countries. This group of countries accounts for 99% of the world’s gross domestic product (GDP). To assess the robustness of our results, we also use recently developed univariate time-series unit root tests that allow for structural breaks and panel unit root tests that accommodate cross-sectional dependence and nonlinearity. Furthermore, an autoregressive wild bootstrap approach is utilized to examine the stationarity of the series. The results are virtually unambiguous in indicating that the reports-based measure of uncertainty is stationary in all countries considered. The results also suggest that uncertainty has a negative impact on the growth rate of GDP. The policy implications of the results are also discussed.

Spurious regression, commonly associated with independent and (nearly) non-stationary time series, has been extensively studied. However, the potential for spurious outcomes in regression involving stationary time series remains largely unexplored, representing a gap in the literature. To address this gap, we propose that regression of stationary time series may yield spurious outcomes and conduct a comprehensive investigation to verify this conjecture. Additionally, we present a remedy algorithm to mitigate spurious effects and improve model interpretability. Through extensive simulations, we validate our conjecture and demonstrate the efficacy of the proposed remedy. A numerical analysis further illustrates the practical utility of our approach. This study offers a fresh perspective on spurious regression and provides a practical solution to enhance the reliability of regression analyses involving stationary time series data.

Thoma's theorem states that a group algebra $C^{\ast }(\mathrm{\Gamma })$ is of type I if and only if $\mathrm{\Gamma }$ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group $\mathrm{\Gamma }$, we have a stationary model of type $\pi :C^{\ast }(\mathrm{\Gamma })\to M_F(C(L))$, with $F$ being a finite quantum group, and with $L$ being a compact group. We discuss then some refinements of these results in the quantum permutation group case, $\widehat{\mathrm{\Gamma }}\subset S_N^{+}$, by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank $\le 1$.

An accurate multiaffine analysis of 23 foreign currency exchange rates has been performed. The roughness exponent H₁ which characterizes the excursion of the exchange rate has been numerically measured. The degree of intermittency C₁ has been also estimated. In the (H₁,C₁) phase diagram, the currency exchange rates are dispersed in a wide region around the Brownian motion value (H₁=0.5,C₁=0) and have a significantly intermittent component (C₁≠0).

Regularization method proposed by Scholtes (2011) has been a recognized approach for deterministic mathematical programs with complementarity constraints (MPCC). Meng and Xu (2006) applied the approach coupled with Monte Carlo techniques to solve a class of one stage stochastic MPCC and presented some promising numerical results. However, Meng and Xu have not presented any convergence analysis of the regularized sample approximation method. In this paper, we fill out this gap. Specifically, we consider a general class of one stage stochastic mathematical programs with complementarity constraint where the objective and constraint functions are expected values of random functions. We carry out extensive convergence analysis of the regularized sample average approximation problems including the convergence of statistical estimators of optimal solutions, C-stationary points, M-stationary points and B-stationary points as sample size increases and the regularization parameter tends to zero.

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

An analytical solution characterizing initial conditions leading to action potential firing in smooth nerve fibers is determined, using the bistable equation. In the first place, we present a nontrivial stationary solution wave, then, using the perturbative method, we analyze the stability of this stationary wave. We show that it corresponds to a frontier between the initiation of the travelling waves and a decay to the resting state. Eventually, this analytical approach is extended to FitzHugh–Nagumo model.

Software effort estimation models are typically developed based on an underlying assumption that all data points are equally relevant to the prediction of effort for future projects. The dynamic nature of several aspects of the software engineering process could mean that this assumption does not hold in at least some cases. This study employs three kernel estimator functions to test the stationarity assumption in five software engineering datasets that have been used in the construction of software effort estimation models. The kernel estimators are used in the generation of nonuniform weights which are subsequently employed in weighted linear regression modeling. In each model, older projects are assigned smaller weights while the more recently completed projects are assigned larger weights, to reflect their potentially greater relevance to present or future projects that need to be estimated. Prediction errors are compared to those obtained from uniform models. Our results indicate that, for the datasets that exhibit underlying nonstationary processes, uniform models are more accurate than the nonuniform models; that is, models based on kernel estimator functions are worse than the models where no weighting was applied. In contrast, the accuracies of uniform and nonuniform models for datasets that exhibited stationary processes were essentially equivalent. Our analysis indicates that as the heterogeneity of a dataset increases, the effect of stationarity is overridden. The results of our study also confirm prior findings that the accuracy of effort estimation models is independent of the type of kernel estimator function used in model development.

We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and Lions. An application of the abstract result is given for Maxwell's equations in random dissipative bianisotropic media.

Electrochemical corrosion processes can be investigated by observation of charge flows between the electrolyte and the corroding metal. Usually, the charge flows are observed as spontaneous current and voltage fluctuations (electrochemical noise) in a three-electrode setup. Different types of corrosion processes can be recognized by electrochemical noise analysis. Uniform corrosion rate can be evaluated by estimation of polarization resistance between the metal and electrolyte. Local corrosion events (breakdowns of the passive layer) that produce characteristic transients observed in noise can be detected as well. Different methods of electrochemical noise analysis are presented in a brief review. The limitations and advantages of the proposed methods for corrosion monitoring and research are underlined. The experimental results are also discussed.

A one-dimensional (1D) point process, if considered as a random measure, can be represented by a sum, at most countable, of Delta Dirac measures concentrated at some random points. The integration with respect to the point process leads to the definition of the continuous wavelet transform of the process itself. As a possible choice of the mother wavelet, we propose the Mexican hat and the Morlet wavelet in order to implement the energy density of the process as a function of two wavelet parameters. Such mathematical tool works as a microscope to process an in-depth analysis of some classes of processes, in particular homogeneous, cluster, and locally scaled processes.

We have read many papers in the literature and found that some papers report results of regressing a stationary time series on a non-stationary time series (we call it the $IOI1$ model). However, very few studies, if there are any, examine the $IOI1$ model and the robustness of inference in such settings remains an open question. To bridge the gap in the literature, in this paper, we investigate whether regressing a stationary time series, $Y_t$, on a non-stationary time series, $X_t$ (that is, $Y_t=\alpha +\beta X_t+u_t$) could get any meaningful result. To do so, we first conduct a simulation and find regressing a stationary time series on a non-stationary time series could be spurious. Thereafter, we develop the estimation and testing theory for the $I0I1$ model and find that the statistics $T_N^{\beta }$ for testing $H_0^{\beta }:\beta ={\beta }_0$ versus $H_1^{\beta }:\beta \ne {\beta }_0$ from the traditional regression model (we call it $IOI0$ model) does not have any asymptote distribution with $E(T_N^{\beta })\to \infty$ and $\mathrm{\text{Var}}(T_N^{\beta })\to \infty$ as $N\to \infty$, and thus, it cannot be used for the $I0I1$ model. We have found other interesting results as shown in our paper. Thus, our paper extends the spurious regression literature to cover a previously unexplored case, thereby contributing to a more comprehensive understanding of time series modeling and inference.

In this chapter, we will show that the asymptotic theory for linear regression models with IID observations carries over to ergodic stationary linear time series regression models with Martingale Difference Sequence (MDS) disturbances. Some basic concepts in time series analysis are introduced, and some tests for serial correlation are described.

This chapter aims to summarize the theories, models, methods and tools of modern econometrics which we have covered in the previous chapters. We first review the classical assumptions of the linear regression model and discuss the historical development of modern econometrics by various relaxations of the classical assumptions. We also discuss the challenges and opportunities for econometrics in the Big data era and point out some important directions for the future development of econometrics.

Electrochemical corrosion processes can be investigated by observation of charge flows between the electrolyte and the corroding metal. Usually, the charge flows are observed as spontaneous current and voltage fluctuations (electrochemical noise) in a three electrode setup. Different types of corrosion processes can be recognized by electrochemical noise analysis. Uniform corrosion rate can be evaluated by estimation of polarization resistance between the metal and electrolyte. Local corrosion events (breakdowns of the passive layer) that produce characteristic transients observed in noise can be detected as well. Different methods of electrochemical noise analysis are presented in a brief review. The limitations and advantages of the proposed methods for corrosion monitoring and research are underlined. The experimental results are also discussed.

We have shown that the least squares estimator for a non-ergodic, first order, self-exciting, threshold autoregressive model is strongly consistent under quite general conditions.

In this chapter, we shall build on the fundamental notions of probability distribution and statistics in the last chapter, and extend consideration to a sequence of random variables. In financial application, it is mostly the case that the sequence is indexed by time, hence a stochastic process. Interesting statistical laws or mathematical theories result when we look at the relationships within a stochastic process. We introduce an application of the Central Limit Theorem to the study of stock return distributions.

Our goal in this paper is to look at stochastic financial models and especially on those properties of the involved distributions which are expressed in terms of the moments. The questions discussed and the results presented reveal the role which the moments play in the analysis of distributions. Interesting conclusions can be derived in both cases when available are finitely many moments and when we know all moments.

Among the results included in the paper are sharp lower and/or upper bounds for option prices in terms of a finite number of moments. However the main attention is paid to the determinacy of distributions by their moments. While any light tailed distribution is uniquely determined by its moments, the uniqueness may fail for heavy tailed distributions. And, here is the point. Heavy tailed distributions are essentially involved in stock market modelling, and most of them are non-unique in terms of the moments. This is why the phenomenon non-uniqueness of distributions deserves a special attention.

We treat distributions on the positive half-line used to describe, e.g., stock prices and option prices, and distributions on the whole real line describing log-returns. For reader's convenience we give a brief and unified general picture of existing results about uniqueness and non-uniqueness of distribution in terms of their moments. Some of the results are classical and well-known. In several cases we provide here new arguments along with presenting new recent results on the moment determinacy of random variables and their non-linear transformations and also of stochastic processes which are solutions to SDEs.