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In this paper, we present how high-order accurate solutions to elliptic partial differential equations can be achieved in arbitrary spatial domains using radial basis function-generated finite differences (RBF-FD) on unfitted node sets (i.e., not adjusted to the domain boundary). In this novel method, we only collocate on nodes interior to the domain boundary and enforce boundary conditions as constraints by means of Lagrange multipliers. This combination enables full geometric flexibility near boundaries without compromising the high-order accuracy of the RBF-FD method. The high-order accuracy and robustness of two formulations of this approach are illustrated by numerical experiments., Comment: 17 pages, 14 figures

We extend a classical result by Triebel on boundedness of bandlimited multipliers on $L^p(\mathbb{R}^n)$, $0

For linear regression models with cross-section or panel data, it is natural to assume that the disturbances are clustered in two dimensions. However, the finite-sample properties of two-way cluster-robust tests and confidence intervals are often poor. We discuss several ways to improve inference with two-way clustering. Two of these are existing methods for avoiding, or at least ameliorating, the problem of undefined standard errors when a cluster-robust variance matrix estimator (CRVE) is not positive definite. One is a new method that always avoids the problem. More importantly, we propose a family of new two-way CRVEs based on the cluster jackknife. Simulations for models with two-way fixed effects suggest that, in many cases, the cluster-jackknife CRVE combined with our new method yields surprisingly accurate inferences. We provide a simple software package, twowayjack for Stata, that implements our recommended variance estimator.

We study cluster-robust inference for binary response models. Inference based on the most commonly-used cluster-robust variance matrix estimator (CRVE) can be very unreliable. We study several alternatives. Conceptually the simplest of these, but also the most computationally demanding, involves jackknifing at the cluster level. We also propose a linearized version of the cluster-jackknife variance matrix estimator as well as linearized versions of the wild cluster bootstrap. The linearizations are based on empirical scores and are computationally efficient. Throughout we use the logit model as a leading example. We also discuss a new Stata software package called logitjack which implements these procedures. Simulation results strongly favor the new methods, and two empirical examples suggest that it can be important to use them in practice.

The design and optimization of antibodies requires an intricate balance across multiple properties. Protein inverse folding models, capable of generating diverse sequences folding into the same structure, are promising tools for maintaining structural integrity during antibody design. Here, we present AntiFold, an antibody-specific inverse folding model, fine-tuned from ESM-IF1 on solved and predicted antibody structures. AntiFold outperforms existing inverse folding tools on sequence recovery across complementarity-determining regions, with designed sequences showing high structural similarity to their solved counterpart. It additionally achieves stronger correlations when predicting antibody-antigen binding affinity in a zero-shot manner, while performance is augmented further when including antigen information. AntiFold assigns low probabilities to mutations that disrupt antigen binding, synergizing with protein language model residue probabilities, and demonstrates promise for guiding antibody optimization while retaining structure-related properties. AntiFold is freely available under the BSD 3-Clause as a web server at https://opig.stats.ox.ac.uk/webapps/antifold/ and and pip installable package at https://github.com/oxpig/AntiFold

Given a matrix-weight $W$ in the Muckenhoupt class $\mathbf{A}_p(\mathbb{R}^n)$, $1\leq p<\infty$, we introduce corresponding vector-valued continuous and discrete $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}(W)$ and $m^{s,\alpha}_{p,q}(W)$ and prove their equivalence through the use of adapted tight frames. Compatible notions of molecules and almost diagonal matrices are also introduced, and an application to the study of pseudo-differential operators on vector valued spaces is given.

We study statistical inference on unit roots and cointegration for time series in a Hilbert space. We develop statistical inference on the number of common stochastic trends embedded in the time series, i.e., the dimension of the nonstationary subspace. We also consider tests of hypotheses on the nonstationary and stationary subspaces themselves. The Hilbert space can be of an arbitrarily large dimension, and our methods remain asymptotically valid even when the time series of interest takes values in a subspace of possibly unknown dimension. This has wide applicability in practice; for example, to the case of cointegrated vector time series that are either high-dimensional or of finite dimension, to high-dimensional factor model that includes a finite number of nonstationary factors, to cointegrated curve-valued (or function-valued) time series, and to nonstationary dynamic functional factor models. We include two empirical illustrations to the term structure of interest rates and labor market indices, respectively.

We study a class of almost diagonal matrices compatible with the mixed-norm $\alpha$-modulation spaces $M_{\vec{p},q}^{s,\alpha}(\mathbb{R}^n)$, $\alpha\in [0,1]$, introduced recently by Cleanthous and Georgiadis [Trans.\ Amer.\ Math.\ Soc.\ 373 (2020), no. 5, 3323-3356]. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm $\alpha$-modulation spaces. We show that the "change of frame" matrix for a pair of time-frequency frames for the mixed-norm $\alpha$-modulation consisting of suitable molecules is almost diagonal. As examples of applications, we use the almost diagonal matrices to construct compactly supported frames for the mixed-norm $\alpha$-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm $\alpha$-modulation spaces.

Subsampling of node sets is useful in contexts such as multilevel methods, computer graphics, and machine learning. On uniform grid-based node sets, the process of subsampling is simple. However, on node sets with high density variation, the process of coarsening a node set through node elimination is more interesting. A novel method for the subsampling of variable density node sets is presented here. Additionally, two novel node set quality measures are presented to determine the ability of a subsampling method to preserve the quality of an initial node set. The new subsampling method is demonstrated on the test problems of solving the Poisson and Laplace equations by multilevel radial basis function-generated finite differences (RBF-FD) iterations. High-order solutions with robust convergence are achieved in linear time with respect to node set size., Comment: 21 pages, 12 figures

We provide computationally attractive methods to obtain jackknife-based cluster-robust variance matrix estimators (CRVEs) for linear regression models estimated by least squares. We also propose several new variants of the wild cluster bootstrap, which involve these CRVEs, jackknife-based bootstrap data-generating processes, or both. Extensive simulation experiments suggest that the new methods can provide much more reliable inferences than existing ones in cases where the latter are not trustworthy, such as when the number of clusters is small and/or cluster sizes vary substantially. Three empirical examples illustrate the new methods.

The overwhelming majority of empirical research that uses cluster-robust inference assumes that the clustering structure is known, even though there are often several possible ways in which a dataset could be clustered. We propose two tests for the correct level of clustering in regression models. One test focuses on inference about a single coefficient, and the other on inference about two or more coefficients. We provide both asymptotic and wild bootstrap implementations. The proposed tests work for a null hypothesis of either no clustering or ``fine'' clustering against alternatives of ``coarser'' clustering. We also propose a sequential testing procedure to determine the appropriate level of clustering. Simulations suggest that the bootstrap tests perform very well under the null hypothesis and can have excellent power. An empirical example suggests that using the tests leads to sensible inferences.

We characterize digital cash as the digital equivalent of physical cash: secure, fungible, decentralized, directly controlled, privacy-preserving; but enhanced with qualitatively new functionality. It is extremely efficiently transferable and, most importantly, transactional or, more generally, contract-backed. This facilitates fully automated, guaranteed transactional execution of atomic resource exchanges and more complex contracts, without a multitude of intermediaries and expensive or slow semi-manual processes. A didactic objective is separating money characteristics from technology aspects such as specific blockchain and distributed ledger systems to help disentangle discussions of digital money design from implementation techniques. We finally discuss the power and role of programmable (contract-backed) digital money in case studies: tokenization of invoice debt using smart contracts on Ethereum, with stablecoins serving as digital money; smart contracts for disbursing payments transparently and reliably in accordance with social legislation; and a Danish e-krone for crowdfunding public and private community projets. These contributions are made in independent chapters by participants of the Working Group on Digital Cash at Copenhagen FinTech in 2018 and 2019, which have not been published before. Collectively, the contributions illustrate the design space and potential of digital money when powered by smart digital contracts that effectively eliminate both counterparty risk (somebody does not pay or does not deliver) and settlement risk (a trade fails and needs to be aborted) orders of magnitude faster than in current financial practice., Comment: Editor: Fritz Henglein. Authors: Fritz Henglein (Chapters 1, 2, 3, 5), Fritz Henglein and Christian Olesen (Chapter 4), Gert Sylvest (Chapter 6), S{\o}ren Debois (Chapter 7), Morten C. Nielsen and Christian Olesen (Chapter 8)

In this chapter we present an overview of the main ideas and methods in the fractional integration and cointegration literature. We do not attempt to give a complete survey of this enormous literature, but rather a more introductory treatment suitable for a researcher or graduate student wishing to learn about this exciting field of research. With this aim, we have surely overlooked many relevant references for which we apologize in advance. Knowledge of standard time series methods, and in particular methods related to nonstationary time series, at the level of a standard graduate course or advanced undergraduate course is assumed.

It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter $d$ converges weakly to fractional Brownian motion for $d>1/2$. We show that, for any non-negative integer $M$, derivatives of order $m=0,1,\dots,M$ of the normalized fractional process with respect to the fractional parameter $d$, jointly converge weakly to the corresponding derivatives of fractional Brownian motion. As an illustration we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.

We consider bootstrap inference for estimators which are (asymptotically) biased. We show that, even when the bias term cannot be consistently estimated, valid inference can be obtained by proper implementations of the bootstrap. Specifically, we show that the prepivoting approach of Beran (1987, 1988), originally proposed to deliver higher-order refinements, restores bootstrap validity by transforming the original bootstrap p-value into an asymptotically uniform random variable. We propose two different implementations of prepivoting (plug-in and double bootstrap), and provide general high-level conditions that imply validity of bootstrap inference. To illustrate the practical relevance and implementation of our results, we discuss five examples: (i) inference on a target parameter based on model averaging; (ii) ridge-type regularized estimators; (iii) nonparametric regression; (iv) a location model for infinite variance data; and (v) dynamic panel data models.

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel-Lizorkin spaces in an anisotropic setting on $\bR^d$. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear $m$-term nonlinear approximation with the constructed basis in the mixed-norm setting, where the behaviour, in general, for $d\geq 2$, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for $m$-term approximation can still be derived.

We introduce a new Stata package called summclust that summarizes the cluster structure of the dataset for linear regression models with clustered disturbances. The key unit of observation for such a model is the cluster. We therefore propose cluster-level measures of leverage, partial leverage, and influence and show how to compute them quickly in most cases. The measures of leverage and partial leverage can be used as diagnostic tools to identify datasets and regression designs in which cluster-robust inference is likely to be challenging. The measures of influence can provide valuable information about how the results depend on the data in the various clusters. We also show how to calculate two jackknife variance matrix estimators efficiently as a byproduct of our other computations. These estimators, which are already available in Stata, are generally more conservative than conventional variance matrix estimators. The summclust package computes all the quantities that we discuss., Comment: The summclust package is available here: https://github.com/mattdwebb/summclust

Methods for cluster-robust inference are routinely used in economics and many other disciplines. However, it is only recently that theoretical foundations for the use of these methods in many empirically relevant situations have been developed. In this paper, we use these theoretical results to provide a guide to empirical practice. We do not attempt to present a comprehensive survey of the (very large) literature. Instead, we bridge theory and practice by providing a thorough guide on what to do and why, based on recently available econometric theory and simulation evidence. To practice what we preach, we include an empirical analysis of the effects of the minimum wage on labor supply of teenagers using individual data., Comment: Accepted at Journal of Econometrics

Mixed-norm $\alpha$-modulation spaces were introduced recently by Cleanthous and Georgiadis [Trans.\ Amer.\ Math.\ Soc.\ 373 (2020), no. 5, 3323-3356]. The mixed-norm spaces $M^{s,\alpha}_{\vec{p},q}(\mathbb{R}^n)$, $\alpha\in [0,1]$, form a family of smoothness spaces that contain the mixed-norm Besov spaces as special cases. In this paper we prove that a pseudodifferential operator $\sigma(x,D)$ with symbol in the H\"ormander class $S^b_{\rho}$ extends to a bounded operator $\sigma(x,D)\colon M^{s,\alpha}_{\vec{p},q}(\mathbb{R}^n) \rightarrow M^{s-b,\alpha}_{\vec{p},q}(\mathbb{R}^n)$ provided $0<\alpha\leq \rho\leq 1$, $\vec{p}\in (0,\infty)^n$, and $0

In this article we construct orthonormal bases compatible with bi-variate homogeneous $\alpha$-modulation spaces and the associated spaces of Triebel-Lizorkin type. The construction is based on generating a separable $\alpha$-covering and using carefully selected tensor products of univariate brushlet functions with regards to this covering. We show that the associated systems form an unconditional bases for the homogeneous $\alpha$-spaces of Triebel-Lizorkin type.