Research

Some of the research work I'm presently working on:

1) PROBABILITY AND MATHEMATICAL ANALYSIS

"Solution to stochastic Loewner equation with several complex variables using Nevanlinna theory"

Becker (1973) studied solutions to the Loewner differential equation in one complex variable using Carathéodory class of holomorphic functions. However, in several complex variables point singularities are removable and other approaches necessary. Pfaltzgraff generalized to higher dimensions the Loewner differential equation and developed existence and uniqueness theorems for its solutions. The existence and regularity theory has been considered by several authors, and applications given to the characterization of subclasses of biholomorphic mappings, univalence criteria, growth theorems and coefficient bounds for restricted classes of biholomorphic mappings. Duren et al. (2010) studied general form of solutions to the Loewner differential equation under common assumptions of holomorphicity and uniquely determined univalent subordination chains. To our knowledge, to date stochastic Loewner equation has not been studied in a several complex variable setting. We solve the equation in its radial and chordal ordinary stochastic differential form by firstly appropriately defining Loewner chain, evolution family and Herglotz vector field in $\mathbb{C}^n$. We use previous work on invariance in the connections of Nevanlinna theory and stochastic processes (e.g. Atsuji, 1995; Dong et al., 2019) to include stochastic element (appropriately scaled Brownian motion) in Herlotz-Nevanlinna function in $\mathbb{C}^n$ which is the core element of the $\mathbb{C}^n$ Herglotz vector field representation of the corresponding Loewner chain as representation of solution to stochastic Loewner ODE in $\mathbb{C}^n$. This represents the solution to the equation which is in present knowledge of Loewner theory on $\mathbb{C}^n$ limited to spirallike mappings on the unit ball $B^n$ in $\mathbb{C}^n$. We study value distribution of our meromorphic solution using generalizations of Nevanlinna theory for several complex variables (Noguchi and Winkelman, 2013). Solution allows to study stochastic phenomena such as Schramm-Loewner evolution and scaling limits of stochastic processes in $d>2$. Connection between Nevanlinna theory and Brownian motions and its generalization to several complex variables promises to provide a path to solving stochastic differential equations with several complex variables (linear, non-linear or algebraic) in future, something not done or possible so far systematically. Future work in probability theory would have to consider possibilities of connections to complex analysis in several variables on $\mathbb{C}^n$ manifolds and objects.

"Level densities for general β-ensembles: An operator-valued free probability perspective", joint with Miroslav Verbič

Random point processes corresponding to β-ensembles for arbitrary β > 0, or, equivalently, log gases at inverse temperature β, are being subject to intense study. The orthogonal, unitary, and symplectic ensembles (β = 1, 2, or 4, respectively) are now well understood, but other values of β are believed to also be relevant in theory (e.g. relevant for the study of self-adjoint and Schrödinger operators) and applications (e.g. in logistics). For certain rational values of β, β-ensembles are related to Jack polynomials, but for general β much less is known. In a seminal article, Dumitriu and Edelman (2002) constructed tridiagonal random matrix models for general β-Hermite and β-Laguerre ensembles and defined open problems for research on general β-ensembles, including finding a unified formula for the level density in general β-case. In general, level density is defined as distribution of a random eigenvalue of an ensemble (by the Wigner semicircular law, the limiting distribution of the eigenvalue is semicircular). We exploit the product nature of Dumitriu and Edelman's construction of tridiagonal random matrix models and derive the formula for the level density in the general β-case depending on the multivariate Fuss-Narayana polynomials and concepts from operator-valued free probability theory. We study perturbation invariability of the level densities (Wang and Yan, 2005; Kozhan, 2017) and discuss extensions to problems of sampling general β-ensembles (referring to Li and Menon, 2013; Olver et al., 2015; Srakar and Verbic, 2020) and limiting entropy in β-ensembles related point processes (Mészaros, 2020). Perspective we are developing for the article allows to study open problems in general β-ensembles, one of cornerstones of random matrix theory, in a free probability perspective which promises to provide feasible solutions to many of them.

"Sampling orthogonal and symplectic invariant ensembles", joint with Miroslav Verbič

In random matrix theory (RMT), in order to test a given algorithm for universality one needs to sample various ensembles. This is a standard matter if one is dealing with Wigner ensembles and the entries are independent, but when the entries are dependent, as in the case for general invariant ensembles, this is a non-trivial matter. Main existing algorithms have been developed by Li and Menon (2013) for sampling invariant ensembles based on simulating Dyson Brownian motion, and by Olver et al. (2015) for invariant unitary ensembles based on computing the orthogonal polynomials naturally associated with the ensembles. To present, it has been open to extend the method of Olver et al. to orthogonal and symplectic invariant ensembles (Deift, 2017). We present a logical extension of their work exploiting the fact that the eigenvalues of both orthogonal and symplectic ensembles can be represented as Pfaffian point process whose kernel is given in terms of skew orthogonal polynomials. Olver et al.'s algorithm is extended using skew transformations based on Lie algebras of the respective orthogonal groups exploring the validity of the algorithm and its convergence. Efficient computation of the associated skew orthogonal polynomials is implemented also using algebraic statistics and Buchberger algorithm (La Scala and Levandovskyy, 2012) and the approach is compared with more common Monte Carlo based algorithm. We include examples of several experiments and applications and discuss possible extensions. By facilitating efficient and accurate sampling of non-classical matrix ensembles and solving an open problem in RMT, the algorithm can aid in the experimentation-based inference involving eigenvalue statistics that might presently be unamenable to theoretical analysis.

"Approximating the Ising model on fractal lattices of dimension two or greater than two"

The exact solutions of the Ising model in one and two dimensions are well known, but much less is known about solutions on fractal lattices. In an important contribution, Codello, Drach and Hietanen (2015) construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two using a generalization of combinatorial method of Feynman (1972) and Vdovichenko (1965). We generalize their approach to fractal lattices of dimensions 2 and greater than 2, in particular of Koch curve variety (e.g. quadratic and von Koch surface). To this end we combine combinatorial optimization and transfer matrix approaches, referring to earlier works of Andrade and Salinas (1984), including works on random walks on fractal graphs (e.g. Sun, 2013). We compute approximate estimates for the critical temperatures and compare them to more usual Monte Carlo estimates. Similar as in Codello et al., we compute the correlation length as a function of the temperature and extract the relative critical exponent. The method allows generalizations to any fractal lattice, as well as to approach solutions for other interacting particle systems on fractal lattices. Extensions to Potts models are considered.

"Scaling limits for parking on Frozen Erdős Rényi Cayley trees with heavy tails"

In a recent contribution, Contat and Curien (2021; 2023) studied parking problem on uniform rooted Cayley tree with n vertices and m cars arriving sequentially, independently, and uniformly on its vertices. In a previous contribution, Lackner and Panholzer (2016) established a phase transition for this process when m ≈ n/2 . Contat and Curien couple this model with a variant of the classical Erdos–Renyi random graph process which enables describing the phase transition for the size of the components of parked cars using a ”frozen” modification of the multiplicative coalescent, which freezes components with surplus. They showed scaling limit convergence in Skorokhod topology towards the frozen multiplicative coalescent and growth-fragmentation trees canonically associated to the 3/2-stable process (Bertoin, 2017). We study their model in the presence of group arrival of cars with power-law tail, and derive the appropriate metric space scaling limits in a Gromov-weak and Gromov-Hausdorff-Prokhorov topologies using results on configuration model (Bhamidi et al., 2018; Broutin et al., 2020; Conchon-Kerjan and Goldschmidt, 2020; Dhara et al., 2020). Our scaling limit is a novel stochastic process which combines additive and multiplicative coalescent (Aldous, 1997). We compare the results to more commonly studied Bienayme–Galton–Watson trees, as well as study extensions to generalized frozen process (Contat and Curien, 2023) and discuss extensions to moderate and high dimensions and analysis of moderate and large deviations for the novel process.

"Spectral CLTs with long memory for large language models"

Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for Yt have been constantly refined, extended and applied to an increasing number of diverse situations. In recent years, fourth moment theorem CLTs, quantitative CLTs, Breuer-Major and Dobrushin-Major CLTs, de Jong CLTs, functional CLTs and others have been developed. Recently, Maini and Nourdin (2024) extended this to spectral central limit theorems valid for additive functionals of isotropic and stationary Gaussian fields. Their work uses Malliavin-Stein method and Fourier analysis techniques to situations where $Y_t$ admits Gaussian fluctuations in a long memory context. In another recent article, Wang et al. (2023) augmented existing language models with long-term memory. Namely, existing large language models (LLMs) can only afford fix-sized inputs due to the input length limit, preventing them from utilizing rich long-context information from past inputs. They proposed a framework of Language Models Augmented with Long-Term Memory, which enables LLMs to memorize long history. In our article we develop spectral central limit theorems in a context of augmented large language models of Wang and coauthors. Our analysis is put in a mean-field analysis context to derive appropriate limiting theorems in the usual two part scheme: a nonlinear partial differential and linear stochastic partial differential equation, to take into account the mean field limit and CLT. Analysis is set in a stochastic Ising model interacting particle systems perspective to account for the Transformer structure of the LLM. We present applications on datasets from finance and medical imaging. In conclusion we discuss possible Bayesian extensions, as well as implications for statistical estimation and inference in a natural language processing context.

2) OBJECT-ORIENTED AND NONPARAMETRIC STATISTICS

"Covariance and autocovariance estimation on a Liouville quantum gravity sphere in a functional context"

Research on spherical random fields and their applications has become an important part of probability, statistics and mathematical physics. Approaches are extended to study anisotropic spherical random fields previously unaddressed in this context area in random geometry, namely Liouville quantum gravity (LQG) spheres. The quantum Liouville theory was introduced in 1981 as a model for quantizing the bosonic string in the conformal gauge and gravity in two space-time dimensions. Liouville measure is formally the exponential of the Gaussian free field (GFF), and it is possible to study in depth its properties about SLE curves or geometrical objects in the plane that can be constructed out of the GFF. The problem of estimation of Green-type covariance is studied, and autocovariance functions of a continuous Gaussian free field are defined on an LQG sphere. Their estimators are proposed within a functional data analysis context and study their asymptotics, including their computational aspects. In an application, data on sea surface temperature anomalies (temperature and salinity of the upper 2000 m of the ocean) is studied and recorded by Argo floats. In conclusion, extensions are discussed in many areas of studying spherical random fields and their relationship to probability and random geometry in a functional context, such as Dirichlet and Neumann GFF, random combinatorial objects, and random hyperbolic surfaces on Teichmüller space.

"Composite empirical likelihood for histogram-valued variables"

Continuing advances in measurement technology and information storage are leading to the creation of increasingly large and complex datasets, which inevitably brings new inferential challenges. Symbolic data analysis (SDA), a relatively new field in statistics, has been developed as one way of addressing these issues. It aggregates the micro-data into a much smaller number of distributional summaries, each summarising a portion of the larger dataset (Billard and Diday, 2006; Le-Rademacher and Billard, 2013; Dias and Brito, 2015). Likelihood-based methods for distributional data were introduced by Le-Rademacher and Billard (2010). More recently, Zhang et al. (2019) and Beranger et al. (2018) developed likelihood functions for observed random rectangles and histograms that directly accounts for the process of constructing the symbols from the underlying micro-data. Whitaker et al. (2019) address this problem by extending the likelihood-based approach of Beranger et al. (2018) to the composite-likelihood setting. Finally, Emilion (2018; 2019) analyzes the problem with Dirichlet models, using nonparametric Bayesian extensions to Dirichlet Process Mixtures. In our contribution we extend existing and developing approaches by considering nonparametric, empirical likelihood which seems natural to apply to aggregated data and avoids distributional assumptions. In order to avoid computational issues we follow Jaeger and Lazar (2015) and Whitaker et al. (2019) and construct composite empirical likelihood estimators for histogram valued variables. We explore the asymptotic behaviour of histogram composite empirical MLE and its attainment of asymptotically consistent approximation of the standard composite empirical likelihood estimator and provide exploration of its variance consistency. Finally, we provide simulation studies, varying the number of bins, observations, histograms and marginal histogram dimensions, as well as applications using SHARE (Survey of Health, Ageing and Retirement in Europe) and data on COVID-19 pandemic from Johns Hopkins University.

"Combinatorial regression in simplicial complexes", joint with Miroslav Verbič

Regression analysis with compositional data has so far been limited to regressions on a single simplex space. We extend this to regression in a simplicial complex (as a family set of simplicial objects), developing a novel regression perspective, labelled combinatorial regression, based on combining ntuplets of sampling units into groups and treating them on a simplicial complex (Korte, Lovasz and Schrader, 1991) as the regression sample space. The novel perspective is estimated in two stages: in the first (estimating initial regression output), combining Multivariate Distance Matrix Regression (McArdle and Anderson, 2001) and Plackett-Luce approaches, and in the second extending random walk perspectives on simplicial complexes with the recent regression simplicial complex (neural) network perspective (Firouzi et al., 2020). It allows extensive number of perspectives in the analysis of, for example, triplets, quadruplets or quintuplets (or any n-tuplet) and using as measure of disparity between the units (to construct regressors) different distance and/or divergence measures. It also allows applications to very small datasets as the number of units in the new model can be expressed in terms of generalized factorial products (Dedekind numbers) of units of original sample. Computational issues, prone to statistical and probabilistic work on simplicial complexes are solved using approaches of computational topology. In this article (referring to short version in Srakar and Verbic, 2021), we provide the analysis of new approach for different n-tuple combinations using Jensen-Shannon and generalized Jensen-Shannon divergence measures, provide the asymptotic limits of the approach and explore its properties in a Monte Carlo simulation study. In a short application we present analysis of sessile hard-substrate marine organisms image data from Italian coast areas which allows to explore the new approach in relative abundance data setting.

"Wavelet regression for compositional data", joint with Tim Fry

Regression for symplectic, i.e. compositional data has so far been largely considered only from a parametric point of view. Some work adapted nonparametric regression to non-Euclidean manifolds (Di Marzio et al., 2013 for circular data; Di Marzio et al., 2014 for spherical data). In recent articles, Di Marzio, Panzera and Venieri (2015) extended this to nonparametric situations, introducing local constant and local linear smoothing for regression with compositional data; and Machalová, Hron and Talská (2019) to simplicial (compositional functional) splines. In our analysis, we extend their analysis to wavelet regressions, constructing wavelet transforms referring to previous work of Lounsbery et al. (1997), Yu et al. (1997) and Dey and Wang (2015) which constructed wavelets specifically for triangles, deriving father and motherwavelets using Legendre polynomial based sequential approach to orthogonalization. We extend their perspective for wavelet construction in any topological and symplectic space, enabling its usage for modelling compositional data of any dimension. To derive the wavelet regression estimator we refer to previous work studying Bayesian approach to regression with compositional data which used simple hierarchical Bayes models (Shimizu et al., 2015; van den Merwe, 2018) which we extend to multivariate wavelet (specifically multivariate Laplace and multivariate Gaussian) priors, evaluating the fit with the recent Stein-based procedure of Ghaderinezhad and Ley (2018). We extend this further to nonparametric Bayes perspective using often quoted approach for compositional data using random Bernstein polynomials (Barrientos, Jara and Quintana, 2014). The new regression estimators are derived for all three cases: simplicial-real; simplicial-simplicial; and real-simplicial regression. Performance of the estimators is studied in delta-type asymptotic analysis and simulation study, which show that in most cases the new estimators outperform more commonly used parametric and nonparametric ones in efficiency. We apply the findings to two economic datasets for inference on income inequality and international trade.

"Instrumental variable estimation in compositional regression"

In an increasing number of empirical studies, the dimensionality (i.e. size of the parameter space) can be very large. In functional data analysis, appropriate setting to analyze such problems is functional linear model in which the covariates belong to Hilbert spaces. Florens and Van Bellegem (2015) extended this to case where covariates are endogenous (functional instrumental variables/FIV). Our paper extends this to compositional (with extensions also to histogram, i.e. empirical distributional) data setting where either or both independent and dependent variables are compositional. We show there exist two ways of deriving compositional IV’s, one which follows basic derivation of compositional regressions with isometric log-ratio transform and Chesher et al. (2013)’s instrumental variable model of multiple discrete choice; and another deriving from the recent literature on compositional functional data in Bayes spaces (based on compositional splines, for more see e.g. Machalova et al., 2021). We show that estimation in second case resembles the original Florens and van Bellegem one and leads to an ill-posed inverse problem with a data-dependent operator. We use and extend the notion of instrument strength to compositional setting and discuss generalized versions of the estimators when the problem is premultiplied by an instrument-dependent operator. We establish appropriate functional CLT’s and study the finite sample performance in a Monte Carlo simulation setting. Our application studies the relationship between long term care provision to relatives and paid work, using recent time use survey from Survey of Health, Ageing and Retirement in Europe (SHARE). In conclusion we discuss extensions to other causal inference approaches in the line of recent distributional synthetic control approach.

3) BAYESIAN STATISTICS:

"Pitman-Yor mixtures for BART: A novel nonparametric prior for Bayesian causal inference"

Recently authors pointed to Bayesian causal inference as important pathway for future development of causal methodologies (Li, 2022; Imbens, 2022). Bayesian additive regression trees (BART) perspective has been developed by Chipman et al. (2010) and popularized in recent years in its usage in regression and causal inference problems (for example Tan and Roy, 2019; Hahn, Murray and Carvalho, 2020). Commonly, BART uses a specific regularization prior, sometimes combined with Gaussian, Dirichlet, Dirichlet Process Mixture (DPM) and semiparametric perspectives (Tan and Roy, 2019). Despite the success of BART, there has been a growing number of papers that point out its limitations. Hahn, Murray and Carvalho have developed Bayesian Causal Forests (BCF - Hahn, Murray and Carvalho, 2020) as a novel regularization approach for nonlinear models geared specifically towards situations with small effect sizes, heterogeneous effects, and strong confounding, to improve on the earlier BART perspective. We develop a novel nonparametric regularization prior for BART based on Pitman-Yor Mixture (PYM) partition-based process which has to date to our knowledge rarely been used in causal inference (but is suggested for classification and mixture modelling). Pitman–Yor process mixtures (Ishwaran and James, 2001; 2003) are a generalization of DPMs based on the Pitman–Yor process, also known as the two-parameter Poisson–Dirichlet process. Our novel BART perspective is studied in more detail for several different causal perspectives: regression discontinuity design; causal maximally oriented partially directed acyclic graph (MPDAG); direct causal clause; and causal mediation. Our results on simulated and real data confirm improved properties as compared to earlier BART priors and the performance is similar to the Hahn et al. BCF model, but improved in the presence of strong confounding. Performance of the prior is different for causal mediation and we provide suggestions for future work in this perspective. We address computational issues by using importance sampling with the integrated nested Laplace approximation (Outzen Berild et al., 2021). In conclusion we discuss extensions to endogeneity corrections in the line of BCF-IV approach of Bargagli Stoffi et al. (2022) and extensions to Single World Intervention Graph perspective (Richardson and Robins, 2013; 2014).

"Bayesian probabilistic numerical method with product-Whittle-Matérn-Yasuda kernel for Rosen's hedonic regression"

Hedonic regression has featured an extensive amount of applications. Originally, following Sherwin Rosen’s contribution (Rosen, 1974) it is estimated in a spatial equilibrium context in two stages which leads to a nonlinear Euler ordinary or partial differential equation framework. To date, its Bayesian extensions have not adequately addressed features of its original proposal. We develop an approximate Bayesian probabilistic numerical method with product-Whittle-Matern-Yasuda kernel, which extends literature in several aspects and is able to address different features of the original proposal: it is developed for nonlinear differential equations, is based on spatial kernels and a Gaussian process regression framework, and is applicable to any hedonic regression specification. Usage of product-Whittle-Matern-Yasuda spatial kernel addresses computational issues of the approach. We use Bernstein-von Mises type asymptotic theorems to assess performance of the approach (Frazier et al., 2018) and study asymptotic normality of the posterior mean and Bayesian consistency. Different possibilities of Markov chain Monte Carlo (MCMC) and Quasi-MC sampling schemes are considered. Using Bayesian model comparison approaches, we compare its performance to several parametric and nonparametric Bayesian priors and apply it to simulated and real data examples from the areas of real estate and retail. We study its properties for different possible regression specifications and discuss prior elicitation possibilities (Mikkola et al., 2023). Extensions of the approach discuss other differential equation numerical resolvers, extension to stochastic Euler differential equation, extensions to other hyperbolic differential equation contexts and implicit price frameworks, and deep Gaussian process extensions.

"Approximate Bayesian algorithm for tensor robust principal component analysis"

Recently proposed Tensor Robust Principal Component Analysis (TRPCA) aims to exactly recover the low-rank and sparse components from their sum, extending the earlier Low-Rank Tensor Completion model representation. We construct a Bayesian approximate inference algorithm for TRPCA, based on regression adjustment methods suggested in the literature to correct for high-dimensional nature of the problem and sequential Monte Carlo approach with adaptive weights. Our results are compared to previous studies which used variational Bayes inference for matrix and tensor completion.

"Approximate Bayesian Computation for high-dimensional data using Itakura-Saito divergence"

Approximate Bayesian Computation (ABC) is a family of methods for approximate inference when likelihoods are impossible or impractical to evaluate numerically but simulating datasets from the model of interest is straightforward. Despite the recent interest in Approximate Bayesian Computation, high-dimensional data still remain a bottleneck. We extend the conditional density estimation (ABC-CDE) framework of Izbicki et al. (2018), intended for nonparametric conditional density estimation using ABC methods using functional Bregman divergence between posterior distributions as a measure of the posterior surrogate loss (Frigyik, 2008; Goh and Dey, 2014). Functional Bregman divergence is a generalized Bayesian model diagnostic tool nesting many divergence measures. We limit to special type of functional Bregman divergence, Itakura-Saito, but study also the general case. Specifically, we study the new approach for functional and spatiotemporal epidemiological data, comparing the performance of several high-dimensional epidemiological models estimated by the novel ABC approach: a nonseasonal homogeneous Susceptible-Exposed-Infectious-Recovered (SEIR) model; a homogeneous SEIR model with seasonality in transmission; an age-structured SEIR model; a multiplex network-based model; and an agent-based simulator. In conclusion, we discuss extensions to other functional Bregman divergence measures, Bayesian synthetic likelihood and variational approaches.

"Assessing Bayesian priors for hedonic regression", joint with Marilena Vecco

To estimate art returns, two methods have been commonly used: the repeat sales method and the hedonic pricing model. The hedonic pricing model (Rosen, 1974), is estimated in regression context where dependent variable is the (logarithm of the) price of each painting and the independent variables are each one of its characteristics or attributes. Despite different regression perspectives used to estimate art returns, in particular in hedonic regression perspective, possibilities of Bayesian modelling have been almost completely neglected. This situation persists despite clear advantages of Bayesian methods over their frequentist counterparts in many modelling situations. In our contribution we present a comparison of different Bayesian priors (parametric and nonparametric) for hedonic regression modelling of art returns, using a sample dataset of surrealist paintings. We compare the priors using common Bayesian model comparison approaches and find that best fit is provided by the general nonparametric Bayesian additive regression trees (BART) prior of Chipman et al. (2010). We provide an explanation and suggest possibilities for future modelling of art prices. The article has relevance for hedonic regression models in general where Bayesian possibilities have not been studied adequately so far.

4) ECONOMETRICS WITH A FOCUS ON CAUSAL INFERENCE AND MACHINE LEARNING:

"Linear regression with Bouchaud's stochastic aging"

Aging is an out-of-equilibrium physical phenomenon gaining interest in physics and mathematics. Bouchaud has proposed the following toy model to study the phenomenon (Bouchaud, 1998). Let $ G = (\mathcal{V,E)}$ be a graph, and let $ E = \{ E_{i}\}_{i\in\mathcal{V}}$ be the collection of i.i.d. random variables indexed by vertices of this graph. The continuous-time Markov chain $ X(t)$ is considered with state space $\mathcal{V}$. The transition rates $ w_{ij}$ are defined by $ w_{ij} = \nu \exp\left(-\beta\left(\left(1-a\right)E_{i}-aE_{j}\right)\right)$. Proving an aging result consists in finding a two-point function $ F(t_{w},t_{w}+t)$ such that a nontrivial limit $ \lim_{t\rightarrow \infty ,\frac{t}{t_{w}} = \theta }F(t_{w},t_{w}+t = F(\theta )$ exists. His model has later been explored and extended to spin glasses in probability literature (Rinn et al., 2000; Dembo et al., 2001; Fontes et al., 2002; Ben Arous, 2002; Ben Arous et al., 2001; 2002; 2003; 2006). This contribution introduces aging in a linear regression model for cross-section and panel data. Regression specification based on regression regularization is proposed as well as maximum likelihood-based estimation and inference. Asymptotic results rely on earlier literature on trap models for random walks. In an application, the effects of childhood book reading and diseases on the health status of old-age individuals using retrospective panel models are studied. Including aging in regression models is novel and opens many unexplored possibilities. As Bouchaud's trap models are also underexplored in probability theory, this promises interesting avenues for research in econometrics and probability, with significant practical applications for regression modelling in the economics of health and aging.

"Causal inference for distributional treatments"

We build on contributions from Gunsilius (2020) on distributional synthetic controls and Pollmann (2022) who develops spatial treatment estimators in building distributional treatments, i.e. causal variables of distributional nature. We develop explicit formulas for average treatment effect, average treatment effect for compliers and local average treatment effect. We study the performance of our approach in asymptotic analysis and simulation study, with an application using SHARE data. We transform the problem in three regression contexts for distributional data. In Billard-Diday (2006) formula for the regression model is just a simple linear regression for histograms. In Dias and Brito (2011) model each variable is represented by two quantile functions, original and inverse symmetrical one. Irpino and Verde (2012) utilize a decomposition of squared Wasserstein distance in two components, average value of histogram variable and its centered quantile function. We use toolkit from asymptotic analysis, in particular Donsker type central limit theorems, as well as Monte Carlo simulation to analyze the performance of the novel approach. In Billard-Diday model, the analysis reduces to the basic OLS algebra from linear regressions for numeric data. In Dias-Brito model estimation transforms into a constrained OLS problem and the optimization problem is well defined and has optimal solutions. In Irpino-Verde model we derive an explicit form for the estimator and show it is consistent and asymptotically normal. We apply the approach to causal relationship between health indicators and decision to retire, using longitudinal dataset of Survey of Health, Ageing and Retirement in Europe (SHARE), Waves 1-6. Our analysis develops one of first causal inference estimators for distributional data. The results allow richer assessment of causal effects for any causal analysis in future as many problems in health field and economics in general are distributional by nature. Compared to other recent articles developing causal estimators for distributional data our article takes into account the problem of negative weights in quantile function regressions - other contributions might be biased in this aspect.

"Asymptotic properties of multiple indicators multiple causes model in cross-sectional and time series perspective", joint with Miroslav Verbič

Multiple Indicators Multiple Causes (MIMIC) model is a latent variable approach to confirm the influence of a set of exogenous causal variables on the latent variable, and also the effect of the latent variable on observed indicator variables. It is most commonly used in economics for modelling the shadow economy. Its mathematical structure relies on original contribution of Jöreskog and Goldberger (1975), but its asymptotic properties have to date not been studied. We study its maximum likelihood estimation perspective by transforming it into the general analysis of covariance structures (Jöreskog, 1970; 1973) and analyze separately the cases for cross-sectional, dynamic (Engle et al., Aigner et al., 1988) and error-correction (Buehn and Schneider, 2008) specification. Consistency, efficiency and asymptotic normality for the maximum likelihood estimation are developed for all three cases. In conclusion, we discuss asymptotics for factor-analytic and econometric estimation, originally discussed by Jöreskog and Goldberger and generalizability to other latent variable approaches and causal schemes.

"Latent variable causal modelling with instrumental variables in static and error correction framework: A 2SLS-MIMIC approach", joint with Marilena Vecco and Miroslav Verbič

Economic aggregates such as shadow economy and hidden job market can be modeled as latent constructs. To model them, different latent variable estimation approaches have been proposed in the literature. In our contribution we apply novel latent variable estimators in the instrumental variable framework to model a highly disputed problem in museum management, estimation of the extent of deaccessioning, i.e. selling of museum artworks. We develop a Grossman-Hart type principal-agent model able to prove and extend the Jensen’s conjecture on agency costs of free cash flow for nonprofit firms. We use multiple indicators and multiple causes (MIMIC) modelling framework deriving from the latent variable modelling literature and develop three new estimators to account for endogeneity problems, frequently present in such framework. Our estimators are valid for static and error correction situations and are based on noniterative Bollen’s 2SLS instrumental variable estimator (and its extension to dynamic generalized instrumental variable framework) and Jöreskog’s analysis of covariance structures. We show the new estimators are consistent and asymptotically normal and explore this in a Monte Carlo simulation study. Using the error corrected estimator we model the relative extent of deaccessioning and study its features. Using various microeconometric models, we demonstrate its dependence upon the size of the museum and macroeconomic conditions. We find that deaccessioning has not risen in the US in times of the financial crisis, which is an interesting result that has to be explored in further analysis.

"Semiparametric dynamic panel causal mediation with iterative kernel and Bayesian dynamic estimation", joint with Boris Majcen and Tjaša Bartolj

The paper addresses causal relationship between long term care and health care utilization of the elderly. The expansion of long-term care (LTC) may improve health system efficiency by reducing hospitalisations, and pave the way for the implementation of health and social care coordination plans. We draw upon the longitudinal evidence from Survey of Health, Ageing and Retirement in Europe (SHARE), Waves 4-8, to derive causal estimates of the effects of receiving different types of LTC on health care utilization. We analyze the causal problem with health indicators as mediators. To solve for multiple reverse causality we utilize cross-lagged panel models, a form of longitudinal mediation analysis. As latter are based on strong Gaussianity assumptions we construct two novel estimators for cross-lagged panel models: semiparametric, based on iterative kernel estimation of dynamic panel mediation using semiparametric sieves based on Laguerre polynomials as consistent initial estimators (Su and Lu, 2013; Kreiss and Van Keilegom, 2022) and a Bayesian semiparametric, based on autoregressive Dirichlet process mixtures for longitudinal data (Quintana et al., 2016) used in combination with a dynamic Bayesian modelling approach of Kim and colleagues (2019). We provide results on the asymptotic behaviour of the estimators and several simulation experiments. Empirical results confirm significant effects of LTC provision on reducing health care utilization and we provide estimates of the reduction of costs in several Central and Eastern European countries (e.g. Slovenia, Croatia, Hungary, Czechia) health care systems due to proposed measures in long-term care. The article provides a novel methodological possibility to analyze relationship between long term care and health care, development of two novel estimators for research in longitudinal causal mediation and important information to decision makers in different countries for their future measures in long term care and health care. Extensions to fixed and random effects linear panel dynamic causal mediation as well as nonlinear models are considered.

"Fuzzy Wald ratio spatial and spatio-temporal difference-in-differences and changes-in-changes", joint with Marilena Vecco, Renata Erker and Miroslav Verbič

Difference-in-differences (DiD) literature is a fast growing field in econometrics. Topics such as presence of serial correlation, aggregating data, clustered standard errors, arbitrary covariance structures, parallel growth assumption, synthetic control, multiple and continuous treatments and staggered treatment adoption have been subject to recent research. In previous contributions DiD has been extended to spatial data using Hadamard product conditioned calculus. We extend this in treatment effect with network interference context by controlling for violated stable unit treatment values assumption (SUTVA), inherent for such analysis (but seldom controlled so far) and in a spatiotemporal autoregressive setting. We develop a time-corrected Wald ratio DiD estimator based on fuzzy DiD approach with extensions to changes-in-changes estimation. We provide asymptotic analysis using moment-based approach for graphon stochastic processes, and a Monte Carlo simulation analysis. In an application, we study causal effects of the yearly Venice carnival, being able to isolate the effect respective to other competing large events in Venice in the studied period. The application uses three stage approach of using ARIMA models in the first stage, Frechet mean and median based derivation of spatial matrices in the second stage, and our new estimator in the final third stage. In conclusion, we consider extensions using spillover double robust DiD and Bayesian approaches, as well as tests for parallel trends (and corrections for its violation, see Rambachan and Roth, 2021; Roth and Sant'Anna, 2023) in our mathematical context.

"Panel nonparametric regression with tensorial long short-term memory recurrent neural networks"

Neural networks and deep learning are current state-of-the-art in machine learning and artificial intelligence. Seldom have they been applied to panel data estimation, in particular as replacement for nonparametric fixed or random effects specification. This paper builds on a previous contribution by Crane-Droesch (2017) upgrading his approach using tensorial long short-term memory recurrent neural networks based on tensor Tucker decomposition. Neural networks are fitted to panel nonparametric fixed effects specification. Loss function and backpropagation are defined in accordance with Kolda and Bader (2009). The model is estimated by a variant of minibatch gradient descent using approximate inference. Model performance is compared to tensorial gated recurrent units. To evaluate the novel estimator we use non-asymptotic approaches for neural network estimation and Monte Carlo type simulation evidence. Extension of the approach to nonparametric random effects specification is discussed. The approach is applied to the prediction of agricultural yields from weather data, relevant for short-term economic forecasts as well as for longer-range climate change impact assessment. The model is general and could be used for high-dimensional regression adjustment, general nonparametric regression problems, heterogeneous treatment effect estimation, and forecasting with longitudinal data.

"Robust Malliavin-Stein confidence intervals for multiple shrinkage problems"

Inference for shrinkage problems has been claimed by Bruce E. Hansen (2016) as a key open problem of shrinkage models in econometrics with very few existing econometric contributions. In recent articles, Armstrong and Kolesár (2019) and Armstrong, Kolesár and Plagborg-Møller (2020) constructed robust confidence intervals through empirical Bayes methods and approximate moment condition (GMM-type) models. Our article goes into another direction suggested already by Hansen and constructs robust confidence intervals for shrinkage problems using Stein and Malliavin-Stein approaches (Stein, 1956; James and Stein, 1961; Nualart and Peccati, 2005; Nourdin and Peccati, 2009; Beran, 2010). The intervals are centered at the usual James-Stein estimator, but use a critical value accounting for shrinkage. We construct Stein confidence balls following Beran (2010) and extend this using Malliavin calculus as suggested previously in Privault and Réveillac (2008) and recently in Moustaaid and Ouassou (2021). Furthermore, we extend the approach to multiple shrinkage and multiple affine shrinkage (Beran, 2008) by applying Malliavin calculus to the analysis of bounded variation shrinkage (Beran and Dümbgen, 1998). The coverage of our confidence intervals is improved as compared to Armstrong, Kolesár and Plagborg-Møller's empirical Bayes coverage regardless of the means distribution. But they significantly outperform them for multiple and multiple affine shrinkage in particular when the dimension grows large. While the article relies on parametric assumptions (Gaussian processes), extensions to nonparametric and nonparametric Bayes (again contrasting the EB approach of AKPM) directions are suggested, as well as generalization to adaptive symmetric linear estimators. Our empirical applications consider structural changes in a large dynamic factor model for the Eurozone and relationship of mobility in US neighbourhoods on Covid-19 prevalence.

"Granger mediation with multiple multilevel and functional mediators", joint with Miroslav Verbič

Mediation analysis is a popular statistical approach for many social and scientific studies. It aims to assess the role of an intermediate variable or mediator sitting in the pathway from a treatment variable to an outcome variable. Causal mediation analysis, widely studied in the statistical literature, was developed to infer the causal effects in mediation models, see a review in Imai et al. (2010). Most of the existing methods, however, cannot be applied to time series data, because clearly the independence assumption is violated. In this paper, we focus on the time series data for variables which can be multilevel or functional. We extend the analysis of Zhao and Luo (2017) which proposed a Granger-type mediation analysis for variables which are time series and can be multilevel, to multiple mediators setting. Zhao and Luo proposed a framework allowing modeling time series for all the three variables in mediation analysis. We generalize their approach for multiple mediators, deriving conditional likelihood expressions for estimators and their asymptotic properties. Our approach is based on interventional indirect effect approach of VanderWeele et al. (2014) and Wei Loh et al. (2019). We derive also conditional likelihood estimates for mixed (i.e. multilevel) models and when either the treatment, outcome and/or mediator are continuous functions – all of this for the framework of multiple mediators. To derive asymptotic bounds and behaviour, we follow Caponera and Marinuci (2019) and establish rates of consistency and a quantitative version of the Central Limit Theorem in Wasserstein distance by means of Malliavin-Stein methods (Nualart and Peccati, 2005; Nourdin and Peccati, 2008; 2009). We demonstrate the performance of new estimators in a simulation study and two short applications: to the estimation of direct and indirect effects of education on economic growth (Phoong and Phoong, 2018); and to the estimation of effects of direct and indirect effects of public funding cuts on organizational behaviour.