Raffaello Seri | Università degli Studi dell'Insubria (original) (raw)

Papers by Raffaello Seri

Research paper thumbnail of The problem of determining the number of runs in your simulation

Research paper thumbnail of Power analysis for agent-based modeling

Research paper thumbnail of On the quest for defining organisational plasticity: a community modelling experiment

Evidence-based HRM: a Global Forum for Empirical Scholarship, 2020

PurposeThis viewpoint article is concerned with an attempt to advance organisational plasticity (... more PurposeThis viewpoint article is concerned with an attempt to advance organisational plasticity (OP) modelling concepts by using a novel community modelling framework (PhiloLab) from the social simulation community to drive the process of idea generation. In addition, the authors want to feed back their experience with PhiloLab as they believe that this way of idea generation could also be of interest to the wider evidence-based human resource management (EBHRM) community.Design/methodology/approachThe authors used some workshop sessions to brainstorm new conceptual ideas in a structured and efficient way with a multidisciplinary group of 14 (mainly academic) participants using PhiloLab. This is a tool from the social simulation community, which stimulates and formally supports discussions about philosophical questions of future societal models by means of developing conceptual agent-based simulation models. This was followed by an analysis of the qualitative data gathered during th...

Research paper thumbnail of Settore bancario e frontiere di efficienza

From the 1980s the architecture of the Italian banking system underwent a process of reform aimed... more From the 1980s the architecture of the Italian banking system underwent a process of reform aimed at introducing higher degrees of competition among intermediaries and at enhancing their efficiency, in order to enable it to comply with the European integration process and to face the new challenges of rapidly globalizing markets. While in the 1980s the banking regulation slowly adapted to European rules and to a more liberalized contest, in the 1990s privatizations, liberalizations, and higher competition fueled concentration in the banking system. The paper studies these transformations using a dataset gathering balance sheet and loss-and-profit data, at firm level, for a wide number of banks, covering almost the entire system, between 1979 and 2000. The analysis is based on nonparametric estimation of the technology set through the DEA (Data Envelopment Analysis) and tries to determine the effects of new regulation and institutional frameworks on the efficiency of the banking system and of individual banks

Research paper thumbnail of Statistical properties of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>-adic diaphonies

Mathematics of Computation, 2016

The aim of this paper is to derive the asymptotic statistical properties of a class of discrepanc... more The aim of this paper is to derive the asymptotic statistical properties of a class of discrepancies on the unit hypercube called b-adic diaphonies. They have been introduced to evaluate the equidistribution of quasi-Monte Carlo sequences on the unit hypercube. We consider their properties when applied to a sample of independent and uniformly distributed random points. We show that the limiting distribution of the statistic is an infinite weighted sum of chi-squared random variables, whose weights can be explicitly characterized and computed. We also describe the rate of convergence of the finite-sample distribution to the asymptotic one and show that this is much faster than in the classical Berry-Esséen bound. Then, we consider in detail the approximation of the asymptotic distribution through two truncations of the original infinite weighted sum, and we provide explicit and tight bounds for the truncation error. Numerical results illustrate the findings of the paper, and an empirical example shows the relevance of the results in applications.

Research paper thumbnail of A non-recursive formula for the higher derivatives of the Hurwitz zeta function

Journal of Mathematical Analysis and Applications, 2015

In this paper, we provide an asymptotic formula for the higher derivatives of the Hurwitz zeta fu... more In this paper, we provide an asymptotic formula for the higher derivatives of the Hurwitz zeta function with respect to its first argument that does not need recurrences. As a by-product, we correct some formulas that have appeared in the literature.

Research paper thumbnail of A Tight Bound on the Distance Between a Noncentral Chi Square and a Normal Distribution

IEEE Communications Letters, 2015

We provide a nonasymptotic bound on the distance between a noncentral chi square distribution and... more We provide a nonasymptotic bound on the distance between a noncentral chi square distribution and a normal approximation. It improves on both the classical Berry-Esséen bound and previous distances derived specifically for this situation. First, the bound is nonasymptotic and provides an upper limit for the real distance. Second, the bound has the correct rate of decrease and even the correct leading constant when either the number of degrees of freedom or the noncentrality parameter (or both) diverge to infinity. The bound is applied to some probabilities arising in energy detection and Rician fading.

Research paper thumbnail of Analytic Hierarchy Process, A Psychometric Approach

2 Satisficing Games for Multiple-Participant …, 2001

... at DR Insua R. Jiroušek Rey Juan Carlos University Academy of Sciences Spain Czech Republic d... more ... at DR Insua R. Jiroušek Rey Juan Carlos University Academy of Sciences Spain Czech Republic drios@ escet. urjc. es radim@ utia. cas. cz M. Kárný J. Kracık Academy of Sciences Academy of Sciences Czech Republic Czech Republic school@ utia. cas. cz kracik@ utia. cas. ...

Research paper thumbnail of An Analysis of Experienced Discrimination in Italy. The Case of Home Mortgages

files.uwlax.edu

The paper addresses the issue of home mortgage discrimination in one of the richest areas of nort... more The paper addresses the issue of home mortgage discrimination in one of the richest areas of northern Italy. Since no official data is available, workers of the Varese city hospital have been interviewed to figure out if they have asked for a mortgage and if the ...

Research paper thumbnail of Stochastic Boundedness In Biological Models

… modelling & computing in biology and …, 2003

Stochastic Boundedness in Biological Models Raffaello Seri1 and Christine Choirat2 1 CREST-LFA, Ti... more Stochastic Boundedness in Biological Models Raffaello Seri1 and Christine Choirat2 1 CREST-LFA, Timbre J320, 15 bd Gabriel Peri, 92245 Malakoff Cedex, France http://www. crest. fr/pageperso/lfa/seri/seri. htm seri@ ensae. fr 2 Centre de Recherche Viabilite, ...

Research paper thumbnail of Generic Consistency for Approximate Stochastic Programming and Statistical Problems

SIAM Journal on Optimization, 2019

In stochastic programming, statistics, or econometrics, the aim is in general the optimization of... more In stochastic programming, statistics, or econometrics, the aim is in general the optimization of a criterion function that depends on a decision variable theta\thetatheta and reads as an expectation with ...

Research paper thumbnail of Randomness, Emergence and Causation: A Historical Perspective of Simulation in the Social Sciences

Complexity and Emergence

This book series features volumes composed of selected contributions from workshops and conferenc... more This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the

Research paper thumbnail of Nonparametric Estimation and Inference in Psychological and Economic Experiments

arXiv: Econometrics, 2019

The goal of this paper is to provide some statistical tools for nonparametric estimation and infe... more The goal of this paper is to provide some statistical tools for nonparametric estimation and inference in psychological and economic experiments. We consider a framework in which a quantity of interest depends on some primitives through an unknown function fff. An estimator of this unknown function can be obtained from a controlled experiment in which nnn subjects are gathered, and a vector of stimuli is administered to each subject who provides a set of TTT responses. We propose to estimate fff nonparametrically using the method of sieves. We provide conditions for consistency of this estimator when either nnn or TTT or both diverge to infinity, and when the answers of each subject are correlated and this correlation differs across subjects. We further demonstrate that the rate of convergence depends upon the covariance structure of the error term taken across individuals. A convergence rate is also obtained for derivatives. These results allow us to derive the optimal divergence r...

Research paper thumbnail of How many times should my simulation run?': Power analysis for agent-based modeling

Research paper thumbnail of A complex adaptive approach to the gender pay gap

Research paper thumbnail of Nonparametric Estimation and Inference in Economic and Psychological Experiments

arXiv: Econometrics, 2019

The goal of this paper is to provide some tools for nonparametric estimation and inference in psy... more The goal of this paper is to provide some tools for nonparametric estimation and inference in psychological and economic experiments. We consider an experimental framework in which each of nnnsubjects provides TTT responses to a vector of TTT stimuli. We propose to estimate the unknown function fff linking stimuli to responses through a nonparametric sieve estimator. We give conditions for consistency when either nnnor TTTor both diverge. The rate of convergence depends upon the error covariance structure, that is allowed to differ across subjects. With these results we derive the optimal divergence rate of the dimension of the sieve basis with both nnn and TTT. We provide guidance about the optimal balance between the number of subjects and questions in a laboratory experiment and argue that a large nnnis often better than a large TTT. We derive conditions for asymptotic normality of functionals of the estimator of TTTand apply them to obtain the asymptotic distribution of the Wald...

Research paper thumbnail of Asymptotic Properties of the Plug-in Estimator of the Discrete Entropy Under Dependence

IEEE Transactions on Information Theory, 2021

We consider the estimation of the entropy of a discretely-supported time series through a plug-in... more We consider the estimation of the entropy of a discretely-supported time series through a plug-in estimator. We provide a correction of the bias and we study the asymptotic properties of the estimator. We show that the widely-used correction proposed by Roulston (1999) is incorrect as it does not remove the Oleft(N−1right)O\left ({N^{-1}}\right)Oleft(N1right) part of the bias while ours does. We provide the asymptotic distribution and we show that it differs when the values taken by the marginal distribution of the process are equiprobable (a situation that we call degeneracy) and when they are not. We introduce estimators of the bias, the variance and the distribution under degeneracy and we study the estimation error. Finally, we propose a goodness-of-fit test based on entropy and give two motivations for it. The theoretical results are supported by specific numerical examples.

Research paper thumbnail of Model calibration and validation via confidence sets

Econometrics and Statistics, 2020

The issues of calibrating and validating a theoretical model are considered, when it is required ... more The issues of calibrating and validating a theoretical model are considered, when it is required to select the parameters that better approximate the data among a finite number of alternatives. Based on a user-defined loss function, Model Confidence Sets are proposed as a tool to restrict the number of plausible alternatives, and measure the uncertainty associated to the preferred model. Furthermore, an asymptotically exact logarithmic approximation of the probability of choosing a model via a multivariate rate function is suggested. A simple numerical procedure is outlined for the computation of the latter and it is shown that the procedure yields results consistent with Model Confidence Sets. The illustration and implementation of the proposed approach is showcased in a model of inquisitiveness in ad hoc teams, relevant for bounded rationality and organizational research. 1

Research paper thumbnail of What are we estimating when we fit Stevens’ power law?

Journal of Mathematical Psychology, 2016

h i g h l i g h t s • Estimates in Stevens' power laws often display sensitivities to experimenta... more h i g h l i g h t s • Estimates in Stevens' power laws often display sensitivities to experimental design. • These so-called 'contextual effects' concern range, location and averaging. • This paper links them with the separable representation model of Luce and Narens. • Theoretical results are illustrated using data from papers of R. Duncan Luce.

Research paper thumbnail of Asymptotic Distributions of Covering and Separation Measures on the Hypersphere

Discrete & Computational Geometry

We consider measures of covering and separation that are expressed through maxima and minima of d... more We consider measures of covering and separation that are expressed through maxima and minima of distances between points of an hypersphere. We investigate the behavior of these measures when applied to a sample of independent and uniformly distributed points. In particular, we derive their asymptotic distributions when the number of points diverges. These results can be useful as a benchmark against which deterministic point sets can be evaluated. Whenever possible, we supplement the rigorous derivation of these limiting distributions with some heuristic reasonings based on extreme value theory. As a by-product, we provide a proof for a conjecture on the hole radius associated to a facet of the convex hull of points distributed on the hypersphere.

Research paper thumbnail of The problem of determining the number of runs in your simulation

Research paper thumbnail of Power analysis for agent-based modeling

Research paper thumbnail of On the quest for defining organisational plasticity: a community modelling experiment

Evidence-based HRM: a Global Forum for Empirical Scholarship, 2020

PurposeThis viewpoint article is concerned with an attempt to advance organisational plasticity (... more PurposeThis viewpoint article is concerned with an attempt to advance organisational plasticity (OP) modelling concepts by using a novel community modelling framework (PhiloLab) from the social simulation community to drive the process of idea generation. In addition, the authors want to feed back their experience with PhiloLab as they believe that this way of idea generation could also be of interest to the wider evidence-based human resource management (EBHRM) community.Design/methodology/approachThe authors used some workshop sessions to brainstorm new conceptual ideas in a structured and efficient way with a multidisciplinary group of 14 (mainly academic) participants using PhiloLab. This is a tool from the social simulation community, which stimulates and formally supports discussions about philosophical questions of future societal models by means of developing conceptual agent-based simulation models. This was followed by an analysis of the qualitative data gathered during th...

Research paper thumbnail of Settore bancario e frontiere di efficienza

From the 1980s the architecture of the Italian banking system underwent a process of reform aimed... more From the 1980s the architecture of the Italian banking system underwent a process of reform aimed at introducing higher degrees of competition among intermediaries and at enhancing their efficiency, in order to enable it to comply with the European integration process and to face the new challenges of rapidly globalizing markets. While in the 1980s the banking regulation slowly adapted to European rules and to a more liberalized contest, in the 1990s privatizations, liberalizations, and higher competition fueled concentration in the banking system. The paper studies these transformations using a dataset gathering balance sheet and loss-and-profit data, at firm level, for a wide number of banks, covering almost the entire system, between 1979 and 2000. The analysis is based on nonparametric estimation of the technology set through the DEA (Data Envelopment Analysis) and tries to determine the effects of new regulation and institutional frameworks on the efficiency of the banking system and of individual banks

Research paper thumbnail of Statistical properties of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>-adic diaphonies

Mathematics of Computation, 2016

The aim of this paper is to derive the asymptotic statistical properties of a class of discrepanc... more The aim of this paper is to derive the asymptotic statistical properties of a class of discrepancies on the unit hypercube called b-adic diaphonies. They have been introduced to evaluate the equidistribution of quasi-Monte Carlo sequences on the unit hypercube. We consider their properties when applied to a sample of independent and uniformly distributed random points. We show that the limiting distribution of the statistic is an infinite weighted sum of chi-squared random variables, whose weights can be explicitly characterized and computed. We also describe the rate of convergence of the finite-sample distribution to the asymptotic one and show that this is much faster than in the classical Berry-Esséen bound. Then, we consider in detail the approximation of the asymptotic distribution through two truncations of the original infinite weighted sum, and we provide explicit and tight bounds for the truncation error. Numerical results illustrate the findings of the paper, and an empirical example shows the relevance of the results in applications.

Research paper thumbnail of A non-recursive formula for the higher derivatives of the Hurwitz zeta function

Journal of Mathematical Analysis and Applications, 2015

In this paper, we provide an asymptotic formula for the higher derivatives of the Hurwitz zeta fu... more In this paper, we provide an asymptotic formula for the higher derivatives of the Hurwitz zeta function with respect to its first argument that does not need recurrences. As a by-product, we correct some formulas that have appeared in the literature.

Research paper thumbnail of A Tight Bound on the Distance Between a Noncentral Chi Square and a Normal Distribution

IEEE Communications Letters, 2015

We provide a nonasymptotic bound on the distance between a noncentral chi square distribution and... more We provide a nonasymptotic bound on the distance between a noncentral chi square distribution and a normal approximation. It improves on both the classical Berry-Esséen bound and previous distances derived specifically for this situation. First, the bound is nonasymptotic and provides an upper limit for the real distance. Second, the bound has the correct rate of decrease and even the correct leading constant when either the number of degrees of freedom or the noncentrality parameter (or both) diverge to infinity. The bound is applied to some probabilities arising in energy detection and Rician fading.

Research paper thumbnail of Analytic Hierarchy Process, A Psychometric Approach

2 Satisficing Games for Multiple-Participant …, 2001

... at DR Insua R. Jiroušek Rey Juan Carlos University Academy of Sciences Spain Czech Republic d... more ... at DR Insua R. Jiroušek Rey Juan Carlos University Academy of Sciences Spain Czech Republic drios@ escet. urjc. es radim@ utia. cas. cz M. Kárný J. Kracık Academy of Sciences Academy of Sciences Czech Republic Czech Republic school@ utia. cas. cz kracik@ utia. cas. ...

Research paper thumbnail of An Analysis of Experienced Discrimination in Italy. The Case of Home Mortgages

files.uwlax.edu

The paper addresses the issue of home mortgage discrimination in one of the richest areas of nort... more The paper addresses the issue of home mortgage discrimination in one of the richest areas of northern Italy. Since no official data is available, workers of the Varese city hospital have been interviewed to figure out if they have asked for a mortgage and if the ...

Research paper thumbnail of Stochastic Boundedness In Biological Models

… modelling & computing in biology and …, 2003

Stochastic Boundedness in Biological Models Raffaello Seri1 and Christine Choirat2 1 CREST-LFA, Ti... more Stochastic Boundedness in Biological Models Raffaello Seri1 and Christine Choirat2 1 CREST-LFA, Timbre J320, 15 bd Gabriel Peri, 92245 Malakoff Cedex, France http://www. crest. fr/pageperso/lfa/seri/seri. htm seri@ ensae. fr 2 Centre de Recherche Viabilite, ...

Research paper thumbnail of Generic Consistency for Approximate Stochastic Programming and Statistical Problems

SIAM Journal on Optimization, 2019

In stochastic programming, statistics, or econometrics, the aim is in general the optimization of... more In stochastic programming, statistics, or econometrics, the aim is in general the optimization of a criterion function that depends on a decision variable theta\thetatheta and reads as an expectation with ...

Research paper thumbnail of Randomness, Emergence and Causation: A Historical Perspective of Simulation in the Social Sciences

Complexity and Emergence

This book series features volumes composed of selected contributions from workshops and conferenc... more This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the

Research paper thumbnail of Nonparametric Estimation and Inference in Psychological and Economic Experiments

arXiv: Econometrics, 2019

The goal of this paper is to provide some statistical tools for nonparametric estimation and infe... more The goal of this paper is to provide some statistical tools for nonparametric estimation and inference in psychological and economic experiments. We consider a framework in which a quantity of interest depends on some primitives through an unknown function fff. An estimator of this unknown function can be obtained from a controlled experiment in which nnn subjects are gathered, and a vector of stimuli is administered to each subject who provides a set of TTT responses. We propose to estimate fff nonparametrically using the method of sieves. We provide conditions for consistency of this estimator when either nnn or TTT or both diverge to infinity, and when the answers of each subject are correlated and this correlation differs across subjects. We further demonstrate that the rate of convergence depends upon the covariance structure of the error term taken across individuals. A convergence rate is also obtained for derivatives. These results allow us to derive the optimal divergence r...

Research paper thumbnail of How many times should my simulation run?': Power analysis for agent-based modeling

Research paper thumbnail of A complex adaptive approach to the gender pay gap

Research paper thumbnail of Nonparametric Estimation and Inference in Economic and Psychological Experiments

arXiv: Econometrics, 2019

The goal of this paper is to provide some tools for nonparametric estimation and inference in psy... more The goal of this paper is to provide some tools for nonparametric estimation and inference in psychological and economic experiments. We consider an experimental framework in which each of nnnsubjects provides TTT responses to a vector of TTT stimuli. We propose to estimate the unknown function fff linking stimuli to responses through a nonparametric sieve estimator. We give conditions for consistency when either nnnor TTTor both diverge. The rate of convergence depends upon the error covariance structure, that is allowed to differ across subjects. With these results we derive the optimal divergence rate of the dimension of the sieve basis with both nnn and TTT. We provide guidance about the optimal balance between the number of subjects and questions in a laboratory experiment and argue that a large nnnis often better than a large TTT. We derive conditions for asymptotic normality of functionals of the estimator of TTTand apply them to obtain the asymptotic distribution of the Wald...

Research paper thumbnail of Asymptotic Properties of the Plug-in Estimator of the Discrete Entropy Under Dependence

IEEE Transactions on Information Theory, 2021

We consider the estimation of the entropy of a discretely-supported time series through a plug-in... more We consider the estimation of the entropy of a discretely-supported time series through a plug-in estimator. We provide a correction of the bias and we study the asymptotic properties of the estimator. We show that the widely-used correction proposed by Roulston (1999) is incorrect as it does not remove the Oleft(N−1right)O\left ({N^{-1}}\right)Oleft(N1right) part of the bias while ours does. We provide the asymptotic distribution and we show that it differs when the values taken by the marginal distribution of the process are equiprobable (a situation that we call degeneracy) and when they are not. We introduce estimators of the bias, the variance and the distribution under degeneracy and we study the estimation error. Finally, we propose a goodness-of-fit test based on entropy and give two motivations for it. The theoretical results are supported by specific numerical examples.

Research paper thumbnail of Model calibration and validation via confidence sets

Econometrics and Statistics, 2020

The issues of calibrating and validating a theoretical model are considered, when it is required ... more The issues of calibrating and validating a theoretical model are considered, when it is required to select the parameters that better approximate the data among a finite number of alternatives. Based on a user-defined loss function, Model Confidence Sets are proposed as a tool to restrict the number of plausible alternatives, and measure the uncertainty associated to the preferred model. Furthermore, an asymptotically exact logarithmic approximation of the probability of choosing a model via a multivariate rate function is suggested. A simple numerical procedure is outlined for the computation of the latter and it is shown that the procedure yields results consistent with Model Confidence Sets. The illustration and implementation of the proposed approach is showcased in a model of inquisitiveness in ad hoc teams, relevant for bounded rationality and organizational research. 1

Research paper thumbnail of What are we estimating when we fit Stevens’ power law?

Journal of Mathematical Psychology, 2016

h i g h l i g h t s • Estimates in Stevens' power laws often display sensitivities to experimenta... more h i g h l i g h t s • Estimates in Stevens' power laws often display sensitivities to experimental design. • These so-called 'contextual effects' concern range, location and averaging. • This paper links them with the separable representation model of Luce and Narens. • Theoretical results are illustrated using data from papers of R. Duncan Luce.

Research paper thumbnail of Asymptotic Distributions of Covering and Separation Measures on the Hypersphere

Discrete & Computational Geometry

We consider measures of covering and separation that are expressed through maxima and minima of d... more We consider measures of covering and separation that are expressed through maxima and minima of distances between points of an hypersphere. We investigate the behavior of these measures when applied to a sample of independent and uniformly distributed points. In particular, we derive their asymptotic distributions when the number of points diverges. These results can be useful as a benchmark against which deterministic point sets can be evaluated. Whenever possible, we supplement the rigorous derivation of these limiting distributions with some heuristic reasonings based on extreme value theory. As a by-product, we provide a proof for a conjecture on the hole radius associated to a facet of the convex hull of points distributed on the hypersphere.

Research paper thumbnail of An Analysis of Experienced Discrimination in Italy. The Case of Home Mortgages

The paper addresses the issue of home mortgage discrimination in one of the richest areas of nort... more The paper addresses the issue of home mortgage discrimination in one of the richest areas of northern Italy. Since no official data is available, workers of the Varese city hospital have been interviewed (n = 278); we group workers by (a) gender, (b) origin/ethnicity, (c) income/profession, and (d) age. The study is developed on perceptions and experiences of those interviewed. The analysis is structured in two steps. In particular, as a check for discrimination we test whether the probability of acceptance of a mortgage depends only on creditworthiness or also on non-economic variables. We found evidence for discrimination based on origin and conclude with managerial and policy issues.

Research paper thumbnail of Tax Evasion: Does the tax burden matter?

Abstract We study the effects of the tax burden on tax evasion both theoretically and experimenta... more Abstract We study the effects of the tax burden on tax evasion both theoretically and experimentally. We develop a model of tax evasion decision that is based on two ideas from behavioral economics: 1) taxpayers are endowed with reference dependent preferences that are subject to hedonic adaptation; and 2) in making their choices, taxpayers are affected by ethical concerns. The model generates new predictions about the effects of a change in the tax rate on the decision to evade taxes.