Kenneth Judd - Profile on Academia.edu (original) (raw)

Papers by Kenneth Judd

Asymptotic Expansion Methods for Dynamic Models with Incomplete Asset Markets

RePEc: Research Papers in Economics, Jul 1, 2002

O curse of dimensionality, where is thy sting?

Computing in Economics and Finance, 2006

ABSTRACT Economic analysis often leads to multidimensional numerical problems. The {\em Curse of ... more ABSTRACT Economic analysis often leads to multidimensional numerical problems. The {\em Curse of Dimensionality\/} often leads researchers to adopt methods designed for very high-dimension problems, but inefficient for problems of intermediate dimension. However, a little mathematics can greatly help dealing with the {\em Curse\/}. We will survey methods from approximation theory, numerical quadrature, and symbolic computation that have helped economists tackle high-dimensional problems, and current work that will further reduce the computational cost of multidimensional problems.

Teaching Numerical Methods to Economics Students

Computing in Economics and Finance, 2004

ABSTRACT Numerical methods are increasingly used in graduate student research. I will discuss the... more ABSTRACT Numerical methods are increasingly used in graduate student research. I will discuss the problems of how to teach the necessary skills and the challenges of incorporating such teaching into a graduate program

Comments on Prof. Mirowski's “Markets Come to Bits: Evolution, Computation and Markomata in Economic Science”

Journal of Economic Behavior and Organization, Jun 1, 2007

ABSTRACT Prof. Mirowski has written a provocative discussion of new ways to model markets that fo... more ABSTRACT Prof. Mirowski has written a provocative discussion of new ways to model markets that focus on their algorithmic aspects. Many of his substantive points about the weaknesses of standard theory are widely accepted; in particular, we have little idea of how prices are formed, we agree that economic agents are not infinitely intelligent, and it is clear that the markets in any modem economy take many forms. Prof. Mirowski correctly notes that there has been some movement towards more detailed analyses of markets, such as in the literatures on mechanism design, auction design, "Zero-Intelligence Agent" models, market microstructure, engineering economics, and applications of artificial intelligence. All economists welcome further work on detailed analysis of how markets work and how they evolve over time. Prof. Mirowski helps the reader greatly by anchoring his presentation in the following definition of a market. For the purposes of our present argument, we shall define a market as a formal automaton, in the sense of the field of mathematics pioneered by John von Neumann, and now taught as basic computational theory. Intuitively, we shall characterize a particular market as a specialized piece of software, which both calculates and acts upon inputs, comprised of an integrated set of algorithms that perform the following functions: Data dissemination and communications, plus rules of exclusion. Order routing through time and space. Order queuing and execution. Price discovery and assignment. Custody and delivery arrangement. Clearing and settlement, including property rights assignment. Record-keeping.

Computational Public Economics

Journal of Public Economic Theory, May 1, 2004

Efficient Likelihood Ratio Confidence Intervals using Constrained Optimization

SSRN Electronic Journal, 2019

Using constrained optimization, we develop a simple, efficient approach (applicable in both uncon... more Using constrained optimization, we develop a simple, efficient approach (applicable in both unconstrained and constrained maximum-likelihood estimation problems) to computing profile-likelihood confidence intervals. In contrast to Wald-type or score-based inference, the likelihood ratio confidence intervals use all the information encoded in the likelihood function concerning the parameters, which leads to improved statistical properties. In addition, the method does no suffer from the computational burdens inherent in the bootstrap. In an application to Rust's (1987) bus-engine replacement problem, our approach does better than either the Wald or the bootstrap methods, delivering very accurate estimates of the confidence intervals quickly and efficiently. An extensive Monte Carlo study reveals that in small samples, only likelihood ratio confidence intervals yield reasonable coverage properties, while at the same time discriminating implausible values.

Big Data Techniques as a Solution to Theory Problems

Conquering Big Data with High Performance Computing, 2016

This chapter proposes a general approach for solving a broad class of difficult optimization prob... more This chapter proposes a general approach for solving a broad class of difficult optimization problems using big data techniques. We provide a general description of this approach as well as some examples. This approach is ideally suited for solving nonconvex optimization problems, multiobjective programming problems, models with a large degree of heterogeneity, rich policy structure, potential model uncertainty, and potential policy objective uncertainty. In our applications of this algorithm we use Hierarchical Database Format (HDF5) distributed storage and I/O as well as message passing interface (MPI) for parallel computation of a large number of small optimization problems.

Numerical methods in economics

© 1998 Massachusetts Institute of Technology All rights reserved. No part of this book may be rep... more © 1998 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the ...

Comments on Marcet, Rust and Pakes

Advances in Econometrics: Sixth World Congress Vol II

Handbooks in Economics - Computational Economics

Data Archiving and Networked Services (DANS), 2014

This chapter of the Handbook of Computational Economics is mostly about research on active learni... more This chapter of the Handbook of Computational Economics is mostly about research on active learning and is confined to discussion of learning in dynamic models in which the systems equations are linear, the criterion function is quadratic and the additive noise terms are Gaussian. Though there is much work on learning in more general systems, it is useful here to focus on models with these specifications since more general systems can be approximated in this way and since much of the early work on learning has been done with these quadraticlinear-gaussian systems. We begin with what has been learned about learning in dynamic economic models in the last few decades. Then we progress to a discussion of what we hope to learn in the future from a new project that is just getting underway. However before doing either of these it is useful to provide a short description of the mathematical framework that will be used in the chapter.

AMPL and Matlab Code for \Constrained Optimization Approaches to Estimation of Structural Models

Tipping Points in a Dynamic Stochastic IAM

Social Science Research Network, 2012

ABSTRACT We use a dynamic stochastic general equilibrium model of integrated climate and economy ... more ABSTRACT We use a dynamic stochastic general equilibrium model of integrated climate and economy (DSICE) to account for abrupt and irreversible climate change. We model a climate shock in the form of a stochastic tipping point. We investigate the impact of the tipping point externality on optimal mitigation policy.We conclude that the optimal mitigation policy depends on the dynamic pattern of the impact. In the case of abrupt and irreversible climate change with a permanent impact, the optimal policy implies a constant anti-tipping effort to prevent the catastrophe, calling for immediate limitations on emissions.

Constrainted Optimization Approaches to Estimation of Structural Models

RePEc: Research Papers in Economics, 2008

ABSTRACT Maximum likelihood estimation of structural models is often viewed as computationally di... more ABSTRACT Maximum likelihood estimation of structural models is often viewed as computationally difficult. This impression is due to a focus on the Nested Fixed-Point approach. We present a direct optimization approach to the general problem and show that it is significantly faster than the NFXP approach when applied to the canonical Zurcher bus repair model. The NFXP approach is inappropriate for estimating games since it requires finding all Nash equilibria of a game for each parameter vector considered, a generally intractable computational problem. We formulate the problem of maximum likelihood estimation of games as a constrained optimization problem that is qualitatively no more difficult to solve than standard maximum likelihood problems. The direct optimization approach is also applicable to other structural estimation methods such as methods of moments, and also allows one to use computationally intensive bootstrap methods to calculate inference. The MPEC approach is also easily implemented on software with high-level interfaces. Furthermore, all the examples in this paper were computed using only free resources available on the web.

Solving Continuous-Time Markov-Perfect Nash Equilibria

Computing in Economics and Finance, 2004

ABSTRACT We explore methods for solving continuous-time, finite-state dynamic games, and document... more ABSTRACT We explore methods for solving continuous-time, finite-state dynamic games, and document the advantages of such models over comparable discrete-time models. In particular, continuous-time models are more flexible since they avoid lumpiness in time, and computational methods for continuous-time models avoid a curse of dimensionality that arises in discrete-time models.

Non-Redundant Derivatives in a Dynamic General Equilibrium Economy

SSRN Electronic Journal, 2002

2010), “Solving the Multi-Country Real Business Cycle Model with Ergodic Set Methods,”Journal of Economic Dynamics and Control, this issue

We use the stochastic simulation algorithm, described in Judd, Maliar and Maliar (2009), and the ... more We use the stochastic simulation algorithm, described in Judd, Maliar and Maliar (2009), and the cluster-grid algorithm, developed in Judd, Maliar and Maliar (2010a), to solve a collection of multi-country real business cycle models. The following ingredients help us reduce the cost in high-dimensional problems: an endogenous grid enclosing the ergodic set, linear approximation methods, fixed-point iteration and efficient integration methods, such as non-product monomial rules and Monte Carlo integration combined with regression. We show that high accuracy in intratemporal choice is crucial for the overall accuracy of solutions and offer two approaches, precomputation and iterationon-allocation, that can solve for intratemporal choice both accurately and quickly. We also implement a hybrid solution algorithm that combines the perturbation and accurate intratemporal-choice methods.

Capital Market Imperfections and Tax Policy Analysis in the Life Cycle Model

Annales d'Économie et de Statistique

Structural estimation of discrete-choice games of incomplete information with multiple equilibria

Proceedings of the Behavioral and Quantitative Game Theory on Conference on Future Directions - BQGT '10, 2010

ABSTRACT Estimation of games with multiple equilibria has received much attention in the recent e... more ABSTRACT Estimation of games with multiple equilibria has received much attention in the recent econometrics literature. Unlike other estimation problems such as single-agent dynamic decision models or demand estimation, in which there is a unique solution in the underlying structural models, games usually admit multiple equilibria and the number of equilibria in a game can vary for different structural parameters. This fact makes the estimation of games far more challenging because the likelihood function or other criterion function defined in the space of structural parameters can be discontinuous or non-differentiable. Two-step estimators by Bajari et al. (2007) and Pesendorfer and Schmidt-Dengler (2008) and Nested Pusedo Likelihood (NPL) estimators by Aguirregabiria and Mira (2007) are proposed to address this problem. We recast the estimation problem as a constrained optimization problem with the Bayesian-Nash equilibrium condition being the constraints. The advantage of our formulation is that the likelihood function, now defined in the equilibrium probability space, is continuous and smooth. This allows researchers to use state-of-the-art optimization software to solve the estimation problem. In a Monte Carlo study, we compare the performance of a two-step estimator, NLP estimator, and our constrained optimization estimator.

Pruning and Higher-Order Perturbation Solutions

ABSTRACT This paper evaluates the pruning procedure proposed by Kim, Kim, Schaumburg, and Sims (2... more ABSTRACT This paper evaluates the pruning procedure proposed by Kim, Kim, Schaumburg, and Sims (2008) that ensures that higher-order perturbation solutions are not explo-sive. This procedure induces substantial distortions even when the simulation is not on an explosive trajectory. In fact, the procedure turns the policy functions into policy correspondences. An accuracy procedure is proposed to evaluate the severity of the exposed disadvantages of pruning for the problem at hand. A simple alternative to pruning is proposed. for useful comments.

Analysis of Taxation in a Model with Perfect Foresight

Asymptotic Expansion Methods for Dynamic Models with Incomplete Asset Markets

RePEc: Research Papers in Economics, Jul 1, 2002

O curse of dimensionality, where is thy sting?

Computing in Economics and Finance, 2006

ABSTRACT Economic analysis often leads to multidimensional numerical problems. The {\em Curse of ... more ABSTRACT Economic analysis often leads to multidimensional numerical problems. The {\em Curse of Dimensionality\/} often leads researchers to adopt methods designed for very high-dimension problems, but inefficient for problems of intermediate dimension. However, a little mathematics can greatly help dealing with the {\em Curse\/}. We will survey methods from approximation theory, numerical quadrature, and symbolic computation that have helped economists tackle high-dimensional problems, and current work that will further reduce the computational cost of multidimensional problems.

Teaching Numerical Methods to Economics Students

Computing in Economics and Finance, 2004

ABSTRACT Numerical methods are increasingly used in graduate student research. I will discuss the... more ABSTRACT Numerical methods are increasingly used in graduate student research. I will discuss the problems of how to teach the necessary skills and the challenges of incorporating such teaching into a graduate program

Comments on Prof. Mirowski's “Markets Come to Bits: Evolution, Computation and Markomata in Economic Science”

Journal of Economic Behavior and Organization, Jun 1, 2007

ABSTRACT Prof. Mirowski has written a provocative discussion of new ways to model markets that fo... more ABSTRACT Prof. Mirowski has written a provocative discussion of new ways to model markets that focus on their algorithmic aspects. Many of his substantive points about the weaknesses of standard theory are widely accepted; in particular, we have little idea of how prices are formed, we agree that economic agents are not infinitely intelligent, and it is clear that the markets in any modem economy take many forms. Prof. Mirowski correctly notes that there has been some movement towards more detailed analyses of markets, such as in the literatures on mechanism design, auction design, "Zero-Intelligence Agent" models, market microstructure, engineering economics, and applications of artificial intelligence. All economists welcome further work on detailed analysis of how markets work and how they evolve over time. Prof. Mirowski helps the reader greatly by anchoring his presentation in the following definition of a market. For the purposes of our present argument, we shall define a market as a formal automaton, in the sense of the field of mathematics pioneered by John von Neumann, and now taught as basic computational theory. Intuitively, we shall characterize a particular market as a specialized piece of software, which both calculates and acts upon inputs, comprised of an integrated set of algorithms that perform the following functions: Data dissemination and communications, plus rules of exclusion. Order routing through time and space. Order queuing and execution. Price discovery and assignment. Custody and delivery arrangement. Clearing and settlement, including property rights assignment. Record-keeping.

Computational Public Economics

Journal of Public Economic Theory, May 1, 2004

Efficient Likelihood Ratio Confidence Intervals using Constrained Optimization

SSRN Electronic Journal, 2019

Using constrained optimization, we develop a simple, efficient approach (applicable in both uncon... more Using constrained optimization, we develop a simple, efficient approach (applicable in both unconstrained and constrained maximum-likelihood estimation problems) to computing profile-likelihood confidence intervals. In contrast to Wald-type or score-based inference, the likelihood ratio confidence intervals use all the information encoded in the likelihood function concerning the parameters, which leads to improved statistical properties. In addition, the method does no suffer from the computational burdens inherent in the bootstrap. In an application to Rust's (1987) bus-engine replacement problem, our approach does better than either the Wald or the bootstrap methods, delivering very accurate estimates of the confidence intervals quickly and efficiently. An extensive Monte Carlo study reveals that in small samples, only likelihood ratio confidence intervals yield reasonable coverage properties, while at the same time discriminating implausible values.

Big Data Techniques as a Solution to Theory Problems

Conquering Big Data with High Performance Computing, 2016

This chapter proposes a general approach for solving a broad class of difficult optimization prob... more This chapter proposes a general approach for solving a broad class of difficult optimization problems using big data techniques. We provide a general description of this approach as well as some examples. This approach is ideally suited for solving nonconvex optimization problems, multiobjective programming problems, models with a large degree of heterogeneity, rich policy structure, potential model uncertainty, and potential policy objective uncertainty. In our applications of this algorithm we use Hierarchical Database Format (HDF5) distributed storage and I/O as well as message passing interface (MPI) for parallel computation of a large number of small optimization problems.

Numerical methods in economics

© 1998 Massachusetts Institute of Technology All rights reserved. No part of this book may be rep... more © 1998 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the ...

Comments on Marcet, Rust and Pakes

Advances in Econometrics: Sixth World Congress Vol II

Handbooks in Economics - Computational Economics

Data Archiving and Networked Services (DANS), 2014

This chapter of the Handbook of Computational Economics is mostly about research on active learni... more This chapter of the Handbook of Computational Economics is mostly about research on active learning and is confined to discussion of learning in dynamic models in which the systems equations are linear, the criterion function is quadratic and the additive noise terms are Gaussian. Though there is much work on learning in more general systems, it is useful here to focus on models with these specifications since more general systems can be approximated in this way and since much of the early work on learning has been done with these quadraticlinear-gaussian systems. We begin with what has been learned about learning in dynamic economic models in the last few decades. Then we progress to a discussion of what we hope to learn in the future from a new project that is just getting underway. However before doing either of these it is useful to provide a short description of the mathematical framework that will be used in the chapter.

AMPL and Matlab Code for \Constrained Optimization Approaches to Estimation of Structural Models

Tipping Points in a Dynamic Stochastic IAM

Social Science Research Network, 2012

ABSTRACT We use a dynamic stochastic general equilibrium model of integrated climate and economy ... more ABSTRACT We use a dynamic stochastic general equilibrium model of integrated climate and economy (DSICE) to account for abrupt and irreversible climate change. We model a climate shock in the form of a stochastic tipping point. We investigate the impact of the tipping point externality on optimal mitigation policy.We conclude that the optimal mitigation policy depends on the dynamic pattern of the impact. In the case of abrupt and irreversible climate change with a permanent impact, the optimal policy implies a constant anti-tipping effort to prevent the catastrophe, calling for immediate limitations on emissions.

Constrainted Optimization Approaches to Estimation of Structural Models

RePEc: Research Papers in Economics, 2008

ABSTRACT Maximum likelihood estimation of structural models is often viewed as computationally di... more ABSTRACT Maximum likelihood estimation of structural models is often viewed as computationally difficult. This impression is due to a focus on the Nested Fixed-Point approach. We present a direct optimization approach to the general problem and show that it is significantly faster than the NFXP approach when applied to the canonical Zurcher bus repair model. The NFXP approach is inappropriate for estimating games since it requires finding all Nash equilibria of a game for each parameter vector considered, a generally intractable computational problem. We formulate the problem of maximum likelihood estimation of games as a constrained optimization problem that is qualitatively no more difficult to solve than standard maximum likelihood problems. The direct optimization approach is also applicable to other structural estimation methods such as methods of moments, and also allows one to use computationally intensive bootstrap methods to calculate inference. The MPEC approach is also easily implemented on software with high-level interfaces. Furthermore, all the examples in this paper were computed using only free resources available on the web.

Solving Continuous-Time Markov-Perfect Nash Equilibria

Computing in Economics and Finance, 2004

ABSTRACT We explore methods for solving continuous-time, finite-state dynamic games, and document... more ABSTRACT We explore methods for solving continuous-time, finite-state dynamic games, and document the advantages of such models over comparable discrete-time models. In particular, continuous-time models are more flexible since they avoid lumpiness in time, and computational methods for continuous-time models avoid a curse of dimensionality that arises in discrete-time models.

Non-Redundant Derivatives in a Dynamic General Equilibrium Economy

SSRN Electronic Journal, 2002

2010), “Solving the Multi-Country Real Business Cycle Model with Ergodic Set Methods,”Journal of Economic Dynamics and Control, this issue

We use the stochastic simulation algorithm, described in Judd, Maliar and Maliar (2009), and the ... more We use the stochastic simulation algorithm, described in Judd, Maliar and Maliar (2009), and the cluster-grid algorithm, developed in Judd, Maliar and Maliar (2010a), to solve a collection of multi-country real business cycle models. The following ingredients help us reduce the cost in high-dimensional problems: an endogenous grid enclosing the ergodic set, linear approximation methods, fixed-point iteration and efficient integration methods, such as non-product monomial rules and Monte Carlo integration combined with regression. We show that high accuracy in intratemporal choice is crucial for the overall accuracy of solutions and offer two approaches, precomputation and iterationon-allocation, that can solve for intratemporal choice both accurately and quickly. We also implement a hybrid solution algorithm that combines the perturbation and accurate intratemporal-choice methods.

Capital Market Imperfections and Tax Policy Analysis in the Life Cycle Model

Annales d'Économie et de Statistique

Structural estimation of discrete-choice games of incomplete information with multiple equilibria

Proceedings of the Behavioral and Quantitative Game Theory on Conference on Future Directions - BQGT '10, 2010

ABSTRACT Estimation of games with multiple equilibria has received much attention in the recent e... more ABSTRACT Estimation of games with multiple equilibria has received much attention in the recent econometrics literature. Unlike other estimation problems such as single-agent dynamic decision models or demand estimation, in which there is a unique solution in the underlying structural models, games usually admit multiple equilibria and the number of equilibria in a game can vary for different structural parameters. This fact makes the estimation of games far more challenging because the likelihood function or other criterion function defined in the space of structural parameters can be discontinuous or non-differentiable. Two-step estimators by Bajari et al. (2007) and Pesendorfer and Schmidt-Dengler (2008) and Nested Pusedo Likelihood (NPL) estimators by Aguirregabiria and Mira (2007) are proposed to address this problem. We recast the estimation problem as a constrained optimization problem with the Bayesian-Nash equilibrium condition being the constraints. The advantage of our formulation is that the likelihood function, now defined in the equilibrium probability space, is continuous and smooth. This allows researchers to use state-of-the-art optimization software to solve the estimation problem. In a Monte Carlo study, we compare the performance of a two-step estimator, NLP estimator, and our constrained optimization estimator.

Pruning and Higher-Order Perturbation Solutions

ABSTRACT This paper evaluates the pruning procedure proposed by Kim, Kim, Schaumburg, and Sims (2... more ABSTRACT This paper evaluates the pruning procedure proposed by Kim, Kim, Schaumburg, and Sims (2008) that ensures that higher-order perturbation solutions are not explo-sive. This procedure induces substantial distortions even when the simulation is not on an explosive trajectory. In fact, the procedure turns the policy functions into policy correspondences. An accuracy procedure is proposed to evaluate the severity of the exposed disadvantages of pruning for the problem at hand. A simple alternative to pruning is proposed. for useful comments.

Analysis of Taxation in a Model with Perfect Foresight