Likelihood approximation by numerical integration on sparse grids

Heiß, Florian; Winschel, Viktor

doi:10.1016/j.jeconom.2007.12.004

Cited by 352 publications

(273 citation statements)

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“…In our computations, we use the following numerical integration techniques: pseudo-Monte Carlo (pMC), Gaussian Hermite product rule, sparse grid integration (SGI) (Heiss and Winschel, 2008), and Stroud monomial rule 11-1 (Stroud, 1971). Because we have assumed that the mixing distribution of the random coefficients is Normal, we compute the pMC nodes by drawing between 1, 000 and 10, 000 nodes from a standard Normal distribution using MATLAB TM 's randn function.…”

Section: Methodsmentioning

confidence: 99%

“…The set of nodes, then, is all possible zs which are on the lattice formed from the Kronecker product of the one-dimensional nodes. See Figure 1 in Heiss and Winschel (2008). The weights are the product of the weights which correspond to the one-dimensional nodes.…”

Section: Gaussian Product Rulementioning

confidence: 99%

“…In addition, the nodes and weights for higher levels of exactness are easier to compute for SGI than for monomial rules. We use a Konrad-Patterson rule for choosing nodes as explained in Heiss and Winschel (2008). Our experiments show that SGI is very competitive with monomial rules in many cases.…”

Section: Sparse Grid Integrationmentioning

confidence: 99%

“…Our paper builds on Heiss and Winschel (2008) which showed that sparse grid integration outperforms simulation for likelihood estimation of a mixed logit model of multiple alternatives. However, we make several new contributions including that simulation introduces false local optima and excessively tight standard errors; that polynomial-based rules approximate both the level and derivatives of integrals better than pMC; that quadrature rules affect solver convergence; that polynomial rules outperform Monte Carlo for low to moderate degree monomials but both are poor for higher degrees; and, that monomial rules provide a lower cost alternative to SGI.…”

mentioning

confidence: 99%

“…In general, polynomial-based methods are both more efficient and more accurate. Heiss and Winschel (2008) warn that polynomial-based methods poorly approximate functions with large flat 9 This notation is based on Cools (2002).…”

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confidence: 99%

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High performance quadrature rules: how numerical integration affects a popular model of product differentiation

Judd

Skrainka

2011

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract Efficient, accurate, multi-dimensional, numerical integration has become an important tool for approximating the integrals which arise in modern economic models built on unobserved heterogeneity, incomplete information, and uncertainty. This paper demonstrates that polynomialbased rules out-perform number-theoretic quadrature (Monte Carlo) rules both in terms of efficiency and accuracy. To show the impact a quadrature method can have on results, we examine the performance of these rules in the context of Berry, Levinsohn, and Pakes (1995)'s model of product differentiation, where Monte Carlo methods introduce considerable numerical error and instability into the computations. These problems include inaccurate point estimates, excessively tight standard errors, instability of the inner loop 'contraction' mapping for inverting market shares, and poor convergence of several state of the art solvers when computing point estimates. Both monomial rules and sparse grid methods lack these problems and provide a more accurate, cheaper method for quadrature. Finally, we demonstrate how researchers can easily utilize high quality, high dimensional quadrature rules in their own work. Terms of use: Documents in

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Section: Methodsmentioning

confidence: 99%

Section: Gaussian Product Rulementioning

confidence: 99%

Section: Sparse Grid Integrationmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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High performance quadrature rules: how numerical integration affects a popular model of product differentiation

Judd

Skrainka

2011

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The Sigma Point Class: The Gauss–Hermite Kalman Filter

Haug¹

2012

Bayesian Estimation and Tracking

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A Stochastic Optimization approach for the design of Individualized Dosage Regimens

Laínez-Aguirre

Reklaitis

2013

AIChE Journal

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Quantitative methods for individualizing and optimizing the dosage regimen and clinically monitoring each patient are desirable to insure that each patient can obtain effective therapeutic benefit while minimizing undesirable side effects. This is of special concern for medicines that are expensive or whose toxic side effects are severe (e.g., oncological agents). The optimal dosage regimen for an individual is a combination of dose amount and/or dosing interval (i.e., time between doses) which minimizes the risk that the drug exposure deviates from the desired therapeutic window. The therapeutic window is defined as the range of drug exposure (e.g., blood concentration, area under the curve concentration‐time) which is below a threshold defining an acceptable toxic side effect and above a threshold defining a minimum acceptable level of therapeutic efficacy. In this work, the dosage regimen optimization problem defined in terms of general pharmacometric models (i.e., described by differential‐algebraic equations) is presented and a solution approach outlined which uses a scenario‐based stochastic optimization formulation that minimizes a risk metric. The scenarios are derived from the posterior joint probability distribution of the individual's pharmacometric parameters which is obtained following an approximate Bayesian inference approach. A Smolyak rule is used for the selection of the scenarios (i.e., combination of pharmacometric parameters) to be considered and for computing the approximation to the risk metric. Two case studies, gabapentin and cyclophosphamide, are presented to elucidate the advantages and limitations of the proposed approach. The numerical results demonstrate that low risk optimal solutions can be generated via the proposed stochastic optimization; while significantly reducing the computational burden in comparison with the conventional Markov chain Monte Carlo—grid search approach. This partially alleviates implementation issues preventing the deployment of dosage regimen individualization in clinical practice. Since stochastic optimization has been extensively used in other domains, the approach for uncertainty characterization proposed in this work may have general relevance beyond the pharmacometrics domain. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3296–3307, 2013

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Likelihood approximation by numerical integration on sparse grids

Cited by 352 publications

References 30 publications

High performance quadrature rules: how numerical integration affects a popular model of product differentiation

High performance quadrature rules: how numerical integration affects a popular model of product differentiation

The Sigma Point Class: The Gauss–Hermite Kalman Filter

A Stochastic Optimization approach for the design of Individualized Dosage Regimens

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