Table Of Content

Recursive Models of Dynamic Linear Economies The Gorman Lectures in Economics Richard Blundell, Series Editor Lawlessness and Economics: Alternative Modes of Governance, Avinash K. Dixit Rational Decisions, Ken Binmore Forward-Looking Decision Making: Dynamic-Programming Models Applied to Health, Risk, Employment, and Financial Stability, Robert E. Hall Recursive Models of Dynamic Linear Economies, Lars Peter Hansen and Thomas J. Sargent A series statement appears at the back of the book Recursive Models of Dynamic Linear Economies Lars Peter Hansen Thomas J. Sargent Princeton University Press • Princeton and Oxford (cid:2)c 2014 byPrinceton University Press Published byPrinceton University Press, 41 William Street,Princeton, New Jersey 08540 In theUnited Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire, OX20 1TW press.princeton.edu Jacket Art: Camille Pissarro (1830-1903). Rue de l’E´picerie, Rouen. 1898. Oil on canvas, 32 x 25 5/8 in. (81.3 x 65.1 cm). Purchase, Mr. and Mrs. Richard J. Bernhard Gift, 1960 (60.5). The Metropolitan Museum of Art, New York,NY,USA. Image copyright (cid:2)c The Metropolitan Museum of Art. Image Source: Art Resource, NY. All RightsReserved Library of Congress Cataloging-in-Publication Data Hansen, Lars Peter, 1952- Recursivemodels of dynamic linear economies / Lars Peter Hansen and Thomas J. Sargent. pages cm. – (The Gorman lectures in economics) Includesbibliographical references and index. ISBN 978-0-691-04277-0 (hardcover: alk. paper) 1. Economics–Mathematical models. I. Sargent,Thomas J. II.Title. HB135.H36 2013? 330.01’511352–dc23 2013016059 British Library Cataloging-in-Publication Data is available The publisherwould like to acknowledge theauthors of this volume for providing thecamera- ready copy from which this book was printed. This book has been composed in TEX Printed on acid-free paper ∞ Printed in theUnited States of America 10 9 8 7 6 5 4 3 2 1 To a project at the University of Minnesota that in the 1970s used economic theory to restrict stochastic processes and inform econometrics Contents Preface xiii Acknowledgments xv Part I: Overview 1. Theory and Econometrics 3 1.1. Introduction.1.2. AClassofEconomies.1.3. ComputerPrograms. 1.4. Organization.1.5. Recurring Mathematical Ideas. Part II: Tools 2. Linear Stochastic Difference Equations 15 2.1. Introduction. 2.2. Notation and Basic Assumptions. 2.3. Predic- tion Theory.2.4. TransformingVariables to Uncouple Dynamics.2.4.1. Deterministic Seasonals. 2.4.2. Indeterministic Seasonals. 2.4.3. Uni- variate Autoregressive Processes. 2.4.4. Vector Autoregressions. 2.4.5. Polynomial Time Trends. 2.4.6. Martingales with Drift. 2.4.7. Covari- ance Stationary Processes. 2.4.8. Multivariate ARMA Processes.2.4.9. Prediction of a Univariate First-Order ARMA. 2.4.10. Growth. 2.4.11. A Rational Expectations Model. 2.4.12. Method of Undetermined Co- efficients. 2.5. Concluding Remarks. – vii – viii Contents 3. Efficient Computations 33 3.1. Introduction. 3.2. The Optimal Linear Regulator Problem. 3.3. TransformationstoEliminateDiscountingandCross-Products.3.4. Sta- bility Conditions. 3.5. Invariant Subspace Methods. 3.5.1. Px as La- grange Multiplier. 3.5.2. Invariant Subspace Methods. 3.6. Doubling Algorithm. 3.7. Partitioning the State Vector. 3.8. Periodic Optimal Linear Regulator. 3.9. A Periodic Doubling Algorithm. 3.9.1. Parti- tioning the State Vector. 3.10. Linear ExponentialQuadratic Gaussian Control. 3.10.1. Doubling Algorithm for a Risk-Sensitive Problem. A. Concepts of Linear Control Theory. B. Symplectic Matrices. C. Alter- native Forms of the Riccati Equation. Part III: Components of Economies 4. Economic Environments 61 4.1. Information. 4.2. Taste and Technology Shocks. 4.3. Production Technologies. 4.4. Examples of Production Technologies. 4.4.1. Other Technologies.4.5. HouseholdTechnologies.4.6. ExamplesofHousehold Technologies. 4.7. Square Summability. 4.8. Summary. 5. Optimal Resource Allocations 79 5.1. Planning Problem. 5.2. Lagrange Multipliers. 5.3. Dynamic Pro- gramming. 5.4. Lagrange Multipliers as Gradients of Value Function. 5.5. Planning Problem as Linear Regulator. 5.6. Allocations for Five Economies. 5.6.1. Brock-Mirman (1972) or Hall (1978) Model. 5.6.2. A Growth Economy Fueled by Habit Persistence. 5.6.3. Lucas’s Pure Exchange Economy. 5.6.4. An Economy with a Durable Consumption Good. 5.6.5. Computed Examples. 5.7. Hall’s Model. 5.8. Higher Ad- justmentCosts.5.9. AlteredGrowthCondition.5.10. AJones-Manuelli (1990) Economy. 5.11. Durable Consumption Goods. 5.12. Summary. A. Synthesizinga LinearRegulator.B.ABrock-Mirman(1972)orHall (1978) Model. 5.B.1. Uncertainty. 5.B.2. Optimal Stationary States. Contents ix 6. A Commodity Space 125 6.1. Valuation. 6.2. Price Systems as Linear Functionals. 6.3. A One- PeriodModel under Certainty.6.4. One Periodunder Uncertainty.6.5. AnInfinite NumberofPeriodsandUncertainty.6.5.1. ConditioningIn- formation. 6.6. Lagrange Multipliers. 6.7. Summary. A. Mathematical Details. 7. Competitive Economies 131 7.1. Introduction. 7.2. Households. 7.3. Type I Firms. 7.4. Type II Firms. 7.5. Competitive Equilibrium. 7.6. Lagrangians. 7.6.1. House- hold Lagrangian. 7.6.2. Type I Firm Lagrangian. 7.6.3. Type II Firm Lagrangian. 7.7. Equilibrium Price System. 7.8. Asset Pricing. 7.9. TermStructure ofInterestRates.7.10. Reopening Markets.7.11. Non- Gaussian Asset Prices. 7.12. Asset Pricing Example. Part IV: Representations and Properties 8. Statistical Representations 153 8.1. The Kalman Filter. 8.2. Innovations Representation. 8.3. Con- vergence. 8.3.1. Computation of Time-Invariant Kalman Filter. 8.4. FactorizationofLikelihoodFunction.8.4.1. InitializationAssumptions. 8.4.2. Possible Nonexistence of Stationary Distribution. 8.5. Spectral Factorization Identity. 8.6. Wold and Autoregressive Representations. 8.7. Frequency Domain Estimation. 8.8. Approximation Theory. 8.9. Aggregation over Time. 8.10. Simulation Estimators. A. Initialization of Kalman Filter. B. Zeros of Characteristic Polynomial. C. Serially CorrelatedMeasurementErrors.D.Innovationsin yt+1 asFunctionsof wt+1 and ηt+1. E. Innovations in a Permanent Income Model. 9. Canonical Household Technologies 191 9.1. Introduction. 9.2. Definition of a Canonical Household Technol- ogy. 9.3. Dynamic Demand Functions. 9.3.1. Wealth and the Multi- plier μw. 9.3.2. Dynamic Demand System. 9.3.3. GormanAggregation 0 and Engel Curves. 9.3.4. Reopened Markets. 9.4. Computing Canon- ical Representations. 9.4.1. Basic Idea. 9.4.2. An Auxiliary Problem Induces a Canonical Representation. 9.5. An Operator Identity. 9.6.