Recursive Models of Dynamic Linear Economies

419 Pages·2013·13.343 MB·English

by  Lars Peter Hansen, Thomas J. Sargent

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,
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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.
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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.