Recursive Models of Dynamic Linear Economies Recursive Models of Dynamic Linear Economies Lars Hansen University of Chicago Thomas J. Sargent New York University and Hoover Institution c Lars Peter Hansen and Thomas J. Sargent 21 March 2005 Contents Acknowledgements xii Preface xiii Part I: Components of an economy 1. Introduction 3 1.1. Introduction. 1.2. Computer Programs. 1.3. Organization. 2. Stochastic Linear Difference Equations 9 2.1. Introduction. 2.2. Notation and Basic Assumptions. 2.3. Predic- tion Theory. 2.4. Transforming Variables to Uncouple Dynamics. 2.5. Examples. 2.5.1. Deterministic seasonals. 2.5.2. Indeterministic season- als. 2.5.3. Univariate autoregressive processes. 2.5.4. Vector autoregres- sions. 2.5.5. Polynomial time trends. 2.5.6. Martingales with drift. 2.5.7. Covariance stationary processes. 2.5.8. Multivariate ARMA processes. 2.5.9. Prediction of a univariate first order ARMA. 2.5.10. Growth. 2.5.11. A rational expectations model. 2.6. The Spectral Density Ma- trix. 2.7. Computer Examples. 2.7.1. Deterministic seasonal. 2.7.2. Indeterministic seasonal, unit root. 2.7.3. Indeterministic seasonal, no unit root. 2.7.4. First order autoregression. 2.7.5. Second order autore- gression. 2.7.6. Growth with homoskedastic noise. 2.7.7. Growth with heteroskedastic noise. 2.7.8. Second order vector autoregression. 2.7.9. A rational expectations model. 2.8. Conclusion. 3. The Economic Environment 39 3.1. Information. 3.2. Taste and Technology Shocks. 3.3. Technologies. 3.4. Examples of Technologies. 3.4.1. Other technologies. 3.5. Prefer- ences and Household Technologies. 3.6. Examples of Household Tech- nology Preference Structures. 3.7. Constraints to Keep the Solutions “Square Summable”. 3.8. Summary. – v – vi Contents 4. Optimal Resource Allocation 57 4.1. Planning problem. 4.2. Lagrange Mmultipliers. 4.3. Dynamic pro- gramming. 4.4. Lagrange multipliers as gradients of value function. 4.5. Planning problem as linear regulator. 4.6. Solutions for five economies. 4.6.1. Preferences. 4.6.2. Technology. 4.6.3. Information. 4.6.4. Brock- Mirman model. 4.6.5. A growth economy fueled by habit persistence. 4.6.6. Lucas’s pure exchange economy. 4.6.7. An economy with a durable consumption good. 4.7. Hall’s model. 4.8. Higher Adjustment Costs. 4.9. Altered ‘growth condition’. 4.10. A Jones-Manuelli economy. 4.11. Durable consumption goods. 4.12. Summary. A. Synthesizing the linear regulator. B. A Brock-Mirman model. 4.B.1. Uncertainty. 4.B.2. Optimal Stationary States. 5. The Commodity Space 105 5.1. Valuation. 5.2. Price systems as linear functionals. 5.3. A one period model under certainty. 5.4. One period under uncertainty. 5.5. An infinite number of periods and uncertainty. 5.5.1. Conditioning in- formation. 5.6. Lagrange multipliers. 5.7. Summary. A. Appendix. 6. A Competitive Economy 113 6.1. Introduction. 6.2. The Problems of Households and Firms. 6.2.1. Households. 6.2.2. Firms of type I. 6.2.3. Firms of type II. 6.3. Compet- itive Equilibrium. 6.4. Lagrangians. 6.4.1. Households. 6.4.2. Firms of type I. 6.4.3. Firms of type II. 6.5. Equilibrium Price System. 6.6. Asset Pricing. 6.7. Term Structure of Interest Rates. 6.8. Re-opening Mar- kets. 6.8.1. Recursive price system. 6.8.2. Non-Gaussian asset prices. 6.9. Summary of Pricing Formulas. 6.10. Asset Pricing Example. 6.10.1. Preferences. 6.10.2. Technology. 6.10.3. Information. 6.11. Exercises. 7. Applications 139 7.1. Introduction. 7.2. Partial Equilibrium Interpretation. 7.2.1. Par- tial equilibrium investment under uncertainty. 7.3. Introduction. 7.4. A Housing Model. 7.4.1. Demand. 7.4.2. House producers. 7.5. Cattle Cycles. 7.5.1. Mapping cattle farms into our framework. 7.5.2. Pref- erences. 7.5.3. Technology. 7.6. Models of Occupational Choice and Pay. 7.6.1. A one-occupation model. 7.6.2. Skilled and unskilled work- ers. 7.7. A Cash-in-Advance Model. 7.7.1. Reinterpreting the household technology. 7.8. Taxation in a Vintage Capital Model. A. Decentralizing the Household. Contents vii 8. Efficient Computations 157 8.1. Introduction. 8.2. The Optimal Linear Regulator Problem. 8.3. Transformations to eliminate discounting and cross-products. 8.4. Sta- bility Conditions. 8.5. Invariant Subspace Methods. 8.5.1. P x as La- grange multiplier. 8.5.2. Invariant subspace methods. 8.5.3. Distorted Economies. 8.5.4. Transition Dynamics. 8.6. The Doubling Algorithm. 8.7. Partitioning the State Vector. 8.8. The Periodic Optimal Linear Regulator. 8.9. A Periodic Doubling Algorithm. 8.9.1. Partitioning the state vector. 8.10. Linear Exponential Quadratic Gaussian Control. 8.10.1. Doubling algorithm. A. Concepts of Linear Control Theory. B. Symplectic Matrices. C. Alternative forms of Riccati equation. viii Contents Part II: Representations and Properties 9. Representation and Estimation 187 9.1. The Kalman Filter. 9.2. Innovations Representation. 9.3. Conver- gence results. 9.3.1. Time-Invariant Innovations Representation. 9.4. Serially Correlated Measurement Errors. 9.5. Combined System. 9.6. Recursive Formulation of Likelihood Function. 9.6.1. Initialization. 9.6.2. Non-existence of a stationary distribution. 9.6.3. Serially cor- related measurement errors. 9.7. Wold Representation. 9.8. Vector Au- toregression for {y t }. 9.8.1. The factorization identity. 9.8.2. Location of zeros of characteristic polynomial. 9.8.3. Wold and autoregressive representations (white measurement errors). 9.8.4. Serially correlated measurement errors. 9.9. Innovations in y t+1 as Functions of Innova- tions w t+1 and η t+1 . 9.10. Innovations in the y t ’s and the w t ’s in a Permanent Income Model. 9.10.1. Preferences. 9.10.2. Technology. 9.10.3. Information. 9.11. Frequency Domain Estimation. 9.12. Ap- proximation Theory. 9.13. Aggregation Over Time. 9.14. Simulation Estimators. A. Initialization of the Kalman Filter. 10. Semiparametric Estimation with Limited Information 227 10.1. Introduction. 10.2. Underlying Economic Model. 10.3. Econome- trician’s information and the implied orthogonality conditions. 10.4. An Adjustment Cost Example. 10.5. A Slightly Simpler Estimation Prob- lem. 10.5.1. Scalar Parameterizations of B. 10.6. Multidimensional Parameterizations of B . 10.7. Nonparametric Estimation of B . 10.8. Back to the Adjustment Cost Model. 11. Representation of Demand 239 11.1. Introduction. 11.2. Canonical Representations of Services. 11.3. Dynamic Demand Functions for Consumption Goods. 11.3.1. The mul- tiplier µ w 0 . 11.3.2. Dynamic Demand System. 11.3.3. Foreshadow of Gorman aggregation. 11.4. Computing Canonical Representations. 11.4.1. Heuristics. 11.4.2. An auxiliary problem that induces a canonical rep- resentation. 11.5. Operator Identities. 11.6. Becker-Murphy Model of Rational Addiction. A. Fourier transforms. 11.A.1. Primer on trans- forms. 11.A.2. Time reversal and Parseval’s formula. 11.A.3. One sided Contents ix sequences. 11.A.4. Useful properties. 11.A.5. One sided transforms. 11.A.6. Discounting. 11.A.7. Fourier transforms. 11.A.8. Verifying Equivalent Valuations. 11.A.9. Equivalent representations of prefer- ences. 11.A.10. First term: factorization identity. 11.A.11. Second term. 11.A.12. Third term. 12. Gorman Heterogeneous Households 265 12.1. Introduction. 12.2. A Digression on Gorman Aggregation. 12.3. An Economy with Heterogeneous Consumers. 12.4. Allocations. 12.4.1. Consumption sharing rules. 12.5. Risk Sharing Implications. 12.6. Im- plementing the Allocation Rule with Limited Markets. 12.7. A Com- puter Example. 12.8. Exercises. 12.8.1. Part one. 12.8.2. Part two. 12.9. Economic integration. 12.9.1. Preferences:. 12.9.2. Technology. 12.9.3. Information. 13. Permanent Income Models 287 13.1. Technology. 13.2. Two Implications. 13.3. Solution. 13.4. Deter- ministic Steady States. 13.5. Cointegration. 13.6. Constant Marginal Utility of Income. 13.7. Consumption Externalities. 13.8. Tax Smooth- ing Models. 14. Non-Gorman Heterogeneity Among Households 307 14.1. Introduction. 14.2. Households’ Preferences. 14.2.1. Technol- ogy. 14.3. A Pareto Problem. 14.4. Competitive Equilibrium. 14.4.1. Households. 14.4.2. Firms of type I and II. 14.4.3. Definition of compet- itive equilibrium. 14.5. Computation of Equilibrium. 14.5.1. Candidate equilibrium prices. 14.5.2. A Negishi algorithm. 14.6. Mongrel Aggrega- tion. 14.6.1. Static demand. 14.6.2. Frequency domain representation of preferences. 14.7. A Programming Problem for Mongrel Aggregation. 14.7.1. Factoring S  S . 14.8. Summary of Findings. 14.9. The Mon- grel Preference Shock Process. 14.9.1. Interpretation of ˆs t component. 14.10. Choice of Initial Conditions. x Contents Part III: Extensions 15. Equilibria with Distortions 331 15.1. Introduction. 15.2. A Representative Agent Economy with Dis- tortions. 15.2.1. a. Consumption externalities. 15.2.2. b. Production externalities. 15.2.3. c. Taxes. 15.3. Households. 15.4. Firms. 15.5. Information. 15.6. Equilibrium. 15.7. Heterogeneous Households with Distortions. 15.7.1. Households. 15.7.2. Firms of type I. 15.7.3. Firms of type II. 15.7.4. Government. 15.7.5. Definition of equilibrium. 15.7.6. Equilibrium computation. 15.8. Government Deficits and Debt. 15.9. Examples. 15.9.1. A production externality. 15.9.2. Consumption tax only. 15.9.3. Machinery investment subsidy. 15.9.4. ‘Personal’ habit persistence. 15.9.5. ‘Social’ habit persistence. 15.10. Conclusions. A. Invariant subspace equations for first specification. 15.A.1. Household’s Lagrangian. 15.A.2. Firm’s first order conditions. 15.A.3. Representa- tiveness conditions. B. Invariant subspace equations for heterogeneous agent model. 16. Recursive Risk Sensitive Control 367 16.1. Introduction. 16.2. A Control Problem. 16.3. Pessimistic Inter- pretation. 16.4. Recursive Preferences. 16.4.1. Endowment economy. 16.5. Asset Pricing. 16.6. Characterizing the Pricing Expectations Op- erator. 16.7. Production Economies. 16.8. Risk-Sensitive Investment under Uncertainty. 16.9. Equilibrium Prices in the Adjustment Cost Economies. 17. Periodic Models of Seasonality 385 17.1. Introduction. 17.2. A Periodic Economy. 17.3. Asset Pricing. 17.4. Prediction Theory. 17.5. The Term Structure of Interest Rates. 17.6. Conditional Covariograms. 17.7. The Stacked and Skip-Sampled System. 17.8. Covariances of the Stacked, Skip Sampled Process. 17.9. The Tiao-Grupe Formula. 17.9.1. A state space realization of the Tiao- Grupe formulation. 17.10. Some Calculations with a Periodic Hall Model. 17.11. Periodic Innovations Representations for the Periodic Model. A. A Model of Disguised Periodicity. 17.13. A1. Two Illustra- tions of Disguised Periodicity. 17.14. A2. Mathematical Formulation of Disguised Periodicity. Contents xi Part IV: Economies as Objects 18. Introduction to Objects 425 18.1. Matlab Objects. 18.1.1. Definitions. 18.1.2. Matlab Specifics. 18.1.3. How to Define a Matlab Class. 18.2. Summary. 19. Economies as Matlab Objects 431 19.1. Introduction. 19.2. Parent Classes: Information. 19.2.1. Struc- ture. 19.2.2. Functions. 19.3. Parent Classes: Technology. 19.3.1. Struc- ture. 19.3.2. Functions. 19.4. Parent Classes: Preferences. 19.4.1. Structure. 19.4.2. Functions. 19.5. Child Class: Economy. 19.5.1. Struc- ture. 19.5.2. Fields containing the history of the economy. 19.5.3. Functions. 19.5.4. Constructing the object and changing parameters. 19.5.5. Analyzing the economy. 19.6. Working with economies. 19.6.1. The built-in economies. 19.6.2. Mixing and matching built-in parent objects. 19.6.3. Building your own economy. 19.7. Tutorial. 20. MATLAB Programs 441 20.1. Matlab programs. 21. References 495 22. Index 509 23. Author Index 513 24. Matlab Index 515 Acknowledgements – xii – [...]... Components of an economy Chapter 1 Introduction 1.1 Introduction This book views many apparently disparate dynamic economic models as examples of a single class of models that can be adapted and specialized to study diverse economic phenomena The class of models was created by using recent advances in (i) the theory of recursive dynamic competitive economies; 1 (ii) methods for estimating and interpreting... powerful vocabulary and a convenient structure that liberate time and energy from programming, and thereby spur creative application of linear control theory Our goal has been to create a class of models that merge recursive economic theory and with dynamic econometrics Systems of autoregressions and of mixed autogregressive, moving average processes are a dominant setting for dynamic econometrics... econometrics We constructed our economic models by adopting a version of recursive competitive theory in which an outcome of theorizing is a vector autoregression We formulated this class of models because practical difficulties of computing and estimating recursive equilibrium models still limit their use as a tool for thinking about applied problems in economic dynamics Recursive competitive equilibria were... special case of the Arrow-Debreu competitive equilibrium, both to restrict the range of outcomes possible in the 1 2 3 4 This work is summarized by Harris (1987) and Stokey, Lucas, and Prescott (1989) See Sims (1980), Hansen and Sargent (1980, 1981, 1990) For example, see Kwakernaak and Sivan (1972), and Anderson and Moore (1979) See the MATLAB manual –3– 4 Introduction Arrow-Debreu setting and to create... covariance of forecast error distributions impinge on equilibrium decision rules – it does so in a way that preserves linear equilibrium laws of motion, and retains calculation of equilibria and asset prices via simple modifications of our standard formulas These preferences are a version of those studied by Epstein and Zin ( ) and Weil ( ) Chapter 17 describes how to adapt our setup to include features of. .. vector autoregression A cost of the approach is that it does not accommodate many specifications that we would like to be able to analyze The purpose of this book is to display the versatility and tractability of our class of models Versions of a wide range of models from modern capital theory and asset pricing theory can be represented within our framework The equilibria of these models can be computed so... speed the computation of expectations of geometric sums of quadratic forms, and help to solve dynamic programming problems Chapter 11 describes alternative ways to represent demand It identifies an equivalence class of preference specifications that imply the same demand functions, and characterizes a special subset of them as canonical household preferences Canonical representations of preferences are... in typewriter font 6 You will get much more out of this book if you use and modify our programs as you read 1.3 Organization This book is organized as follows Chapter 10 describes the first order linear vector stochastic difference equation, and shows how special cases of it are formed by a variety of models of time series processes that have been studied by economists This difference equation will be used... view of equilibrium computation of prices and aggregate quantities, adequately stands in for the household of chapters 3–6 The allocations to individual consumers require additional computations, which this chapter describes Chapter 13 uses our model of preferences to represent multiple goods versions of permanent income models along the lines of Robert Hall’s (1978) We retain Hall’s specification of. .. determining the equilibrium of the model, and possibly also from parameters of measurement error processes, we are completing a key step needed to permit econometric estimation of the model’s free Organization 7 parameters Chapter 9 also treats two other topics intimately related to econometric implementation of the models; aggregation over time, and the theory of approximation of one model by another Chapter . Recursive Models of Dynamic Linear Economies Recursive Models of Dynamic Linear Economies Lars Hansen University of Chicago Thomas J. Sargent New York University and Hoover Institution c Lars. time and energy from programming, and thereby spur creative application of linear control theory. Our goal has been to create a class of models that merge recursive economic theory and with dynamic. summarized by Harris (1987) and Stokey, Lucas, and Prescott (1989). 2 See Sims (1980), Hansen and Sargent (1980, 1981, 1990). 3 For example, see Kwakernaak and Sivan (1972), and Anderson and Moore