Recursive Models of Dynamic Linear Economies

Transcription

1 Recursive Models of Dynamic Linear Economies

2 Recursive Models of Dynamic Linear Economies Lars Hansen University of Chicago Thomas J. Sargent New York University and Hoover Institution

3 Contents Acknowledgements x Preface xi Part I: Overview 1. Theory and Econometrics Introduction A Class of Economies Computer Programs Organization Recurring Mathematical Ideas. Part II: Tools 2. Linear Stochastic Difference Equations Introduction Notation and Basic Assumptions Prediction Theory Transforming Variables to Uncouple Dynamics Deterministic Seasonals Indeterministic Seasonals Univariate Autoregressive Processes Vector Autoregressions Polynomial Time Trends Martingales with Drift Covariance Stationary Processes Multivariate ARMA Processes Prediction of a Univariate First-Order ARMA Growth A Rational Expectations Model Method of Undetermined Coefficients Concluding Remarks. v

4 vi Contents 3. Efficient Computations Introduction The Optimal Linear Regulator Problem Transformations to Eliminate Discounting and Cross-Products Stability Conditions Invariant Subspace Methods P x as Lagrange Multiplier Invariant Subspace Methods Doubling Algorithm Partitioning the State Vector Periodic Optimal Linear Regulator A Periodic Doubling Algorithm Partitioning the State Vector Linear Exponential Quadratic Gaussian Control Doubling Algorithm for a Risk-Sensitive Problem. A. Concepts of Linear Control Theory. B. Symplectic Matrices. C. Alternative Forms of Riccati Equation. Part III: Components of Economies 4. Economic Environments Information Taste and Technology Shocks Production Technologies Examples of Production Technologies Other Technologies Household Technologies Examples of Household Technologies Square Summability Summary. 5. Optimal Resource Allocations Planning Problem Lagrange Multipliers Dynamic Programming Lagrange Multipliers as Gradients of Value Function Planning Problem as Linear Regulator Allocations for Five Economies Brock-Mirman (1972) or Hall (1978) Model A Growth Economy Fueled by Habit Persistence Lucas s Pure Exchange Economy An Economy with a Durable Consumption Good Computed Examples Hall s Model Higher Adjustment Costs Altered Growth Condition A Jones- Manuelli (1990) Economy Durable Consumption Goods Summary. A. Synthesizing a Linear Regulator. B. A Brock-Mirman (1972) or Hall (1978) Model. 5.B.1. Uncertainty. 5.B.2. Optimal Stationary States.

5 Contents vii 6. A Commodity Space Valuation Price Systems as Linear Functionals A One Period Model Under Certainty One Period Under Uncertainty An Infinite Number of Periods and Uncertainty Conditioning Information Lagrange Multipliers Summary. A. Mathematical Details. 7. Competitive Economies Introduction Households Type I Firms Type II Firms Competitive Equilibrium Lagrangians Household Lagrangian Type I Firm Lagrangian Type II Firm Lagrangian Equilibrium Price System Asset Pricing Term Structure of Interest Rates Re-Opening Markets Non-Gaussian Asset Prices Asset Pricing Example Exercises. Part IV: Representations and Properties 8. Statistical Representations The Kalman Filter Innovations Representation Convergence Computation of Time-Invariant Kalman Filter Factorization of Likelihood Function Initialization Assumptions Possible Non-Existence of Stationary Distribution Spectral Factorization Identity Wold and Autoregressive representations Frequency Domain Estimation Approximation Theory Aggregation Over Time Simulation Estimators. A. Initialization of Kalman Filter. B. Zeros of Characteristic Polynomial. C. Serially Correlated Measurement Errors. D. Innovations in y t+1 as Functions of w t+1 and η t+1. E. Innovations in a Permanent Income Model. 9. Canonical Household Technologies Introduction Definition of a Canonical Household Technology Dynamic Demand Functions Wealth and the Multiplier µ w Dynamic Demand System Gorman Aggregation and Engel curves Re-Opened Markets Computing Canonical Representations Basic Idea An Auxiliary Problem Induces a Canonical Representation An Operator Identity. 9.6.

6 viii Contents Becker-Murphy Model of Rational Addiction. A. Fourier Transforms. 9.A.1. Primer on z-transforms. 9.A.2. Time Reversal and Parseval s Formula. 9.A.3. One-Sided Sequences. 9.A.4. Useful Properties. 9.A.5. One-Sided Transforms. 9.A.6. Discounting. 9.A.7. Fourier Transforms. 9.A.8. Verifying Equivalent Valuations. 9.A.9. Equivalent Representations of Preferences. 9.A.10. First Term: Factorization Identity. 9.A.11. Second Term. 9.A.12. Third Term. 10. Examples Partial Equilibrium The Setup Equilibrium Investment Under Uncertainty A Housing Model Demand House Producers Cattle Cycles Mapping Cattle Farms into our Framework Preferences Technology Models of Occupational Choice and Pay A One- Occupation Model Skilled and Unskilled Workers. A. Decentralizing the Household. 11. Permanent Income Models Technology Two Implications Allocation Rules Deterministic Steady States Cointegration Constant Marginal Utility of Income Consumption Externalities. A. Exotic Tax Smoothing Models. 12. Gorman Heterogeneous Households Introduction Gorman Aggregation (Static) An Economy with Heterogeneous Consumers Allocations Consumption Sharing Rules Risk Sharing Implementing the Allocation Rule with Limited Markets. A. Computer Example Exercises. 13. Non-Gorman Heterogeneous Households Introduction Preferences and Household Technologies Production Technology A Pareto Problem Competitive Equilibrium Households Firms of Types I and II Computation of Equilibrium Candidate Equilibrium Prices A Negishi Algorithm Mongrel Aggregation Static Demand Frequency Domain Representation of Preferences A Programming Problem for Mongrel Aggregation

7 Contents ix Factoring S S Summary of Findings The Mongrel Preference Shock Process Interpretation of ŝ t Component Initial Conditions. 14. Periodic Models of Seasonality Three Models of Seasonality A Periodic Economy Asset Pricing Prediction Theory Term Structure of Interest Rates Conditional Covariograms A Stacked and Skip- Sampled System Covariances of the Stacked and Skip Sampled Process Tiao-Grupe Formula State-Space Realization of Tiao-Grupe Formula Periodic Hall Model Periodic Innovations Representations for a Periodic Model. A. Disguised Periodicity. A. MATLAB Programs References Subject Index Author Index MATLAB Index 417

8 Acknowledgements For helpful discussions and research assistance, we thank Saki Bigio, Ross Doppelt, Casey Mulligan, Martin Schneider, Thomas Tallarini, Pierre-Olivier Weill, François Velde, and Neng Wang. We thank Bilal Kamal for excellent help in proof reading and criticizing the manuscript. We thank François Velde for writing TEXmacros that we used to typeset this book. x

9 Preface In 1992, Carolyn Sargent said You are writing the second edition of your book before you have published the first. We plead guilty. We wrote ninety percent of this book between 1988 and 1994, but completed it only in Richard Blundell s invitation to deliver the Gorman Lectures at University College London convinced us to abandon our earlier intentions to publish this book only after we had implemented the many improvements that we still know are possible. For us, among the most interesting parts of this book are two chapters on different ways of aggregating heterogeneous household preferences and the senses in which they do or don t rationalize a representative household. These chapters extend ideas of W.M. Gorman to a particular dynamic setting. We attempt to honor W.M. Gorman by thinking hard about his ideas. xi

10 Part I Overview

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12 Chapter 1 Theory and Econometrics Complete market economies are all alike... Robert E. Lucas, Jr., (1989) 1.1. Introduction Economic theory can identify patterns that unite apparently diverse subjects. Consider the following list of economic models: 1. Ryoo and Rosen s (2004) partial equilibrium model of the market for engineers; 2. Rosen, Murphy, and Scheinkman s (1994) model of cattle cycles; 3. Lucas s (1978) model of asset prices; 4. Brock and Mirman s (1972) and Hall s (1978) model of the permanent income theory of consumption; 5. Time-to-build models of business cycles; 6. Siow s (1984) model of occupational choice; 7. Rosen and Topel s (1988) model of the dynamics of house prices and quantities; 8. Theories of dynamic demand curves; 9. Theories of dynamic supply curves; 10. Lucas-Prescott s (1972) model of investment under uncertainty,... and many more. These models have identical structures because all describe competitive equilibria with complete markets. This is the meaning of the remark by Robert E. Lucas, Jr., with which we have chosen to begin this book. 1 Lucas refers to the fact that complete markets models are cast in terms of a common set of objects and a common set of assumptions about how those objects fit together, namely: 1. Descriptions of flows of information over time, of endowments of resources, and of commodities that can be traded. 1 Unity goes only so far. The words comprising the... in Lucas s sentence are but each incomplete market economy is incomplete in its own individual way. 3

13 4 Theory and Econometrics 2. A technology for transforming endowments into commodities and an associated set of feasible allocations. 3. A list of people and their preferences over feasible allocations. 4. An assignment of endowments to people, a price system, and a single budget budget constraint for each person An equilibrium concept that adjusts prices to reconcile the decisions of diverse price-taking people. This book is about constructing and applying competitive equilibria for a class of linear-quadratic-gaussian dynamic economies with complete markets. For us, an economywill consist ofalist of matrices that describe peoples household technologies, their preferences over consumption services, their production technologies, and their information sets. Competitive equilibrium allocations and prices satisfy some equations that are easy to write down and solve. These competitive equilibrium outcomes have representations that are convenient to represent and estimate econometrically. Practical and analytical advantages flow from identifying an underlying structure that unites a class of economies. Practical advantages come from recognizing that apparently different applications can be formulated and estimated using the same tools simply by replacing one list of matrices with another. Analytical advantages and deeper understandings come from appreciating the roles played by key assumptions such as completeness of markets and structures of heterogeneity. 2 A single budget constraint for each person is a tell-tale sign marking a complete markets model.

14 A Class of Economies A Class of Economies We constructed our class of economies by using (i) a theory of recursive dynamic competitive economies; 3 (ii) linearoptimal controltheory; 4 (iii) methods for estimating and interpreting vector autoregressions; 5 and (iv) a computer language for rapidly manipulating linear systems. 6 Our economies have competitive equilibria with representations in terms of vector autoregressions that can be swiftly computed, simulated, and estimated econometrically. The models thus merge economic theory with dynamic econometrics. The computer language MATLAB implements the computations. It has a structure and vocabulary that economize time and effort. Better yet, the development of dynare has immensely improved, accelerated, and eased practical applications. We formulated this class of models because practical difficulties of computing and estimating more general recursive competitive equilibrium models continue to limit their use as tools for thinking about applied problems. Recursive competitive equilibria were developed as special cases of the Arrow-Debreu competitive equilibrium model in order to study applied problems in dynamic economics. Relative to the more general Arrow-Debreu setting, the great advantage of recursive competitive equilibria is that they can be computed by solving discounted dynamic programming problems. Further, under some additional conditions, a competitive equilibrium can be represented as a Markov process. When that Markov process has a unique invariant distribution, there exists a vector autoregressive representation. Thus, the theory of recursive competitive equilibria holds out the promise of making easier contact with econometric theory than did previous formulations of equilibrium theory. Two computational difficulties continue to leave some of this promise unrealized. The first is a curse of dimensionality that makes dynamic programming a costly procedure with even small numbers of state variables. The second is that after a dynamic program has been solved and an equilibrium Markov process computed, an implied vector autoregression has to be computed by applying least-squares projection formulas involving a large number of moments from the 3 This work is summarized by Harris (1987) and Stokey, Lucas, and Prescott (1989). 4 For example, see Kwakernaak and Sivan (1972), and Anderson and Moore (1979). 5 See Sims (1980) and Hansen and Sargent (1980, 1981, 1990). 6 See the MATLAB manual.

15 6 Theory and Econometrics model s invariant probability distribution. Typically, each of these computational steps can be solved only approximately. Good research along several lines has been directed at improving these approximations. 7 The need to approximate originates in the fact that for general functional forms for objective functions and constraints, even one iteration on the key functional equation of dynamic programming (named the Bellman equation after Richard Bellman) cannot be performed analytically. It so happens that the functional forms economists would most like to use are ones for which the Bellman equation cannot be iterated on analytically. Linear control theory studies the most important special class of problems for which iterations on the Bellman equation can be performed analytically, namely, problems having a quadratic objective function and a linear transition function. Application of dynamic programming leads to a system of well understood and rapidly solvable equations known as the matrix Riccati difference equation. The philosophy of this book is to swallow hard and to accept up front primitive descriptions of tastes, technology, and information that satisfy the assumptions of linear optimal control theory. This approach facilitates computing competitive equilibria that automatically take the form of a vector autoregression. A cost of the approach is that it does not accommodate specifications that we sometimes prefer. A 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. Competitive equilibria can be computed so easily that we hope that the reader will soon be thinking of new models. We provide formulas and software for the reader to experiment; and for many of our calculations, dynare offers even better software. 7 See Marcet (1988), Marcet and Marshall (1994), Judd (1996), Judd (1998), Coleman (1990), Miranda and Fackler (2004), and Tauchen (1986).

16 Organization Computer Programs In writing this book, we put ourselves under a restriction that we should supply the reader with a computer program that implements every equilibrium concept and mathematical representation. The programs are written in MATLAB, and are described throughout the book. 8 When a MATLAB program is referred to in the text, we place it in typewriter font. Similarly, all computer codes appear in typewriter font. 9 You will get much more out of this book if you use and modify our programs as you read Organization This book is organized as follows. Chapter 2 describes the first-order linear vector stochastic difference equation and shows how special cases of it can represent a variety of models of time series processes popular with economists. We use this difference equation to represent the information flowing to economic agents and also to represent competitive equilibria. Chapter 3 describes fast ways to compute equilibria. We describe doubling algorithms that accelerate computation of expectations of geometric sums of quadratic forms and solve dynamic programming problems. Chapter 4 defines an economic environment in terms of preferences of a representative agent, a technology for producing consumption and investment goods, stochastic processes of shocks to preferences and technologies, and an information structure. The stochastic processes fit into the model introduced in chapter 2, while the preferences, technology, and information structure are specified with an eye toward making competitive equilibria computable with linear control theory. Chapter 5 describes a planning problem that generates competitive equilibrium allocations. We formulate the planning problem in two ways, first as a variational problem using stochastic Lagrange multipliers, then as a dynamic 8 These programs are referred to in a special index at the end of the book. They can be downloaded from < >. 9 To run our programs, you will need MATLAB s Control Toolkit in addition to the basic MATLAB software. 10 The dynare suite of MATLAB programs is also very useful for analyzing and estimating our models.

17 8 Theory and Econometrics programming problem. We describe how to solve the dynamic programming problem with formulas from linear control theory. The solution of the planning problem is a first-order vector stochastic difference equation of the form studied in chapter 2. We also show how to use the value function for the planning problem to compute Lagrange multipliers associated with constraints on the planning problem. Chapter 6 describes a commodity space and a price system that support a competitive equilibrium. We use a formulation that lets the values that appear in agents budget constraints and objective functions be represented as conditional expectations of geometric sums of streams of future prices times quantities. Chapter 6 relates these prices to Arrow-Debreu state-contingent prices. Chapter 7 describes a decentralized economy and its competitive equilibrium. Competitive equilibrium quantities solve a planning problem. The price system can be deduced from the stochastic Lagrange multipliers associated with the chapter 5 planning problem. Chapter 8 describes links between competitive equilibria and autoregressive representations. We show how to obtain an autoregressive representation for observable variables that are error-ridden linear functions of state variables. In describing how to deduce an autoregressive representation from a competitive equilibrium and parameters of measurement error processes, we complete a key step that facilitates econometric estimation of free parameters. An autoregressive representation is naturally affiliated with a recursive representation of a likelihood function for the observable variables. More precisely, a vector autoregressive representation implements a convenient factorization of the joint density of a complete history of observables (i.e., the likelihood function) into a product of densities of time t observables conditioned on histories of those observables up to time t 1. Chapter 8 also treats two other topics intimately related to econometric implementation: aggregation over time and the theory of approximation of one model by another. Chapter 9 describes household technologies that describe the same preferences and dynamic demand functions. It characterizes a special subset of them as canonical. Canonical household technologies are useful for describing economies with heterogeneity among household s preferences because of how they align linear spaces consisting of histories consumption services, on the one hand, and consumption rates, on the other.

18 Organization 9 Chapter 10 describes some applications in the form of versions of several dynamic models that fit easily within our class of models. These include models of markets for housing, cattle, and occupational choice. Chapter 11 uses our model of preferences to represent multiple goods versions of permanent income models. We retain Robert Hall s (1978) specification of a storage technology for accumulating physical capital and also a restriction on the discount factor, depreciation rate, and gross return on capital that in Hall s simple setting made the marginal utility of consumption a martingale. In more general settings, adopting Hall s specification of the storage technology imparts a martingale to outcomes, but it is concealed in an index whose increments drive demands for multiple consumption goods that themselves are not martingales. This permanent income model forms a convenient laboratory for thinking about sources in economic theory of unit roots and co-integrating vectors. Chapter 12 describes a type of heterogeneity among households that allows us to aggregate preferences in a sense introduced by W.M. Gorman. Affine Engel curves of common slopes across agents give rise to a representative consumer. This representative consumer is easy to find, and from the point of view of computing equilibrium prices and aggregate quantities, adequately stands in for the representative household of chapters 4 7. Finding competitive equilibrium allocations to individual consumers requires additional computations that this chapter also describes. Chapter 13 describes a setting with heterogeneity among households preferences of a kind that violates the conditions for Gorman aggregation. Households Engel curves are still affine, but dispersion of their slopes prevents Gorman aggregation. However, there is another sense in which there is a representative household whose preferences are a peculiar kind of average over the preferences of different types of households. We show how to compute and interpret this preference ordering over economy-wide aggregate consumption. This non-gorman preference ordering cannot be computed until one knows the distribution of wealth evaluated at equilibrium prices, so it is less useful than the one produced by Gorman aggregation. Chapter 14 adapts our setups to include features of periodic models of seasonality studied by Osborne (1988) and Todd (1983, 1990). Appendix A is a manual of the MATLAB programs that we have prepared to implement the calculations described in this book.

19 10 Theory and Econometrics 1.5. Recurring Mathematical Ideas Duality between control problems and filtering problems underlies the finding that recursive filtering problems have the same mathematical structure as recursive formulations of linear optimal control problems. Both problems ultimately lead to matrix Riccati equations. 11 We use the duality of recursive linear optimal control and linear filtering repeatedly both in chapter 8 (for representing equilibria econometrically) and chapters 9, 12, and 13 (for representing and aggregating preferences). In chapter 8, we state a spectral factorization identity from linear control theory to characterize the link between the state-space representation for a competitive equilibrium and the vector autoregression for observables. This is by way of obtaining the innovations representation that achieves a recursive representation of a Gaussian likelihood function or quasi likelihood function. In another guise, the factorization identity is also a key tool in constructing what we call a canonical representation of a household technology in chapter 9. In more detail: 1. We use a linear state space system to represent information flows that drive shocks to preferences and technologies (chapter 2). 2. We use a linear state space system to represent observable quantities and scaled Arrow-Debreu prices associated with competitive equilibria(chapters 5 and 7). 3. We coax scaled Arrow-Debreu prices from Lagrange multipliers associated with a planning problem (chapters 5 and 7). 4. We derive formulas for scaled Arrow-Debreu prices from gradients of the value function for a planning problem (chapters 5 and 7). 5. We use another linear state space system called an innovations representation to deduce a recursive representation of a Gaussian likelihood function or quasi-likelihood function associated with competitive equilibrium quantities and scaled Arrow-Debreu prices (chapter 8). a. We use a Kalman filter to deduce an innovations representation associated with competitive equilibrium quantities and scaled Arrow-Debreu prices. 11 We expand on this theme in Hansen and Sargent (2008, ch. 4).

20 Recurring Mathematical Ideas 11 b. To build an innovations representation, we use a Kalman filter to construct a sequence of densities of time t observables conditional on a historyoftheobservablesuptotime t 1. Thissequenceofconditional densities is an essential ingredient of a recursive representation of the likelihood function (also known as the joint density of the observables over a history of length T). c. The innovations in the innovation representation are square summable linear functions of the history of the observables. Thus, the innovations representation is said to be invertible, while the original state-state space representation is in general not invertible. d. The limiting time-invariant innovations representation associated with a fixed point of the Kalman filtering equations implements a spectral factorization identity. 6. Intimate technical relationships prevail between the innovations representation of chapter 8 and what in chapter 9 we call a canonical representation of preferences. a. An innovations representation is invertible in the sense that it expresses the innovations in observables at time t as square-summable linear combinations of the history up to time t. b. A canonical representation of a household technology is invertible in the sense that it can be used to express a flow of consumption services as a square-summable linear combination of the history of consumption services. c. A canonical representation of a household technology allows us to express dynamic demand curves for consumption flows. d. A canonical representation of a household technology can be constructed using a version of the same spectral factorization identity encountered in chapter We describe two sets of conditions that allow us to aggregate heterogeneous consumers into a representative consumer. a. Chapter 12 describes a dynamic version of Gorman s (1953) conditions for aggregation, namely, that Engel curves be linear with common slopes across consumers. These conditions allow us to incorporate settings with heterogeneity in preference shocks and endowment processes, but they require that households share a common household

21 12 Theory and Econometrics technology for converting flows of purchases of consumption goods into consumption services. b. When the chapter 12 conditions for Gorman aggregation hold, it is possible to compute competitive equilibrium prices and aggregate quantities without simultaneously computing individual consumption allocations. Without knowing allocations across heterogeneous agents, knowing prices and aggregate quantities is enough for many macroeconomic applications. 12 c. Chapter 13 describes a weaker (non-gorman) sense in which there exists a representative consumer. Here, consumers have diverse household technologies for converting flows of consumption goods into the consumption services that enter their utility functions. The household technology that converts aggregate consumption flows into the service flows valued by a representative consumer is a weighted average of the household technologies of individual consumers, an average best expressed in the frequency domain. To construct a canonical representation of the household technology of the representative consumer requires using the spectral factorization identity. d. To construct a non-gorman representative agent requires knowing the vector of Pareto weights associated with a competitive equilibrium allocation. It has to be constructed simultaneously with and not before finding a competitive equilibrium allocation. Therefore, non-gorman aggregation is less useful than Gorman aggregation for practical computations of competitive equilibrium prices and aggregate quantities. 8. The spectral factorization identity makes yet another appearance in chapter 14 where we study models with hidden periodicity. A population vector autoregression, not conditioned on the period, can be constructed by an appropriate application of the factorization identity to an appropriate average of as many conditional spectral densities as there are seasons. 9. Much of our reasoning and mathematics extend to risk-sensitive and robust economies that allow households to express their distrust of an approximating statistical model. Hansen and Sargent (2008) describes some of these extensions James Tobin once defined macroeconomics as a discipline that neglects distribution effects. 13 See especially chapters 12 and 13 of Hansen and Sargent (2008).

22 Part II Tools

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24 Chapter 2 Linear Stochastic Difference Equations 2.1. Introduction This chapter introduces the vector first-order linear stochastic difference equation. 1 We use it first to represent information flowing to economic agents, then again to represent competitive equilibria. The vector first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric application. Ease of analysis has prompted us to adopt economic specifications that cause competitive equilibria to have representations as vector first-order linear stochastic difference equations. Because it expresses next period s vector of state variables as a linear function of this period s state vector and a vector of random disturbances, a vector first-order vector stochastic difference equation is recursive. Disturbances that form a martingale difference sequence are basic building blocks used to construct time series. Martingale difference sequences are easy to forecast, a fact that delivers convenient recursive formulas for optimal predictions of time series Notation and Basic Assumptions Let {x t : t = 1,2,...} be a sequence of n-dimensional random vectors, i.e. an n-dimensional stochastic process. Let {w t : t = 1,2,...} be a sequence of N-dimensional random vectors. We shall express x t as the sum of two terms. The first is a moving average of past w t s. The second describes the effects of an initial condition. The {w t } generates a sequence of information sets {J t : t = 0,1,...}. Let J 0 be generated by x 0 and J t be generated by 1 See Hansen and Sargent (2014) and Ljungqvist and Sargent(2012, ch. 2)forpresentations of related material and extensions. 15

25 16 Linear Stochastic Difference Equations x 0,w 1,...,w t, which means that J t consists of the set of all measurable functionsof {x 0,w 1,...,w t }. 2 Theprocess {w t+1 } t=0 isassumedtobeamartingale difference sequence adapted to this sequence of information sets. Definition 1: The sequence {w t : t = 1,2,...} is said to be a martingale difference sequence adapted to {J t : t = 0,1,...} if E(w t+1 J t ) = 0 for t = 0,1,.... In addition, we assume that the {w t : t = 1,2,...} process is conditionally homoskedastic, a phrase whose meaning is conveyed by Definition 2: The sequence {w t : t = 1,2,...} is said to be conditionally homoskedastic if E(w t+1 w t+1 J t) = I for t = 0,1,.... It is an implication of the law of iterated expectations that {w t : t = 1,2,...} is a sequence of (unconditional) mean zero, serially uncorrelated random vectors. 3 In addition, the entries of w t are assumed to be mutually uncorrelated. The process {x t : t = 1,2,...} is constructed recursively using an initial random vector x 0 and a time-invariant law of motion: x t+1 = Ax t +Cw t+1, for t = 0,1,..., (2.2.1) where A is an n by n matrix and C is an n by N matrix. Representation (2.2.1) will be a workhorse in this book. First, we will use it to model the information upon which economic agents base their decisions. Information will consist of variables that drive shocks to preferences and to technologies. Second, we shall specify the economic problems faced by the agents in our models and the economic arrangement through which agents decisions 2 The phrase J0 is generated by x 0 means that J 0 can be expressed as a measurable function of x 0. 3 Where φ1 and φ 2 are information sets with φ 1 φ 2, and x is a random variable, the law of iterated expectations states that E(x φ 1 ) = E(E(x φ 2 ) φ 1 ). Letting φ 1 be the information set corresponding to no observations on any random variables, letting φ 2 = J t, and applying this law to the process {w t }, we obtain E ( w t+1 ) = E ( E ( wt+1 J t )) = E(0) = 0.

26 Prediction Theory 17 are reconciled (competitive equilibrium) so that the state of the economy has a representation of the form ( 2.2.1) Prediction Theory A tractable theory of prediction is associated with ( 2.2.1). We use this theory extensively both in computing a competitive equilibrium and in representing that equilibrium in the form of (2.2.1). The optimal forecast of x t+1 given current information is E(x t+1 J t ) = Ax t, (2.3.1) and the one-step-ahead forecast error is x t+1 E(x t+1 J t ) = Cw t+1. (2.3.2) The covariance matrix of x t+1 conditioned on J t is E(x t+1 E(x t+1 J t ))(x t+1 E(x t+1 J t )) = CC. (2.3.3) A nonrecursive expression for x t as a function of x 0,w 1,w 2,...,w t can be found by using (2.2.1) repeatedly to obtain x t = Ax t 1 +Cw t = A 2 x t 2 +ACw t 1 +Cw t = A τ Cw t τ ]+A t x 0. [t 1 τ=0 (2.3.4) Representation (2.3.4) is one type of moving-average representation. It expresses {x t : t = 1,2,...} as a linear function of current and past values of the process {w t : t = 1,2,...} and an initial condition x 0. 4 The list ofmovingaverage coefficients {A τ C : τ = 0,1,...} in representation (2.3.4) is often called an 4 Slutsky (1937) argued that business cycle fluctuations could be approximated by moving average processes. Sims (1980) showed that a fruitful way to summarize correlations between time series is to calculate an impulse response function. In chapter 8, we study the relationship between the impulse response functions based on the vector autoregressions calculated by Sims (1980) and the impulse response function associated with ( 2.3.4).

27 18 Linear Stochastic Difference Equations impulse response function. An impulse response function depicts the response of current and future values of {x t } to a random shock w t. In representation (2.3.4), the impulse response function is given by entries of the vector sequence {A τ C : τ = 0,1,...}. 5 Shift (2.3.4) forward in time: j 1 x t+j = A s Cw t+j s +A j x t. (2.3.5) s=0 Projecting both sides of (2.3.5) on the information set {x 0,w t,w t 1,...,w 1 } gives 6 E t x t+j = A j x t. (2.3.6) where E t ( ) E[( ) x 0,w t,w t 1,...,w 1 ] = E( ) J t, and x t isin J t. Equation ( 2.3.6) gives the optimal j-step-ahead prediction. It is useful to obtain the covariance matrix of the j-step-ahead prediction error j 1 x t+j E t x t+j = A s Cw t s+j. (2.3.7) Evidently, s=0 E t (x t+j E t x t+j ) (x t+j E t x t+j ) j 1 = A k CC A k v j. k=0 Note that v j defined in (2.3.8a) can be calculated recursively via (2.3.8a) v 1 = CC v j = CC +Av j 1 A, j 2. (2.3.8b) For j 1, v j is the conditional covariance matrix of the errors in forecasting x t+j on the basis of time t information x t. To decompose these covariances into parts attributable to the individual components of w t, we let i τ be an 5 Given matrices A and C, the impulse response function can be calculated using the MATLAB program dimpulse.m. 6 For an elementary discussion of linear least squares projections, see Sargent (1987b, ch. XI).

28 Transforming Variables to Uncouple Dynamics 19 N-dimensional column vector of zeroes except in position τ, where there is a one. Define a matrix υ j,τ by j 1 υ j,τ = A k Ci τ i τc A k. k=0 (2.3.8c) Note that N τ=1 i τi τ = I, so that from (2.3.8a) and (2.3.8c) we have N υ j,τ = υ j. τ=1 Evidently, the matrices {υ j,τ,τ = 1,...,N} give an orthogonal decomposition of the covariance matrix of j-step-ahead prediction errors into the parts attributable to each of the components τ = 1,...,N. 7 The innovation accounting methods of Sims (1980) are based on ( 2.3.8). Sims recommends computing the matrices v j,τ in (2.3.8) for a sequence j = 0,1,2,... This sequence represents the effects of components of the shock process w t on the covariance of j-step ahead prediction errors for each series in x t Transforming Variables to Uncouple Dynamics It is sometimes useful to uncouple the dynamics of x t by using the distinct eigenvalues of the matrix A. The Jordan decomposition of the matrix A is A = TDT 1, (2.4.1) where T isanonsingularmatrixand D isanothermatrixtobeconstructed. The eigenvalues of A are the zeroes of the polynomial det(ζi A). This polynomial has n zeroes because A is n by n. Not all of these zeroes are necessarily distinct. 8 Suppose that there are m n distinct zeroes of this polynomial, 7 For given matrices A and C, the matrices vj,τ and v j are calculated by the MATLAB program evardec.m. 8 In the case in which the eigenvalues of A are distinct, D is taken to be the diagonal matrix whose entries are the eigenvalues and T is the matrix of eigenvectors corresponding to those eigenvalues.

29 20 Linear Stochastic Difference Equations denoted δ 1,δ 2,...,δ m. For each δ j, we construct a matrix D j that has the same dimension as the number of zeroes of det(ζi A) that equal δ j. The diagonal entries of D j are δ j and the entries in the single diagonal row above the main diagonal are all either zero or one. The remaining entries of D j are zero. Then the matrix D is block diagonal with D j in the j th diagonal block. Transform the state vector x t as follows: Substituting into ( 2.2.1) gives x t = T 1 x t. (2.4.2) x t+1 = Dx t +T 1 Cw t+1. (2.4.3) Since D is block diagonal, we can partition x t according to the diagonal blocks of D or, equivalently, according to the distinct eigenvalues of A. In the law of motion (2.4.3), partition j of x t+1 is linked only to partition j of x t. In this sense, the dynamics of system ( 2.4.3) are uncoupled. To calculate multi-period forecasts and dynamic multipliers, we must raise the matrix A to integer powers (see (2.3.6)). It is straightforward to verify that A τ = T(D τ )T 1. (2.4.4) Since D is block diagonal, D τ is also block diagonal, where block j is just (D j ) τ. The matrix (D j ) τ is upper triangular with δj τ on the diagonal, with all entries of the k th upper right diagonal given by (δ j ) τ k τ!/[k!(τ k)!] for 0 k τ, (2.4.5) and zeroes elsewhere. Consequently, raising D to an integer power involves raising the eigenvalues to integer powers. Some of the eigenvalues of A may be complex. In this case, it is convenient to use the polar decomposition of the eigenvalues. Write eigenvalue δ j in polar form as where ρ j = δ j. Then δ j = ρ j exp(iθ j ) = ρ j [cos(θ j )+isin(θ j )] (2.4.6) δ τ j = (ρ j ) τ exp(iτθ j ) = (ρ j ) τ [cos(τθ j )+isin(τθ j )]. (2.4.7)

30 Transforming Variables to Uncouple Dynamics 21 We shall often assume that ρ j is less than or equal to one, which rules out instability in the dynamics. Whenever ρ j is strictly less than one, the term (ρ j ) τ decays to zero as τ. When θ j is different from zero, eigenvalue j induces an oscillatory component with period (2π/ θ j ). Next we consider some examples of processes that can be represented as (2.2.1) Deterministic Seasonals Wecanuse(2.2.1)torepresent y t = y t 4. Let n = 4,C = 0, x t = (y t,y t 1,y t 2,y t 3 ),x 0 = ( ), A = , C = 0 0. (2.4.8) Here the A matrix has four distinct eigenvalues and the absolute values of each of these eigenvalues is one. Two eigenvalues are real (1, 1) and two are imaginary (i, i), and so have period four. 9 The resulting sequence {x t : t = 1,2,...} oscillates deterministically with period four. It can be used to model deterministic seasonals in quarterly time series Indeterministic Seasonals We want to use (2.2.1) to represent y t = α 4 y t 4 +w t, (2.4.9) where w t is a martingale difference sequence and α 4 1. We define x t = [y t, y t 1, y t 2, y t 3 ], n = 4, α 4 1 A = , C = For example, note that from representation (2.4.6), i = exp(π/2) + isin(π/2), so the period associated with i is 2π π 2 = 4.

31 22 Linear Stochastic Difference Equations With these definitions, (2.2.1) represents (2.4.9). This model displays an indeterministic seasonal. Realizations of ( 2.4.9) display recurrent, but aperiodic, seasonal fluctuations Univariate Autoregressive Processes We can use (2.2.1) to represent the model y t = α 1 y t 1 +α 2 y t 2 +α 3 y t 3 +α 4 y t 4 +w t, (2.4.10) where w t isamartingaledifferencesequence. Weset n = 4,x t = [y t y t 1 y t 2 y t 3 ], A = α 1 α 2 α 3 α , C = The matrix A has the form of the companion matrix to the vector [α 1 α 2 α 3 α 4 ] Vector Autoregressions Reinterpret (2.4.10) as a vector process in which y t is a (k 1) vector, α j a (k k) matrix, and w t a k 1 martingale difference sequence. Then (2.4.10) is termed a vector autoregression. To map this into (2.2.1), we set n = k 4, A = α 1 α 2 α 3 α 4 I I I 0 where I is the (k k) identity matrix., C = I 0 0 0

32 Transforming Variables to Uncouple Dynamics Polynomial Time Trends Let n = 2,x 0 = [0 1], and A = [ ] 1 1, C = 0 1 [ ] 0. (2.4.11) 0 Notice that D = A in the Jordan decomposition of A. It follows from (2.4.5) that [ ] 1 t A t =. (2.4.12) 0 1 Hence x t = [t 1], so that the first component of x t is a linear time trend and the second component is a constant. It is also possible to use (2.2.1) to representpolynomial trends of anyorder. For instance, let n = 3,C = 0,x 0 = [0 0 1], and A = (2.4.13) Again, A = D in the Jordan decomposition of A. It follows from (2.4.5) that 1 t t(t 1)/2 A t = 0 1 t. (2.4.14) Then x t = [t(t 1)/2 t 1], so that x t contains linear and quadratic time trends.

33 24 Linear Stochastic Difference Equations Martingales with Drift We modify the linear time trend example [ by ] making C[ nonzero. ] Suppose that t N is one and C = [1 0]. Since A = and A t =, it follows that A τ C = [ ] 1. (2.4.15) 0 Substituting into the moving-average representation ( 2.3.4), we obtain (2.25) t 1 x 1t = w t τ +[1 t]x 0 τ=0 where x 1t is the first entry of x t. The first term on the right side of the preceding equation is a cumulated sum of martingale differences, and is called a martingale, while the second term is a translated linear function of time Covariance Stationary Processes Next we consider specifications of x 0 and A that imply that the first two unconditional moments of {x t : t = 1,2,...} are replicated over time. Let A satisfy [ ] A11 A 12 A =, (2.4.16) 0 1 where A 11 is an (n 1) (n 1) matrix with eigenvalues that have moduli strictly less than one and A 12 is an (n 1) 1 column vector. In addition, let C = [C 1 0]. We partition x t = [x 1t x 2t ], where x 1t has n 1 entries. It follows from (2.2.1) that x 1t+1 = A 11 x 1t +A 12 x 2t +C 1 w t+1 (2.4.17) x 2t+1 = x 2t. (2.4.18) By construction, the second component, x 2t, simply replicates itself over time. For convenience, take x 20 = 1 so that x 2t = 1 for t = 1,2,... We can use (2.4.17) to compute the first two moments of x 1t. Let µ t = Ex 1t. Taking unconditional expectations on both sides of (2.4.17) gives µ t+1 = A 11 µ t +A 12. (2.4.19)

34 Transforming Variables to Uncouple Dynamics 25 We can solve the nonstochastic difference equation ( ) for the stationary value of µ t. Define µ as the stationary value of µ t, and substitute µ for µ t and µ t+1 in (2.4.19). Solving for µ gives µ = (I A 11 ) 1 A 12. Therefore, if Ex 10 = (I A 11 ) 1 A 12, (2.4.20) then Ex 1t will be constant over time and equal to the value on the right side of (2.4.20). Further, if the eigenvalues of A 11 are less than unity in modulus, then starting from any initial value of µ 0, µ t will converge to the stationary value (I A 11 ) 1 A 12. Next we use (2.4.17) to compute the unconditional covariances of x t. Subtracting (2.4.19) from (2.4.17) gives From (2.4.21) it follows that (x 1t+1 µ t+1 ) = A 11 (x 1t µ t )+C 1 w t+1. (2.4.21) (x 1t+1 µ t+1 )(x 1t+1 µ t+1 ) = A 11 (x 1t µ t )(x 1t µ t ) A 11 +C 1 w t+1 w t+1c 1 +C 1 w t+1 (x 1t µ t ) A 11 +A 11 (x 1t µ t )w t+1c 1. The law of iterated expectations implies that w t+1 is orthogonal to (x 1t µ t ). Therefore, taking expectations on both sides of the above equation gives V t+1 = A 11 V t A 11 +C 1C 1, where V t E(x 1t µ t )(x 1t µ t ). Evidently, the stationary value V of the covariance matrix V t must satisfy V = A 11 VA 11 +C 1 C 1. (2.4.22) It is straightforward to verify that V is a solution of (2.4.22) if and only if V = j=0 A j 11 C 1C 1 Aj 11. (2.4.23) The infinite sum (2.4.23) converges when all eigenvalues of A 11 are less in modulus than unity. 10 If the covariance matrix of x 10 is V and the mean of 10 Equation (2.4.22) is known as the discrete Lyapunov equation. Given the matrices A11 and C 1, this equation is solved by the MATLAB programs dlyap.m and doublej.m.

35 26 Linear Stochastic Difference Equations x 10 is (I A 11 ) 1 A 12, then the covariance and mean of x 1t remain constant over time. In this case, the process is said to be covariance stationary. If the eigenvalues of A 11 are all less than unity in modulus, then V t V as t, starting from any initial value V 0. From (2.3.8) and (2.4.23), notice that if all eigenvalues of A 11 are less than unity in modulus, then lim j v j = V. That is, the covariance matrix of j-step ahead forecast errors converges to the unconditional covariance matrix of x as the horizon j goes to infinity. 11 The matrix V can be decomposed to display contributions of each entry of the process {w t }. Let ι τ be an N-dimensional column vector of zeroes except in position τ, where there is a one. Then I = N ι τ ι τ. (2.4.24) τ=1 Define Ṽ τ (A 11 ) j C 1 ι τ ι τc 1(A 11 ) j. (2.4.25) j=o Wehave,byanalogyto(2.4.22)and(2.4.23), that Ṽτ satisfies Ṽτ = A 11 Ṽ τ A 11 + C 1 i τ i τ C 1. In light of (2.4.24), (2.4.25), and (2.4.23), we have that V = N Ṽ τ. (2.4.26) τ=1 The matrix Ṽτ is the contribution to V of the τ th component of the process {w t : t = 1,2,...}. Hence, (2.4.26) gives a decomposition of the covariance matrix V into parts attributable to each of the underlying economic shocks. Next, consider the autocovariances of {x t : t = 1,2,...}. From the law of iterated expectations, it follows that E[(x 1t+τ µ)(x 1t µ) ] = E{E[(x 1t+τ µ) J t ](x 1t µ) } = E[A τ 11(x 1t µ)(x 1t µ) ] = A τ 11V. (2.4.27) 11 The doubling algorithm described in chapter 3 can be used to compute the solution of ( ) via iterations that approximate ( ). The algorithm is implemented in the MATLAB programs doublej.m and doublej2.m.

36 Transforming Variables to Uncouple Dynamics 27 Notice that this expected cross-product or autocovariance does not depend on calendartime, onlyonthegap τ betweenthetimeindices. 12 Independencefrom calendar time of means, covariances, and autocovariances defines covariance stationary processes. For the particular class of processes we are considering, if the covariance matrix does not depend on calendar time, then none of the autocovariance matrices does Multivariate ARMA Processes Specification (2.2.1) assumes that x t contains all the information available at time t to forecast x t+1. In many applications, vector time series are modelled as multivariate autoregressive moving-average (ARMA) processes. Let y t be a vector stochastic process. An ARMA process {y t : t = 1,2,...} has a representation: y t = α 1 y t 1 +α 2 y t 2 + +α k y t k +γ 0 w t +γ 1 w t 1 + +γ k w t k. (2.4.28) where E[w t y t 1,y t 2, y t k+1,w t 1,w t 2, w t k+1 ] = 0. The requirement that the same number of lags of y enter (2.4.28) as the number of lags of w is not restrictive because some coefficients can be set to zero. Hence, we can think of k as being the greater of the two lag lengths. A representation such as (2.4.28) can be shown to satisfy (2.2.1). To see this, we define x t = y t α 2 y t 1 +α 3 y t 2 +α k y t k+1 +γ 1 w t +γ 2 w t 1 +γ k 1 w t k+2 +γ k w t k+1 α 3 y t 1 +α k y t k+2 +γ 2 w t +γ k 1 w t k+3 +γ k w t k+2. α k y t 1 +γ k 1 w t +γ k w t 1 γ k w t (2.4.29) 12 Equation (2.4.27) shows that the matrix autocovariogram of x1t (i.e., Γ τ E[(x1t+τ µ)(x 1t µ) ] taken as a function of τ) satisfies the nonrandom difference equation Γ t+1 = A 11 Γ t.