Numerical Strategies for Solving the Nonlinear Rational Expectations Commodity Market Model

In this paper, I compare the accuracy, efficiency and stability of different numerical strategies for computing approximate solutions to the nonlinear rational expectations commodity market model. I find that polynomial and spline function collocation methods are superior to the space discretization, linearization and least squares curve-fitting methods that have been preferred by economists in the past.     

Analytic Derivatives for Linear Rational Expectations Models

This paper sets out the analytic solution for the calculation of exact derivatives in linear rational expectations models with reference to the optimal simple rule problem. We argue that there are substantial computational advantages of using analytic derivatives and compare the likely computational costs of using approximate and exact derivatives when calculating optimal coefficients for simple feedback rules. A specific algorithm for finite time optimization is also outlined, which will reduce the computational time required and is simple to implement. We discuss modifications to allow for stochastic models.     

Optimal Control and Stochastic Simulation of Large Nonlinear Models with Rational Expectations

This paper presents a computationally fesible procedure for the optimalcontrol and stochastic simulation of large nonlinear models with rationalexpectations under the assumption of certainty equivalence.     

Solving Linear Rational Expectations Models

We describe methods for solving general linear rational expectations models in continuous or discrete timing with or without exogenous variables. The methods are based on matrix eigenvalue decompositions.     

Solution of Nonlinear Rational Expectations Models with Applications toFinite-Horizon Life-Cycle Models of Consumption

This paper considers the solution of nonlinear rationalexpectations models resulting from the optimality conditions of afinite-horizon intertemporal optimization problem satisfying Bellman'sprinciple of optimality (and possibly involving inequality constraints). Abackward recursive procedure is used to characterize and solve thetime-varying optimal decision rules generally associated with these models.At each stage of these backward recursions, either an analytical ornumerical solution of the optimality conditions is required. When ananalytical solution is not possible, a minimum weighted residual approach isused. The solution technique is illustrated using a life-cycle model ofconsumption under labor income and interest rate uncertainties (and possiblyinvolving liquidity constraints). Approximate numerical solutions areprovided and compared with certainty-equivalent solutions and, whenpossible, with exact solutions.     

Solving and Estimating Dynamic Models under Rational Expectations

This paper combines recent developments in methods for solving and estimatingrationalexpectations dynamic models. These developments are applied to a model oflabor-market search,where firms operate under uncertainty. We assess the ability of the structuralmodel to mimicnonlinear features found in the data. The solution to the model is obtainedusing a method ofweighted residuals. The model is then estimated using an auxiliary modeltechnique. Our resultsindicate that the simple theoretical representation of the labor market wepropose is able tomatch the overall behavior of US hours worked along various dimensions. Beyondthey show theusefulness of this combined approach to study dynamic models under rationalexpectations.     

A Homotopy Approach to Solving Nonlinear Rational Expectation Problems

Many numerical methods have been developed in an attempt to find solutions to nonlinear rational expectations models. Because these algorithms are numerical in nature, they rely heavily on computing power and take sizeable cycles to solve. In this paper we present a numerical tool known as homotopy theory that can be applied to these methods. Homotopy theory reduces the computing time associated with an iterative algorithm by using a rational expectation problem with known solutions and transforming it into the problem at hand. If this transformation is performed slowly, homotopy theory also helps the global convergence properties of the numerical algorithm. We apply homotopy theory to Den Haan and Marcet's Parameterized Expectation Approach to show how homotopies improve the computing speed and global convergence properties of this algorithm.     

Information Aggregation in Double Auctions: Rational Expectations and the Winner's Curse

This paper inquires about the ability of double auction institutions to aggregate information in the context of a common value information structure that is known to produce the winner's curse in sealed bid environments. While many fundamental features of the economic trading mechanism are different from those studied in the context of sealed bids, the pattern of information distributed to the population of traders is the same. This gives us an opportunity to determine if the behaviors reported in sealed bid environments can be detected in the more active market environment. As such, the experiments are also a test of the robustness of earlier experiments that demonstrate that in economies with homogeneous preferences single compound securities organized by double auctions are able to aggregate information. The basic result is that a severe winner's curse is not observed. The irrationality observed in sealed bids does not extend itself to the double auction environment. Information aggregation is observed and the rational expectations model receives support.     

Solving Rational-Expectations Models through the Anderson-Moore Algorithm: An Introduction to the Matlab Implementation

The Anderson-Moore algorithm provides a well-established solution method for systems of linear rational expectations equations. The purpose of this paper is to support a wider use of the algorithm by describing two sets of Matlab routines that allow its practical implementation. The emphasis is on the structures that should be modified to tailor the programs to one’s needs. I present the application of the algorithm for the solution of a version of [Coenen, G. and V. Wieland, ECB Working Paper, No. 30 (2000)]’s macroeconometric model of the Euro area.JEL Classification: C88; E17; C63     

Using Parallelization to Solve a Macroeconomic Model: A Parallel Parameterized Expectations Algorithm

Solving nonlinear macroeconomic models with rational expectations can be time-consuming. This paper shows how the parameterized expectations algorithm (PEA) can be parallelized to reduce the time needed to solve a simple model by more than 80%. The general idea of using parallelization applies naturally to other algorithms, as well. This paper is illustrative of the speedup that can be obtained, and it provides computer code that may serve as an example for parallelization of other algorithms. For those who would like to use the parallelized PEA, the implementation does not confront end users with the details of parallelization. To solve a model, it is only necessary to provide ordinary serial code that simulates data from the model. All needed code is available, on a standalone basis, or pre-installed on ParallelKnoppix (Creel, J Appl Economet 22:215–223, 2007).      

The Linearisation and Optimal Control of Large Non-Linear Rational Expectations Models by Persistent Excitation

Since the onset of the rational expectations revolution in macroeconomics some 30 or more years ago, a variety of techniques have evolved for the solution of rational expectations models. The first generation of methods were for linear models starting with the method of Blanchard and Kahn (1980). Because the models are large and usually non-linear, methods for solving (and optimising) such models have evolved in parallel (Holly and Zarrop, 1983; Finan and Tetlow, 1999; Fair, 2003). In contrast to recent methods that apply secon-order approximations (Schmitt-Grohe and Uribe, 2004; Sims, 2002b) in this paper we describe some computationally simple methods for linearising a non-linear model with rational expectations using persistent excitation. Each instrument, exogenous variable and expectational term is excited with a white noise process. Given superimposition, each input process is orthogonal so each equation can be estimated by OLS. Once the linear form is obtained and the time-consistent optimal feedback rule computed by dynamic programming, we apply the rational expectations solution of Anderson and Moore (1985) which is particularly suited when the leading structural matrix is singular. We apply the method to a nonlinear model of the UK Economy and report a series of impulse responses for output, inflation, the exchange rate and the short term interest rate.     

Is It Worth Refining Linear Approximations to Non-Linear Rational Expectations Models?

We characterize the balanced growth path of the basic neoclassical growth economy using standard numerical solution methods which solve a linear or log-linear approximation to the economic model, as well as methods which preserve the nonlinearity in the original model. We also apply the same methods adding indivisible labor to the basic model, and to a monetary version of that economy, subject to a cash-in-advance constraint. In a unified framework, we show that log-linear approximations should generally be preferred to linear approximations. We also provide evidence that preserving the original nonlinear structure of the model when computing the numerical solution generally yields minor gains in accuracy. Methods that use either a linear or a log-linear approximation to the model can produce solutions as accurate as the parameterized expectations method. However, in extreme parametric cases, the solution may be rather sensible to small numerical errors, and even a log-linear approximation may then be inappropriate. Methods using the nonlinear structure of the original model can then perform significantly better.     

A Log-Linear Homotopy Approach to Initialize the Parameterized Expectations Algorithm

In this paper I present a proposal to obtain appropriate initial conditions while solving general equilibrium rational expectations models with the Parameterized Expectations Algorithm. The proposal is based on a log-linear approximation for the model under study, so that it can be a particular variant of the homotopy approach. The main advantages of the proposal are: (i) it guarantees the ergodicity of the initial time series used as an input to the Parameterized Expectations Algorithm; (ii) it performs well in regard to the speed of convergence when compared to some homotopy alternatives; (iii) it is easy to implement. The claimed advantages are successfully illustrated in the framework of the Cooley and Hansen (1989) model with indivisible labor and money demand motivated via a cash-in-advance constraint, as compared to a procedure based on the standard implementation of homotopy principles.     

Rational Error Correction

Under general conditions, linear decision rules of agents with rational expectations are equivalent to restricted error corrections. However, empirical rejections of rational expectation restrictions are the rule, rather than the exception, in macroeconomics. Rejections often are conditioned on the assumption that agents aim to smooth only the levels of actions or are subject to geometric random delays. Generalizations of dynamic frictions on agent activities are suggested that yield closed-form, higher-order decision rules with improved statistical fits and infrequent rejections of rational expectations restrictions. Properties of these generalized `rational' error corrections are illustrated for producer pricing in manufacturing industries.     

Solving Rational Expectations Models with Informational Subperiods: A Perturbation Approach

This paper presents an algorithm to solve up to the second order of approximation rational expectations models with informational subperiods, and provides simple examples to demonstrate how the algorithm works.     

Nonlinear Versus Linear Learning Devices: A Procedural Perspective

We provide a discussion of bounded rationality learning behind traditional learning mechanisms, i.e., Recursive Ordinary Least Squares and Bayesian Learning . These mechanisms lack for many reasons a behavioral interpretation and, following the Simon criticism, they appear to be substantively rational. In this paper, analyzing the Cagan model, we explore two learning mechanisms which appear to be more plausible from a behavioral point of view and somehow procedurally rational: Least Mean Squares learning for linear models and Back Propagation for Artificial Neural Networks . The two algorithms look for a minimum of the variance of the error forecasting by means of a steepest descent gradient procedure. The analysis of the Cagan model shows an interesting result: non-convergence of learning to the Rational Expectations Equilibrium is not due to the restriction to linear learning devices; also Back Propagation learning for Artificial Neural Networks may fail to converge to the Rational Expectations Equilibrium of the model.     

A Simple Solver for Linear Equations Containing Nonlinear Operators

This paper presents a simple equation solver. The solver finds solutions for sets of linear equations extended with several nonlinear operators, including integer division and modulus, sign extension, and bit slicing. The solver uses a new technique called {\em balancing}, which can eliminate some nonlinear operators from a set of equations before applying Gaussian elimination. The solver's principal advantages are its simplicity and its ability to handle some nonlinear operators, including nonlinear functions of more than one variable. The solver is part of an application generator that provides encoding and decoding of machine instructions based on equational specifications. The solver is presented not as pseudo code but as a literate program, which guarantees that the code shown in the paper is the same code that is actually used. Using real code exposes more detail than using pseudocode, but literate-programming techniques help manage the detail. The detail should benefit readers who want to implement their own solvers based on the techniques presented here.     

A New Time-Based Iterative Solver for Linear Standing-Wave Problems

We explore time-based solvers for linear standing-wave problems, especially the oscillatory Helmholtz equation. Here, we show how to accelerate the convergence properties of timestepping. We introduce a new time-based solver that we call phase-adjusted time-averaging (PATA), which we couple to timestepping to form the PATA-TS solver. Numerical experiments indicate that the PATA-TS solver is faster than the PATA solver and timestepping by a factor of 1.2 and 1.5 or more, respectively. We also explain why the PATA-TS solver is robust, efficient, and easy to program for a variety of practical applications.