Linear Rational Expectations Research Articles

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.

Winding number criterion for existence and uniqueness of equilibrium in linear rational expectations models

This paper develops a simple geometric criterion for the existence and uniqueness of equilibrium in linear rational expectations (LRE) models. The class of models considered is very general and includes models with infinitely many lags and/or leads of endogenous variables. I show that the criterion is tightly related to the familiar Blanchard and Kahn (Econometrica 48(5) (1980) 1305), Hansen and Sargent (Rational Expectations and Econometric Practice, 1981), and Whiteman (Linear Rational Expectations Models: A User's Guide, 1983) conditions for the existence and uniqueness of equilibrium in LRE models. The new criterion can be used to extend the policy rule robustness analysis of Onatski and Stock (Macroeconomic Dynamics 6(1) (2002) 85) from purely backward-looking models to forward-looking models.

Convergence Rates to Purchasing Power Parity for Traded and Nontraded Goods

This article estimates the speed of the adjustment coefficient in structural error-correction models. We use a system method for real exchange rates of traded and nontraded goods by combining a single-equation method with Hansen and Sargent's instrumental variables methods for linear rational expectations models. We apply these methods to a modified version of Mussa's model. Our results show that the half-lives of purchasing power parity deviations for the rates of traded goods are less than 1 year and are shorter than those for general price and for nontraded goods in most cases, implying a faster adjustment speed to parity.

Computing Second-Order-Accurate Solutions for Rational Expectation Models Using Linear Solution Methods

This paper shows how to compute a second-order-accurate solution of a non-linear rational expectation model using algorithms developed for the solution of linear rational expectation models. The result is a state-space representation for the realized values of the variables of the model. This state-space representation can easily be used to compute impulse responses as well as conditional and unconditional forecasts.

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.

Robust Expectations and Uncertain Models - A Robust Control Approach with Application to the New Keynesian Economy

This paper extends Svensson and Woodford's (2003) partial information framework by allowing the private agents to achieve robustness against incomplete information about the structure of the economy by distorting their expectations in a particular direction. It shows how a linear rational expectations equilibrium under concern for robustness can be solved by exploiting the recursive structure of the problem and appropriately modifying the Bellman equations in their framework. The standard Kalman filter is then used for information updating under imperfect measurement of the state variables. The standard New Keynesian model is used for illustrating how concern for modelling errors interacts with imperfect information. Agents achieve robustness by simultaneously over-estimating the persistence of exogenous shocks, but underestimating the policy response to the output gap. This under-estimation, combined with imperfect measurement, leads to larger and more persistent responses of private consumption to government expenditure shocks under robust expectations.

On the relationship between determinate and MSV solutions in linear RE models

This paper considers the possibility that, in linear rational expectations (RE) models, all determinate (uniquely non-explosive) solutions coincide with the minimum state variable (MSV) solution, which is unique by construction. In univariate specifications of the form y(t) = AE(t)y(t+1) + Cy(t-1) + u(t) that result holds: if a RE solution is unique and non-explosive, then it is the same as the MSV solution. Also, this result holds for multivariate versions if the A and C matrices commute and a certain regularity condition holds. More generally, however, there are models of this form that possess unique non-explosive solutions that differ from their MSV solutions. Examples are provided and a strategy for easily constructing others is outlined.

Determinacy in linear rational expectations models

The purpose of this paper is to assess the relevance of rational expectations solutions to the class of linear univariate models where both the number of leads in expectations and the number of lags in predetermined variables are arbitrary. It recommends to rule out all the solutions that would fail to be locally unique, or equivalently, locally determinate. So far, this determinacy criterion has been applied to particular solutions, in general some steady state or periodic cycle. However solutions to linear models with rational expectations typically do not conform to such simple dynamic patterns but express instead the current state of the economic system as a linear difference equation of lagged states. The innovation of this paper is to apply the determinacy criterion to the sets of coefficients of these linear difference equations. Its main result shows that only one set of such coefficients, or the corresponding solution, is locally determinate. This solution is commonly referred to as the fundamental one in the literature. In particular, in the saddle point configuration, it coincides with the saddle stable (pure forward) equilibrium trajectory.

The California Rockfish Conservation Area and Groundfish Trawlers at Moss Landing Harbor

This article uses a bioeconomic model and data for groundfish trawlers at Moss Landing Harbor in Central California to analyze effects of spatial closures that were implemented recently by West Coast fishery managers to reduce bycatch of overfished groundfish stocks. The model has a dynamic linear rational expectations structure, and estimates of its parameters exhibit spatial variation in microeconomic and ecological factors that affect decisions about where and when to fish. Test results show that variation in marginal costs of crowding externalities and biological rates of stock productivity are the most significant factors to consider in the spatial management of groundfish trawlers at Moss Landing.

Open loop time consistency for linear rational expectations models

We derive an open loop time consistent equilibrium for linear rational expectations control problems. It can be solved using standard methods. We show how it relates to other consistent solutions.

Stable Noisy K-state Markov Sunspots

We consider a linear stochastic univariate rational expectations model, with a predetermined variable, and consider solutions driven by an extraneous finite state Markov process as well as by the fundamental noise. We obtain conditions for existence of noisy k-state sunspot equilibria (noisy k-SSEs) and, for the case k=2, of noisy k-state dependent sunspot equilibria (noisy k-*SDSs). k-*SDSs are driven by a finite state sunspot but have an infinite range of values even in the nonstochastic model. Stability under econometric learning is analyzed using representations that nest both types of solution. For the case k=2, we find that noisy 2-SSEs and noisy 2-*SDSs are learnable for parameters in proper subsets of the regions of their existence.

UNIQUENESS OF BUBBLE-FREE SOLUTION IN LINEAR RATIONAL EXPECTATIONS MODELS

One usually identifies bubble solutions to linear rational expectations models by extra components (irrelevant lags) arising in addition to market fundamentals. Although there are still many solutions relying on a minimal set of state variables, i.e., relating in equilibrium the current state of the economic system to as many lags as initial conditions, there is a conventional wisdom that the bubble-free (fundamentals) solution should be unique. This paper examines the existence of endogenous stochastic sunspot fluctuations close to solutions relying on a minimal set of state variables, which provides a natural test for identifying bubble and bubble-free solutions. It turns out that only one solution is locally immune to sunspots, independently of the stability properties of the perfect-foresight dynamics. In the standard saddle-point configuration for these dynamics, this solution corresponds to the so-called saddle stable path.

Computing sunspot equilibria in linear rational expectations models

We provide a computationally simple method of analyzing the effects of fundamental and sunspot shocks in linear rational expectations models when the equilibrium is indeterminate. Under indeterminacy sunspots can affect model dynamics through endogenous forecast errors. Moreover, the effect of fundamental shocks on forecast errors is not uniquely determined. The solution method is illustrated with a New Keynesian dynamic stochastic equilibrium model that can be solved analytically. Under a passive interest-rate rule, the response of inflation to an unanticipated interest rate cut is ambiguous: there are some equilibria in which inflation increases and others in which prices fall.

Stable Sunspot Equilibria in Models with Predetermined Variables

We consider a linear stochastic univariate rational expectations model, with a predetermined variable, and provide alternative representations of SSEs (stationary sunspot equilibria). For a strict subset of the parameter space there exist SSEs that are locally stable under least squares learning provided agents use a common factor representation for their estimated law of motion. These results indicate that for some parameter regions SSEs are more likely to arise under private agent learning than previously recognized.

Inflation Targeting, Exchange Rate Volatility and International Policy Coordination

In a linear rational expectations two–country model, using an aggregate demand, aggregate supply framework, we analyse the effects of the adoption of an inflation–targeting regime on exchange rate volatility and the possible scope for policy coordination. This analysis is conducted using optimized interest rate policy rules within a calibrated model. Rules for interest rates that respond either to exchange rates or to portfolio shocks give improved performance and permit gains from international coordination. Optimized Taylor rules perform relatively well.

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.

Computing Sunspots in Linear Rational Expectations Models

We provide a computationally simple method of including and analyzing the effects of sunspot shocks in linear rational expectations models when the equilibrium is indeterminate. Under non uniqueness sunspots can affect model dynamics through endogenous forecast errors that do not completely adjust to fundamental shocks alone. We show that sunspot shocks can ire modeled as exogenous belief shocks which can be included in the set of fundament als. By means of a simple example we illustrate that the exact specification of the transmission mechanism of the belief shocks is irrelevant for the solution of the model.

El Niño, Expectations, and Fishing Effort in Monterey Bay, California

This article uses vector autoregressions (VARs) and a dynamic linear rational expectations model to analyze the effects of sea surface temperatures and catch prices on fishing effort for albacore tuna, Chinook salmon, sablefish, and squid in Monterey Bay, California. The VAR results show that fluctuations in sea surface temperatures, including El Niño events, have significant effects on effort and prices for these fisheries. Estimation of the model shows that El Niño has positive effects on the abundance in Monterey Bay of albacore and negative effects on Chinook, sablefish, and squid. Testing of the model supports rational expectations by harvesters of albacore, Chinook, and sablefish but not squid. Test results also suggest that dynamic adjustment costs are not significant for albacore, Chinook, or sablefish fisheries.

A Higher-Order Taylor Expansion Approach to Simulation of Stochastic Forward-Looking Models with an Application to a Nonlinear Phillips Curve Model

We propose to apply to the simulation of general nonlinear rational-expectation models a method where the expectation functions are approximated through a higher-order Taylor expansion. This method has been advocated by Judd (1998) and others for the simulation of stochastic optimal-control problems and we extend its application to more general cases. The coefficients for the first-order approximation of the expectation function are obtained using a generalized eigenvalue decomposition as it is usual for the simulation of linear rational-expectation models. Coefficients for higher-order terms in the Taylor expansion are then obtained by solving a succession of linear systems. In addition, we provide a method to reduce a bias in the computation of the stochastic equilibrium of such models. These procedures are made available in DYNARE, a MATLAB and GAUSS based simulation program. This method is then applied to the simulation of a macroeconomic model embodying a nonlinear Phillips curve. We show that in this case a quadratic approximation is sufficient, but different in important ways from the simulation of a linearized version of the model.

Using the generalized Schur form to solve a multivariate linear rational expectations model

In this paper, I show how to use the generalized Schur form to solve a system of linear expectational difference equations (a multivariate linear rational expectations model). The method is simple to understand and to use, and is applicable to a large class of rational expectations models. The only hard part is taken care of by just two standard algorithms, both of which are available as freeware on the Internet as part of LAPACK. Like other matrix decomposition based methods, it is also very fast to execute.