Linear Rational Expectations Models Research Articles

The Forward Method as a Solution Refinement in Rational Expectations Models

This paper generalizes the standard forward method of recursive substitution to a general class of linear Rational Expectations models with potentially multiple fundamental solutions. We propose a key property embedded in the forward solution -- the no-bubble condition -- as an economically sensible solution refinement in the class of fundamental solutions. In the literature, the no-bubble condition has been assumed to rule out non-fundamental bubble solutions. However, since the condition involves expectations of the future endogenous variables, it must be verified for every Rational Expectations equilibrium. We show that the forward solution is the only fundamental solution satisfying the no-bubble condition and that it is hard to justify economically fundamental solutions violating this condition. We provide several economic examples where the fundamental solutions obtained by other solution methods and refined by other solution selection criteria violate the no-bubble condition.

Solving Linear Rational Expectations Models with Lagged Expectations Quickly and Easily

A solution method is derived in this paper for solving a system of linear rationalexpectations equation with lagged expectations (e.g., models incorporating sticky information) using the method of undetermined coefficients for the infinite MA representation. The method applies a combination of a Generalized Schur Decomposition familiar elsewhere in the literature and a simple system of linear equations when lagged expectations are present to the infinite MA representation. Execution is faster, applicability more general, and use more straightforward than with existing algorithms. Current methods of truncating lagged expectations are shown to not generally be innocuous and the use of such methods are rendered obsolete by the tremendous gains in computational efficiency of the method here which allows for a solution to floating-point accuracy in a fraction of the time required by standard methods. The associated computational application of the method provides impulse responses to anticipated and unanticipated innovations, simulations, and frequency-domain and simulated moments.

Structural Error Correction Models: A System Method for Linear Rational Expectations Models and an Application to an Exchange Rate Model

This paper develops a system instrumental variable method to estimate the speed of adjustment coefficient in the long‐run equilibrium of structural error correction models for a class of linear rational expectations models. This method is applied to an exchange rate model with sticky prices, in which the speed of adjustment coefficient governs the half‐life of the real exchange rate. Compared to single equation methods, the system method gives smaller half‐life estimates with sharper standard errors.

Solving Linear Rational Expectations Models: A Horse Race

This paper compares the generality, accuracy and computational speed of alternative approaches to solving linear rational expectations models, including the procedures of Sims (Solving linear rational expectations models, 1996), Anderson and Moore (Unpublished manuscript, 1983), Binder and Pesaran (Multivariate rational expectations models and macroeconometric modelling: A review and some new results, 1994), King and Watson (International Economic review, 39, 1015---1026, 1998), Klein (Journal of Economic Dynamics and Control, 24, 1405---1423, 1999), and Uhlig (A toolkit for analyzing nonlinear dynamic stochastic models easily, 1999). While all six procedures yield equivalent results for models with a unique stationary solution, the algorithm of Anderson and Moore (Unpublished manuscript, 1983) is the fastest and provides the highest accuracy; furthermore, the speed advantage increases with the size of the model.

How equilibrium prices reveal information in a time series model with disparately informed, competitive traders

Accommodating asymmetric information in a dynamic asset pricing model is technically challenging due to the problems associated with higher-order expectations. That is, rational investors are forced into a situation where they must forecast the forecasts of other agents. In a dynamic setting, this problem telescopes into the infinite future and the dimension of the relevant state space approaches infinity. By using the frequency domain approach of Whiteman [Linear Rational Expectations Models: A User's Guide, University of Minnesota Press, Minneapolis, 1983] and Kasa [Forecasting the forecasts of others in the frequency domain, Rev. Econ. Dynam. 3 (2000) 726–756], this paper demonstrates how information structures previously believed to preserve asymmetric information in equilibrium, converge to a symmetric information, rational expectations equilibrium. The revealing aspect of the price process lies in the invertibility of the observed state space, which makes it possible for agents to infer the economically fundamental shocks and thus eliminating the need to forecast the forecasts of others.

Optimal Monetary Policy in Markov-Switching Models with Rational Expectations Agents

In this paper we consider the optimal control problem of models with Markov regime shifts and forward-looking agents. These models are very general and flexible tools for modelling model uncertainty. An algorithm is devised to compute the solution of a linear rational expectations model with random parameters or regime shifts. This algorithm can also be applied in the optimisation of any arbitrary instrument rule. A second algorithm computes the time-consistent policy and the resulting Nash-Stackelberg equilibrium. Similar methods can be easily employed to compute the optimal policy under commitment. Furthermore, the algorithms can also handle the case in which the policymaker and the private sector hold different beliefs. We apply these methods to compute the optimal (non-linear) monetary policy in a small open economy subject to random structural breaks in some of its key parameters.

E-stability vis-a-vis determinacy results for a broad class of linear rational expectations models

It is argued that learnability/E-stability is a necessary condition for a RE solution to be plausible. A class of linear models considered by Evans, G.W. and Honkapohja, S. [2001. Learning and Expectations in Macroeconomics, Princeton University Press.] is shown to include all models of the form used by King, R.G. and Watson, M.W. [1998. The solution of singular linear difference systems under rational expectations. International Economic Review 39, 1015–1026] and Klein, P. [2000. Using the generalized Schur form to solve a multivariate linear rational expectations model. Journal of Economic Dynamics and Control 24, 1405–1423], which permits any number of lags, leads, and lags of leads. For this broad class it is shown that, if current-period information is available in the learning process, determinacy is a sufficient condition for E-stability. It is not a necessary condition, however; there exist cases with more than one stable solution in which the solution based on the decreasing-modulus ordering of the system's eigenvalues is E-stable. If in such a case the other stable solutions are not E-stable, then the condition of indeterminacy may not be crucial for practical issues.

Matlab Toolbox for Solving Regular Linear Rational Expectations Models: The Algorithm

We present a computationally effective method of solving regular linear dynamic systems based on Schur decomposition. One of the advantage of presented method is possibility of problem dimension reduction. This allows for efficient solution of models with very large set of endogenous variables and moderate set of state variables. Presented method allows for considering models with sunspots. We also derive first differential of matrices describing model dynamics with respect to model parameters. The toolbox lrem_solve implements methods described in this paper.

Economic activity, foreign exchange rate, and the interest rate during the Asian crisis

In a simple linear rational expectations model in which monetary authorities act to stabilize the exchange rate and also to mitigate adverse effects on real activity, a rise in interest rate is followed by exchange rate depreciation. In daily data across VAR and VECM models a rise in interest rate is associated with significant exchange rate depreciation for Thailand, the Philippines, and Korea. Depreciation and increases in the interest rate raise business failures that intensify crisis. Crisis management of the exchange rate must consider damaging effect of restrictive policy on financially constrained banking sector and non-financial firms.

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.

A Method to Generate Structural Impulse-Responses for Measuring the Effects of Shocks in Structural Macro Models

We develop a technique for analyzing the response dynamics of economic variables to structural shocks in linear rational expectations models. Our work differs fromstandard SVARs since we allow expectations of future variables to enter structural equations. We show how to estimate the variance-covariance matrix of fundamental and non-fundamental shocks and we construct point estimates and confidence bounds for impulse response functions. Our technique can handle both determinate and indeterminate equilibria. We provide an application to U.S. monetary policy under pre and post Volcker monetary policy rules. JEL Classification: C39, C62, D51, E52, E58

Solving Linear Rational Expectations Models: A Horse Race

This paper compares the functionality, accuracy, computational efficiency, and practicalities of alternative approaches to solving linear rational expectations models, including the procedures of (Sims, 1996), (Anderson and Moore, 1983), (Binder and Pesaran, 1994), (King and Watson, 1998), (Klein, 1999), and (Uhlig, 1999). While all six prcedures yield similar results for models with a unique stationary solution, the AIM algorithm of (Anderson and Moore, 1983) provides the highest accuracy; furthermore, this procedure exhibits significant gains in computational efficiency for larger-scale models.

Solving Linear Rational Expectations Models: A Horse Race

This paper compares the functionality, accuracy, computational efficiency, and practicalities of alternative approaches to solving linear rational expectations models, including the procedures of (Sims, 1996), (Anderson and Moore, 1983), (Binder and Pesaran, 1994), (King and Watson, 1998), (Klein, 1999), and (Uhlig, 1999). While all six procedures yield similar results for models with a unique stationary solution, the AIM algorithm of (Anderson and Moore, 1983) provides the highest accuracy; furthermore, this procedure exhibits significant gains in computational efficiency for larger-scale models.

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.

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.

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.