Rational Expectations Solution Research Articles

Under the same (Chole)sky: DNK models, timing restrictions and recursive identification of monetary policy shocks

Recent structural VAR studies of the monetary transmission mechanism have voiced concerns about the use of recursive identification schemes based on short-run exclusion restrictions. This paper characterizes the effects on impulse propagation of informational constraints embodying classical Cholesky-type timing restrictions in otherwise standard DSGE models. We formally show that timing restrictions can produce non-trivial moving average components of rational expectations solutions, or even serve as an independent source of model-based nonfundamentalness, thereby hampering impulse response analysis via VAR procedures. We then derive population conditions for existence of VAR representations of DSGE economies exhibiting timing restrictions, and numerically explore their bearing on shock identification in a range of monetary models of the business cycle. Our analysis reveals that dynamic New Keynesian models admit invertible equilibrium representations as well as fast-converging VAR coefficient matrices under empirically tenable parameterizations. This alleviates concerns about identification and lag truncation bias: low-order Cholesky-VARs do well at retrieving the true aggregate effects of monetary policy shocks in a Cholesky world.

Refining linear rational expectations models and equilibria

This paper develops the case for forward convergence as a model refinement scheme for linear rational expectations models and an associated no-bubble condition as a solution selection criterion. We relate these two concepts to determinacy and characterize the complete set of economically relevant rational expectations solutions to the linear rational expectations models under determinacy and indeterminacy. Our results show (1) why a determinate solution is economically cogent in most, but not all, cases, and (2) that those models that are not forward-convergent have no economically relevant solutions.

Solving generalized multivariate linear rational expectations models

We generalize the linear rational expectations solution method of Whiteman (1983) to the multivariate case. This facilitates the use of a generic exogenous driving process that must only satisfy covariance stationarity. Multivariate cross-equation restrictions linking the Wold representation of the exogenous process to the endogenous variables of the rational expectations model are obtained. We argue that this approach offers important insights into rational expectations models. We give two examples in the paper—an asset pricing model with incomplete information and a monetary model with observationally equivalent monetary-fiscal policy interactions. We relate our solution methodology to other popular approaches to solving multivariate linear rational expectations models, and provide user-friendly code that executes our approach.

The El Farol Problem Revisited

The so-called El Farol problem describes a prototypical situation of interacting agents making binary choices to participate in a non-cooperative environment or to stay by themselves and choosing an outside option. In a much cited paper Arthur (1994) argues that persistent non-converging sequences of rates of participation with permanent forecasting errors occur due to the non-existence of a prediction model for agents to forecast the attendance appropriately to induce stable rational expectations solutions. From this he concludes the need for agents to use boundedly rational rules. This note shows that in a large class of such models the failure of agents to find rational prediction rules which stabilize is not due to a non-existence of perfect rules, but rather to the failure of agents to identify the correct class of predictors from which the perfect ones can be chosen. What appears as a need to search for boundedly rational predictors originates from the non existence of stable confirming self-referential orbits induced by predictors selected from the wrong class. Specifically, it is shown that, within a specified class of the model and due to a structural non-convexity (or discontinuity), symmetric Nash equilibria of the associated static game may fail to exist generically depending on the utility level of the outside option. If they exist, they may induce the least desired outcome while, generically, asymmetric equilibria are uniquely determined by a positive maximal rate of attendance. The sequential setting turns the static game into a dynamic economic law of the Cobweb type for which there always exist nontrivial ǫ−perfect predictors implementing ǫ−perfect steady states as stable outcomes. If zero participation is a Nash equilibrium of the game there exists a unique perfect predictor implementing the trivial equilibrium as a stable steady state. In general, Nash equilibria of the one-shot game are among the ǫ−perfect foresight steady states of the dynamic model. If agents randomize over indifferent decisions the induced random Cobweb law together with recursive predictors becomes an iterated function system (IFS). There exist unbiased predictors with associated stable stationary solutions for appropriate randomizations supporting nonzero asymmetric equilibria which are not mixed Nash equilibria of the one-shot game. However, the least desired outcome remains as the unique stable stationary outcome for ǫ = 0 if it is a Nash equilibrium of the static game.

Response to updated mortality forecasts in life cycle saving and labor supply

Historical evidence shows that demographic forecasts, including mortality forecasts, have often been grossly in error. One consequence of this is that forecasts are updated frequently. How should individuals or institutions react to updates, given that these are likewise expected to be uncertain? We discuss this problem in the context of a life cycle saving and labor supply problem, in which a cohort of workers decides how much to work and how much to save for mutual pensions. Mortality is stochastic and point forecasts are updated regularly. A Markovian approximation for the predictive distribution of mortality is derived. This renders the model computationally tractable, and allows us to compare a theoretically optimal rational expectations solution to a strategy in which the cohort merely updates the life cycle plan to match each updated mortality forecast. The implications of the analyses for overlapping generations modeling of pension systems are pointed out.

Constructing Multi‐Country Rational Expectations Models

AbstractThis article considers some of the technical issues involved in using the global vector autoregression (GVAR) approach to construct a multi‐country rational expectations (RE) model and illustrates them with a new Keynesian model for 33 countries estimated with quarterly data over the period 1980–2011. The issues considered are: the measurement of steady states; the determination of exchange rates and the specification of the short‐run country‐specific models; the identification and estimation of the model subject to the theoretical constraints required for a determinate rational expectations solution; the solution of a large RE model; the structure and estimation of the covariance matrix and the simulation of shocks. The model used as an illustration shows that global demand and supply shocks are the most important drivers of output, inflation and interest rates in the long run. By contrast, monetary or exchange rate shocks have only a short‐run impact in the evolution of the world economy. The article also shows the importance of international connections, directly as well as indirectly through spillover effects. Overall, ignoring global inter‐connections as country‐specific models do, could give rise to misleading conclusions.

OLG Life Cycle Model Transition Paths: Alternate Model Forecast Method

The overlapping generations (OLG) model is an important framework for analyzing any type of question in which age cohorts are affected differently by exogenous shocks. However, as the dimensions and degree of heterogeneity in these models increase, the computational burden imposed by rational expectations solution methods for nonstationary equilibrium transition paths increases exponentially. As a result, these models have been limited in the scope of their use to a restricted set of applications and a relatively small group of researchers. In addition to providing a detailed description of the benchmark rational expectations computational method, this paper presents an alternative method for solving for equilibrium transition paths in OLG life cycle models that is new to this class of model. The key insight is that even naïve limited information forecasts within the model produce aggregate time series similar to full information rational expectations time series as long as the naïve forecasts are updated each period. We find that our alternate model forecast method reduces computation time by 85 percent, and the approximation error is small.

Interpreting the evidence for New Keynesian models of inflation dynamics

We present a framework for interpretation of the empirical results of New Keynesian models of inflation dynamics. Both the rational expectations solution of the structural New Keynesian Phillips curve, NKPC, and the reduced form VAR analysis of the multivariate time series properties give insight about the joint implications of the evidence in the NKPC literature. For example, we show that the unit-root form of non-stationary may be implied for inflation even though the econometric model initially assumed stationarity. The uniqueness and form of a rational expectations solution may depend on whether dynamic (in)homogeneity is present, and on the size of the forward-coefficient in the NKPC.

OLG Life Cycle Model Transition Paths: Alternate Model Forecast Method

The overlapping generations (OLG) model is an important framework for analyzing any type of question in which age cohorts are affected differently by exogenous shocks. However, as the dimensions and degree of heterogeneity in these models increase, the computational burden imposed by rational expectations solution methods for non-stationary equilibrium transition paths increases exponentially. As a result, these models have been limited in the scope of their use to a restricted set of applications and a relatively small group of researchers. In addition to providing a detailed description of the benchmark rational expectations computational method, this paper presents an alternative method for solving for nonstationary equilibrium transition paths in OLG life cycle models that is new to this class of model. We find that our alternate model forecast method reduces computation time to 15 percent of the benchmark time path iteration computation time, and the approximation error is less than 1 percent.

Invertible and non-invertible information sets in linear rational expectations models

Rational expectations solutions are usually derived by assuming that all state variables relevant to forward-looking behaviour are directly observable, or that they are “…an invertible function of observables” (Mehra and Prescott, 1980). Using a framework that nests linearised DSGE models, we give a number of results useful for the analysis of linear rational expectations models with restricted information sets. We distinguish between instantaneous and asymptotic invertibility, and show that the latter may require significantly less information than the former. We also show that non-invertibility of the information set can have significant implications for the time series properties of economies.

News versus Sunspot Shocks in a New Keynesian Model

Abstract Separately, news and sunspot shocks have been shown empirically to be determinants of changes in expectations. This paper considers both of them together in a simple New Keynesian monetary business cycle model. A full set of rational expectations solutions is derived analytically. The analytical characterization allows an explicit comparison of news about future monetary policy and sunspots. The key distinction between the shocks lies in their relation to the realized policy shock. If monetary policy is “passive”, both types of shocks affect model dynamics through forecast errors. The effect of the news on forecast errors is not unique, and the dynamics induced by news and sunspot shocks can be observationally equivalent. If monetary policy is “active”, the sunspots are irrelevant, and the model responses to the news shocks are unique. In both cases, news shocks strengthen the endogenous propagation of the model, since anticipation of future changes prolongs agents’ reaction. Keywords: news shocks, sunspots, expectations, monetary policy, indeterminacy.

AN INTERVIEW WITH ROGER GUESNERIE

Roger Guesnerie is a prominent economic theorist. Among his many honors and distinctions, he is a Foreign Member of the American Academy of Arts and Science and an Honorary Foreign Member of the American Economic Association; he has been a President of the European Economic Association and of the Econometric Society. Currently a Professor at the College de France, he has made fundamental contributions to second-best theory and public economics, information economics, and rational expectation theory.Guesnerie was one of the founding fathers of second-best theory. His contributions to public economics, addressing issues such as nonconvex economies and especially taxation in a second-best context, remain absolute references on these topics; more recently, he has provided an in-depth analysis of issues related to climate change and global warming from a public economics perspective. Regarding contract theory, his work with Jean Jacques Laffont has considerably extended our understanding of the properties of the basic models, as well as the set of tools that economists can use to address these problems. Finally, Guesnerie's work has been influential in the critical reassessment of rational expectation theory that took place over the last decades. On one hand, he has been a prominent figure in the development of so-called “sunspot” models, which reconcile the Keynesian notion of “animal spirits” with the formal requirements of rational expectations. On the other hand, he has provided path-breaking discussions of the theoretical relevance of rational expectation theory, based on a detailed assessment of the coordination between individual expectations that is implicit in the concept. In some classes of models, Guesnerie argues, rational expectation equilibria constitute a natural outcome, because coordination can be achieved either through “eductive” reasoning (for which common knowledge of individual rationality is sufficient) or as the product of simple evolutionary or learning dynamics (and in general in both ways). In other cases, however, such arguments do not appear to support the rational expectations solution, and its relevance is therefore much less obvious. On a more institutional side, Guesnerie has created the doctoral program “Analyse et Politique Economique” at the Ecole des Hautes Etudes en Sciences Sociales; besides this program, in 1989, several related research centers in Paris merged into Delta, which Guesnerie chaired between 1992 and 2000, and then into the Paris School of Economics (PSE), at which Guesnerie is Chairman of the Board.This interview with Roger Guesnerie gives an opportunity to discuss further the current situation of economic theory and economics in general, as well as to compare the European and American traditions in economics.

Another weakness of “determinacy” as a selection criterion for rational expectations models

This paper argues against determinacy as a guide to interpret rational expectations solutions, using an example in which a determinate solution exists but differs sharply in dynamic behavior from that implied by the model considered on a sector-by-sector basis.

Matrix Polynomial Conditions for the Existence of Rational Expectations Solutions

In this article we derive minimal conditions to determine the existence of rational expectations solutions obtained using a Generalized Bzout Theorem. We demonstrate that as long as the matrix polynomial derived from the model is regular, then a monic polynomial factor always exists and from this result we can derive a backward forward solution. For the existence of the multivariate rational expectations solution, real roots are sufficient, but not necessary as has previously been suggested in the literature. Existence can be established by the presence of a simple rank order condition, this less restrictive approach allows researchers to consider a wider class of models with rational expectations.

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.

Learning in linear models with expectational leads

Models with expectational leads typically admit multiple rational expectations solutions. Based on the ordinary least-squares algorithm, this paper provides an adaptive learning scheme which allows a forecasting agent to select a particular solution on economic grounds. Conditions are given under which this scheme converges to rational expectations solutions globally for all initial conditions. We strengthen convergence results in relaxing standard assumptions and in providing conditions ensuring algorithm convergence which are easier to verify and to interpret than those previously known.

VAR-based estimation of Euler equations with an application to New Keynesian pricing

VAR-based estimation of Euler equations exploits cross-equation restrictions that the theory imposes on a vector-autoregressive (VAR) process for market expectations. This paper shows that Sargent's (1979. A note on maximum likelihood estimation of the rational expectations model of the term structure. Journal of Monetary Economics 5, 133–143.) original approach of imposing these restrictions on the endogenous variable of the Euler equation (i.e. computing the rational expectations solution) implies multiple solutions and is therefore impracticable. The paper shows that this identification problem can be circumvented by reverse-engineering the cross-equation restrictions on the forcing variable of the Euler equation. The proposed reverse-engineering approach contrasts with the conventional approach in the literature that avoids the identification problem by constraining the system formed by the Euler equation and the VAR process to yield a unique stable rational expectations equilibrium. This uniqueness condition makes little economic sense because the VAR process is just a reduced-form approximation of true market expectations, and a simulation experiment shows that imposing uniqueness can lead to severe misspecification bias. The severity of this misspecification bias is illustrated in practice with an application to Gali and Gertler's (1999. Inflation dynamics: a structural econometric analysis. Journal of Monetary Economics 44, 195–222.) hybrid New Keynesian Phillips Curve.

Taylor Rules and Macroeconomic Instability or How the Central Bank Can Preempt Sunspot Expectations

This paper derives new results on the effects of employing Taylor rules in economies that are subject to real market imperfections such as production externalities. Policies which respond to output movements can block sunspot equilibria that arise from the increasing returns. The paper also finds that rules which should be chosen (avoided) in perfect market environments often yield (ensure) multiple (unique) rational expectations solutions in alternative settings. Therefore, exact knowledge on the degree of market imperfection may be integral for robust policy advice.

Monetary and Fiscal Theories of the Price Level: The Irreconcilable Differences

The fiscal theory of the price level (FTPL) has attracted much attention but disagreement remains concerning its defining characteristics. Some writers have emphasized implications regarding interest-rate pegging and determinacy of rational expectations solutions, whereas others have stressed its capacity to generate equilibria in which price-level trajectories mimic those of bonds and differ drastically from those of money supplies. We argue that the FTPL attained prominence precisely because it appeared to provide a theory whose implications differ greatly from conventional monetary analysis; accordingly we review monetarist writings to identify the primary distinctions. In addition, we review recent findings concerning learnability--and therefore plausibility--of competing rational expectations equilibria. These indicate that when FTPL and monetarist equilibria differ, the latter are more plausible in the vast majority of cases. Under Ricardian assumptions, necessary for clear distinctions, theoretical analysis indicates that fiscal and monetary coordination is not necessary for macroeconomic stability. Copyright 2005, Oxford University Press.

MEAN VARIANCE PREFERENCES, EXPECTATIONS FORMATION, AND THE DYNAMICS OF RANDOM ASSET PRICES

This paper analyzes the dynamics of an explicit random process of prices and price expectations of finitely many assets in an economy with overlapping generations of heterogeneous consumers. They maximize expected utility with respect to subjective transition probabilities defined by Markov kernels which describe the forecasting behavior of agents. Given such forecasting rules (predictors) and an exogenous process of dividends, the evolution of equilibrium asset prices and expectations is described by a random dynamical system in the sense of Arnold (1998). The paper investigates the long‐run behavior (stationary solutions) by proving the existence and stability of random fixed points for mean‐variance preferences under various predictors, including unbiased predictions, and adaptive, as well as OLS forecasting. An explicit characterization of rational expectations solutions is given, providing a full dynamic characterization of asset price processes for the classical CAPM in the case of stationary OLG economies. Numerical simulations are used to compare the performance of the different predictors under an AR(1) dividend process.