Limited-dependent rational expectations models with future expectations

This paper examines limited-dependent rational expectations (LD-RE) models containing future expectations of the dependent variable. Limited dependence is of a two-limit tobit variety which may, for example, arise as a result of a policy of imposing limits on the movement of the dependent variable by means of marginal as well as intramarginal interventions. We show that when the forcing variables are serially independent the model has an analytical solution which can be computed by backward recursion. With serially correlated forcing variables, we discuss an approximate solution method, as well as a numerically exact method that, in principle, can be implemented by stochastic simulation, although in practice it is limited by available computational capacity. The paper discusses some properties of the approximate solutions and reports the results of a limited number of Monte Carlo experiments in order to illustrate the computational feasibility of using the exact solution when the fundamentals are serially independent and the approximate solution when they are serially correlated.     

Pooling forecasts in linear rational expectations models

Estimating linear rational expectations models in a limited-information setting requires replacing the expectations of future, endogenous variables either with instrumented, actual values or with forecast survey data. Applying the method of Gottfries and Persson [Empirical examinations of the information sets of economic agents. Quarterly Journal of Economics 103, 251–259], I show how to augment these methods with actual, future values of the endogenous variables to improve statistical efficiency. The method is illustrated with an application to the US hybrid new Keynesian Phillips curve, where traditional, lagged instruments and the median forecast from the Survey of Professional Forecasters both appear to miss significant information used by price-setters, so that forecast pooling with actual values improves the statistical fit to inflation.     

Optimal policy in Markov-switching rational expectations models

In this paper we consider the optimal quadratic control problem of Markov-switching linear rational expectation models. These models are general and flexible tools for modelling not only regime but also model or parameter uncertainty. We show, first, how to find the solution of a Markov-switching linear rational expectation model. Based on this solution we then show how to apply dynamic programming to find the optimal time-consistent policy and the resulting Nash-Stackelberg equilibrium. Suitable modifications of the algorithm allow to deal with the (non-RE) case in which the policymaker and the private sector hold different beliefs or probabilities over regime change. We also show how the optimisation procedure can be employed to obtain the optimal policy under commitment. As an illustration we compute the optimal policy in a small open economy subject to stochastic structural breaks in some of its key parameters.     

Minimal state variable solutions to Markov-switching rational expectations models

We develop a new method for deriving minimal state variable (MSV) equilibria of a general class of Markov switching rational expectations models and a new algorithm for computing these equilibria. We compare our approach to previously known algorithms, and we demonstrate that ours is both efficient and more reliable than previous methods in the sense that it is able to find MSV equilibria that previously known algorithms cannot. Further, our algorithm can find all possible MSV equilibria in models. This feature is essential if one is interested in using a likelihood based approach to estimation.     

The role of beliefs in inference for rational expectations models

This paper discusses inference for rational expectations models estimated via minimum distance methods by characterizing the probability beliefs regarding the data generating process (DGP) that are compatible with given moment conditions. The null hypothesis is taken to be rational expectations and the alternative hypothesis to be distorted beliefs. This distorted beliefs alternative is analyzed from the perspective of a hypothetical semiparametric Bayesian who believes the model and uses it to learn about the DGP. This interpretation provides a different perspective on estimates, test statistics, and confidence regions in large samples, particularly regarding the economic significance of rejections in rational expectations models. A suggestive application to the equity premium puzzle is examined.     

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.     

Two-step two-stage least squares estimation in models with rational expectations

The paper develops a two-step estimator for use in rational-expectations models with autocorrelated residuals and predetermined, but not strictly exogenous, instruments. The estimator extends the applicability of McCallum's (1976) error-in-variablesapproach to estimating such models, and is asymptotically efficient in a class of intrumental-variables estimators. As an application we use instrumental-variables techniques to estimate Taylor's (1979) rational-expectations macroeconomic model of the United States.     

Weakly rational expectations

Aumann and Drèze (2008) characterised the set of interim expected payoffs that players may have in rational belief systems, in which there is common knowledge of rationality and a common prior. We show here that common knowledge of rationality is not needed: when rationality is satisfied in the support of an action-consistent distribution (a concept introduced by Barelli (2009)), one obtains exactly the same set of rational expectations, despite the fact that in such ‘weakly rational belief systems’ there may not be mutual knowledge of rationality, let alone common knowledge of rationality. In the special case of two-player zero-sum games, the only expected payoff is the minmax value, even under these weak assumptions.     

Examining bias in estimators of linear rational expectations models under misspecification

Most rational expectations models involve equations in which the dependent variable is a function of its lags and its expected future value. We investigate the asymptotic bias of generalized method of moment (GMM) and maximum likelihood (ML) estimators in such models under misspecification. We consider several misspecifications, and focus more specifically on the case of omitted dynamics in the dependent variable. In a stylized DGP, we derive analytically the asymptotic biases of these estimators. We establish that in many cases of interest the two estimators of the degree of forward-lookingness are asymptotically biased in opposite direction with respect to the true value of the parameter. We also propose a quasi-Hausman test of misspecification based on the difference between the GMM and ML estimators. Using Monte-Carlo simulations, we show that the ordering and direction of the estimators still hold in a more realistic New Keynesian macroeconomic model. In this set-up, misspecification is in general found to be more harmful to GMM than to ML estimators.     

Full information estimation and stochastic simulation of models with rational expectations

A computationally feasible method for the full information maximum-likelihood estimation of models with rational expectations is described in this paper. The stochastic simulation of such models is also described. The methods discussed in this paper should open the way for many more tests of the rational expectations hypothesis within macroeconomic models.     

Determinacy,indeterminacy and dynamic misspecification in linear rational expectations models

This paper proposes a testing strategy for the null hypothesis that a multivariate linear rational expectations (LRE) model may have a unique stable solution (determinacy) against the alternative of multiple stable solutions (indeterminacy). The testing problem is addressed by a misspecification-type approach in which the overidentifying restrictions test obtained from the estimation of the system of Euler equations of the LRE model through the generalized method of moments is combined with a likelihood-based test for the cross-equation restrictions that the model places on its reduced form solution under determinacy. The resulting test has no power against a particular class of indeterminate equilibria, hence the non rejection of the null hypothesis can not be interpreted conclusively as evidence of determinacy. On the other hand, this test (i) circumvents the nonstandard inferential problem generated by the presence of the auxiliary parameters that appear under indeterminacy and that are not identifiable under determinacy, (ii) does not involve inequality parametric restrictions and hence the use of nonstandard inference, (iii) is consistent against the dynamic misspecification of the LRE model, and (iv) is computationally simple. Monte Carlo simulations show that the suggested testing strategy delivers reasonable size coverage and power against dynamic misspecification in finite samples. An empirical illustration focuses on the determinacy/indeterminacy of a New Keynesian monetary business cycle model of the US economy.     

Estimation and solution of linear rational expectations models using a polynomial matrix factorization

A method to solve and estimate multivariate linear rational expectations models is described. The method is based on an iterative factorization of the polynomial matrix that describes the lags and expected leads in the model. Our experience is that the method works well in a variety of applications where other methods are either difficult or expensive to use.     

Dangers of commitment under rational expectations

Within a New Keynesian framework, interest rate rules that respond to public expectations lead to determinate and expectationally stable solutions for any level of commitment, as shown by Waters (Macroecon Dyn 13(4):421–449, 2009). That paper also demonstrates gains to commitment, under least square learning, though over-commitment can lead to some very poor outcomes for some parameter values. This paper shows an identical outcome under rational expectations. The optimal level of commitment is unchanged if there are observation errors in the policymaker’s knowledge of public expectations, which is not the case under learning. However, if there is sufficient policymaker uncertainty about the parameter values, partial commitment is best.     

Are employment decisions based on rational expectations?

Conventional employment functions with partial adjustment to output fitted to quarterly data tend to have positively autocorrelated residuals, to imply implausibly high returns to scale and almost always fail tests for parameter stability. The hypothesis of this paper is that mis-specified expectations are the main cause of these findings and rational and adaptive expectations models are compared. Further, employment is conditioned not on output but on variables which firms can more reasonably take as exogenous. ‘Disequilibrium’ features of labour markets are introduced by making adjustment costs depend upon current and expected labour market tightness.One of the implications of rational expectations is that the revision between points in time t and $\text{t ? 1}$ in the expected value of any variable should be independent of any information available before t and serially uncorrelated. Given a model of a forward looking firm whose hiring decisions are subject to quadratic adjustment costs, an appropriately transformed employment equation can be derived which has a very similar structure to the Koyck transformed employment equation which corresponds to adaptive expectations. Maximum likelihood estimation of the adaptive expectations form gives parameter estimates for quarterly British data for the manufacturing sector which are so unreasonable that this hypothesis can be rejected. Maximum likelihood estimation of the rational expectations form would involve modelling the stochastic processes of all the driving variables. However, conditional upon one parameter, consistent estimates of the remaining parameters can be obtained by OLS and these accord well with economic theory. This is the direct evidence in favour of the rational expectations hypothesis. However, it can also explain why the adaptive expectations form gives such poor results and why conventional employment functions give the unsatisfactory results referred to above. Further, rational expectations provides an explanation for the common finding, particularly in the context of employment and the demand for durable goods, of implausibly low or wrong signed levels effects in more general quarterly time series models with lagged dependent variables.     

Testing rational expectations by the use of overidentifying restrictions

A simple econometric test for rational expectations in the case in which unobservable, rationally expected variables appear in a structural equation is presented. Using McCallum's instrumental variable estimator as a base, a test for rational expectations per se and a joint test of rational expectations and hypotheses about the structural equation are presented. The new test is shown to be a new interpretation of Basmann's test of overidentifying restrictions. As an illustration, the hypothesis that the forward exchange rate is the rationally expected future spot exchange rate is tested and rejected.     

Market analysis with rational expectations: Theory and estimation

In a dynamic market structure the rational expectations hypothesis implies the existence of a ‘fixed point’ or‘equilibrium’ price sequence. For a prototype market structure we develop the This approach displays the relationships between exogenous structure, expectations, and behavior. The resulting framework suggests effective estimation procedures and clarifies analysis. Results for an application are presented and discussed.     

Testing the restrictions implied by the rational expectations hypothesis

The nature and form of the restrictions implied by the rational expectations hypothesis are examined in a variety of models with expectations and the properties of appropriate test statistics are analyzed with Monte Carlo evidence. Specifically, we consider the implications of lagged variables, simultaneous equations, and future period expectations upon the number and functional form of the rational expectations restrictions. Two asymptotically equivalent test statistics — a likelihood ratio and a Wald test — are available for implementing a test of these restrictions. Monte Carlo evidence is offered to provide a comparison between the properties of the alternative test statistics in small samples.     

On the impossibility of insider trade in rational expectations equilibria

Existing no trade results are based on the common prior assumption (CPA). This paper identifies a strictly weaker condition than the CPA under which speculative trade is impossible in a rational expectations equilibrium (REE). As our main finding, we demonstrate the impossibility of speculative asset trade in an REE whenever an insider is involved who knows the asset's true value. To model insider trade as an equilibrium phenomenon an alternative equilibrium concept than the REE is thus required.