Formulating wald tests of the restrictions implied by the rational expectations hypothesis

In this paper, we examine the small sample properties of alternative formulations of Wald tests of non-linear restrictions implied by the rational expectations hypothesis. A Monte Carlo analysis is presented as well as an example using Canadian aggregate time series data. The evidence indicates that Wald test results are extremely sensitive in small samples to the way in which the non-linear restrictions in such models are parameterized, with a multiplicative form yielding tests of most accurate size. Least squares degrees of freedom adjustments also improve the sample performance of the tests.     

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.     

Identification of rational expectations models

This paper deals with the problem of the identification of simultaneous Rational Expectations (RE) models. In the case of RE models with current expectations of the endogenous variables, the necessary and sufficient conditions for the global identification are derived explicitly in terms of the structural parameters and the linear homogenous identifying restrictions. It is shown that in the absence of a priori restrictions on the processes generating the exogenous variables and the disturbances, RE models and general distributed lag models are ‘observationally equivalent’. In the case of RE models with future expectations of the endogenous variables, a general solution that highlights the ‘non-uniqueness’ problem and from which other solutions such as forward or backward solutions can be obtained, is derived. It is shown that untestable and often quite arbitrary restrictions are needed if RE models with future expectations are to be identifiable. Certain order conditions similar to those obtained for the identification of RE models with current expectations are also derived for this case.     

On the solutions in rational expectations models

This paper deals with the solutions to macroeconomic models with rational expectations. A first purpose is to demonstrate that two forms of the efficient-market equation commonly used in the literature may be treated as approximations of the same general equation, of different orders with respect to perturbative forces of the market. Second aim is to show how the perturbative terms can be given explicit forms. Finally, the paper addresses the question of finding contributions which can concur to explain the phenomenon usually referred to asstock price volatility.

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Riassunto Si è interessati alla soluzione di modelli macroeconomici con aspettativa razionale. In primo luogo si vuole dimostrate come due forme differenti dell'equazione del mercato efficiente utilizzate nella letteratura possano essere considerate approssimazioni della stessa equazione generale, di ordini diversi rispetto alle forze perturbative del mercato. Si vuole quindi individuare una tecnica per dare forma esplicita ai termini perturbativi. Infine, si va alla ricerca di contributi che possano concorrere a spiegare il fenomeno noto comevolatilità dei prezzi del mercato azionario.

     

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.     

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.     

Expectation formation of older married couples and the rational expectations hypothesis

This paper tests the Rational Expectations (RE) hypothesis regarding retirement expectations of older married American couples, controlling for sample selection and reporting biases. In prior research we found that individual retirement expectation formation was consistent with the Rational Expectation hypothesis, but in that work spousal considerations were not analyzed. In this research we take advantage of panel data on expectations to test the RE hypothesis among married individuals as well as joint expectations among couples. We find that regardless of whether we assume that married individuals form their own expectations taking spouse's information as exogenous, or the reports of the couple are the result of a joint expectation formation process, their expectations are consistent with the RE hypothesis. Our results support a wide variety of models in economics that assume rational behavior of married couples.     

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 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.     

Structural stability for dynamical economic models

The paper discusses the implications for economics of the notion of structural stability. Definitions are given to characterize dynamical economic models generating movements the qualitative properties of which are conserved under perturbation. Consequences of these definitions, in particular for time evolutions which are globally or locally stable, are then presented through three theorems. Last, two examples — one in micro, one in macroeconomics — illustrate the type of results that can be obtained from such theorems.     

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.     

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 design of feedback rules in linear stochastic rational expectations models

This paper is concerned with optimal and sub-optimal feedback rules for linear stochastic continuous time models with rational expectations. We consider four types of feedback rules: (1) the optimal but time-inconsistent rule which is available if the controller is able to commit himself or herself; (2) quasi-optimal and time-inconsistent rules of the form w = Dy where w is the vector of instruments, y the state vector and D a matrix of constants possibly with constraints; (3) the optimal time-consistent rule which is also linear in y; (4) ‘over-stable’ rules which have ‘too many’ stable roots. We show that rules of type (1) can be expressed and implemented as a form of integral control, all except type (2) satisfy certainty equivalence and that rules of type (4) will always be inferior to the optimal rule (1). These results are demonstrated in two illustrative examples.     

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.     

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.     

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. It is shown that the existence and uniqueness of the well-known forward solution are preserved in a general context. We also 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. We show that the forward solution is the only rational expectations equilibrium satisfying the no-bubble condition and consequently, it is the most relevant fundamental solution within the class of fundamental equilibria. Several economic examples are provided where the fundamental solutions obtained by other solution methods and refined by other solution selection criteria violate the no-bubble condition.     

Alternation of parties in power and economic volatility: testing the rational partisan hypothesis and policy learning hypothesis

Right and left parties have distinct macroeconomic preferences that could create different levels of volatility during their executive tenures. But rational partisan theory argues that, because actors in the economy anticipate ruling party behavior, partisan differences only matter when election outcomes are uncertain. We argue that policy risk from ruling parties extends beyond elections, leading to important variation in growth volatility that occurs during a ruling party’s tenure. Building on theories of policy risk and learning, we argue that after elections, economic actors still face uncertainty about the policies of new ruling parties. With time in power, new ruling parties build policy track records, reducing policy risk and, thus, volatility. We estimate a learning curve model of ruling party duration’s effect on the variation in quarterly GDP growth rates. Using data from 44 democracies between 1981 and 2009, we find that learning processes extend beyond the conclusion of uncertain elections.