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.     

Rationality,specification tests,and macroeconomic models

Rational expectation models embody cross-equation restrictions that are implied by the theory of rational expectations. In this paper we illustrate how tests of these restrictions may be implemented in terms of general macroeconomic models by employing the models of Taylor and Sargent as examples. In addition, the more important issue of the proper interpretation of the results of these tests is addressed. We contend that tests for rationality should become part of the model-building process as they are akin to specification tests for models in which rational expectations is treated as the maintained hypothesis. A procedure is suggested when the restriction are inconsistent with data. Special emphasis is placed upon examining how changes in specifications of the model's exogenous variables can influence test results.     

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.     

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.     

Simulation‐based tests of forward‐looking models under VAR learning dynamics

In this paper we propose a simulation‐based technique to investigate the finite sample performance of likelihood ratio (LR) tests for the nonlinear restrictions that arise when a class of forward‐looking (FL) models typically used in monetary policy analysis is evaluated with vector autoregressive (VAR) models. We consider ‘one‐shot’ tests to evaluate the FL model under the rational expectations hypothesis and sequences of tests obtained under the adaptive learning hypothesis. The analysis is based on a comparison between the unrestricted and restricted VAR likelihoods, and the p‐values associated with the LR test statistics are computed by Monte Carlo simulation. We also address the case where the variables of the FL model can be approximated as non‐stationary cointegrated processes. Application to the ‘hybrid’ New Keynesian Phillips Curve (NKPC) in the euro area shows that (i) the forward‐looking component of inflation dynamics is much larger than the backward‐looking component and (ii) the sequence of restrictions implied by the cointegrated NKPC under learning dynamics is not rejected over the monitoring period 1984–2005. Copyright © 2010 John Wiley & Sons, Ltd.     

Stationary Markov equilibria on a non-compact self-justified set

The existence of stationary processes of temporary equilibria is examined in an OLG model, where there are finitely many commodities and consumers in each period, and endowments profiles and expectations profiles are subject to stochastic shocks. A state space is taken as the set of all payoff-relevant variables, and dynamics of the economy is captured as a stochastic process in the state space. In our model, however, the state space does not necessarily admit a compact-truncation consistent with the intertemporal restrictions because distributions over expectations profiles may have non-compact supports. As shown in Duffie et al. [Duffie, D., Geanakoplos, J., Mas-Colell, A., McLennan, A., 1994. Stationary Markov equilibria. Econometrica 62, 745–781), such a compact-truncation, called a self-justified set, is essential for the existence of stationary Markov equilibria. We extend their existence theorem so as to be applicable to our model.     

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.     

GMM Estimation with Non‐causal Instruments under Rational Expectations*

Lanne and Saikkonen [Oxford Bulletin of Economics and Statistics (2011a) Vol. 73, pp. 581–592], show that the generalized method of moments (GMM) estimator is inconsistent, when the instruments are lags of variables that admit a non‐causal autoregressive representation. This article argues that this inconsistency depends on distributional assumptions, that do not always hold. In particular under rational expectations, the GMM estimator is found to be consistent. This result is derived in a linear context and illustrated by simulation of a nonlinear asset pricing model.     

Weak Identification of Forward‐looking Models in Monetary Economics*

Recently, single‐equation estimation by the generalized method of moments (GMM) has become popular in the monetary economics literature, for estimating forward‐looking models with rational expectations. We discuss a method for analysing the empirical identification of such models that exploits their dynamic structure and the assumption of rational expectations. This allows us to judge the reliability of the resulting GMM estimation and inference and reveals the potential sources of weak identification. With reference to the New Keynesian Phillips curve of Galí and Gertler [Journal of Monetary Economics (1999) Vol. 44, 195] and the forward‐looking Taylor rules of Clarida, Galí and Gertler [Quarterly Journal of Economics (2000) Vol. 115, 147], we demonstrate that the usual ‘weak instruments’ problem can arise naturally, when the predictable variation in inflation is small relative to unpredictable future shocks (news). Hence, we conclude that those models are less reliably estimated over periods when inflation has been under effective policy control.     

Identifiability criteria for Muth-rational expectations models

The authors deal with complete static linear models that contain current Muth-rational expectations. Rank, order and variety conditions for the identifiability of the structural parameter are derived under general restrictions. We also correct statements that appeared in the literature. Our main finding is that, in general, the standard rank and order conditions are sufficient also for the identifiability of the Muth-rational expectations model parameter, whenever there are enough not fully anticipated exogenous variables. If the number of imperfectly forecasted exogenous variables falls short of the number of endogenous variables by g, then g extra restrictions are needed on every equation and the restrictions must meet easily verifiable variety conditions as well as an augmented rank criterion.     

Binary choice under social interactions: an empirical study with and without subjective data on expectations

This paper examines two methods of modeling binary choice with social interactions: models assuming homogeneous rational expectations and models using subjective data on expectations. Exploiting a unique survey conducted during the 1996 US presidential election that was designed to study voting behavior under social context, we find that in various model specifications using subjective expectations consistently improves models' goodness‐of‐fit; and that subjective expectations are not rational as formulated by Brock and Durlauf. Specifically, members' characteristics are individually important in forming expectations. We also include correlated effect in the rational expectation model. This extension provides a remedy to the selection issues that often arise in social interaction models. Copyright © 2008 John Wiley & Sons, Ltd.     

Restricted Kalman filtering revisited

We propose a more compact and general derivation of results concerning the estimation of linear state space models with linear restrictions in the state vector. The resulting methodological contributions are that the restricted Kalman filtering is valid regardless of the type of linear restriction being considered, and that linear restrictions can be carried out by any type of state smoothing.     

Constructing Multi‐Country Rational Expectations Models

This 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.     

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.     

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.     

Steady‐state modeling and macroeconomic forecasting quality

Vector autoregressions (VARs) with informative steady‐state priors are standard forecasting tools in empirical macroeconomics. This study proposes (i) an adaptive hierarchical normal‐gamma prior on steady states, (ii) a time‐varying steady‐state specification which accounts for structural breaks in the unconditional mean, and (iii) a generalization of steady‐state VARs with fat‐tailed and heteroskedastic error terms. Empirical analysis, based on a real‐time dataset of 14 macroeconomic variables, shows that, overall, the hierarchical steady‐state specifications materially improve out‐of‐sample forecasting for forecasting horizons longer than 1 year, while the time‐varying specifications generate superior forecasts for variables with significant changes in their unconditional mean.     

Optimal fiscal policy under learning

This paper characterizes optimal fiscal policy when agents learn about future taxation. A benevolent and fully rational government chooses taxes on labor income and state-contingent bonds to finance public spending, considering that private agents form their expectations through a learning algorithm. Facing a trade-off between distortionary taxes and distorted expectations, the Ramsey planner chooses the policy that minimizes the total cost of distortions. The analysis produces two main results. First, the government will use fiscal variables to manipulate expectations, reducing taxes and issuing debt at times of pessimism and doing the opposite at times of optimism. This speeds up learning. Second, the expectation-dependent fiscal plan is also history-dependent, and it prescribes taxes that are not as smooth and more persistent than under rational expectations. These findings are robust to alternative learning algorithms.     

Switching from complete to incomplete information

We construct an elementary mechanism [Dutta, B., Sen, A., Vohra, R., 1995. Nash implementation through elementary mechanisms in economic environments. Review of Economic Design 1, 173–203] that Nash implements the constrained Walrasian correspondence. We extend it to incomplete and non-exclusive information economies by enlarging the message space of agents. In addition, measurability restrictions on allocations with respect to prices proper to constrained rational expectations equilibria are imposed in the outcome function. We show that by imposing such restrictions, the mechanism Bayesian implements the constrained rational expectations equilibrium correspondence. This result shows game-theoretic connections between these two market equilibrium concepts. However, these connections are obtained at the price of strong restrictions on the behavior of agents.     

Forecasts of US short-term interest rates: A flexible forecast combination approach

This paper develops a flexible approach to combine forecasts of future spot rates with forecasts from time-series models or macroeconomic variables. We find empirical evidence that, accounting for both regimes in interest rate dynamics, and combining forecasts from different models, helps improve the out-of-sample forecasting performance for US short-term rates. Imposing restrictions from the expectations hypothesis on the forecasting model are found to help at long forecasting horizons.     

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.