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.     

Testing Steady‐State Restrictions of Linear Rational Expectations Models when Data are Highly Persistent*

Steady‐state restrictions are commonly imposed on highly persistent variables to achieve stationarity prior to confronting rational expectations models with data. However, the resulting steady‐state deviations are often surprisingly persistent indicating that some aspects of the underlying theory may be empirically problematic. This paper discusses how to formulate steady‐state restrictions in rational expectations models with latent forcing variables and test their validity using cointegration techniques. The approach is illustrated by testing steady‐state restrictions for alternative specifications of the New Keynesian model and shown to be able to discriminate between different assumptions on the sources of the permanent shocks.     

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.     

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.     

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.     

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.     

STABILISATION POLICY,RATIONAL EXPECTATIONS AND PRICE‐LEVEL VERSUS INFLATION TARGETING: A SURVEY

We survey literature comparing inflation targeting (IT) and price‐level targeting (PT) as macroeconomic stabilisation policies. Our focus is on New Keynesian models and areas that have seen significant developments since Ambler's (2009, Price‐level targeting and stabilisation policy: a survey. Journal of Economic Surveys 23(5): 974–997) survey: optimal monetary policy; the zero lower bound; financial frictions and transition costs of adopting a PT regime. Ambler's conclusion that PT improves social welfare in New Keynesian models is fairly robust, but we note an interesting split in the literature: PT consistently outperforms IT in models where policymakers commit to simple Taylor‐type rules, but results in favour of PT when policymakers minimise loss functions are overturned with small deviations from the baseline model. Since the beneficial effects of PT appear to hang on the joint assumption that agents are rational and the economy New Keynesian, we discuss survey and experimental evidence on rational expectations and the applied macro literature on the empirical performance of New Keynesian models. Overall, the evidence is not clear‐cut, but we note that New Keynesian models can pass formal statistical tests against macro data and that models with rational expectations outperform those with behavioural expectations (i.e. heuristics) in direct statistical tests. We therefore argue that policymakers should continue to pay attention to PT.     

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.     

A comparison among partial adjustment,rational expectations and error correction estimates of the canadian demand for money

Starting from a dynamic optimization principle, the currently most popular approaches to modelling money demand functions are derived. The partial adjustment/adaptive expectations, rational expectations, and error correction mechanism formulations are then estimated using a common data set. The error correction mechanism equation is found to dominate the others either because their implicit restrictions are rejected (rational expectations) or by employing the encompassing principle (partial adjustment/adaptive expectations). Surprisingly all three forms have similar long-run solutions. Since the short-run dynamics differ substantially, the results have important implications for the conduct of monetary policy.     

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.     

Learning, monetary policy rules, and macroeconomic stability

Several papers have documented a regime switch in US monetary policy from ‘passive’ and destabilizing in the pre-1979 period to ‘active’ and stabilizing afterwards. These studies typically work with DSGE models with rational expectations.This paper relaxes the assumption of rational expectations and allows for learning instead. Economic agents form expectations from simple models and update the parameters through constant-gain learning. In this setting, the paper aims to test whether monetary policy may have been a source of macroeconomic instability in the 1970s by inducing unstable learning dynamics.The model is estimated by Bayesian methods. The constant-gain coefficient is jointly estimated with the structural and policy parameters in the system.The results show that monetary policy was respecting the Taylor principle also in the pre-1979 period and, therefore, did not trigger macroeconomic instability.     

An Econometric Analysis of Polish Inflation Dynamics with Learning about Rational Expectations

Rational expectations modelling has been criticized for assuming that economic agents can learn quickly about and compute rational price expectations. In response, various authors have studied theoretical models in which economic agents use adaptive statistical rules to develop price expectations. A goal of this literature has been to compare resulting learning equilibria with rational expectations equilibria. The lack of empirical analysis in this literature suggests that adaptive learning makes otherwise linear dynamic models nonlinearly intractable for current econometric technology. In response to the lack of empirical work in this literature, this paper applies to post-1989 monthly data for Poland a new method for modelling learning about price expectations. The key idea of the method is to modify Cagan’s backward-looking adaptive-expectations hypothesis about the way expectations are actually updated to a forward-looking characterization which instead specifies the result of learning. It says that, whatever the details of how learning actually takes places, price expectations are expected to converge geometrically to rationality. The method is tractable because it involves linear dynamics. The paper contributes substantively by analyzing the recent Polish inflation, theoretically by characterizing learning, and econometrically by using learning as a restriction for identifying (i.e., estimating wth finite variance) unobserved price expectations with the Kalman filter. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

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.     

A Reduced Rank Regression Approach to Tests of Asset Pricing

Both the Arbitrage Pricing Theory (APT) and the Capital Asset Pricing Model (CAPM) place restrictions of the cross-sectional variation of conditional expectations of asset returns and of macro indicators. We show that these restrictions imposed on the reference statistical models lead to special cases of the reduced rank regression model. The maximum likelihood problem is solved by canonical correlation analysis. Likelihood ratio tests about the number of factors underlying stock returns are straightforward to calculate, thus allowing discrimination between competing financial theories. Moreover LR tests on the relevance of each macroeconomic indicator within a chosen model can be implemented. Some of the tests are illustrated by an application to Italian stock market data.     

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.     

ON ERROR CORRECTION MODELS: SPECIFICATION, INTERPRETATION, ESTIMATION

Abstract. Error Correction Models (ECMs) have proved a popular organising principle in applied econometrics, despite the lack of consensus as to exactly what constitutes their defining characteristic, and the rather limited role that has been given to economic theory by their proponents. This paper uses a historical survey of the evolution of ECMs to explain the alternative specifications and interpretations and proceeds to examine their implications for estimation. The various approaches are illustrated for wage equations by application to UK labour market data 1855–1987. We demonstrate that error correction models impose strong and testable non-linear restrictions on dynamic econometric equations, and that they do not obviate the need for modelling the process of expectations formation. With the exception of a few special cases, both the non-linear restrictions and the modelling of expectations have been ignored by those who have treated ECMs as merely reparameterisations of dynamic linear regression models or vector autoregressions.     

E-stability and stability of adaptive learning in models with private information

The paper demonstrates how the E-stability principle introduced by Evans and Honkapohja [2001. Learning and Expectations in Macroeconomics. Princeton University Press, Princeton, NJ] can be applied to models with heterogeneous and private information in order to assess the stability of rational expectations equilibria under learning. The paper extends already known stability results for the Grossman and Stiglitz [1980. On the impossibility of informationally efficient markets. American Economic Review 70, 393–408] model to a more general case with many differentially informed agents and to the case where information is endogenously acquired by optimizing agents. In both cases it turns out that the rational expectations equilibrium of the model is inherently E-stable and thus locally stable under recursive least squares learning.     

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.