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.     

Biased Bayesian learning with an application to the risk-free rate puzzle

Based on the axiomatic framework of Choquet decision theory, we develop a closed-form model of Bayesian learning with ambiguous beliefs about the mean of a normal distribution. In contrast to rational models of Bayesian learning the resulting Choquet Bayesian estimator results in a long-run bias that reflects the agent's ambiguity attitudes. By calibrating the standard equilibrium conditions of the consumption based asset pricing model we illustrate that our approach contributes towards a resolution of the risk-free rate puzzle. For a plausible parameterization we obtain a risk-free rate in the range of 3.5–5%. This is 1–2.5% closer to the empirical risk-free rate than according calibrations of the rational expectations model.     

Policy change and learning in the RBC model

What is the impact of surprise and anticipated policy changes when agents form expectations using adaptive learning rather than rational expectations? We examine this issue using the standard stochastic real business cycle model with lump-sum taxes. Agents combine knowledge about future policy with econometric forecasts of future wages and interest rates. Dynamics under learning can have large impact effects and a gradual hump-shaped response, and tend to be prominently characterized by oscillations not present under rational expectations. These fluctuations reflect periods of excessive optimism or pessimism, followed by subsequent corrections.     

Asset pricing with expectation shocks

This paper adds persistent shocks into the adaptive learning expectation formation process in stochastic growth asset pricing production and endowment economies. These expectation shocks, designed to capture psychological elements which can arise from news, changes in sentiment, herding and bandwagon effects, generate waves of optimism and pessimism in equity price forecasts. The paper estimates parameters of the expectation shock and adaptive learning process with the method of simulated moments, and compares simulation results to U.S. economic and financial market stylized facts. Numerical results for both the estimated production and endowment economies show that the expectation shock model matches several of the stylized facts better than does a model that assumes rational expectations or adaptive learning alone.     

Modelling oil price expectations: Evidence from survey data

Using Consensus Forecast survey data on WTI oil price expectations for 3- and 12-month horizons over the period November 1989 to December 2008, we find that the rational expectation hypothesis is rejected and that none of the traditional extrapolative, regressive and adaptive processes fits the data by itself. We suggest a mixed expectation model defined as a linear combination of these traditional processes, which we interpret as the aggregation of individual mixing behavior and of heterogenous groups of agents using these simple processes. This approach is consistent with the economically rational expectations theory. We show that the target oil price included in the regressive component of this model depends on the long-run marginal cost of crude oil production and on short term macroeconomic fundamentals whose effects are subject to structural changes. For the two horizons, estimation results provide evidence for our mixed expectation model incorporating this break-dependent target price.     

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.     

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.     

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.     

Optimal consumption under uncertainty,liquidity constraints,and bounded rationality

I study how boundedly rational agents can learn a “good” solution to an infinite horizon optimal consumption problem under uncertainty and liquidity constraints. Using an empirically plausible theory of learning I propose a class of adaptive learning algorithms that agents might use to choose a consumption rule. I show that the algorithm always has a globally asymptotically stable consumption rule, which is optimal. Additionally, I present extensions of the model to finite horizon settings, where agents have finite lives and life-cycle income patterns. This provides a simple and parsimonious model of consumption for large agent based models.     

On the Epidemiological Microfoundations of Sticky Information*

We estimate and compare two models in which households periodically update their expectations. The first model assumes that households update their expectations towards survey measures. In the second model, households update their expectations towards rational expectations (RE). While the literature has used these specifications indistinguishably, we argue that there are important differences. The two models imply different updating probabilities, and the data seem to prefer the second one. We then analyse the properties of both models in terms of mean expectations, median expectations, and a measure of disagreement among households. The model with periodical updates towards RE also seems to fit the data better along these dimensions.     

Sentiment and the U.S. business cycle

Psychological factors are commonly believed to play a role on cyclical economic fluctuations, but they are typically omitted from state-of-the-art macroeconomic models.This paper introduces “sentiment” in a medium-scale DSGE model of the U.S. economy and tests the empirical contribution of sentiment shocks to business cycle fluctuations.The assumption of rational expectations is relaxed. The paper exploits, instead, observed data on expectations in the estimation. The observed expectations are assumed to be formed from a near-rational learning model. Agents are endowed with a perceived law of motion that resembles the model solution under rational expectations, but they lack knowledge about the solution’s reduced-form coefficients. They attempt to learn those coefficients over time using available time series at each point in the sample and updating their beliefs through constant-gain learning. In each period, however, they may form expectations that fall above or below those implied by the learning model. These deviations capture excesses of optimism and pessimism, which can be quite persistent and which are defined as sentiment in the model. Different sentiment shocks are identified in the empirical analysis: waves of undue optimism and pessimism may refer to expected future consumption, future investment, or future inflationary pressures.The results show that exogenous variations in sentiment are responsible for a sizable (above forty percent) portion of historical U.S. business cycle fluctuations. Sentiment shocks related to investment decisions, which evoke Keynes’ animal spirits, play the largest role. When the model is estimated imposing the rational expectations hypothesis, instead, the role of structural investment-specific and neutral technology shocks significantly expands to capture the omitted contribution of sentiment.     

On the initialization of adaptive learning in macroeconomic models

We review and evaluate methods previously adopted in the applied literature of adaptive learning in order to initialize agents’ beliefs. Previous methods are classified into three broad classes: equilibrium-related, training sample-based, and estimation-based. We conduct several simulations comparing the accuracy of the initial estimates provided by these methods and how they affect the accuracy of other estimated model parameters. We find evidence against their joint estimation with standard moment conditions: as the accuracy of estimated initials tends to deteriorate with the sample size, spillover effects also deteriorate the accuracy of the estimates of the model’s structural parameters. We show how this problem can be attenuated by penalizing the variance of estimation errors. Even so, the joint estimation of learning initials with other model parameters is still subject to severe distortions in small samples. We find that equilibrium-related and training sample-based initials are less prone to these issues. We also demonstrate the empirical relevance of our results by estimating a New Keynesian Phillips curve with learning, where we find that our estimation approach provides robustness to the initialization of learning. That allows us to conclude that under adaptive learning the degree of price stickiness is lower compared to inferences under rational expectations.     

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.     

The dangers of commitment: Monetary policy with adaptive learning

This paper studies a class of interest rate rules, introduced by Evans and Honkapohja (2001a, 2004), that respond to public expectations and to lagged variables. The policymaker commits to the extent that the interest rate responds to lagged output in an effort to influence public expectations. Simulation results show that full commitment, the commitment optimum under rational expectations, is not optimal under adaptive learning for any reasonable parameter values.     

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.     

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.     

Rational expectations models: An approach using forward–backward stochastic differential equations

We show that a general class of continuous time rational expectations models can be reformulated as forward–backward stochastic differential equations (FBSDEs). Using this connection we obtain results on the conditions under which paths leading to, or keeping close to equilibrium exist, as well as their qualitative properties. We also provide a method for the construction of such paths through the connection of FBSDEs with quasilinear partial differential equations (PDEs). The theory is applied to specific macroeconomic models.     

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.     

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.     

Good luck or good policy? An expectational theory of macro volatility switches

In an otherwise unique-equilibrium model, agents are segmented into a few informational islands according to the signal they receive about others' expectations. Even if agents perfectly observe fundamentals, rational-exuberance equilibria (REX) can arise as they put weight on expectational signals to refine their forecasts. Constant-gain adaptive learning can trigger jumps between the equilibrium where only fundamentals are weighted and a REX. This determines regime switching in macro volatility despite unchanged monetary policy and time-invariant distribution of exogenous shocks. In this context, a tight inflation-targeting policy can lower expectational complementarity preventing rational exuberance, although its effect is non-monotone.