Bruce Smith was a brilliant economist who made important contributions in several fields, most notably monetary and financial economics, monetary history, economic growth, business cycle theory, and development economics. Bruce was a prolific author with nearly 100 published papers and about a dozen unpublished papers. I will try in this short article to describe his way of doing economics and being an economist. Even though Bruce's research agenda was wide, his distinctive style was clear on every page of his writings.

The title speaks for itself: This special issue focuses attention on dynamical models in which expectations matter and in which expectational coordination is not taken for granted. This is a subject that has generated a lot of research in the past 20 years; the sample of papers presented here is witness to the present activity and the ongoing progress of our understanding. Here, as in previous research, the rational expectations construct remains the anchor of the analysis of expectational coordination, but it is subject to the additional critical scrutiny for which the present state of our knowledge calls more and more.

We analyze expectations coordination on a stationary sunspot equilibrium (SSE) using an eductive approach in which agents forecast the forecasts of others. This approach is grounded in some common-knowledge assumptions. We show that two institutional contexts may be envisaged; we also distinguish between stationary and nonstationary beliefs. We show that the conditions for eductive stability depend mainly on whether stationarity of beliefs is imposed. We also prove the existence of eductively stable SSEs by means of the Poincaré–Hopf theorem and show that eductively stable SSEs exist in a neighborhood of some indeterminate steady states.

We investigate local strong rationality (LSR) in a one-step-forward-looking univariate model with memory one. Eductive arguments are used to determine when common knowledge (CK) that the solution is near some perfect-foresight path is sufficient to trigger complete coordination on that path (i.e., the path is LSR). Coordination of expectations is shown to depend on three factors: the nature of the CK initial beliefs, the degree of structural heterogeneity, and the information structure. Our sufficient conditions for LSR precisely reflect these features and provide basic consistent justifications for the choice of the saddle-path solution.

Linear models with infinite horizon generally admit infinitely many rational expectations solutions. Consequently, some additional selection devices are needed to narrow the set of relevant solutions. The viewpoint of this paper is that a solution will be more likely to arise if it is locally determinate (i.e., locally isolated), locally immune to sunspots, and locally stable under learning. These three criteria are applied to solutions of linear univariate models along which the level of the state variable evolves through time. In such models the equilibrium behavior of the level of the state variable is described by a linear recursive equation characterized by the set of its coefficients. The main innovation of this paper is to define new perfect-foresight dynamics whose fixed points are these sets of coefficients, thus allowing us to study the property of determinacy of these sets, or, equivalently, of the associated solutions. It is shown that only one solution is locally determinate in the new dynamics. It is also locally immune to sunspots and locally stable under myopic learning. This solution corresponds to the saddle path in the saddle-point case.

Dynamic models in which agents' behavior depends on expectations of future prices or other endogenous variables can have steady states that are stationary equilibria for a wide variety of expectations rules, including rational expectations. When there are multiple steady states, stability is a criterion for selecting among them as predictions of long-run outcomes. The purpose of this paper is to study how sensitive stability is to certain details of the expectations rules, in a simple OLG model with constant government debt that is financed through seigniorage. We compare simple recursive learning rules, learning rules with vanishing gain, and OLS learning, and also relate these to expectational stability. One finding is that two adaptive expectation rules that differ only in whether they use current information can have opposite stability properties.

In their landmark paper, Bray and Savin note that the constant-parameters model used by their agents to form expectations is misspecified and that, using standard econometric techniques, agents may be able to determine the time-varying nature of the model's parameters. Here, we consider the same type of model as employed by Bray and Savin except that our agents form expectations using a perceived model with parameters that vary with time. We assume agents use the Kalman filter to form estimates of these time-varying parameters. We find that, under certain restrictions on the structure of the stochastic process and on the value of the stability parameter, the model will converge to its rational expectations equilibrium. Further, the restrictions on the stability parameter required for convergence are identical to those found by Bray and Savin.

Multiple-equilibria macroeconomic models suggest that consumers' and investors' perceptions about the state of the economy may be important independent factors for business cycles. In this paper, we verify empirically the interrelations between waves of optimism and pessimism and subsequent economic fluctuations. We focus on the behavior of nonfundamental movements in the consumer sentiment index, as a proxy for consumers' sunspots, and in the business formation index, representing investors' animal spirits, around economic turning points. We find that bearish consumers and entrepreneurs were present before the onset of some U.S. economic downturns, sometimes even when the fundamentals were all very strong. In particular, our analysis shows that self-fulfilling pessimism may have played a nontrivial role in the 1969–1970, the 1973–1975, and the 1981–1982 recessions. The results are robust to a range of alternative linear and nonlinear specifications. Our evidence provides some empirical support for the role of nonfundamental rational expectations in economic fluctuations.