Learning and Expectations in Macroeconomics

George W. Evans
Seppo Honkapohja
Pages: 424
https://www.jstor.org/stable/j.ctt7s6t9

1. Front Matter
(pp. i-viii)
(pp. ix-xiv)
3. Preface
(pp. xv-2)
4. Part I. View of the Landscape

• Chapter 1 Expectations and the Learning Approach
(pp. 5-24)

Modern economic theory recognizes that the central difference between economics and natural sciences lies in the forward-looking decisions made by economic agents. In every segment of macroeconomics expectations play a key role. In consumption theory the paradigm life-cycle and permanent income approaches stress the role of expected future incomes. In investment decisions present-value calculations are conditional on expected future prices and sales. Asset prices (equity prices, interest rates, and exchange rates) clearly depend on expected future prices. Many other examples can be given.

Contemporary macroeconomics gives due weight to the role of expectations. A central aspect is that expectations influence...

• Chapter 2 Introduction to the Techniques
(pp. 25-44)

In this second introductory chapter we will introduce the main analytical technique which we will use to study convergence under learning dynamics when the economy is subject to stochastic shocks. We will do this in the context of a simple economic model: the cobweb market model introduced in Chapter 1. Our presentation in this chapter will be heuristic and the techniques will be rigorously developed subsequently. Later chapters will also show how to apply these tools to study the dynamics of learning in numerous macroeconomic models.

In the cobweb model there is a unique rational expectations equilibrium (REE). Even if...

• Chapter 3 Variations on a Theme
(pp. 45-58)

In this chapter we discuss some extensions and variations of econometric learning. Several issues arise naturally. So far we have assumed representative agent learning, although diversity of expectations should surely be treated. One can also consider alternative adaptive learning schemes and the possibility that the agents do not know the true model. In this chapter we show how such issues can be readily addressed in the context of the basic cobweb and asset pricing models.

We also take up learning in nonstochastic frameworks and obtain the key conditions for local stability under adaptive learning of perfect-foresight steady states. Since some...

• Chapter 4 Applications
(pp. 59-84)

In this last introductory chapter we consider adaptive learning in several well-known macroeconomic models, including some standard models which appear in graduate-level textbooks. Stability under learning is of interest even in models with a unique equilibrium, such as the Ramsey growth model and the Real Business Cycle model. Here expectations play a central role in the structure of the model, but they have no independent influence on the paths of the economy. That is, given the current state of the economy, there is a single way to forecast the future under RE: expectations are fully determinate. Still, rational expectations remains...

5. Part II. Mathematical Background and Tools

• Chapter 5 The Mathematical Background
(pp. 87-120)

This chapter is devoted to a presentation of the mathematical concepts and techniques that are needed for a thorough understanding of the material in this book. Its purpose is to provide a convenient reference of the mathematical concepts and results that appear in different parts of the book. This should make the book essentially self-contained. However, we emphasize that the summary is no substitute for a proper study of these mathematical tools. The references cited in Section 5.8 should be consulted for thorough presentations of the mathematical techniques discussed here. At the same time we wish to point out that...

• Chapter 6 Tools: Stochastic Approximation
(pp. 121-146)

In Chapter 2 we analyzed in detail the cobweb model and showed how the analysis of least squares learning in that model formally leads to a stochastic recursive algorithm (SRA). Mathematically, such algorithms are dynamical systems consisting of two parts: (i) dynamics for estimating a vector of parameters, and (ii) dynamics for a vector of state variables. These are nonlinear stochastic systems operating in discrete time, but it turns out that their convergence can be studied by using the so-called ordinary differential equation (ODE) approach. The study of the convergence of adaptive learning behavior in macroeconomic models can generally be...

• Chapter 7 Further Topics in Stochastic Approximation
(pp. 147-170)

In the preceding chapter we provided the central convergence theorems under the assumption that the dynamics of the state variables follows a stationary vector autoregressive process (that is possibly dependent on the vector of parameters${\theta _{t - 1}}$). In this chapter we continue with the techniques for analyzing these algorithms and thereby provide some further results that can be useful in the study of econometric learning behavior in different models. We are interested in obtaining several extensions of the basic local convergence results.

First, we present some convergence results of adaptive algorithms that arise from modeling learning in nonstochastic frameworks. Such setups...

6. Part III. Learning in Linear Models

• Chapter 8 Univariate Linear Models
(pp. 173-204)

Many economic applications of rational expectations (RE) use linear models. These may be either exact formulations, for appropriate specifications of technology and preferences, linear approximations around a nonlinear RE solution, or ad hoc specifications which are taken to be linear for convenience. Frequently, the reduced form makes the endogenous variables of interest depend on expected future values of the endogenous variables (as well as on exogenous variables). This is crucial since the dependence on future expectations leads to the possibility of multiple rational expectations equilibria (REEs). This in turn leads to the issue of which solution should be selected by...

• Chapter 9 Further Topics in Linear Models
(pp. 205-226)

In this chapter we take up a number of further topics in the analysis of learning in univariate linear models. First, we provide an example of learning in a model with a mixture of dates at which expectations are formed. Second, we look at weak and strong stability for the basic “special case” of the previous chapter, as well as for the “extended special case,” also considered last chapter, which incorporates a lagged endogenous variable. These cases cover many models in the literature and thus provide a convenient class of models for discussing the literature on alternative selection criteria for...

• Chapter 10 Multivariate Linear Models
(pp. 227-264)

The techniques described in the previous chapter can be generalized to multivariate models, allowing us to analyze most of the models frequently encountered in macroeconomics, including the standard Real Business Cycle model and irregular versions with sunspot solutions, as well as more traditional IS-LM-Phillips curve models. We will present the general formal techniques after first giving an example.

As an introductory example we consider a fairly standard aggregate demand/aggregate supply model with rational expectations and gradual price adjustmet.

Example 1:An IS–LM–Phillips curve model.

${p_t} - {p_{t - 1}} = {a_0} + {a_1}{q_t} + \left( {E_{t - 1}^*{p_{t + 1}} - E_{t - 1}^*{p_t}} \right) + {v_1}_t,$

${q_t} = {b_0} - {b_1}\left( {{r_t} - E_{t - 1}^*{p_{t + 1}} + E_{t - 1}^*{p_t}} \right) + {v_{2t}},$

${m_t} - {p_t} = {c_0} + {c_1}{q_t} - {c_2}{r_t} + {v_3}_t,$

${m_t} = {d_0} + {d_1}{p_{t - 1}} + {d_2}{q_{t - 1}} + {d_3}{r_{t - 1}} + {d_4}{m_{t - 1}} + {v_{4t}}.$

The first equation is a Phillips curve in which inflation,...

7. Part IV. Learning in Nonlinear Models

• Chapter 11 Nonlinear Models: Steady States
(pp. 267-286)

In Chapter 4 several nonlinear economic models were introduced for the study of learning dynamics. Some of these models have unique equilibria, while in others multiple equilibria may prevail, as was illustrated in that chapter. The equilibria in nonlinear models can take different forms, such as steady states, cycles, and sunspot equilibria, and the models in Chapter 4 provided examples of these types of equilibria. We now start to analyze systematically adaptive learning in nonlinear models. Our emphasis will be on stochastic models, i.e., models which include intrinsic random shocks such as preference or productivity shocks, though we will briefly...

• Chapter 12 Cycles and Sunspot Equilibria
(pp. 287-314)

In this chapter we will continue the analysis of stochastic nonlinear models of the form

${y_t} = H\left( {G{{({y_{t + 1}},{v_{t + 1}})}^e},{v_t}} \right).$(12.1)

${y_t}$is a scalar variable and${v_t}$denotes a possible iid random shock to, say, preferences or technology, which might be present in the model. Here$G{({y_{t + 1}},{v_{t + 1}})^e}$denotes the value of$G({y_{t + 1}},{v_{t + 1}})$expected by agents at time 𝑡, and under rational expectations,$G{({y_{t + 1}},{v_{t + 1}})^e} = {E_t}G({y_{t + 1}},{v_{t + 1}})$. In the previous chapter this framework was introduced and we studied the stability of rational stochastic steady states under learning. As was there noted, models leading to equation (12.1) can have other types of REE besides steady states. These include...

8. Part V. Further Topics

• Chapter 13 Misspecification and Learning
(pp. 317-330)

In Section 3.6 of Chapter 3 we briefly considered the possibility of agents using a misspecified model. We take up this issue here at greater length. We will focus on two examples using models that have been previously analyzed under the assumption that their perceived laws of motion (PLMs) are correctly specified asymptotically. We will show that the same convergence tools used for correctly specified models can be used to show convergence to a restricted perceptions equilibrium in a misspecified model. Depending on the model and the nature of the misspecification, the relevant E-stability conditions which govern convergence may need...

• Chapter 14 Persistent Learning Dynamics
(pp. 331-360)

Throughout most of the book we have focused on the conditions under which adaptive learning rules converge in the limit to an REE. We have seen that this is usually governed, at least locally, by E-stability conditions and that, when there are multiple equilibria, these impose a substantive selection criterion. The previous chapter altered the framework to permit asymptotic misspecification of the law of motion followed by the economy. We saw that under appropriate stability conditions, the parameter estimates of the learning rule still converge asymptotically, but now to a forecast rule which is not fully rational, and which we...

• Chapter 15 Extensions and Other Approaches
(pp. 361-384)

The main body of this book has been devoted to statistical or econometric learning, since the greatest concentration of research has probably been in this approach. In recent years other approaches have also been introduced to model learning behavior in macroeconomic models, and the literature has also considered some topics that we have not covered in the earlier chapters. We rectify these omissions here by providing an overview of other approaches and some further topics.

Several strands of alternative learning models have their origins in computational intelligence. The basic idea is that certain artificial devices have capabilities to memorize and...

• Chapter 16 Conclusions
(pp. 385-388)

This book has focused on macroeconomic models in which expectations of current or future variables play a central role. We have treated a large range of models which are in current use in macroeconomics. These include standard linear setups, such as the cobweb, Cagan, overlapping contracts, and IS-LM-Phillipscurve models, and linearized multivariate structures, such as the Real Business Cycle and Farmer–Guo models. Several nonlinear frameworks, including various versions of overlapping generations models and economies with complementarities and coordination failure, have also been analyzed at some length.

Our approach has been to treat the forecasting agents as statisticians or econometricians...

9. Bibliography
(pp. 389-406)
10. Author Index
(pp. 407-410)
11. Subject Index
(pp. 411-421)