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discussion of the different types of models in which complementarities arise. A number of the models presented in this book incorporate complementarities into a dynamic framework. A simple example is given in Section 4.6. Howitt and McAfee (1992) study learning in a dynamic search model with complementarities.

3.5 Stochastic Gradient Learning

We return to the cobweb model of Chapter 2 which we repeat for convenience:

pt = µ + αpte + δ wt−1 + ηt .

Various alternatives to least squares learning have been proposed in the literature. These include neural networks and genetic algorithms, which we will discuss in Chapter 15.10 Another learning scheme which has been proposed is the stochastic gradient method, and using our techniques it is possible to give formal results.11

Agents are assumed to use the linear forecast rule

The stochastic gradient algorithm adjusts the parameter estimates in accordance with the following scheme:

This algorithm differs from least squares by neglecting the Rt−1 term. It is thus a gradient algorithm rather than a Newton-type algorithm, since the latter also uses information on second moments (as does least squares). Stochastic gradient algorithms have been proposed as a simple alternative to least squares.

Substituting the forecast function into the original model yields

pt = (µ + αat−1) + (δ + αbt−1) wt−1 + ηt

10Time-varying parameter methods have also been suggested. For the cobweb model this has recently been examined by McGough (1999). See also Bullard (1992) and Margaritis (1990).

11See Sargent (1993), Kuan and White (1994), Heinemann (2000b), Barucci and Landi (1997), and Evans and Honkapohja (1998c).

The system (3.7), (3.8) can be combined to yield a stochastic recursive algorithm

φt = φt−1 + γt zt−1 (T (φt−1) − φt−1)zt−1 + ηt .

Following the methods of Chapter 2, the associated ODE can be computed as

of differential equations in φ with constant coefficients. Its coefficient matrix (α − 1)M of the ODE is negative definite iff α < 1, and under this condition the ODE is (globally) asymptotically stable.

It follows from the results on stochastic recursive algorithms that under the E-stability condition α < 1, stochastic gradient learning converges to the RE solution. Thus, for this model the convergence conditions for least squares and stochastic gradient learning are identical. For models of the cobweb type this holds more generally (see Evans and Honkapohja, 1998c), but there appear to be examples in which least squares and stochastic gradient learning do not have identical stability conditions (see Heinemann, 2000b).

3.6 Learning with Misspecification

So far it has been assumed that agents learn using a PLM (perceived law of motion) that is well specified, i.e., nests an REE of interest. However, economic agents, like econometricians, may fail to correctly specify the actual law of motion, even asymptotically. It may still be possible to analyze the resulting learning dynamics.

As an illustration, consider the Muth model with the reduced form

where we use the alternative notation Et−1pt for pte . For simplicity we assume that wt−1 is an iid vector of exogenous variables and ηt is an unobservable white noise shock. In the treatment in Chapter 2, agents were assumed to have a

PLM of the form pt = a + b wt−1 + ηt , corresponding to the REE. Suppose that instead their PLM is pt = a + ηt , so that agents do not recognize the dependence of price on wt−1, and that they estimate a by least squares. Then

at = at−1 + t−1(pt − at−1),

and the PLM at time t − 1 is pt = at−1 + ηt with corresponding forecasts Et−1pt = at−1. Thus the ALM is

pt = µ + αat−1 + δ wt−1 + ηt and the corresponding stochastic recursive algorithm is

at = at−1 + t−1(µ + (α − 1)at−1 + δ wt−1 + ηt ).

The associated ODE is da/dτ = µ + (α − 1)a, and it follows that at → a¯ = (1 − α)−1µ almost surely.12

In this case we have convergence, but it is not to the unique REE which is pt = (1 − α)−1µ + (1 − α)−1δ wt−1 + ηt . Agents make systematic forecast errors since their forecast errors are correlated with wt−1 and they would do better to condition their forecasts on this variable. However, we have ruled this out by assumption: we have restricted PLMs to those which do not depend on wt−1. Within the restricted class of PLMs we consider, agents in fact converge to one which is rational given this restriction. The resulting solution when the forecasts are Et−1pt = a¯ is

pt = (1 − α)−1µ + δ wt−1 + ηt .

We might describe this as a restricted perceptions equilibrium since it is generated by expectations which are optimal within a limited class of PLMs.13 The basic idea of a restricted perceptions equilibrium is that we permit agents to fall short of rationality specifically in failing to recognize certain patterns or correlations in the data.

It is apparent that there are many forms of misspecification that may be of interest. For example, the agents may include only a subset of wt−1 in their forecast rule. Similarly, the model might be nonlinear, while agents forecast using a linear model. We will explore some of these possibilities and others in Part V.

12The ODE da/dτ can also be interpreted as the E-stability equation for the underparameterized class of PLMs here considered.

13The term self-confirming equilibrium is also used in the literature, see Sargent (1999).

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As discussed in the first chapter, our basic approach in this book is to model economic agents as econometricians who estimate the stochastic process of relevant variables and use these estimates to make forecasts. We adopt this approach because it treats the agents as having a degree of rationality comparable to that of the economic analyst: when making forecasts, economists use econometrics and statistical inference. In this chapter we sometimes use simplified versions of the forecast rules in order to avoid technical complications which we will later treat carefully.

4.2 The Overlapping Generations Model

Various overlapping generations models have been popular frameworks for macroeconomic analysis. The basic overlapping generations model, the socalled Samuelson model, provides a simple dynamic model in which expectations play a crucial role and it provides a convenient example to illustrate some basic ideas of adaptive learning in a nonlinear context. We here develop a simple version of the model which we will later extend in various ways. In this version there is production of a single perishable good, using labor alone, under constant returns to scale.

The economy consists of overlapping generations of identical agents each of whom lives for two periods. Population is constant, as the old agents who die at the end of the second period of life are replaced by an equal number of young agents at the start of the next period. Agents work when they are young and consume when old. The utility function of an agent in generation t takes the form U (ct+1) − V (nt ), where ct+1 is consumption at old age and nt is labor supply. We assume that U (·) is concave and V (·) is convex. Moreover, both U (·) and V (·) are taken to be twice continuously differentiable.

Trade is intertemporal, since in each period t, the output produced by the young is sold to the old in a competitive market. In the basic model there is a constant stock of money M which is the only means of saving the revenue obtained from working. There are no capital goods. In the simplest model the production function depends only on labor input and is linear, so that ct = nt after a normalization by choice of units of measurement. With this production function, the wage earned is equal to the price of the consumption good in the same period.

The budget constraints faced by generation t are

pt nt = mt , pt+1ct+1 = mt .

Here mt denotes the nominal saving by the representative agent, and pt denotes the prevailing price in period t. At time t when the agent is deciding on its choice of nt , the current price pt is known, but next period’s price pt+1 is unknown. The agent’s optimization problem is thus

max Et U (ct+1) − V (nt ) subject to pt+1ct+1 = pt nt ,

where Et denotes the (subjective) expectations of the agent at time t. Substituting in the constraint for ct+1, differentiating with respect to nt , and interchanging the order of expectations and derivatives, leads to the first-order condition for interior points

Since this model can have equilibria in which money becomes worthless, it is convenient to reformulate the analysis in terms of the price of money qt = 1/pt , so that we have

The market-clearing condition is M = mt or equivalently qt M = nt at each period t.

For simplicity we postulate parametric forms for the utility functions

where σ, ε > 0. We also assume point expectations of the price level, i.e., that agents treat as certain their expectations of the price of money Et qt+1, which for convenience we will denote qte+1. The assumption of point expectations allows us to bring the expectation operator Et inside the (possibly) nonlinear function, so that the first-order condition can be written as

This relationship is graphed in Figure 4.1 for the case of σ < 1. With perfect foresight, qte+1 = qt+1, and it can be seen that there are two steady states. There

Figure 4.1.

is an interior steady state at qt = M−1 and an “autarkic” steady state at qt = 0. It can also be seen that there exists a continuum of perfect-foresight paths converging toward the autarky solution, indexed by the initial value of qt which can be chosen arbitrarily provided 0 < q0 < M−1.1

We now posit the forecast rule for qte+1. Suppose people form expectations adaptively from past data in the following way:

where 0 < γt < 1 is the gain sequence. Two main cases of interest are γt = t−1 and γt = γ , a constant.

The first case corresponds to agents taking the average of prices qi , i = 0, . . . , t − 1, i.e.,

t−1

qte+1 = t−1 qi ,

i=0

as can be verified by substitution into equation (4.2) with γt = t−1. Thus this forecast method corresponds to a learning rule in which agents estimate an unknown constant by updating the sample mean. Note that in this kind of learning rule, each new data point has a smaller weight with limt→∞ γt = 0. Such gain

1Note that for q0 > M−1, there appear to be dynamic perfect-foresight paths with qt → ∞. However, if an upper bound on labor supply is imposed, these paths are not feasible.

and where rt denotes the nominal interest rate. Finally, the “LM curve” describes the money market equilibrium, in which the demand for real balances is assumed to depend positively on output and negatively on the nominal interest rate

m = cI + pt + cq qt + cr rt + u3t , where cq > 0, cr < 0.

Here m is the logarithm of the money supply, assumed constant. u1t , u2t , u3t are white noise shocks to output supply, output demand, and money demand, respectively.

The model can be solved to yield the reduced form of the price level

where vt is a linear combination of the shocks and therefore satisfies

Et−1vt = 0.

The reduced form parameters are functions of the structural parameters and the key ones are given by

β0 = ap(1 + br cq cr−1) + br / ap(1 + br cq cr−1) + br cr−1

and

β1 = (1 − β0)/ 1 − cr−1 .

These satisfy the restrictions β1 > 0 and β0 + β1 < 1. Equation (4.3) has the stochastic steady-state solution

There are other solutions to the model, but these are stochastically explosive and equation (4.4) is the solution usually chosen in applied work.

To model the learning we suppose that agents perceive the economy to be in a stochastic steady state, i.e., that prices follow the process pt = a + vt , but