MODEL

Suppose that the reduced form of the model is

where {w_t} is an AR(1) exogenous variable with |ρ| < 1 and u_t ∼i.i.d.(0, σ²_u). When writing the expectations, the asterisk is used to denote that they are not necessarily the expectations in the statistical sense.

Let x_t = (y_t, w_t)′ and Φ = (ϕ₁, ϕ₂)′. If agents perceive the law of motion of y_t to be

then the true law of motion of y_t is

where

is a mapping from the perceived law of motion to the true law of motion and

Solving the fixed-point problem

yields the MSV rational expectations solutions

It can be shown that for |α + λ| < 1, the only stationary solution is

. Furthermore, for α and λ such that |α + λ|>1,

and

both solutions are stationary. For the remaining combinations of α and λ, both solutions are either nonstationary or nonreal.²

E-STABILITY AND LEARNABILITY

An REE is expectationally stable or E-stable if it is a locally asymptotically stable equilibrium of the ordinary differential equation

Equivalently, an REE is E-stable if the Jacobian of T(Φ) − Φ evaluated at the REE, that is,

is a stable matrix (i.e., it has eigenvalues with strictly negative real parts).³

A widely used learning algorithm is the recursive least squares, which is expressed as

It has been shown by Marcet and Sargent (1989) that E-stability is a necessary and sufficient condition for convergence of the recursive least-squares algorithm. Because of this property and because least squares is a simple and natural choice for estimating parameters of linear models, E-stability has been widely used as a learnability criterion.

Nevertheless, E-stability does not always imply learnability when the estimation algorithm is stochastic gradient. This learning algorithm is expressed recursively as

The algorithm differs from least squares in that it ignores the second moment matrix when updating.⁴

where

. The local asymptotic stability of an REE

under stochastic gradient learning is again determined by the stability of the Jacobian matrix J^SG(Φ) = d[M(Φ)(T(Φ) − Φ)]/dΦ, evaluated at

. The stochastic gradient algorithm converges locally to the REE if and only if the real parts of the eigenvalues of

are strictly negative.

Using the reduced-form model introduced in the preceding section, it is possible to demonstrate that E-stability, that is, stability of

, does not necessarily imply stability of

. The E-stability conditions [by imposing that the eigenvalues of

have strictly negative real parts] are

For |α + λ| < 1, the unique stationary solution

is always E-stable. For the areas defined by |α + λ|>1, |αλ| < 1/4, and |α| < 1/2, it is possible to find regions for which both solutions are E-stable, only one solution is E-stable, or no solution is E-stable. For example, if ρ>0, in the area defined by α, λ>0, λ < −ρ²α + ρ, and α + λ>1 (shown in Figure 1 as region II, including the shaded area), both solutions are stationary, but only

is E-stable, whereas in the area defined by α, λ>0, λ > − ρ²α + ρ and αλ < 1/4 (shown in Figure 1 as region III), both solutions are stationary, but none is E-stable.

E-stability and learnability regions. In region I, defined by the thicksolid line, the solution $\bar{\Phi}_{-}$ is E-stable and learnable with stochasticgradient. In region III, the solution is neither E-stable nor learnable withstochastic gradient. In region II (which includes the shaded area) thesolution is E-stable, but is learnable with stochastic gradient only forsome parameters. The shaded area corresponds to combinations ofα and λ where the solution isE-stable but not learnable with stochastic gradient.

Next, the conditions for learnability under stochastic gradient are determined by the stability of the matrix

where the matrix

as well as all the derivations of the stability properties are given in Appendixes A and B.

Although it is not possible to present the learnability conditions under stochastic gradient learning in an elegant way, it is straightforward to explore these numerically, by using a finely defined grid of the regions of interest. In this manner, it can be shown, for example, that

is stable under stochastic gradient only for part of region II of Figure 1. In particular, in the shaded part of region II, the E-stable solution

is not learnable by stochastic gradient learning.⁵

is not learnable under stochastic gradient, becomes larger.

Models that can be expressed in the reduced form analyzed here are the loglinearized versions of the real business cycle (RBC) model and, in general, several dynamic stochastic general equilibrium models. An example of how such models can be written in this reduced form can be found in Giannitsarou (2004). Note that for standard versions of the RBC model [e.g., Hansen (1985)], the reduced-form coefficients are in the zone defined by |α + λ| < 1; that is, the model is regular or saddle-point stable, for all reasonable calibrations of the model parameters. In this zone, the unique stationary equilibrium is E-stable and always learnable under stochastic gradient learning. However, for irregular models (i.e., models with indeterminacies), reasonable calibrations can yield solutions in the shaded parameter region of interest shown in Figure 1. In the next section, I provide an example that fits in this category, namely, the model of Schmitt-Grohé and Uribe (1997). Thus the counterexample provided here cannot be characterized as pathological.

Furthermore, the result presented here has implications for the learnability of REEs under heterogeneous learning, in particular, for the case where some proportion of the population estimates using least squares and the rest of the agents use stochastic gradient. This issue is covered extensively by Giannitsarou (2003).

Finally, turning to the general issue of equilibrium selection, the present result is also related to the recent work of McCallum (2002a,b). Basing his analysis on a similar reduced-form model (the difference is that it includes an intercept), McCallum first demonstrates how to select a unique MSV solution and then shows that this solution is always E-stable, thus concluding that it is always (least squares) learnable in real time. This argument is then used to reinforce the appropriateness of his equilibrium selection device. However, as illustrated here, the unique MSV solution (as defined by McCallum) may fail to be learnable if the adaptive algorithm is other than least-squares learning.

EXAMPLE

In this section, the model of Schmitt-Grohé and Uribe (1997) is employed as an example for demonstrating how a plausible RBC model can be mapped into the reduced form (1)–(2). This model is irregular; that is, for certain parameter values, it has an indeterminate steady state, and therefore it is a potential candidate for having parameter values that bring the coefficients α and λ in the shaded area of Figure 1. The model is an extension of the Hansen (1985) model, augmented with a government that maintains a balanced budget and finances its constant expenditures by taxing labor income at an endogenously determined rate.⁶

where K_t, Y_t, C_t, N_t, Θ_t, and G denote capital, output, consumption, labor, labor income tax rate, and government expenditures, respectively. It is assumed that δ, β, Θ_t ∈ (0, 1). Furthermore, the production function has constant returns to scale, s_k + s_n = 1, and s_k, s_n>0. The loglinearized equilibrium conditions are

where s_i = δK/Y, s_c = C/Y, and λ_t is the Lagrange multiplier. Uppercase letters denote the steady-state values of the variables; lowercase letters denote the loglinear variables, that is, x_t = logX_t − logX for any variable X_t of the model. Equations (22)–(25) correspond to equations (12)–(15) of Schmitt-Grohé and Uribe (1997). Using (22) and (25), one can write n_t, by eliminating θ_t, as

Next, replace n_t in (23) and (24) to obtain

The coefficients κ_i, μ_i, i = 1, 2, 3 are stated in Appendix C. Using the above equations, one can eliminate λ_t to obtain the reduced form (1), with

where d = μ₁ + κ₁μ₂ − κ₂. Using the calibration of Schmitt-Grohé and Uribe (1997) for yearly data (δ = 0.1, β = 1/1.04, s_k = 0.3) and setting ρ = 0.9 and σ = 0.01, it can easily be shown that for labor tax rates in the range (0.6660, 0.7194) the solution

is E-stable but not learnable with stochastic gradient learning. Figure 2 summarizes the stability properties for both solutions, for the above calibration and for all steady-state labor tax rates Θ ∈ (0, 1).⁷

E-stability and learnability for the model of Schmitt-Grohé and Uribe (1997).The left column shows the values of the steady-state tax rate Θ that determinethe stability properties of the twosolutions. For the crossed-out areas, the corresponding solutions are eithernonstationary or nonreal. E & SG denotes a solution that isE-stable and learnable with stochastic gradient. A hyphen isused to denote a solution that is neither E-stable nor learnable withstochastic gradient.

CONCLUDING REMARKS

The idea behind the E-stability conditions goes far back to the work of DeCanio (1979) and Evans (1985) and it was originally viewed as a kind of learning process taking place in notional time, without the need of any associated real-time adaptive algorithm. Originally, its appeal drew from explaining how economic agents come to possess rational expectations, thereby justifying the existence of rational expectations equilibria. After Marcet and Sargent (1989) showed the equivalence of E-stability with convergence of real-time adaptive least-squares learning, the E-stability conditions have been extensively used as a criterion of learnability of REEs.

Motivated by this fact, I provided a counterexample that shows that E-stability generally does not imply learnability. Naturally, this result raises an important issue: How robust is adaptive learning as an equilibrium selection device? So far, the literature has been heavily relying on E-stability to make inferences about which equilibria are learnable or not; it appears, however, that which equilibria are learned depends on the selected learning algorithm. This is a somewhat alarming result, given that the suggested alternative algorithm (stochastic gradient) is just a small deviation from the commonly used least-squares algorithm. It would thus be useful to identify general criteria under which E-stability implies or does not imply stability under algorithms other than the least squares.