Comment on "Adaptive Learning and Monetary Policy Design"

This paper helped me understand descriptions by Donald Brash, former Governor of the Reserve Bank of New Zealand,¹ of his operating procedure. Brash was under a legislative mandate to keep inflation in a narrow target zone. Brash said that his policy was recurrently to publish the Bank's forecasts of inflation and other variables and then to pronounce a judgment about whether these forecasts were consistent with the Bank's policy objectives. According to Brash, after that the "market would do the work" by pushing interest rates in a direction likely to make revised forecasts conform to the bank's policy target. I presume that the Bank's staff used a forecasting model that was close to a vector autoregression.

Before I studied Evans and Honkapohja's recent work, culminating in the present paper (Evans and Honkapohja 2003, this issue of JMCB), Brash's account of his operating procedure was a mystery to me.² But Evans and Honkapohja's papers go a long way toward making sense of a policy procedure that has as a crucial component the requirement that the bank sets interest rates to be a function of its measure of the private sector's expectations of inflation and other endogenous variables. Maybe their import what Donald Brash was telling us.

1. "Type I" Determinacy

Evans and Honkapohja's model has the following form:

Equation (1) includes the Euler equations and other forward-looking equations of a macroeconomic model that determines endogenous variables π_t as functions of exogenous variables x_t and policy variables i_t; Equation (3) describes the exogenous variable x; Equation (3) is a "forward-looking" policy rule for setting government policy variables. We can eliminate future values to get expressions for π_t and i_t that depend only on current and past values of x_t. First, solve Equations (1)-(3) to get the following solution for π _t:

A unique solution of this form can be solved if the roots of the characteristic polynomial associated with Equation (1) are half inside and half outside the unit circle. If more than half of the roots are outside the unit circle, there are multiple equilibria of this form. In this case I will say that the equilibrium is indeterminate "of type I" (to distinguish it from another type of indeterminacy that I will call "of type II") that concerns Evans and Honkapohja. Evans and Honkapohja take type I determinacy for granted.

Use Equations (3) and (4) to represent the interest rate rule as

Along the equilibrium path, both equations (Equations 3 and 5) describe the choice of interest rate. But had we expressed the interest rate rule in the form of Equation (5) in the first place, determinacy would be put in question again.

2. "Type II" Determinacy

What I call type II determinacy starts with the particular backward-looking interest rate rule (Equation 5) and investigates whether determinacy would obtain if we now attempted to solve Equations (1), (3), and (5). We have a new system of stochastic difference equations whose stability we must study. By construction, we know that Equation (4) is one solution of this system, but now there may be others too. If there are, we have type II indeterminacy.

Evans and Honkapohja study the (type II) determinacy and learnability of equilibria when interest rate rules are expressed in the backward-looking form of Equation (4). They pose the learnability question in terms of whether equilibria are "E-stable," that is, whether necessary conditions are satisfied for convergence to equilibria when agents do not know Equation (4) but each period estimates it by recursive application of least squares.

3. Their Proposal

Evans and Honkapohja propose to add two constraints to policy design (aka Ramsey) problems, that the government policy (1) be "learnable" and (2) lead to type II determinacy. These are sensible implementability requirements.

3.1 Their Findings

Evans and Honkapohja apply their criteria to a "new synthesis" monetary policy model as their laboratory. They analyze alternative interesting timing protocols, one that gives a time 0 government the authority...