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McCallum, Bennett T.. "Indeterminacy from inflation forecast targeting: problem or pseudo-problem?." Economic Quarterly. Federal Reserve Bank of Richmond. 2009. HighBeam Research. 17 Oct. 2018 <https://www.highbeam.com>.
McCallum, Bennett T.. "Indeterminacy from inflation forecast targeting: problem or pseudo-problem?." Economic Quarterly. 2009. HighBeam Research. (October 17, 2018). https://www.highbeam.com/doc/1G1-202078471.html
McCallum, Bennett T.. "Indeterminacy from inflation forecast targeting: problem or pseudo-problem?." Economic Quarterly. Federal Reserve Bank of Richmond. 2009. Retrieved October 17, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-202078471.html
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Monetary economists have been rather proud about developments in their subject over the past two decades. There has been great progress in formal analysis and also in the actual conduct of monetary policy. Analytically, the profession has developed an approach to policy analysis that centers around a somewhat standardized dynamic model framework that is designed to be structural--respectful of both theory and evidence--and therefore usable in principle for policy analysis. This framework includes a policy instrument that agrees with the one typically used in practice and, in fact, models of this type are being used (in similar ways) by economists in both academia and in central banks, where several economic researchers have gained leading policymaking positions. Meanwhile, in terms of practice, most central banks have been much more successful than in previous decades in keeping inflation low while avoiding major recessions (with a few exceptions) prior to 2008. Furthermore, these improvements have been interrelated: The "inflation targeting" style of policy practice that has been adopted by numerous important central banks--and that arguably has been practiced unofficially by the Federal Reserve (1)--is strongly related in principle to the prevailing framework for analysis. For a recent exposition discussing this development, by an author who has participated both as researcher and policymaker, see Goodfriend (2007).
There are, nevertheless, reasons for concern about current analysis including ongoing disputes about the empirical performance of key relationships in the semi-standard model; about communication and commitment mechanisms in theory and especially in practice; about the relationship of monetary policy to credit, fiscal, and foreign exchange policies; and about a myriad of technical details. (2) Also, there is much uneasiness about current policy approaches in the face of major credit market difficulties and indications of rising inflation.
In this context, the present article will be devoted to one specific problematic feature of the recent analytical literature, namely, a lack of agreement concerning the importance of multiple-solution indeterminacies in the analysis of monetary policy rules. References to "indeterminacy," in the sense of more than one dynamically stable solution, or "determinacy" appear on about 75 different pages of the hugely influential treatise on monetary policy analysis by Woodford (2003a) and are ubiquitous in the literature, (3) with a substantial majority of references expressed from the point of view that takes indeterminacy per se to be a matter of serious concern, e.g., implying that policies leading to model equilibria with that property should be rigorously avoided. The motivation is that indeterminacy should be avoided because it implies both that the policymaker cannot know which candidate equilibrium will prevail and also the possibility that "sunspot" effects may be created so as to greatly increase the volatility of crucial variables. (4) Several writers, however, have expressed the view that indeterminacy per se is not necessarily a problem--that a more appropriate criterion would be based on the concept of learnability of potential equilibria. This latter position has been taken overtly by McCallum (2001, 2003), Bullard and Mitra (2002), and Bullard (2006), and is stated or indirectly implied in a large number of writings by Evans and Honkapohja, including their influential and authoritative treatise (2001).
In the present article I wish to develop the position that indeterminacy is not necessarily problematic in the context of one particular application, namely, inflation forecast targeting in the sense of Taylor-style policy rules that respond, not to current (or past) inflation rates, but to currently expected values of inflation in future periods. That such indeterminacies might be brought about, and be undesirable, by strong responses of this type was first suggested by Woodford (1994) (5), and the argument was further developed by Bernanke and Woodford (1997), Clarida, Gali, and Gertler (2000), and (most thoroughly) Woodford (2003a, 256-61). Subsequently, many other authors have adopted this point of view, which is briefly mentioned in the textbooks of Walsh (2003, 247) and Gali (2008, 79-80). Indeed, it is apparently the prevailing point of view among analysts, despite the positive actual experience of the Bank of England over (say) 1996-2006. (6) I have briefly taken the opposing line of argument, that strong responses to expected future inflation rates will not be problematic, in McCallum (2001) and (2003), but those papers were primarily occupied with more general topics, which prevented a full development of this particular issue. (7)
In what follows, I will begin in Section 1 with an exposition of the nature of the indeterminacy problem in the context of inflation forecast targeting. Section 2 will then be devoted to the concept of learnability of a rational expectations (RE) solution. The position taken here is that the learnability of any particular RE solution should be considered a necessary condition for that solution to be plausible and, therefore, an equilibrium appropriate as a basis for thinking about real-world policy. In Section 3, numerical examples are developed to illustrate the points that have been made more generally, but also more abstractly, in Sections 1 and 2 and in previous writings. Section 4 then takes up the somewhat esoteric topic of "sunspot" solutions, i.e., solutions that include random components that are entirely unrelated to the specified model, including its exogenous variables. Finally, Section 5 provides a brief conclusion.
1. BASIC ANALYSIS
For concreteness, let us now adopt a simple model, representative of the recent literature, in which to discuss the issues at hand. It can be expressed in terms of a familiar three-equation structure as follows:
[y.sub.t] = [E.sub.t] [y.sub.t+1] + b ([R.sub.t] - [E.sub.t] [[pi].sub.t+1]) + [[upsilon].sub.t] b < 0, (1)
[[pi].sub.t] = [beta] [E.sub.t] [[pi].sub.t+1] + k ([y.sub.t] - [[bar.y].sub.t]) + [u.sub.t] k > 0; 1 > [beta] > 0. (2)
[R.sub.t] = (1- [[mu].sub.3]) [(1 + [[mu].sub.1]) [[pi].sub.t] + [[mu].sub.2]/4 ([y.sub.t] - [[bar.y].sub.t])] + [[mu].sub.3] [R.sub.t-1] + [e.sub.t] [[mu].sub.1] [less than or equal to] 0. (3)
Here, [y.sub.t], and [[pi].sub.t] are output and inflation expressed as fractional deviations from steady state and R, is a one-period nominal interest rate that serves as the policy instrument. Thus, (1) is an IS-type relation consisting of a consumption Euler equation in combination with the overall resource constraint, (2) is a Calvo-style price adjustment equation, and (3) is the monetary policy rule. (8) Also, [bar.y.sub.t] is the flexible-price, natural rate of output, assumed to be generated exogenously by an AR (1) process (9) with AR coefficient [[rho].sub.a] and innovation standard deviation SD(a). In the policy rule, the implicit target rate of inflation is zero. We will take the shock processes for [u,sub.t] and e, to be white noise (with standard deviations SD(u) and SD(e)) and the process for [v.sub.t] to be AR( 1) with AR coefficient p and standard deviation SD(v).
In the policy rule, the policy parameters [[mu].sub.1] and [[mu].sub.2] govern the strength of the central bank's policy response to deviations of inflation and output, respectively, from their target values, while [[mu].sub.3] reflects the extent of interest rate smoothing. We begin with the central bank's policy responding to current observed inflation, [[pi].sub.t], and subsequently consider rules with a response to expected future inflation. In what follows, I will, for clarity, typically take [[mu].sub.2] to be zero so that policy is responding only to inflation (usually with considerable smoothing). This practice (i.e., setting [[mu].sub.2] = 0) changes the numerical values at which effects such as indeterminacy occur but does not alter the arguments to be made in any essential manner.
Let us begin the analysis by also setting [[mu].sub.3] = 0 and [[mu].sub.t] 0 so that there is no smoothing and no price-setting shock. Then, substitution of equation (3) into (1) yields
[y.sub.t] = [E.sub.t] [y.sub.t+1] + b [(1 + [[mu].sub.1]) [[pi].sub.t] + [e.sub.t] - [E.sub.t] [[pi].sub.t+1]] + [[upsilon].sub.t], (4)
and we can consider (2) and (4) as a two-equation system determining the evolution of [[pi].sub.i]t and [y.sub.t]. The "fundamentals" or minimum-state-variable (MSV) solution will have these variables determined as linear functions of [[upsilon].sub.t], [e.sub.t] and [[bar.y].sub.t] (10) As one final simplification we take the latter ([bar.y].sub.t]) to be constant and normalize it at zero. Then a fundamentals solution to the model (2)(4) will be of the form
[y.sub.t] = [[empty set].sub.11] [[upsilon].sub.t] + [[empty set] .sub.12] [e.sub.t] and (5)
[[pi].sub.t] = [[empty set].sub.21] [[upsilon].sub.t] + [[empty set].sub.22] [e.sub.t], (6)
with constant terms again omitted only for simplicity.
In this case, the expected values one period ahead are [E.sub.t] [y.sub.t][y.sub.t-1]= [[empty set.11]] [rho][[upsilon].sub.t] and [E.sub.t][[pi]sub.t+1] = [[empty set].sub.21] [rho][[upsilon].sub.t]. Then we can substitute these two expressions plus (5) and (6) into (2) and (4) to obtain undetermined-coefficient relations that express the [[empty set].sub.ij] coefficients of the solution expressions in terms of the parameters of the structural equations (2) and (4). The results of that (tedious but straightforward) exercise are as follows:
[[empty set].sub.11] = [1 - [beta][rho]]/[(1 - [rho] (1 - [beta][rho] - b[kappa](1 + [[mu].sub.1] - [rho])], (7)
[[empty set].sub.12] = [b]/[1 - [kappa]b (1 + [[mu].sub.1])], (8)
[[empty set].sub.21] = [[kappa]]/[(1 - [rho] (1 - [beta][rho]) - b[kappa] (1 + [[mu].sub.1] - [rho]), and (9)
[[empty set].sub.22] = [kappa]b/1 - [kappa]b (1 + [[mu].sub.1]. (10)
Here, the specified signs of the basic parameters imply that [[empty set]. …
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