The Role of “Determinacy”
in Monetary Policy Analysis
Keynote Speech by Bennett T. McCallum

It is well known that the concept of “determinacy”—a single stable solution—plays a major role in contemporary monetary policy analysis. But
while determinacy is desirable, other things equal, it is not necessary
for a solution to be plausible and is not sufficient for a solution to be
desirable. There is a related but distinct criterion of “learnability” that
seems more crucial. This paper argues that recognition of information
feasibility requires that a candidate solution must, to be plausible, bequantitatively learnable on the basis of information generated by the
economy itself. Since a prominent least-squares (LS) learning process
is highly “biased” toward learnability, it is reasonable to regard it asa necessary condition for any specific solution to be relevant. This implies that determinacy is not necessary for policy analysis; there may
be more than one stable solution, but only one that is LS learnable.
Also, determinacy is not sufficient for satisfactory policy analysis; explosive solutions pertaining to nominal variables will not be eliminatedby transversality conditions. For these and other reasons, the role ofdeterminacy in monetary policy analysis should be reconsidered andsubstantially de-emphasized.
Keywords: Determinacy; Learnability; Rational expectations; Multiple
solutions; Monetary policy
JEL Classification: C62, E4, E5, E52

Carnegie Mellon University and National Bureau of Economic Research (E-mail: bmccallum@
cmu.edu)This paper was prepared for the 2009 International Conference, “Financial System and Monetary
Policy Implementation,” held by the Institute for Monetary and Economic Studies, Bank of Japanin Tokyo on May 27–28, 2009.
MONETARY AND ECONOMIC STUDIES/NOVEMBER 2009
DO NOT REPRINT OR REPRODUCE WITHOUT PERMISSION

25

I. Introduction
The conference topic “Financial System and Monetary Policy Implementation” isextremely important as well as highly topical. It is also quite broad, indeed broad
enough to justify my topic, which is the role of determinacy in monetary policy analysis.
This may seem somewhat esoteric, but it is very well known to researchers that theconcepts of “determinacy” and “indeterminacy” play a fundamental role in contemporary policy analysis—and especially in relation to monetary policy implementation.
In terms of statistical evidence regarding research devoted to determinacy issues, a
bit of searching shows that explicit references to this general topic appear on about
75 different pages in Michael Woodford’s hugely influential treatise Interest and Prices(Woodford [2003a]). In addition, the number of new writings (books, articles, and working papers) with both of the phrases “indeterminacy” and “monetary policy” appearing
in their text was 166 over the time span January 1995 through June 2008.1 In thisliterature, the meaning of determinacy is that the system being analyzed—a macroeconomic model plus the central bank’s policy rule—has a single rational expectations(RE) solution that is dynamically stable (i.e., not explosive). “Indeterminacy” is usuallytaken to mean more than one stable solution, so a third possibility is that none of
the RE solutions is stable. Then the standard procedure in policy analysis is to treat
determinacy as a necessary condition for a recommended (or even potentially recommended) policy rule. In other words, model-plus-rule combinations that imply either
indeterminacy or an explosive solution are ruled out as highly undesirable.2 If a modelis taken as given, then any policy rule that leads to indeterminacy is typically viewed
as not worthy of consideration.
For several years, I have taken a minority position based on a belief that the emphasis being given to these particular determinacy/indeterminacy concepts is unwarranted
and occasionally misleading, especially in the design of recommended rules for monetary policy. I would agree that, other things equal, determinacy is desirable; but would
contend that (assuming a given model) determinacy is not necessary for a solution to
be plausible and in any case is not sufficient for a solution to be desirable. There isa related but distinct criterion of “learnability” that is more crucial, and which should
be regarded as a necessary condition for a solution (resulting from a model-plus-rule
combination) to be considered as plausible and thus relevant for policy analysis. This
position, which has also been explicitly or implicitly advanced by a few other scholars,3
leads to different conclusions regarding suitable policy rules in a number of cases thathave received considerable attention in the literature.

1. This figure comes from use of the EBSCOhost search engine.
2. See, for example, Woodford (2003a, p. 45 and p. 77).
3. These include Evans and Honkapohja (1999, 2001), Bullard (2006), and Bullard and Mitra (2002). It might besaid that Woodford, too, supports this position: see Woodford (2003a, pp. 261–276; 2003b, p. 1178).

26 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2009

The Role of “Determinacy” in Monetary Policy Analysis

II. Determinacy and Learnability
Learnability of RE solutions has been discussed for many years (see, e.g., Bray andKreps [1981] or Bray [1982]), but has come to the forefront of monetary policy analysis
fairly recently; in particular, since the publication of the prominent treatise by Evansand Honkapohja (2001).4 The basic idea is that individual agents must obtain theirquantitative knowledge of the dynamic properties of the system—which is necessary
for forming expectations rationally—on the basis of data generated by the economy
itself; they are not given such knowledge by magic or divine revelation. Thus, thisknowledge must be based on some learning process that depends upon past observationsof variables of the system. Then, since random shocks are bound to occur, one needs todetermine whether a system that is slightly disturbed from an RE solution path would
tend to return to that path, and consequently has to examine the dynamics of learning.
If the implied learning process is not stable, the solution under consideration is highly
unlikely to prevail.
The particular learning process that is featured most prominently in the work ofEvans and Honkapohja (E&H) is least-squares (LS) learnability. For this concept, one
imagines that the individual decision-makers in the model economy are continuallyattempting to learn the quantitative features of a forecasting equation for use in forming
expectations about endogenous variables that will prevail (or become known) in the
future. The hypothetical process is that in each time period a typical decision-maker
will use available data generated by the economy to estimate a forecasting model that
includes the relevant endogenous variables. He then makes supply-demand choices
based on these forecasts and interacts with other agents on markets. Markets clear andgenerate new prices and quantities, which then enter datasets for the next period. As
time passes, these estimates may become progressively nearer to being correct, since
the model economy’s structure is assumed to be unchanging. If this process leads, inthe limit as the number of observations increases without bound, to a particular RE solution, then this solution is learnable. Conversely, if the process does not converge to agiven RE solution process, after a small displacement from it, then that solution is notlearnable. There are, of course, many possible learning procedures that could be considered. The one that is emphasized by E&H, LS learnability, adds to the description
above the proviso that the agents’ forecasts are based on an estimated vector autoregression (VAR) model that is correctly specified and estimated by ordinary LS. Thesefeatures, in the context of an ever-increasing sample size, tend to be highly favorable
to the prospects for convergence—that is, “biased” toward a finding of learnability.5
The technical apparatus needed for the study of LS learnability has been provided,
for linear models, by E&H (2001, pp. 173–263). A brief summary of the essential
results is as follows. Almost any linear or linearized model can be written in the form    z
(1)

4. The slightly earlier survey article by the same authors—Evans and Honkapohja (1999)—was also influential.
5. Additional discussion on this point will be presented below.

27





where
is an   vector of endogenous variables and z is an   vector of
exogenous variables generated by a stable first-order autoregressive process:
z

z





(2)

 

Specifically, by the use of auxiliary variables, systems with variables lagged any finite
number of periods into the past can be expressed in this format, and so can variables expected in period to prevail any finite number of periods in the future. Also,
expectational variables can be lagged.
Consider fundamental solutions of the form  z (3)

It can easily be shown, by straightforward undetermined-coefficients reasoning, thatfor any given  there is a unique  that satisfies (1), but  is determined by the matrix
quadratic equation    (4)which has many solutions—    to be exact. If more than one of these isdynamically stable, we have indeterminacy.
For this very broad class of models, E&H (2001, p. 238) have established that thesufficient and generically necessary conditions for LS learnability are that the eigenvalues of the following three matrices have all their real parts with values less than 1.0:   







(5a)



(5b)

  

(5c)



It will be noted that, if there are no lagged endogenous variables in the system, then
implying that and 
. In that special case, the first two conditions
amount to the requirement that the eigenvalues of  all have real parts less than one.

III. Determinacy Is Not Necessary
Consideration of the necessity of learnability for an RE solution to be regarded as plausible, as a description of an economy’s behavior, indicates clearly that determinacy is
not a necessary condition for an RE solution to be satisfactory as the basis of a policydesign exercise. If a situation of indeterminacy—multiple stable solutions—prevails, itis nevertheless possible that all but one of the solutions can be ruled out as implausibleon the basis of non-learnability. Here the point is that determinacy requires that all of

28 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2009

The Role of “Determinacy” in Monetary Policy Analysis

the eigenvalues of the matrices  and  , denotedand, must be less than 1.0
in modulus.6
Suppose then that for one particular solution of form (3) all, but
fails because one
. Then there are two dynamically stable solutions—indeterminacy prevails—but the particular solution under consideration may be learnable.7 Suppose further that there is no other solution—no other grouping of the systemeigenvalues—for which all the conditions for learnability are satisfied. Then I would
contend that only one of the RE solutions is plausible, since any other dynamicallystable solution is not learnable; that is, the relevant learning dynamics needed for theagents to be able to form expectations rationally is expectationally unstable. In thiscase, there is no need to be concerned by the possibility that the system would be foundfollowing the non-learnable solution.
In McCallum (2003), I have argued that this conclusion is relevant for two problematic issues that have been prominent in the monetary policy literature. One of these
involves cases in which the central bank is using a policy rule, for setting each period’s
one-period nominal interest rate, which involves responses to an expected future inflation rate, rather than the current inflation rate. Several analysts, most prominently
Woodford (1994; 2003a, pp. 252–261) and Bernanke and Woodford (1997), have shownthat in such cases indeterminacy can be brought about by excessively strong responses(as well as by ones that are too weak to satisfy the Taylor principle) and have suggested
that policy should therefore avoid very strong responses to expected future inflation.
My 2003 paper argues against the latter conclusion on the ground of a learnability
discussion, as suggested above. Woodford (2003b) points out that my argument does not
eliminate the possibility of additional (undesirable) solutions of the “resonant frequency
sunspot” variety. I believe that these are implausible for reasons that will be hinted at
below, but must admit that my case is not theoretically airtight.
A significant extension of the foregoing case has been provided by Kurozumi and
Van Zandweghe (2008). These authors consider an extension of the basic type of New
Keynesian model so as to explicitly include endogenous investment in physical capital.
Previous analysis by Carlstrom and Fuerst (2005) had indicated that the inclusion of
investment greatly exacerbates the tendency to indeterminacy described in the previous
paragraph, with indeterminacy in fact prevailing for all values of the coefficient representing responses to expected future inflation (assuming that there is no policy responseto the output gap as a distinct variable). Kurozumi and Van Zandweghe (2008) corroborated that finding, but showed that additional policy response to output would reduce
the region of indeterminacy. Moreover, with respect to the matter at hand they showedthat adoption of the LS learnability requirement greatly reduces multiplicity problems;
they state that “the forward-looking policy (i.e., response to 
) generates a locallyunique non-explosive E-stable fundamental rational expectations equilibrium as long as
the policy response to expected future inflation is sufficiently strong” (Kurozumi and
Van Zandweghe [2008, p. 1489]).8






 

 




6. This result is established by McCallum (2007).
7. It will be learnable unless there is some eigenvalue of that is negative and large enough that (5c) is violated.
8. E-stability is an analytical concept that is convenient for establishing LS learnability. It should be recognizedthat Kurozumi and Van Zandweghe (2008) did not consider sunspot solutions.

29A second issue discussed in McCallum (2003) concerns the possibility of zerolower-bound “inflation traps” of the type discussed extensively by Benhabib, SchmittGrohé, and Uribe (2001, 2002) among others. In this context, my 2003 analysis
indicates, for the special case with full price flexibility, that the inflation-trap solution
is not learnable, whereas the solution that results in the central bank’s target rate of
inflation is learnable. There is in this case no disagreement with Woodford (2003b)
with respect to the substantive conclusions concerning outcomes under standard policy,
although he views the learnability analysis as unnecessary.

IV. Determinacy Is Not Sufficient
The argument that determinacy is not necessary for a single RE solution to be plausible
has been developed by several authors over many years.9 Rather recently, Cochrane(2007) has argued that determinacy is not sufficient. This argument can be outlined
easily and briefly, as follows. In the most standard class of New Keynesian models formonetary policy analysis, there is invariably an RE solution with explosive inflation.
Such solutions are usually not considered to be contenders for describing the behaviorof the economy in question; they are considered to be ruled out by the requirement ofdeterminacy. But Cochrane asks, what is the justification for that? How does one knowthat a Taylor-style10 policy rule will not permit explosive inflation? The usual answer isthat explosive solutions typically violate transversality conditions that are necessary for
optimality of individuals’ choices. But those conditions pertain to real variables, suchas the stock of real money balances held, or real bonds, or capital. There is no suchrationale for ruling out solutions in which inflation explodes. In Cochrane’s words,“Nothing in economics rules out explosive or ‘non-local’ nominal paths. Transversality
conditions can rule out real explosions, but not nominal ones” (2007, p. 2). It is mybelief that this position of Cochrane’s is analytically correct. I have disputed, however,
his strongly expressed contention that this conclusion warrants a distinctly negativeevaluation of current mainstream monetary policy analysis. Instead, I suggest, the latteris in many cases justified by a learnability requirement of the type discussed abovein Section III.

V. Information Feasibility
Cochrane’s 2007 discussion mentions but rejects arguments based on learnability.
He says:
A wide variety of almost philosophical principles have been advocated to prune
equilibria. For example, Evans and Honkapohja (2001) advocate criteria basedon least-squares (LS) learnability, and McCallum (2003) advocates a “minimum
state variable criterion,” which he relates to learnability. These refinements go
9. In this regard, the work of E&H (1999, 2001), Bullard (2006), and Bullard and Mitra (2002) is quite notable.
10. Introduced in Taylor (1993).

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The Role of “Determinacy” in Monetary Policy Analysis

beyond the standard definitions of economic equilibria. One may argue thatwhen a model gives multiple equilibria, we need additional selection criteria.
I argue instead that we need a different model.11
I believe that this aspect of Cochrane’s argument is not persuasive. A basic premiseof any serious model is that equilibrium outcomes must be feasible—private and public
consumption and investment cannot exceed the amount produced and drawn down,
and the amount produced is bounded by production possibilities and resource availability. But feasibility applies to information, as well as tangible resources. Thus, in amodel in which there is randomness, agents cannot know future values of prices andother variables; they have to form expectations based on (at most) knowledge of pastand (perhaps) present values of these variables. Standard formal models accordingly
specify information sets assumed to be available to agents in forming expectations,and these invariably include at most current and past values of endogenous and exogenous variables. I think that there should be no dispute over these statements, but would
go further by arguing as follows. To form these expectations rationally, agents musthave quantitative knowledge of the “laws of motion” of relevant variables. But suchknowledge must be based, in reality and therefore in any model to be used to mimic
reality, on data generated by the economy itself. Individuals cannot—as mentioned
above—obtain such information by magic, or by divine revelation. Accordingly, for a
proposed equilibrium to be feasible, it must be the case that that information generated
in the past is sufficient to permit individuals to develop forecasting rules that mimic
the quantitative properties of the actual laws of motion. In this sense, to be plausible,
a proposed equilibrium must be “learnable.”
Some analysts object to the presumption that candidate equilibria must be learnable, on the grounds that “there are many possible learning procedures—how can you
know that the one mentioned above is correct?” My response to that objection is that it
does not distinguish between necessity and sufficiency. In McCallum (2007, p. 1378),
the argument is expressed as follows:
The position that learnability (and thus E-stability) should be regarded as a necessary condition for the relevance of an RE equilibrium begins with the presumption that individual agents must somehow learn the magnitudes of parametersdescribing the economy’s law of motion from observations generated by the economy; they cannot be endowed with such knowledge by magic. Of course any
particular learning scheme might be incorrect in its depiction of actual learning
behavior. But in this regard it is important to note that the LS learning process
in question assumes that (i) agents are collecting an ever-increasing numberof observations on all relevant variables while (ii) the structure is remaining
unchanged. Furthermore, (iii) the agents are estimating the relevant unknown
parameters (iv) with an appropriate estimator (v) in a properly specified model.
Thus if a proposed RE solution is not learnable by the process in question—the
one to which the E&H results pertain—then it would seem highly implausiblethat it could prevail in practice.
11. Cochrane (2007, p. 44).

31

VI. Recent Developments
Additional lines of argument have been developed more recently. In Cho and McCallum(2009) the following argument is put forth. Consider a model in which there are two
sectors, one of which is autonomous. That is, this “first” sector determines the values ofsome of the model’s variables without reference to any variables or activities pertaining
to the “second” sector. The latter, by contrast, determines values of the remaining endogenous variables in a manner that has them dependent upon the values generated in
the first sector. In such a setting, it is possible that the overall system has the property
of determinacy, suggesting that all variables are dynamically stable, whereas analysis
reflecting the block-recursive nature of the system yields quite different conclusions.
Cho and McCallum (2009) present numerical examples for two-sector models of thistype, with each sector having just a single endogenous variable (and) whosevalue in each period depends upon its own lagged value and its own value expected
one period in the future. Sector one is autonomous, but in sector two
depends inpart on the expected value of. In one example, consideration of sector one as
autonomous indicates thatis dynamically stable, while analysis of sector two, withtreated as exogenous, indicates thatwill explode relative to the path of . In asecond example, the variable of the autonomous sector is explosive but the parameters
for the other sector suggest that its endogenous variable will behave in a stable manner
relative to the explosive path of the autonomous sector.12 Thus in both cases, standard
analysis indicates that the bivariate system is determinate, whereas analysis that recognizes the block-recursive nature of the system indicates that in the first case one
variable will be explosive and the other stable, while in the second case both variables
will be explosive.
Very recently, I have proposed a distinct and more radical argument. It goes asfollows. Consider a linear model with  endogenous variables. In any such model,
there are     different fundamental RE solutions, that is, expressions relating
endogenous to predetermined and exogenous variables, which satisfy all orthogonality
conditions for RE. It can be shown, however, that each of these solutions represents
a different specification relating to the model’s state variables; there is a one-to-one
relationship between solutions and state-variable specifications. But any structuralmodel, one that is designed to be policy invariant, is built upon a particular specification regarding state variables. Therefore, any structural model leads to a unique RE
solution of the fundamentals type; the other fundamental solutions represent different assumptions pertaining to the relevant state variables. But what about “sunspot”
solutions, ones that include random components unrelated to any of the fundamental
variables? The answer is that the single fundamental solution that is consistent with themodel’s state variable specification will not support sunspot variables; only the otherfundamental solutions will do so. Thus sunspot solutions represent, apparently, randomvariations from solutions that are inconsistent with a crucial aspect of the model’s basic
specification. This argument suggests, then, that there is an important sense in which




















12. In Cho and McCallum (2009), the notation makes sector two the autonomous sector in this case. Here I havenot followed that notation so as to avoid changing the designation of which sector is autonomous.

32 MONETARY AND ECONOMIC STUDIES/NOVEMBER 2009

The Role of “Determinacy” in Monetary Policy Analysis

RE “solution multiplicities” represent a multiplicity of models rather than a multiplicity
of solutions to a single model. It should be added that this contention is based on
work in progress.

VII. Conclusion
In this paper, I have argued that fundamental recognition of information feasibility
requires that a candidate solution must, to be considered plausible, be learnable on thebasis of information generated by the economy—model economy or actual economy—
itself. Since the LS learning process is highly “biased” toward a finding of learnability,
it is reasonable to regard LS learnability as a necessary condition for any specific solution to be relevant for policy consideration. This implies that determinacy is not necessary for policy analysis; there may be more than one dynamically stable solution but
only one that is LS learnable. Determinacy is also not sufficient for satisfactory policy
analysis; there needs to be some logical argument for ruling out explosive solutions
pertaining to nominal variables; these are not necessarily eliminated by transversality
conditions. Furthermore, consideration of models with block-recursive structures suggests that determinacy results pertaining to the overall system may be inconsistent withconclusions based on sectoral analysis that recognizes the block-recursive structure of
the system. For these reasons, I believe that the role of determinacy in monetary policy
analysis should be reconsidered and substantially deemphasized or replaced.

33

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