BENNETT T. MCCALLUM

*Carnegie Mellon University and National Bureau of Economic Research, Pittsburgh, PA 15213*

## Abstract

This paper concerns the minimal-state-variable (MSV) criterion for selection among solutions in rational expectations models that feature a multiplicity of paths that satisfy all of the model’s conditions. It compares the MSV criterion with others, including the widely used saddle-path (dynamic stability) criterion. It is emphasized that the MSV criterion can be viewed as a scientifically useful classification scheme that delineates the unique solution that is free of bubble components. In the process of demonstrating uniqueness for a broad class of linear models, the paper exposits a convenient computational procedure. Applications to current issues are outlined.

## I. Introduction

It is well known that Bob Flood’s second paper with Peter Garber (Flood and Garber, 1980b) was an influential pioneering work in empirical testing for the existence of bubble phenomena in rational expectations macroeconomics. It is not so well known, by contrast, that his paper with Burmeister and Garber (Burmeister, Flood, and Garber, 1983) provided one of the earliest steps toward a useful and general classification of rational expectations solutions, theirs focussing on the distinction between bubble and bubble-free (or fundamentals) solutions.^{1} The present paper amounts to an extension of this type of classificational analysis, together with an attempt to establish the scientific merits of one particular scheme.

For many years now it has been commonplace knowledge that many dynamic models with rational expectations (RE) feature a multiplicity of paths that satisfy all of the conditions for intertemporal equilibrium. Indeed, most dynamic RE models that are not based on explicit optimization analysis of individuals’ behavior fall into that category and so do some that involve full-fledged general equilibrium analysis with optimizing agents.^{2} But in many applications the analyst is not specifically concerned with this multiplicity—often interpreted as the possible existence of “bubbles”—and wishes to focus attention on one particular path that is presumed to be of economic relevance, e.g., if bubbles were absent.^{3} Consequently, several alternative criteria have been proposed for selection of the path on which to focus. Among these are Taylor’s (1977) “minimum-variance” criterion, the “expectational-stability” criterion of Evans (1985, 1986), the “minimal-state variable” criterion made explicit in McCallum (1983), and the popular “saddle path” or “stability” criterion. The latter is favored by Sargent (1987), Whiteman (1983), Blanchard and Kahn (1980), Blanchard and Fischer (1989), and many others, and is often used in computation algorithms such as King and Watson (1995) or Klein (1997).

In practice, analysts are often unclear as to which of the criteria is being utilized, when attention is focused on a single solution, because in many cases the last three of the four above-listed criteria all point to the same solution. Some analysts are explicit, however, and a sampling of the literature suggests that the most frequently adopted of the criteria, in these cases of explicit justification, is that of stability or non-explosiveness. The stability criterion has been recommended, moreover, in the influential textbooks of Sargent (1987, pp. 197-9, 306-7) and Blanchard and Fischer (1989, pp. 225, 260).

One purpose of the present paper is to consider the strengths and weaknesses for scientific research of these alternative criteria. In particular, it will be argued that the stability and minimum-variance criteria are inherently unsatisfactory. By contrast, the minimal-state-variable (MSV) criterion is scientifically attractive, according to our argument, for it provides a classificational scheme that is designed to be useful in terms of positive analysis. The criterion of expectational stability, finally, will be characterized as reflecting a substantive behavioral hypothesis rather than a classification scheme, so its attractiveness is an empirical issue rather than a question of constructive scientific practice.

A second purpose of the paper is to emphasize that the minimal-state-variable (MSV) criterion generally identifies a single solution that can reasonably be interpreted as the unique solution that is free of bubble components, i.e., the fundamentals solution. It can accordingly be used as the basis for tests of a substantive hypothesis to the effect that bubble solutions are not of empirical relevance. This hypothesis would remain of interest, moreover, even if the association of the MSV criterion with the bubble-free property were not accepted.

In conducting this argument, it will be expositionally useful to provide illustrations in the context of a particular example. Consequently, one will be developed in Section II. The unsatisfactory nature of the minimum variance and stability criteria will then be argued in Section III. Section IV will make the case for the MSV criterion, with attention being devoted to a critical argument of Froot and Obstfeld (1991), and Section V will consider the “expectational stability” criterion of Evans (1985, 1988). Next, Section VI will demonstrate how unique MSV solutions can be defined and calculated in a very wide class of linear rational expectations models, after which Section VII will describe the relevance of the foregoing analysis for some prominent recent research. Finally, Section VIII will provide a brief summary.

## II. An Illustrative Model

As a vehicle for illustrating several of the points to be made below, consider the familiar Cagan money demand function

where *m _{t}* and

*p*are logs of an economy’s nominal money stock and its price level. Also,

_{t}*E*is defined as

_{t}{·)*E{·|Ω*, where Ω

_{t})_{t}includes

*m*, ...,

_{t}, m_{t-1}*p*, ...., and ξ

_{t}, p_{t-1}_{t},ξ

_{t-1},..... The disturbance ξ

_{t}, which reflects random behavioral demand shifts, will be assumed to be a random walk variate so that Δξ

_{t}=

*u*is white noise. For our purposes it is of no consequence whether or not one conceives of (1) as resulting from an explicit maximization problem, since there are such models that give rise to multiple solutions and our points are designed to be relevant for any model with multiple solutions—with correct account being taken of all non-negativity requirements, transversality conditions, and anything else that might eliminate some paths from contention as solutions.

_{t}To represent policy behavior that generates the money supply, we will adopt a rule of the following form:

Thus the money stock growth rate in each period is related to inflation in the previous period. One would expect sensible policy behavior to involve a negative value of μ_{1}, so that money creation is slowed when recent inflation has been rapid, and a value that is not too large (so as to avoid instrument instability). But for the present we shall adopt only the restriction μ_{1} ≤ (α—l)^{2}/(—4α), which is necessary (as we shall see) for the Δ*p _{t}* solution values to involve real (i.e., non-complex) numbers. It would of course be possible to include a random disturbance term in (2) as well as (1), but nothing would be gained and clutter would be added. To complete the model, it needs to be specified that it pertains to all periods

*t*= 1, 2,... with

*m*and Δ

_{o}*p*given. The specified type of policy behavior can therefore only be adopted after an economy is already in existence so that Δ

_{o}*p*and

_{o}*m*will be well defined. Inserting (2) into the first difference of (1) yields

_{o}and for present purposes it will suffice to consider solutions of the form^{4}

The latter implies *E _{t} ΔP_{t+1}* = π

_{o}+ π

_{1}(π

_{o}+ π

_{1}

*ΔP*+ π

_{t-1}_{2}

*u*) so substitution into (3) yields

_{t}Thus for (4) to be a solution it must be true that

The second of these clearly implies that^{5}

Once it is decided whether to add or subtract the positive term *d* = [(α—1)^{2} + 4αμ_{1}]^{1/2}, the values of π_{o} and π_{2} will be defined uniquely. But that decision is crucial for determining the model’s implied behavior of Δ*p _{t}*. That fact is illustrated in Figure 1, where π

_{1}

^{+}= (α—1 +

*d*)/2α and π

_{1}

^{-}= (α—1—

*d*)/2α are plotted for α =—4 (representative for all α <—1) against μ

_{1}. Clearly, values of π

_{1}

^{+}(the lower branch) and π

_{1}

^{-}(the upper branch) lie both within and outside of the range—1 < π

_{1}< 1 that is necessary for dynamic stability. (In the somewhat unrealistic case with—1 < α < 0, not illustrated in Figure 1, π

_{1}

^{-}exceeds 1.0 for all μ

_{1}that give real roots.)

A particularly simple and transparent special case of this example occurs when μ_{1} = 0 in (2), so that the money stock growth rate is constant. In that case one might expect Δ*p _{t-1}* to be absent from (4), since it does not appear in the model and can affect the value of Δ

*p*only if it is (arbitrarily) expected by the economy’s participants to affect Δ

_{t}*p*. Thus we are led to look for solutions of the form Δ

_{t}*p*= π

_{t}_{o}+ π

_{2}

*u*in this case, and we find that Δ

_{t}*p*= μ

_{t}_{0}—

*u*. This result is of course consistent with our more general example. Indeed, the solutions in (7) for π

_{t}_{1}are π

_{1}

^{+}= 0 and π

_{1}

^{-}= (α—l)/α when μ

_{1}= 0, the first of which implies the absence of Δ

*p*from (4) and duplicates the solution just found.

_{t-1}^{6}The second value, π

_{1}

^{-}= (α—l)/α, is with α < 0 unambiguously greater than 1.0, so it implies an explosive, dynamically unstable path. Furthermore, this value π

_{1}

^{-}will support an infinity of unstable paths. This may be seen by supposing that π

_{3}

*u*is added to the conjectured solution in (4) and then verifying that this expression is consistent with all of the model’s equations for any value of π

_{t-1}_{3}(upon which π

_{2}depends).

^{7}If π

_{1}

^{+}= 0 is taken as the relevant value for π

_{1}, however, it is implied that π

_{3}= 0 and π

_{2}=—1.

Note that in the special case in which μ_{1} = 0, the solution involving π_{1}^{+} (i.e., Δ*p _{t}* = μ

_{0}—

*u*) is clearly the one that would be regarded as the bubble-free or fundamentals solution by Burmeister, Flood, and Garber (1983).

_{t}^{8}Indeed, analogous solutions are so regarded quite generally in the literature in examples similar to our special case. By contrast, the solutions involving π

_{1}

^{-}would generally be regarded, in this special case, as bubble solutions—i.e., solutions that add bubble components to the fundamentals solution. McCallum (1983, pp. 147, 161) proposed a general extension of the bubble vs. bubble-free terminology to cases analogous to those in which μ

_{1}≠ 0 in the example at hand; that extension will be utilized below.

## III. The Stability and Minimum Variance Criteria

As it happens, extensive utilization of the foregoing example will be briefly delayed, for our argument concerning the stability and minimum-variance criteria can be developed without reference to any particular model. Let us begin with Taylor’s (1977) minimum-variance criterion. According to the latter, the choice among multiple solutions should be dictated by the unconditional variance of a variable analogous to Δ*p _{t}* in the foregoing example. But there are two serious flaws with this proposal, the first of which is its ambiguity. Specifically, in many models there will be more than one endogenous variable of interest. (In fact, even in the example of Section II—despite the appearance of equation (3)—there are two endogenous variables, Δ

*p*and Δ

_{t}*m*.) But in such cases the minimum variance criterion will not be well specified, because the various endogenous variables may indicate different solutions. Indeed, in some cases there may even exist some ambiguity as to whether the (possibly detrended) level or first difference of a given variable is relevant. Second, the minimum-variance criterion is presumably intended to pertain to the solution path that would be empirically relevant. But that would of course suggest that the modeled economy’s agents are motivated to choose the minimum-variance solution over others, and it is not the case that agents will typically be so motivated. Indeed, the minimum-variance criterion evidently pertains to some social desideratum, not anything that could be affected by any single agent’s choice. Consequently, the model’s agents will have no incentive to select this solution path, so there is no particular reason to believe that it would in fact be empirically relevant.

_{t}Turning now to the case of the stability criterion, our argument is quite different. Here the problem is that the criterion is, to a significant extent, self-defeating. For the criterion is precisely that the selected solution path must be non-explosive—dynamically stable—under the natural presumption that exogenous driving variables (such as shocks and policy instruments) are non-explosive. Yet one important objective of dynamic economic analysis is to determine whether particular hypothetical policy rules—or institutional arrangements—would lead to desirable economic performance, which will usually require stability. Or, to express the point somewhat differently, the purpose of a theoretical analysis will often be to determine the conditions under which a system will be dynamically stable and unstable. But, obviously, the adoption of the stability criterion for selection among solutions would be logically incompatible with use of the models’ solution to determine if (or under what conditions) instability would be forthcoming. To the extent, then, that this objective of analysis is important, the stability criterion is inherently unsuitable. One cannot use a model to determine whether property “A” would be forthcoming, if the model includes a requirement that “A” must not obtain.

In addition, there are a substantial number of cases in which there exists an infinity of solution paths all of which are stable. In such cases, then, the stability criterion fails to select a single path on which to focus as the bubble-free or fundamentals path.^{9} That failure would be defensible if it were true that no single path has special characteristics that justify labeling it as bubble-free, but it is not. Even in these cases the MSV criterion provides a clear demarcation between one path and the others. To develop that argument is the purpose of the next section.

## IV. The MSV Criterion

The MSV criterion is designed to yield a single bubble-free solution *by construction*. Its definition begins by limiting solutions to those that are linear^{10} functions—analogous to (4) in the example of Section II—of a minimal set of “state variables,” i.e., predetermined or exogenous determinants of current endogenous variables. For a set of state variables to be minimal, it must be “one from which it is impossible to delete … any single variable, or group of variables, while continuing to obtain a solution valid for all admissible parameter values” (McCallum, 1983, p. 145). Here the language is somewhat convoluted because there is not in general a unique minimal set of state variables, even though there is a unique MSV solution. Two or more different sets of variables may span the same space, of course, with neither being a proper subset of the other.

But relying upon a minimal set of state variables is not the only requirement (in addition to linearity) for a MSV solution. In cases in which the minimal set includes a lagged value of an endogenous variable there will typically be more than one solution to the undetermined-coefficient identities analogous to equations (6) above. So one part of the definition of the MSV solution is a rule for selection of the appropriate solution. That rule is that the solution continues to be based on a minimal set of state variables for all special cases of the parameter values. Typically, some admissible sets of parameter values will include zero coefficients in all structural equations for a lagged endogenous variable. But in any such case, this lagged value will not be part of a minimal set, so its solution-equation coefficient analogous to π_{1} will be zero for the MSV solution in that special case. Thus the MSV solution must be, to pertain for all admissible parameter values, the one that is the MSV solution in that special case.

To illustrate this determination, consider the choice between π_{1}^{+} and π_{1}^{-} in the example of Section II. In the special case in which μ_{1} = 0 in (2), the variable Δ*p _{t-1}* does not appear in model (and in fact appears to be an irrelevant bygone). Thus Δ

*p*can in this case affect the value of Δ

_{t-1}*p*only if it is—arbitrarily—expected by the economy’s participants to affect Δ

_{t}*p*. Thus it does not appear in the minimal set of state variables in this special case with μ

_{t}_{1}= 0, so π

_{1}= 0 is implied. But from the perspective of the general case, it is π

_{1}

^{+}that yields the value 0 in this special case, π

_{1}

^{-}instead being equal to (α—l)/α. Consequently, it is the solution to equations (6) with π

_{1}= π

_{1}

^{+}that makes (4) the MSV solution expression for Δ

*p*in this model.

_{t}It is important to recognize that this definition for the MSV solution involves a procedure that makes it unique by construction. It is logically possible to dispute whether this solution warrants being termed the bubble-free or fundamentals solution, although the answer seems to the present writer to be a clear “yes.”^{11} But it makes no logical sense to argue that the MSV solution is not unique.^{12}

In that regard, Froot and Obstfeld (1991) have suggested that the MSV solution is not unique by demonstrating an example in which there is a non-linear function of the single state variable that constitutes a minimal set. That demonstration does not provide a valid counterexample to the claim of the last paragraph above, however, because linearity of the solution expressions such as (4) is required for the MSV solution. It is not surprising, it should be said, that Froot and Obstfeld would have misinterpreted the definition given in McCallum (1983), because the latter mistakenly took it for granted that only linear expressions would provide solutions in the class of linear models considered. But the outlined procedure, which defines the MSV solution, was expressly designed to yield a unique solution. So the restriction of linearity would have been explicitly included if the author had realized that it was needed.

The example presented in Section II was chosen, as one would expect, to illustrate points concerning the contrast between MSV and other solution criteria. In particular, for values of μ_{1} < 2α—1, the MSV solution features dynamic instability since π_{1}^{+} <—1. Thus this case demonstrates that the set of solutions selected by the MSV criterion, but ruled out by the stability criterion, is not empty. It is, moreover, intuitively plausible that instability would obtain in this case, as it reflects a very strong application of policy feedback response—which when excessive induces “instrument instability.” Indeed, this is an example of the type of determination that a dynamic model should be able to provide—i.e., the conditions under which feedback is destabilizing. Alternatively, the example of Section II also illustrates the possibility of non-exploding bubble solutions, which occur when 1 < μ_{1} < (α—1)^{2}/(—4α).

At this point in the discussion it should be clear that the MSV criterion may be regarded as a classification scheme, i.e., a technique for delineating the solution that is of a bubble-free or fundamental nature from those that include bubble components. This scheme is intended to be scientifically useful, by providing a single solution that the researcher may focus upon if he/she is engaged in an investigation such that the possibility of bubbles is deliberately excluded at the outset. In addition, the classification scheme serves a second scientific purpose by providing the basis for a substantive hypothesis to the effect that market outcomes in actual economies are generally of the bubble-free variety. Even though RE general equilibrium analysis provides no general theoretical basis for ruling out bubble solutions, it is a coherent plausible substantive hypothesis that such solutions do not occur in practice.

The plausibility of that hypothesis is emphasized by the undetermined-coefficient method of deriving the MSV solution. The relevant point is that, in the space of π_{1} values, the bubble-free value π_{1}^{+} is of measure 1/2. And this continues to be true in the special case with μ_{1} = 0 in which there is an infinity of non-MSV bubble paths. In that case, the MSV solution features π_{1}^{+} = 0, yielding Δ*p _{t}* = μ

_{0}—

*u*. Use of the value π

_{t}_{1}

^{-}= (α—l)/α, however, gives rise in this case to an infinity of solutions of the form

where the multiplicity arises because any value of π_{3} will satisfy the model when π_{1} equals (α—l)/α (given that μ_{1} = 0). For some researchers, it is a common practice in such cases to presume that the outcome—the particular path realized in the market—is determined by an “initial condition” Δ*p _{o}* that serves to pin down 7r2. From that perspective there is only a single value of Δ

*p*that will imply π

_{o}_{2}=—1, and also that π

_{1}= π

_{3}= 0, thereby yielding the bubble-free solution.

^{13}In the space of initial conditions, then, the bubble-free outcome is of measure zero. But is entirely unclear which of these spaces is relevant to the market’s solution outcome. It is thus a plausible hypothesis that bubble-free solutions will obtain generally.

^{14}The generation of that hypothesis is the second scientific contribution of the MSV solution criterion.

## V. The Expectational Stability Criterion

The last alternative criterion to be explicitly discussed is that of “expectational stability” as developed by Evans (1985, 1986).^{15} The basic idea is to determine whether there is convergence of an iterative procedure toward a RE solution; if there is such convergence the RE solution approached is the one selected by this criterion. It is not entirely clear whether the steps in the iterative process are supposed to reflect sequential positions in calendar time or in some type of conceptual meta-time, but to this reader the latter seems more appropriate. In any event, the sequence of calculations begins with a function, analogous to the expression just below (4), that determines expectations—but with coefficients that differ somewhat from those implied by RE in the model at hand. Then the model and this expectation function imply a “law of motion” for the model’s endogenous variables. This law of motion, which may not be fully consistent with the expectation function used in its derivation, is then adopted as the basis for a revised expectation function to be used (in the same way) in the next round of the iterative process. Expectational stability obtains when this process converges to the RE solution under consideration.^{16} In fact there are two variants: weak expectational stability obtains if the original expectation function is specified so as to include the same determining “state variables” as the RE solution under consideration, whereas strong expectational stability obtains when additional variables are permitted in the expectations function.

The process can be illustrated with the model of Section II. With a RE solution of form (4), whether or not it is the MSV solution, expectations will conform to *E _{t} Δp_{t+1}* = π

_{0}+ π

_{1}Δ

*p*so the iterative procedure assumes that expectations at

_{t}*t*of Δ

*p*satisfy

_{t+1}where *n* indexes the iterations. Now with (9) prevailing, Δ*p _{t}* will be determined by the analog of (3), namely, μ

_{0}+ μ

_{1}Δ

*p*= Δ

_{t-1}*p*+ α(Φ

_{t}^{n}

_{0}+ Φ

^{n}

_{1}Δ

*p*)—αΔ

_{t}*p*, where Δ

^{e}_{t}+ u_{t}*p*is given from the past.

^{e}_{t}^{17}The last equation can be written as

and it suggests that expectations for Δ*p _{t+1}* should satisfy

since *u _{t}* is white noise. Then writing the right-hand side of the latter in form (9) gives

which defines an iterative process for the values.

From the second of expressions (12), we see that the stationary values for Φ_{1} are the same as the two roots in (7). The expectational stability analysis selects the one—if there is one—for which the difference equation in Φ^{n}_{1} is dynamically stable, i.e., the one that would be approached by the iterative process. From plots of Φ^{n+1}_{1} vs. Φ^{n}_{1} such as those in Figure 2, we can see that the root Φ^{+}_{1} is (locally) stable, since the slope is less than 1.0 in absolute value for all μ_{1} < (α—l)^{2}/(—4α). At the root Φ^{-}_{1}, by contrast, the slope will exceed 1.0 in absolute value so the iterative process will not be convergent. With Φ_{1} = Φ^{+}_{1}, moreover, the behavior of Φ^{n}_{1} is stable for all parameter values yielding real roots in (6b). In this example, then, the expectational stability criterion points to the same solution as does the MSV criterion as long as μ_{1} < 1.

It is not entirely clear, however, just how much emphasis should be placed on that agreement. One reason is that Evans and Honkapohja (1992) argue that there are some cases in which expectational stability does not point to the MSV solution. I am not entirely persuaded that these cases include any well-motivated economic models, but in any event that is not the main point. The point, instead, is that if the analysis calls for focus on the bubble-free solution, then that would still be accomplished by means of the MSV criterion. If expectational stability provides an accurate guide to the behavior in actual economies, then a non-MSV bubble solution would prevail in such cases. But that would provide no reason for changing the classification of bubble vs. non-bubble solutions. And it is far from certain that expectational stability does provide a guide for actual economic behavior, for that hypothesis requires that this particular iterative process, among all those that could be conjectured, is empirically relevant. Nevertheless, while the amount of warranted emphasis is unclear, it is the case that in most—if not all—sensible models the expectational stability criterion does point to the MSV solution.^{18}

## VI. General Derivation in Linear Models

The argument above has relied on the proposition that there is a unique MSV solution in a wide class of linear models; the main purpose of this section is to demonstrate the validity of that claim. In addition, a second purpose is to present a compact and easily understood exposition of a convenient and practical computational procedure for solving linear rational expectations models. This procedure, which is applicable to a class of models that is broad enough to include most cases of practical interest, can be implemented by means of a MATLAB routine provided by Paul Klein (1997).^{19} The present exposition departs from Klein’s, however, by relying upon the elementary undetermined-coefficients (UC) approach used throughout the present paper. In a sense, the current exposition could be viewed as merely an extension to the appendix of McCallum (1983). It is an extension that is nontrivial, however, and essential for practical (i.e., computational) purposes. Here it is accomplished by use of the generalized Schur decomposition theorem discussed by Klein. The UC reasoning utilized here is, however, much more elementary mathematically than Klein’s.^{20}

Let *y _{t}* be a

*M*x 1 vector of non-predetermined endogenous variables,

*k*be a

_{t}*K*x 1 vector of predetermined endogenous variables, and

*u*be a

_{t}*N*x 1 vector of exogenous variables

The model can then be written as

where *A _{11}* and

*B*are square matrices while ε

_{11}_{t}is a

*N*x 1 white noise vector.

^{21}Thus

*u*is formally a first-order autoregressive process, which can of course be defined so as to represent AR processes of higher orders for the basic exogenous variables. Also, for the predetermined variables we assume

_{t}If only once-lagged values of *y _{t}* were included in

*k*, then we would have

_{t}*B*, and

_{21}= I, B_{22}= 0*C*, but the present setup is much more general. Crucially, the matrices

_{2}= 0*A*, and

*11*, B*21**B*may be singular; that is what makes the setup convenient in practice.

_{22}In this setting a UC solution will be of the form

where the Ω, Γ, Π_{1}, and Π_{2} matrices are real. Therefore, *E _{t}y_{t+1}* = Ω

*E*+ ΓΩ

_{t}k_{t+1}*E*= Ω(Π

_{t}u_{t+1}_{1}

*k*+ Π

_{t}_{2}

*u*) + Γ

_{t}*Ru*. Substitution into (14) and (16) then yields

_{t}and

Collecting terms in *k _{t}* it is implied by UC reasoning that

whereas the terms in *u _{t}* imply

Let *A* and *B* denote the two square matrices in (21), and assume that |*B*—λ*A*| is nonzero for some complex number λ. This last condition will not hold if the model is poorly formulated (i.e., fails to place any restriction on some endogenous variable); otherwise it will be satisfied even with singular *A _{11}, B_{21}, B_{22}*.

^{22}Then the generalized Schur decomposition theorem guarantees the existence of unitary (therefore invertible) matrices

*Q*and

*Z*such that

*QAZ = S*and

*QBZ = T*, where

*S*and

*T*are triangular.

^{23}The ratios

*t*are generalized eigenvalues of the matrix pencil

_{ii}/s_{ii}*B—λA*;

^{24}they can be rearranged without contradicting the foregoing theorem. Such rearrangements correspond to selection of different UC solutions as discussed in McCallum (1983, pp. 145-147 and 165-166). We shall return to this topic below; for the moment let us assume that the eigenvalues

*t*(and associated columns of

_{ii}/s_{ii}*Q*and

*Z*) are arranged in order of their moduli with the largest values first.

Nowpremultiply (21) by *Q* and define *H ≡ Z ^{-1}* Then since

*QA = SH*and

*QB = TH*, the resulting equation is

and its first row can be written as

The latter will be satisfied for Ω such that

where the second equality results because *HZ = I*. Thus we have a solution for Ω, provided that *Z*_{22}^{-1} exists.^{25}

Next, writing out the second row of (24) we get

Then using (26) and *HZ = I* we can simplify this to

so since *S*^{-1}_{22} exists by construction^{26} we have

To find Γ and Π_{2} we return to (22) and (23). Combining them we have

where *G ≡ A _{11}ΩB_{21}*—B

_{11}and

*F ≡ C*. If G

_{1}—A_{11}ΩC_{2}_{-l}exists, which it typically will with nonsingular

*B*, the latter becomes

_{11}This can be solved for Γ by the steps given in McCallum (1983, p. 163) or can be obtained as

as in Klein (1997, p. 28).^{27} Finally, Π_{2} is obtained from (23). In sum, the UC solution for a given ordering of the eigenvalues is given sequentially by equations (26), (29), (33), and (23).

Different values of Ω, and thus different solutions, will be obtained for different orderings of the generalized eigenvalues *t _{ii}/s_{ii}*. What ordering should be used to obtain the economically relevant solution? Many writers, following Blanchard and Kahn (1980), arrange them in order of decreasing modulus and conclude that a unique solution obtains if and only if the number with modulus less than 1.0 (“stable roots”) equals

*K*, the number of predetermined variables. The minimal-state-variable (MSV) procedure, by contrast, is to choose the arrangement that would yield Ω = 0 if it were the case that

*B*

_{12}= 0—this step relying upon the continuity of eigenvalues with respect to parameters.

^{28}Uhlig (1997, p. 17) correctly notes that this procedure is difficult to implement and also that in many cases it will lead to the same solution as the Blanchard-Kahn stability criterion. Adoption of the decreasing-value arrangement will therefore often be attractive, even for MSV adherents. In such cases it seems unnecessary, however, to limit one’s attention to problems in which there are exactly

*K*stable roots. If there are fewer than

*K*stable roots, the MSV criterion will produce a single explosive solution whereas if there are more than

*K*stable roots, it will yield the single stable solution that is bubble-free—both of these being solutions that may be of particular scientific interest. In those exceptional cases in which an MSV analyst suspects that the Blanchard-Kahn and MSV criteria would call for different solutions, he/she could replace

*B*

_{12}with α

*B*

_{12}, plot eigenvalues for various values of α between 1 and 0, and then adjust the ordering if necessary.

## VII. Relevance for Recent Issues

The example of Section II is simple and clearly related to much of the existing bubble literature, but may seem remote from most monetary policy discussions of the late 1990s. To show that such is not the case—that the example is in fact highly relevant—is the purpose of the present section.

Let us begin by considering the following model, in which *y _{t}* denotes the log of output relative to capacity,

*R*is a nominal interest rate, and

_{t}*ν*is a white-noise disturbance:

_{t}Here (33) is a textbook-style IS function,^{29} (34) is a price-adjustment relation that with 0 ≤ θ < 1 can represent either the specification of Calvo (1983) and Rotemberg (1982) or the Fuhrer-Moore (1995) setup, and (35) is an interest-rate policy rule that can reflect pure inflation targeting (with μ_{2} = 0) or a rule of the more general Taylor (1993) variety.

Substitution of (35) into (33) and elimination of *y _{t}* then yields a linear equation that includes the variables Δ

*p*,

_{t}*E*Δ

_{t}*p*, Δ

_{t+1}*p*, and

_{t-1}*ν*. That list differs from the one pertaining to equation (5) by not including

_{t}*E*Δ

_{t-1}*p*, but that difference is of no consequence for the issues at hand because the distinction between

_{t+1}*E*Δ

_{t}*p*and

_{t+1}*E*Δ

_{t-1}*p*is irrelevant for the condition analogous to (6b) that determines the value of the crucial coefficient on Δ

_{t+1}*p*in the RE solution expression. Indeed, it can be verified that for some admissible parameter values the system has two stable solutions.30 Interestingly, large values of μ

_{t-1}_{1}do not generate explosive MSV solutions with the policy rule (35), but if Δ

*p*is entered in place of

_{t-1}*E*Δ

_{t}*p*then large μ

_{t+1}_{1}values will induce instability, just as in the example of Section II.

An issue that has attracted considerable attention recently is the so-called “Woodford warning” of possible solution “indeterminacy” when policy feedback rules relate to market expectations of inflation or some other target variable, a problem emphasized by Woodford (1994), Kerr and King (1996), Bernanke and Woodford (1997), Clarida, Gali, and Gertler (1997), and Svensson (1998). An example can be presented in the following system, which is adapted from Clarida, Gali, and Gertler (1997, p. 16):

Here we have an expectational IS function, a Calvo-Rotemberg price adjustment specification, and a pure inflation forecast targeting rule.31 For simplicity, constants are eliminated by normalization and *ν _{t}* is again taken to be white noise. In this system there are no predetermined variables so the MSV solution is of the form

*y*=

_{t}*Φ*, Δ

_{1}ν_{t}*p*=

_{t}*Φ*. Trivial calculations show that Φ

_{2}ν_{t}_{1}= 1, Φ

_{2}= α so the solution is

*y*=

_{t}*ν*, Δ

_{t}*p*= α

_{t}*ν*. The policy coefficient μ

_{t}_{1}does not appear in the solution equations because policy is responding to the expected future inflation rate, which is a constant (normalized to zero). A caveat must be applied to the foregoing, however: the MSV solution is defined only for μ

_{1}> 1.0. Values of μ

_{1}< 1.0 are inadmissible for “process consistency” reasons, introduced by Flood and Garber (1980a) and discussed in McCallum (1983, pp. 159-160).

But suppose that the researcher looks for solutions of the form

Then *E _{t}y_{t+1}* = Φ

_{11}(Φ

_{21}Δ

*p*,

_{t-1}1Φ_{22}ν_{t}*E*= Φ

_{t}Δp_{t+1}_{21}(Φ

_{21}Δp

_{t-1}+ Φ

_{22}ν

_{t}), and the undetermined-coefficient conditions analogous to (6) are

From the first and third of these we obtain the crucial requirement

Clearly, one root of the foregoing is Φ_{21} = 0, which implies Φ_{11} = 0 and consequently gives the MSV solution. But (42) is also satisfied by values of Φ_{21} such that

Here δ^{2} - 4β is positive for μ_{1} < 1 and μ_{1} > 1 + [2β^{1/2} - (1 + β)]/(-b_{1}α).^{32} So for those values, there are non-zero real roots for Φ_{21} and thus solutions in addition to the MSV solution. That this possibility obtains for large values of μ_{1} represents a problem for monetary policy, according to the non-MSV analysis of the authors mentioned above. But under the hypothesis that the MSV solution prevails, large values of μ_{1} pose no problem: the solution remains *y _{t}* = ν

_{t}, Δp

_{t}= αν

_{t}. Since μ

_{1}→ ∞ is conceptually akin to setting Rt such that

*E*Δ

_{t}*p*= 0, where 0 is the implicit target rate of inflation, the MSV hypothesis seems more consistent with the inflation forecast targeting prescription of Svensson (1998) than does the non-MSV analysis of Bernanke and Woodford (1997) or Clarida, Gali, and Gertler (1997). This conclusion pertains, I conjecture, to this entire body of analysis, not just the single (and extreme) case considered above. In any event, it should be emphasized that if a multiplicity of solutions is found by considering non-MSV procedures, it has nothing to do with the phenomenon of “nominal indeterminacy”—i.e., cases in which a model determines values of real variables but not nominal variables. For a recent discussion of this distinction, see McCallum (1997).

_{t+1}Finally, we might also mention the “fiscal theory of price level determination,” due principally to Woodford (1995) and Sims (1994), which has been attracting a good bit of attention. In this regard, the argument presented in Section 7 of McCallum (1997) indicates that adoption of the fiscal theory of price level determination, in contrast to the more traditional “monetarist” approach, amounts to acceptance of the hypothesis that a non-MSV or bubble solution is empirically relevant. The MSV solution is also available,^{33} however, and implies fully traditional price level-money stock relationships and behavior.

## VIII. Conclusions

Let us conclude with a brief summary. This paper has been concerned with the minimalstate- variable (MSV) criterion for selection among solutions in linear rational expectations models that feature a multiplicity of paths that satisfy all conditions for equilibrium. The paper compares the MSV criterion with others that have been proposed, including Taylor’s (1977) minimum-variance criterion, the expectational stability criterion of Evans (1985, 1986), and the saddle-path or non-explosiveness (i.e., dynamic stability) criterion favored by Blanchard and Kahn (1980), Blanchard and Fischer (1989), Sargent (1987), and Whiteman (1983) and utilized in practice by a large number of researchers. It is emphasized that the MSV criterion can be viewed as a classification scheme, one that delineates the unique solution that is of a bubble-free nature—i.e., reflecting only market fundamentals—from those that include bubble or bootstrap components.

It is argued that the MSV classification scheme is of scientific value in two ways. First, it provides a unique solution upon which a researcher may focus attention if the project at hand suggests or permits the a priori exclusion of bubble solutions. Second, it provides the basis for a substantive hypothesis to the effect that market outcomes in actual economies are generally of a bubble-free nature. In describing the latter role, the paper argues that the possibility that bubble-free solutions dominate empirically is much more plausible than is suggested by solution approaches that parameterize different solutions by (possibly irrelevant) initial conditions rather than by undetermined-coefficient parameter values. It also explains the basis of McCallum’s (1983) “subsidiary principle” that is used to make the MSV solution unique by construction.

In the process of demonstrating the uniqueness of the MSV solution, the paper presents a convenient and practical computational procedure for solving linear rational expectations models of a very broad class. This exposition, which utilizes the generalized Schur decomposition theorem, is developed by means of the mathematically simple undeterminedcoefficients approach. In addition, examples are provided that illustrate the applicability and importance of the MSV criterion to issues of current concern in the analysis of monetary policy rules.

Finally, it should be recognized that some readers may be unwilling to accept the paper’s interpretation of the MSV solution as the bubble-free or fundamentals solution. In that case, it remains true that the MSV approach provides a unique solution upon which a researcher may focus attention, if desired, and provides the basis for a substantive hypothesis to the effect that actual outcomes generally conform to the MSV solution. If this hypothesis is in fact trae, then several classes of problems discussed in the literature are empirically irrelevant.

### Acknowledgments

I am indebted to Edwin Burmeister, Bob Flood, Robert King, and Edward Nelson for comments on previous drafts.

^{}1

What, it might be asked, is the definition of a bubble in a rational expectations model? The basic idea of Burmeister, Flood, and Garber (1983) is that a bubble is an extra component that arises in addition to the component that reflects “market fundamentals,” an important implication of which is that bubble components are not necessarily explosive. Unfortunately, the identification of market fundamentals has to be made on a model-specific basis, although there is rarely any disagreement. Below it will be argued that the MSV solution procedure is constructed so as to yield the market fundamentals solution, thereby providing a method for defining bubbles in particular cases.

^{}2

Leading examples of the latter type include real asset price bubbles in overlapping-generations models, as demonstrated by Calvo (1978) and Woodford (1984), and price level bubbles in infinite-horizon monetary models, as in Brock (1975), Flood and Garber (1980b), Gray (1986), and Obstfeld and Rogoff (1983).

^{}3

Although empirical testing is attractive in principle, this practice is in fact extremely common. This seems to be recognized by Blanchard and Fischer (1989, p. 260).

^{}4

This point will be explained below, in Section IV.

^{}6

Note that π_{1}^{+} = 0 because (α - 1) + [(α -)_{2}]_{1/2} = (α - 1) - (α - 1) since [(α -)_{2}]_{1/2} is by convention a positive number and α—1 is in the present case negative

^{}7

The undetermined-coefficient conditions are (6a), (6b), 0 = π_{2} + απ_{1}π_{2} + απ_{3} + 1, and 0 = π_{3} + απ_{1}π_{3} - απ_{3}. With π_{1} = (α—l)/α, the last of these is satisfied for any π_{3} and the next to last relates π_{2} to π_{3}

^{}8

Burmeister, Flood, and Garber (1983) work in the context of a Cagan-style model similar to (1), except with a white-noise rather than a random-walk disturbance, and define the bubble-free or fundamentals solution as the one that depends only upon “current and expected future values of money and the disturbance” (1983, p. 312).

^{}9

Because of their use of the stability criterion, Blanchard and Fischer (p. 260) suggest that if bubble paths are explosive then “unless the focus is specifically on bubbles, assume that the economy chooses the [stable] path, which is the fundamental [bubble-free] solution”—and do so even if there is no aspect of the model that explicitly disqualifies the explosive paths. But then in cases in which the bubble paths are not explosive, they are unable to recommend among various courses of action. Instead, they retreat to a hope—a “working assumption”—that “the conditions needed to generate stable multiplicities of equilibria are not met in practice” (p. 261). But we know that in various cases this hope is not justified.

^{}11

The reason, of course, is that all other solutions involve—at least in special case—“extraneous” state variables, ones not in a minimal set. Thus the solution values involve variables that do not appear in the model’s structural equations and therefore affect the endogenous variables only because they are (arbitrarily) expected to do so. I would also claim that the MSV solution corresponds to the bubble-free or fundamental values in all the standard, non-contentious examples in the literature. This claim cannot be proved correct, of course, but I am happy to put it forth as a refutable conjecture.

^{}12

Recall that our argument is presuming a linear model. It is possible to distinguish MSV solutions in some nonlinear models, but no general analysis has yet been developed.

^{}13

Recall that we are discussing the case with π_{1} = 0. The solution value for π_{3} when π_{1} = {α—l)/α is undetermined.

^{}14

Application to the striking argument of Woodford (1994, pp. 105-111), Bernanke and Woodford (1997, pp. 669-675), and Clarida, Gali, and Gertler (1997, pp. 20-23), is considered below in Section VII.

^{}15

For more recent developments see Evans (1989) and Evans and Honkapohja (1992, 1997).

^{}16

Actually, it is shown by Evans (1989) and Evans and Honkapohja (1997) that expectational stability obtains when the differential equation analog of this difference equation converges. This will be the case under a somewhat broader set of conditions, so convergence of the iterative procedure is sufficient but not necessary for expectational stability. This result draws on Marcet and Sargent (1989).

^{}17

It is not entirely clear whether Evans and Honkapohja (1992, 1997) would agree with this derivation, as their examples do not include expectations formed at different times. But in the present model, Δ*P ^{e}_{t}* is clearly meant to represent the expectation of Δ

*P*formed in period

_{t}*t*—1. So it is not what the iterative procedure at

*t*is concerned with! Thus it would seem incorrect to write Φ

^{n-1}

_{0}+ Φ

^{n-1}

_{1}Δ

*P*in place of Δ

_{t-1}*P*in (10).

^{e}_{t}^{}18

The example in Evans and Honkapohja (1992) is an exception but is not, I would suggest, as well motivated as the model in the present paper, which differs in its assumptions regarding the times (actually, information sets) relevant for forming expectations of Δ*p _{t}* and Δ

*p*.

_{t+1}^{}19

Klein’s (1997) approach builds upon earlier contributions of King and Watson (1995) and Sims (1996). Other significant recent contributions are Uhlig (1997) and Binder and Pesaran (1995), which use UC analysis. The Uhlig paper also features a useful procedure for linearizing models that include nonlinear relationships.

^{}20

An earlier draft of this paper included a demonstration that closed-form representations of MSV solutions can be obtained by means of formulae developed by Whiteman (1983). This demonstration was illustrated in the context of the simple example of Section II, in line with the much more extensive analysis in McCallum (1985). That analysis was more tedious and less useful that that of the present section, however, since the latter is based on a convenient computational algorithm. The present discussion is taken in large part from McCallum (1998).

^{}21

Here, as above, *E _{t}y_{t+1}* is the expectation of

*y*conditional upon an information set that includes all of the model’s variables dated

_{t+1}*t*and earlier.

^{}23

See Golub and Van Loan (1996, p. 377).

^{}24

Or, in the terminology used by Uhlig (1997), are eigenvalues of *B* with respect to *A*.

^{}25

This is the same condition as that required by Klein (1997, p. 13) and King and Watson (1995, pp. 9-11). It appears to provide no difficulties in practice. The King and Watson example of a system in which the condition does not hold is one in which *B*_{12} = 0 in my notation so the MSV solution has Ω = 0 and the other solution matrices follow easily.

^{}26

By the arrangement of generalized eigenvalues, *S*_{22} has no zero elements on the diagonal (and is triangular).

^{}27

This uses the identity that if *A, B, C* are real conformable matrices, vec*(ABC)* = (*C’ ⊗ A*) vec (*B*). See Golub and Van Loan (1996, p. 180).

^{}28

With *B _{12}* = 0,

*k*does not appear in the system (14) (19), in this case so

_{t}*k*represents extraneous variables of a bootstrap, bubble, or sunspot nature.

_{t}^{}29

It would be more desirable theoretically to use an expectational IS relation, as argued in McCallum and Nelson (1997) and elsewhere, but that would lead to a cubic equation for the coefficient on Δp_{t-1} in the MSV solution without altering the basic message.

^{}30

Two stable solutions exist if the parameters are α = 0.2, b_{1} = 0.5, θ = 0.2, and μ_{1} = 0.5.

^{}31

Clarida, Gali, and Gertler (1997) also include terms involving *y _{t}* and

*R*on the right-hand side of (38). They are omitted here only to keep the example as simple and transparent as possible.

_{t-1}## References

Bernanke, B. S., and M. Woodford. (1997). “Inflation Forecasts and Monetary Policy.”

29, pp.653–684.*Journal of Money, Credit, and Banking*Binder, M., and M. H. Pesaran. (1995).

*“Multivariate Rational Expectations Models: A Review and Some New Results.” In*M. H. Pesaran and M. Wickens (eds.),. Oxford: Basil Blackwell.*Handbook of Applied Econometrics*Blanchard, O. J., and C. M. Kahn. (1980). “The Solution of Linear Difference Models Under Rational Expectations.”

48, 1305–1311.*Econometrica*Blanchard, O. J., and S. Fischer. (1980).

. Cambridge, MA: MIT Press.*Lectures on Macroeconomics*Blanchard, O. J., and C. M. Kahn. (1980). “The Solution of Linear Difference Models Under Rational Expectations.”

48, 1305–1311.*Econometrica*Brock, W. A. (1975). “A Simple Perfect Foresight Monetary Model.”

1, 133–150.*Journal of Monetary Economics*Burmeister, E., R. P. Flood, and P. M. Garber. (1983). “On the Equivalence of Solutions in Rational Expectations Models.”

5, 311–321.*Journal of Economic Dynamics and Control*Calvo, G. A. and (1978). “On the Indeterminacy of Interest Rates and Wages with Perfect Foresight.”

19, 321–337.*Journal of Economic Theory*Calvo, G. A., and (1983). “Staggered Prices in a Utility-Maximizing Framework.”

12, 383–398.*Journal of Monetary Economics*Clarida, R., J. Gali, and M. Gertler. (1997). “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory.”

*NBER Working Paper 6442*.Evans, G. W. (1985). “Expectational Stability and the Multiple Equilibria Problem in Linear Rational Expectations Models.”

100, 1217–1233.*Quarterly Journal of Economics*Evans, G. W., (1986). “Selection Criteria for Models with Non-Uniqueness.”

18, 147–157.*Journal of Monetary Economics*Evans, G. W., (1989). “The Fragility of Sunspots and Bubbles.”

23, 297–317.*Journal of Monetary Economics*Evans, G. W., and S. Honkapohja. (1992). “On the Robustness of Bubbles in Linear RE Models.”

33, 1–14.*International Economic Review*Evans, G. W., and (1997). “Learning Dynamics.”

*Working Paper*.Flood, R. P., and P. M. Garber. (1980a). “An Economic Theory of Monetary Reform.”

88, 24–58.*Journal of Political Economy*Flood, R. P., and (1980b). “Market Fundamentals vs. Price Level Bubbles: The First Tests.”

88, 745–770.*Journal of Political Economy*Froot, K. A., and M. Obstfeld. (1991). “Intrinsic Bubbles: The Case of Stock Prices.”

81, 1189–1214.*American Economic Review*Fuhrer, J. C, and G. R. Moore. (1995). “Inflation Persistence.”

109, 127–159.*Quarterly Journal of Economics*Golub, G. H., and C. F. Van Loan. (1996).

(3rd ed.). Baltimore: Johns Hopkins.*Matrix Computations*Gray, J. A. (1984). “Dynamic Instability in Rational Expectations Models: An Attempt to Clarify.”

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82 (2), 47–76.*Federal Reserve Bank of Richmond Economic Quarterly*King, R. G., and M. W Watson. (1995). “The Solution of Singular Linear Difference Systems Under Rational Expectations.”

*Working paper*, University of Virginia.Klein, P. (1997). “Using the Generalized Schur Form to Solve a System of Linear Expectational Difference Equations.”

*Working paper*, Stockholm University (IIES).Marcet, A., and T. J. Sargent. “Convergence of Least Squares Learning Mechanisms in Self Referential Linear Stochastic Models.”

48, 337–368.*Journal of Economic Theory*McCallum, B. T. (1983). “On Non-Uniqueness in Rational Expectations Models: An Attempt at Perspective.”

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*Working Paper*, Carnegie Mellon University.McCallum, B. T., (1997). “Issues in the Design of Monetary Policy Rules.”

*NBER Working paper 6016*. Also forthcoming in J. B. Taylor M. Woodford (eds.),.*Handbook of Macroeconomics*McCallum, B. T, and E. Nelson. (1997). “An Optimizing IS-LM Specification for Monetary Policy and Business Cycle Analysis.”

*NBER Working Paper 5875*. Also forthcoming in*Journal of Money, Credit, and Banking*.McCallum, B. T. (1998) “Solutions to Linear Rational Expectations Models: A Compact Exposition.”

61, 143–147.*Economics Letters*Obstfeld, M., and K. Rogoff. (1983). “Speculative Hyperinflations in Maximizing Models: Can We Rule Them Out?”

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(2nd ed.). New York: Academic Press.*Macroeconomic Theory*Sims, C. A. (1994). “A Simple Model for Study of the Determination of the Price Level and the Interaction of Monetary and Fiscal Policy.”

4, 381–399.*Economic Theory*Sims, C. A., (1996). “Solving Linear Rational Expectations Models.”

*Working paper*, Yale University.Svensson, L. E. O. (1998). “Inflation Targeting as a Monetary Policy Rule.”

*NBER Working Paper 6790*.Taylor, J. B. (1977). “Conditions for Unique Solutions in Stochastic Macroeconomic Models with Rational Expectations.”

45, 1377–1385.*Econometrica*Taylor, J. B., (1993) “Discretion versus Policy Rules in Practice.”

39, 195–214.*Carnegie-Rochester Conference Series on Public Policy*Uhlig, H. (1997). “A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily.”

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. Minneapolis: University of Minnesota Press.*Linear Rational Expectations Models: A User’s Guide*Woodford, M. (1984). “Indeterminacy of Equilibrium in the Overlapping Generations Model: A Survey.”

*Working Paper*, Columbia University.Woodford, M., (1994).

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43, 1–46.*Carnegie-Rochester Conference Series on Public Policy*

**Comment**

BY EDWIN BURMEISTER

## Preliminaries

When someone expressed amazement to me that Tiger Woods was able to hit an eight iron 175 yards, I replied that I had seen Bob Flood hit an eight iron that far, or almost that far, many times. Unfortunately, on some of these occasions Bob’s target was only 140 yards away.

His work in economics, however, has always been right on target, and it is a great pleasure for me to participate in this conference honoring him. This occasion has also brought back many, many fond memories of my days at the University of Virginia where both Bob and Ben McCallum were my colleagues and friends.

The general subject addressed by Ben McCallum’s paper has been of interest to me since about 1964 when I was a graduate student at M.I.T. This general subject is the dual instability of dynamic economic models in which there is more than one way to hold wealth. The specific class of linear rational expectations models addressed by Ben may be viewed as one very important example.

## Agreements

First I want to list points about which I believe Ben and I are in complete agreement.

Ben’s problem is very important.

His Minimal State Variable Criterion provides an elegant and computationally efficient algorithm for finding the solution to linear rational expectations models. As a bonus, Ben has convincingly argued that his approach is economically useful for other reasons as well.

Taylor’s Minimum Variance Criterion seems economically flawed.

## Potential Agreements

My second list covers points about which I believe Ben and I could be in complete agreement, especially if he has become more mellow with age.

The general problem arises in both discrete time and continuous time formulations. Discrete time is the obvious choice for empirical applications, but the fundamental problem arises in continuous time formulations as well.

Ben states clearly that his procedure is only for linear models. While this is true, he gives away too much. The standard approach to studying the dynamics of a nonlinear system is to first linearize it around the dynamic rest point. Provided the relevant Jacobian matrix is nonsingular (and there are no purely imaginary characteristic roots), the resulting linear system will correctly describe the nonlinear behavior in a neighborhood of the rest point. Thus, Ben’s Minimum State Variable Criterion is quite relevant for studying the local behavior of nonlinear models. Often one can find some additional argument to show that the local behavior obtains globally as well.

The general problem has nothing to do with uncertainty. In rational expectations models, once expectations are taken we are left with a set of deterministic (nonstochastic) dynamic equations. Ben’s Minimum State Variable Criterion is applied to these deterministic equations and the stochastic solution follows. In perfect foresight models, we simply have a set of deterministic equations to begin with to which Ben’s Minimum State Variable Criterion can be applied.

It is economically interesting to build and analyze

*descriptive*economic models. This is an important point; with sufficient optimizing behavior on the part of economic agents, generally (though not always) the stable path will be the only admissible solution, and in this way the dual instability problem “is solved.” However, these optimizing models are of dubious value for the type of macroeconomic policy questions that Ben and others are interested in studying.

## Brief History of the General Dual Instability Property

The general problemis very old. Perhaps the first references are Solow (1959) and Jorgenson (1960). The fact that the same problem arose in models with neoclassical production functions was demonstrated by Hahn (1966). It was also known early on that optimizing behavior would pick out the stable path. One classic example is Samuelson’s (1959) Turnpike Theorem.

The identical dual instability problem then surfaced in monetary growth models, with numerous contributions by Frank Hahn, Karl Shell, M. Sidrauski, T. Sargent, N. Wallace, D. Foley, J. Stiglitz, W. Brock, and many others.

It was once conjectured that all of these descriptive models could be reduced in an “as if” manner to an optimizing model, and that this would provide a way to determine the correct stable solution. But this conjecture was proved false by Burmeister and others (1973). We showed that a growth model with *n* capital goods could have a convergent manifold of dimension (*n* + 1). This means that one can take as given the *n* capital stocks (state variables) and any one of the capital good prices (costate variables) and then uniquely determine the remaining (*n*—1) prices such that the system will converge to its unique dynamic rest point. This is inconsistent with optimizing behavior.

Professional attention soon shifted to new rational expectations models, displacing the monetary growth models of the late 1960’s and early 1970’s. Now at last we had genuine stochastic models. And essentially the identical dual instability (saddlepoint) property emerged as a characteristic of the dynamic equilibria paths, though the recognition of this fact was slow to arrive. Numerous papers were published touting “new” solutions, many of them misleading at best. In view of the fact that the dynamic difference equations arising from these rational expectations models shared much of the same mathematical structure with the Solow and Jorgenson models studied some 20 years earlier, it is surprising how much effort was spent proving “new” results.

The moral for Bob Flood on this occasion is that getting old has some advantages. You do not need to search the literature when you have lived through its development.

## What Behavior Do We Want to Model?

The rational expectations models to which Ben has so elegantly applied his Minimal State Variable Criterion are well suited for addressing certain questions, certainly for the questions studied by Ben and others.

However, I am more convinced than ever that eventually we will need to do more. My fear is that actual economies may spend part of their time on paths that are not stable. If this is the case, sometimes we may be providing inaccurate or even misleading policy advice. And perhaps this happens precisely when good advice is most needed.

We might prefer models with heterogeneous agents having different information sets, different individual models, and different expectations. In particular, different agents have different time horizons (and probably none of them is infinity). Aggregation of each of these individual models (one for each heterogeneous economic agent) would give rise to the macro model. In particular, note that even when each heterogeneous agent exhibits maximizing behavior, the aggregate macro model probably could not be explained in terms of the maximizing behavior of a “representative agent.” I conjecture that such an aggregated macro model would not resemble any of the current linear macroeconomic models, except perhaps under certain idealistic circumstances.

One key to how such a model might work could involve a subset of agents with unrealistic expectations, say overly optimistic earnings forecasts for firms. Eventually earnings disappointments would reveal the false expectations, they would be revised, and the economy might then return to a stable Minimum State Variable solution. The modeling of how such heterogeneous expectations get revised as new information is processed will probably be a key ingredient for any such story.

The Expectational Stability Criterion seems to me to represent one small step in this direction. Accordingly, I would view it with somewhat more promise than does Ben. I recognize, however, that the issue here really involves what economic questions we wish to address, and Ben’s approach is perfectly appropriate for the specific questions he has singled out.

## Concluding Remark

It is not by accident that you have not seen the B word in any of the above. To me, the B word prejudges important economic phenomena as frivolous. I much prefer the bull and bear Wall Street terminology, or, if we must be pretentious, we could talk about “dynamic equilibria paths that lie off the convergent manifold.”

Of course, we must recognize that science advances by attacking the easier problems first. It is therefore perfectly appropriate that we begin with linear rational expectations models, and Ben’s Minimum State Variable Criterion is an extraordinarily valuable tool for helping us to understand them. But I am afraid that much work lies ahead if we are ever to obtain a deeper understanding of how economies actually behave, especially in times when things may have wandered off track.

## References

Burmeister, Edwin, Christopher Caton, A. Rodney Dobell, and Stephen A. Ross. (1973)

*“The ‘Saddlepoint Property’ and the Structure of Dynamic Heterogeneous Capital Good Models.”*, January.*Econometrica*Hahn, Frank H. (1966).

*“Equilibrium Dynamics with Heterogeneous Capital Goods.”*, November, 633–646.*The Quarterly Journal of Economics*Jorgenson, Dale W. (1960).

*“A Dual Stability Theorem.”*, October, 892–899.*Econometrica*Samuelson, Paul A. (1959).

*“Efficient Paths of Capital Accumulation in Terms of the Calculus of Variations.” In*K. Arrow, S. Karlin, and P. Suppes (eds.),. Stanford: Stanford University Press, 1960, pp. 77–78.*Mathematical Methods in the Social Sciences*Solow, Robert M. (1959).

*“Competitive Valuation in a Dynamic Input-Output System.”*, January, 30–53.*Econometrica*

## General Discussion

**Jinill Kim** commented on the solution methodology, suggesting that the reshuffling of eigenvalues could be embedded automatically into the procedure. **Lars Svensson** added that, indeed, Paul Klein’s algorithm, which was based on the Schur decomposition theorem, included the sorting of eigenvalues. **McCallum** responded that the difficulty was to arrange the eigenvalues in the order that his method requires. Although this was usually the same as ordering them by decreasing modulus, the orderings are different for cases in which the minimal state variable and saddle point stability conditions differ. Klein’s algorithm yields the Blanchard-Kahn solution. But an algorithm that gave the minimal state variable solution had not yet been written because it was too hard for McCallum to do (although he could describe how to do so in words) and nobody else had been interested in doing so.

**Michael Mussa** expressed a general concern about procedures that picked out a solution automatically on the basis of a purely technical exercise. To illustrate his concerns, he presented phase diagrams for several variants of a modified Dornbusch overshooting model. This was a two-equation reduced-form system in which there were two key parameters: the semi-elasticity of money demand and a price adjustment parameter. For the case in which the two key parameters had “normal” signs, the system had one positive characteristic root and one negative characteristic root, and the standard criteria (whether McCallum’s minimal state variable criterion or the nonexplosiveness or saddle path criterion associated with Blanchard-Kahn) picked out a unique stable solution. However, the standard criteria also picked out a unique stable solution when both of the key parameters had the “wrong” signs; this was a case in which both the model and the solution were nonsense. Moreover, for a third case in which the semi-elasticity parameter had the “wrong” sign while the price adjustment parameter had the “normal” sign, the system had an infinity of stable solutions; but once again, the model was complete nonsense, so the multiplicity of stable solutions was not reassuring. Mussa shared McCallum’s view that we needed to apply economic good sense in choosing an economically-relevant solution to such models. While that, unfortunately, could be quite difficult in a highly complicated model, we needed to be very careful about just automatically applying a mathematical routine, since that could generate solutions that were just garbage.

Flood noted that Burmeister’s aversion to the B-word evoked memories of many harangues on the golf course when they had been colleagues at the University of Virginia. On dealing with the multiplicity problem, he and Peter Garber had come up with what they thought was a pretty simple procedure—namely, to estimate the model and then look at the reduced form equations to see which solution they reflected. He still didn’t see what was wrong with that approach. All of the models contain behavioral equations. These can be estimated, the reduced forms can be estimated separately, and one can then look at whether or not certain cross equation restrictions hold. McCallum was focusing on the issue of what kind of null hypothesis one wanted to test with respect to the reduced form, and that was fine. But the key point, in Flood’s view, was that one should test that null hypothesis, or see what the reduced form looked like, in the context of actual empirical models.

In responding to the various comments, **McCallum** indicted that he thought he agreed with what Mussa had said and felt that the type of procedure to which Mussa had objected was associated with the Blanchard-Kahn criterion, not his. Mussa’s second example was similar to one that McCallum had considered in his 1983 paper, where he had reached the conclusion that the model was economic nonsense on grounds that Flood and Garber had discussed in one of their papers. In light of such examples, McCallum had described the minimal state variable criterion as a rule for obtaining a solution valid for all “admissible parameter values.” For a given model specification, there could be some parameter values that resulted in what Flood and Garber had referred to as “process inconsistency,” and such parameter values were simply not admissible.

Burmeister had listed four points of potential agreement. **McCallum** was happy to note that he and Burmeister were actually in full agreement on the first three and in partial agreement on the fourth. His only disagreement with Burmeister’s comments was with statements in which Burmeister seemed to imply that the minimal state variable criterion always finds stable linear solutions. This was a slight misstatement about the objective of the criterion, or about how it works. As McCallum had argued in his paper, there may be cases in which the system is not going to be dynamically stable, and he did not want to restrict himself to stable solutions.