Were There Regime Switches in U.S. Monetary Policy? by

Were There Regime Switches in U.S. Monetary Policy?

Christopher A. Sims, Princeton University and NBER

Tao Zha, Federal Reserve Bank of Atlanta

CEPS Working Paper No. 110

May 2005

The first author’s work on this paper was supported in part by CEPS and by National Science

Foundation grant number SES 0350686. The views expressed here are not necessarily those of

the Federal Reserve Bank of Atlanta or the Federal Reserve System. The authors can be

contacted at [email protected] and [email protected], respectively.

WERE THERE REGIME SWITCHES IN US MONETARY POLICY?

ABSTRACT. A multivariate model, identifying monetary policy and allowing for simul-

taneity and regime switching in coefﬁcients and variances, is confronted with US data since

1959. The best ﬁt is with a version that allows time variation in structural disturbance vari-

ances only. Among versions that allow for changes in equation coefﬁcients also, the best ﬁt

is for a one that allows coefﬁcients to change only in the monetary policy rule. That version

allows switching among three main regimes and one rarely and brieﬂy occurring regime.

The three main regimes correspond roughly to periods when most observers believe that

monetary policy actually differed, but the differences among regimes are not large enough

to account for the rise, then decline, in inﬂation of the 70’s and 80’s. In versions that insist

on changes in the policy rule, the estimates imply monetary targeting was central in the

early 80’s, but also important sporadically in the 70’s.

I. INTRODUCTION

It is widely thought that US monetary policy changed a great deal, and for the better,

between the 1970’s and the 1980’s. Clarida, Galí, and Gertler (2000) (CGG) and Lubik and

Schorfheide (2004) ﬁnd that the policy rule apparently followed in the 70’s was one that,

when embedded in a stochastic general equilibrium models, would imply non-uniqueness

of the equilibrium and hence vulnerability of the economy to “sunspot” ﬂuctuations of

arbitrarily large size. Their estimated policy rule for the later period, on the other hand,

implied no such indeterminacy. These results apparently provide an explanation of the

volatile and rising inﬂation of the 70’s and of its subsequent decline.

There are other interpretations of the evidence, however. Primiceri (2003) and Sargent,

Williams, and Zha (2004) estimate models that ﬁnd only modest changes in policy in the

past four decades. Bernanke and Mihov (1998) and Leeper and Zha (2003) perform several

Date: April 14, 2004, revised March 27, 2005.

Key words and phrases. Counterfactuals; Lucas critique; policy rule; monetary targeting; simultaneity;

volatility; model comparison.

JEL classiﬁcation. E52; E47; C53.

REGIME SWITCHES 2

econometric tests and do not ﬁnd strong evidence against stability of coefﬁcients. An earlier

version of this paper (entitled “Macroeconomic Switching”) and subsequent studies (Kim

and Nelson, 2004; Cogley and Sargent, 2004; Primiceri, in press; Canova and Gambetti,

2004) show little evidence in favor of the view that the monetary policy rule has changed

drastically.

This paper follows the structural VAR literature in making explicit identifying assump-

tions to isolate estimates of monetary policy behavior and its effects on the economy, while

keeping the model free of the many additional restrictive assumptions needed to give every

parameter and equation a behavioral interpretation or to allow structural interpretation of a

single-equation model. We use a model that allows explicitly for changes in policy regime,

including as special cases both short-lived oscillating policy changes and unidirectional,

persistent shifts toward improved policy. We compare versions of the model with Bayesian

posterior odds ratios, a method that automatically penalizes models with unneeded free

parameters.

Our most important empirical ﬁnding is that the version of our model that ﬁts best is one

that shows no change at all in coefﬁcients either of the policy rule or of the private sector

block of the model. What changes across “regimes” is only the variances of structural

disturbances. That is, this version of the model explains differences in the behavior of the

economy between periods as reﬂecting variation in the sources of economic disturbances,

not as variation in the dynamics of the effects of a given disturbance on the economy. The

Volcker reserves-targeting period emerges as a period of high variance in disturbances of

the policy rule. This ﬁnding lends empirical support to the common practice in the literature

of combining the samples before and after reserve-targeting period to estimate the model,

as long as heteroskedasticity is properly taken into account.

We consider also models in which parameters do change. We have looked at models

where all parameters in all equations can change, where only non-monetary-policy coefﬁ-

cients change, and where only monetary-policy coefﬁcients can change. In these cases we

allow structural variances to shift size at the same time coefﬁcients change, and we have

also tried models in which the times of coefﬁcient changes are stochastically independent

of the times of variance changes. We have allowed the number of regimes to vary, including

REGIME SWITCHES 3

the case of a single regime, and we have considered speciﬁcations in which regime change

is constrained to be monotonic, so that old regimes are constrained never to recur. None

of these models ﬁt nearly as well as the best-ﬁtting model in which only residual variances

change across regimes. Particularly ill-ﬁtting are the models with a single regime and the

model that constrains regime changes to be monotonic.

The best ﬁtting model among those that do allow coefﬁcients to change is one that con-

strains the changes to occur only in the monetary policy equation, while coefﬁcients in the

other equations remain constant. Like Cogley and Sargent (2004) and Primiceri (in press),

we ﬁnd that the point estimates of the changes are not trivial, even though the data leave

their magnitudes uncertain. The model ﬁnds the best ﬁt with four regimes. One occurs

in only a few brief spans of months, one of which is September-October 2001, and has

very high residual variance in money demand. Another corresponds to the Volcker reserve-

targeting period and shows clearly the targeting of monetary aggregates, rather than interest

rates, in that regime. Another regime has been in place through nearly all of the years of the

Greenspan Fed chairmanship — but also was in place through most of the 60’s. A fourth

regime occurred in several multi-year episodes in the late 60’s and early 70’s. Though

it does not show as strong a monetary-aggregate-targeting ﬂavor as the Volcker regime,

it does tend much more strongly in that direction than the “Greenspan” regime. We call

this fourth regime the “Burns” regime, even though the “Greenspan” regime was in place

though approximately the same proportion of the Burns chairmanship as was the “Burns”

regime. (For most of this paper we drop the quotes on the regime names, hoping the reader

can bear in mind that the correspondence of the regimes to chairmanship terms is rough.)

We display counterfactual simulations of history with alternate monetary policy regimes.

If we simulate history with the estimated time series of shocks, but the coefﬁcients of the

policy rule set at the estimated Greenspan policy throughout the period 1961-1987, the rise

and fall of inﬂation follows the historical path extremely closely. This is not because the

model is incapable of showing an effect of monetary policy. If we instead use a policy rule

that uses the Greenspan coefﬁcients, except that it doubles the coefﬁcients on inﬂation, the

counterfactual historical simulation shows much lower inﬂation throughout the 70’s and

early 80’s — at the cost of considerably lower output growth through that period. A similar

REGIME SWITCHES 4

lower inﬂation path emerges if we ﬁx the policy rule at the point estimate for the Volcker

reserve-targeting regime.

Although the estimated differences in policy behavior and their effects on the economy

in this four-state model are substantively interesting and consistent with the results from

the recent learning literature (Primiceri, 2003; Sargent, Williams, and Zha, 2004), they are

not as drastic as what is implied by the sunspot-equilibrium model. In particular, for all

of the three main regimes our estimates imply that with high probability monetary policy

responses to inﬂation were strong enough to guarantee a determinate equilibrium price

level.

There are a number of likely explanations for the contrast between our ﬁnding here and

the ﬁndings in some other empirical papers. Perhaps the most important is that rather than

aiming at ﬁnding some model we can interpret that is not rejected by the data, we aim

at fully characterizing the uncertainty about our results. When we run our counterfactual

historical simulations by drawing from the posterior distribution of the coefﬁcients of the

policy rule instead of ﬁxing the coefﬁcients at particular values, we can see that the shape

of uncertainty about these policy rules differs more than do their most likely values. When

we simulate history with the Greenspan, Burns and Volcker rule distributions, the median

paths for inﬂation and output show visible differences, with the Volcker and Greenspan

median paths similar and lower than the Burns median path. The Volcker and Greenspan

distributions show a risk of deﬂation, while the Burns distribution does not, and the Volcker

and Greenspan paths show a risk of periods of output growth below -5 per cent at an annual

rate, while the Burns path does not. The output growth rate along the median Burns path is

slightly above the historical growth rate, while it is notably below (

to 1 per cent at annual

rate) the historical rate along the Greenspan and Volcker medians. The Burns distribution

shows a risk of inﬂation not coming down at all in the 80’s, while neither the Volcker nor the

Greenspan path shows such a risk. In other words, even though the data are best explained

by a model with no change at all in policy rule coefﬁcients, if one looks for changes, and

one is willing to consider policy rules that are unlikely but not impossible, one can tell

a story consistent with the view that the Burns policy, had it persisted (instead of ending

around 1977, as the model estimates it did), would have failed to end inﬂation.

REGIME SWITCHES 5

There are also substantive differences between our model and the rest of the literature

that may contribute to our ﬁnding that there is little evidence of policy change. Of particular

note is the fact that, unlike much previous work, which ﬁts a “Taylor rule” to the whole

period, we include a monetary aggregate in our policy reaction function. The Federal

Reserve is by law required to provide the target paths for various monetary aggregates, and

during the 70’s the behavior of these aggregates was central to discussions of monetary

policy. We show that constraining the monetary aggregate not to appear in our monetary

policy equation greatly worsens the model’s ﬁt to the historical data, and we argue that it is

likely that excluding the aggregate from the equation was a source of bias in earlier work.

However, while excluding money might have led to a spurious ﬁnding of a violation of the

“Taylor principle”, including money in our framework improves the relative ﬁt of models

that allow variation in the policy rule.

We think our results have implications for future research on theoretical models with

more detailed behavioral structure:

• The Taylor rule formalism, valuable as it may be as a way to characterize policy in

the last 20 years, can be seriously misleading if we try to use it to interpret other

historical periods, where monetary aggregate growth was an important factor in the

thinking of policy-makers.

• It is time to abandon the idea that policy change is best modeled as a once-and-for-

all, non-stochastic regime switch. Policy changes, if they have occurred, have not

been monotonic, and they have been difﬁcult to detect. Both the rational public in

our models and econometricians must treat the changes in policy probabilistically,

with a model of how and when the policy shifts occur and with recognition of the

uncertainty about their nature and timing.

II. THE DEBATE OVER MONETARY POLICY CHANGE

The literature in this area is large enough that we will not try to discuss papers in it one

by one, but we lay out what seem to us a few of the most important reasons why our results

differ from much of the previous empirical work in the area.

REGIME SWITCHES 6

(i) As we pointed out above, our speciﬁcation includes a monetary aggregate in the

reaction function. Most of the previous literature does not. We think this is a

possibly important source of bias in estimates of the reaction function.

(ii) Much of the previous literature either makes no allowance for heteroskedasticity or

allows only implausibly restricted forms of heteroskedasticity. Particularly com-

mon have been speciﬁcations in which there is a single change in residual variance

in the sample, and speciﬁcations that generate “robust standard errors” by allowing

for heteroskedasticity that is a function of right-hand side variables. It is clear to the

eye, and apparent in our estimation results, that residual variances in the reaction

function rose sharply at the end of 1979, then dropped back a few years later. A sin-

gle shift in variance cannot capture this fact. And the persistent shifts in variances

that we ﬁnd could not be well modeled as functions of right-hand-side variables. As

we have already noted, failing to allow properly for heteroskedasticity can strongly

bias statistical tests in favor of ﬁnding signiﬁcant shifts in coefﬁcients. This is ap-

parent from the contrast between the results of Cogley and Sargent (2001) and the

later version Cogley and Sargent (2004) that does allow for a fairly general form of

heteroskedasticity.

(iii) Identiﬁcation in these models is fragile. This is particularly true for the forward-

looking Taylor rule speciﬁcation of CGG, for two reasons.

One is that estimating this single equation is based on claiming that a list of

instrumental variables is available that can be used to control for the endogeneity

of expected future inﬂation and and output. But these instruments are available

only because of a claim that we know a priori that they do not enter directly into the

reaction function — they can affect monetary policy only through their effects on

expected future variables. We ﬁnd it inherently implausible that, for example, the

monetary authority reacts to an expected future 3 per cent inﬂation rate in exactly

the same way, whether the recent past level of inﬂation has been 1.5 per cent or 6

per cent.

The other problem with this speciﬁcation is that the Fisher relation is always

lurking in the background. The Fisher relation connects current nominal rates to

REGIME SWITCHES 7

expected future inﬂation rates and to real interest rates, which are in turn plausibly

determined by expected output growth rates. So one might easily ﬁnd an equa-

tion that had the form of the forward-looking Taylor rule, satisﬁed the identifying

restrictions, but was something other than a policy reaction function.

Multivariate models allow a check on the identifying assumptions via exami-

nation of the impulse responses to monetary policy shocks. Single equation ap-

proaches obviously do not. It seems to us that empirical work that has been based

on multivariate models and has included checks for plausibility of responses to

monetary policy shocks has tended to ﬁnd less evidence of changing monetary pol-

icy.

(iv) It is interesting to consider changes in monetary policy and to connect estimated

changes to historical events. Indeed, we do some of that in this paper, with a model

we do not think is our best. As a result, abstracts, introductions and conclusions

often seem to support the idea that there have been changes in monetary policy

even when a look at plotted conﬁdence or probability bands around time paths of

coefﬁcients or functions of them can be seen to include constant paths. So in some

cases there is more contrast between the abstracts of papers in the literature and our

abstract than there is in the detailed results.

III. CLASS OF MODELS

The general framework is described by nonlinear stochastic dynamic simultaneous equa-

tions of the form:

) = x

) +

, t = 1,... ,T, (1)

Pr(s

= i | s

t−1

= k) = p

, i,k = 1, ...,h, (2)

where s is an unobserved state, y is an n× 1 vector of endogenous variables, x is an m× 1

vector of exogenous and lagged endogenous variables, A

is an n× n matrix of parameters,

is an m×n matrix of parameters, T is a sample size, and h is the total number of states.

REGIME SWITCHES 8

Denote the longest lag length in the system of equations (1) by

. The vector of right-

hand variables, x

, is ordered from the n endogenous variables for the ﬁrst lag down to the

n variables for the last (

) lag with the last element of x

being the constant term.

For t = 1, ... ,T, denote

= {y

,. ..,y

We treat as given the initial lagged values of endogenous variables Y

= {y

1−

,. ..,y

Structural disturbances are assumed to have the distribution:

(

| Y

t−1

) = N

n×1

where N(a,b) refers to the normal pdf with mean a and covariance matrix b and I

is an

n× n identity matrix. Following Hamilton 1989 and Chib 1996, we impose no restrictions

on the transition matrix P = [p

The reduced-form system of equations implied by (1) is:

= x

B(s

) + u

), t = 1,... ,T; (3)

where

B(s

) = A

−1

), (4)

) = A

0 −1

)

, (5)

E(u

)

) =

)

−1

. (6)

In the reduced form (4)-(6), B(s

) and u

) involve the structural parameters and shocks

across equations, making it impossible to distinguish regime shifts from one structural

equation to another. In contrast, the structural form (1) allows one to identify each structural

equation, such as the policy rule, for regime switches.

If we let all parameters vary across states, it is relatively straightforward to apply the

existing methods of Chib 1996 and Sims and Zha 1998 to the model estimation because

) and A

) in each given state can be estimated independent of the parameters in

other states. But with such an unrestricted form for the time variation, if the system of

equations is large or the lag length is long, the number of free parameters in the model

becomes impractically large. For a typical monthly model with 13 lags and 6 endogenous

REGIME SWITCHES 9

variables, for example, the number of parameters in A

) is of order 468 for each state.

Given the post-war macroeconomic data, however, it is not uncommon to have some states

lasting for only a few years and thus the number of associated observations is far less than

468. It is therefore essential to simplify the model by restricting the degree of time variation

in the model’s parameters.

We rewrite A

)

m×n

= D(s

)

m×n

+ S

m×n

)

n×n

. (7)

where

S =





(m−n)×n





If we place a prior distribution on D(s

) that has mean zero, our prior is centered on the

same reduced-form random walk model that is the prior mean in existing Bayesian VAR

models (Sims and Zha 1998). As can be seen from (4)-(7), this form of prior implies that

smaller A

−1

values, and thus smaller reduced form residual variances, are associated with

tighter concentration of the prior about the random walk form of the reduced form. On

the other hand, small values of D are also associated with tighter concentration of the prior

about the random walk reduced form, without any corresponding effect on reduced form

residual variances.

Note that this setup centers the prior on models in which the moving average represen-

tation

has the form

∞

∑

s=0

t−s

−1

In all the models studied here, we incorporate the Litterman (1986)’s lag-decay prior that effectively

dampens the unreasonable inﬂuence of long lags. Thus the over-parameterization problems associated with

typical VAR’s do not apply here. In addition, the marginal likelihood or the Schwarz criterion used in this

paper as a measure of ﬁt, by design, would penalize an excessive number of parameters that over-ﬁt the data.

Of course the expression we give here for the MAR is only valid if the innovations are not stationary

inﬁnitely far back into the past, but instead are, e.g., zero before some startup date. Or the expression can be

though of as the limit as

→ 1 of stationary MAR’s with coefﬁcients of the form

(1−

L)A

−1

REGIME SWITCHES 10

This ties our beliefs about lagged effects of structural innovation i on variable j to our

beliefs about contemporaneous effects of innovation i on variable j. Any prior that cen-

ters on a random walk reduced form while leaving beliefs about reduced form residual

covariances independent of beliefs about reduced form coefﬁcients, will have the same ef-

fect. For example the standard “Minnesota prior” on the reduced form, combined with any

identiﬁcation scheme based on restrictions on contemporaneous coefﬁcients, will center on

MAR’s of this form. If one thinks of the model as a discrete approximation to an underlying

continuous-time system, this type of prior is reasonable. It is implausible that the effects of

structural innovations show sharp discontinuities across lags.

We consider the following three cases of restricted time variation for A

) and D(s

0, j

),d

ij,`

),c

) =











¯a

0, j

ij,`

, ¯c

Case I

¯a

0, j

ij,`

), ¯c

) Case II

0, j

ij,`

),c

) Case III

, (8)

where

) is a scale factor for the j

structural equation, a

0, j

) is the j

column of

), d

) is the j

column of D(s

), d

ij,`

) is the element of d

) for the i

variable

at the `

lag, the last element of d

), c

), is the constant term for equation j. The

parameter

) changes with variables but does not vary across lags. This allows long

run responses to vary over time, while constraining the dynamic form of the responses

to vary only through

, which can be though of as indexing the degree of inertia in the

variable interpreted as the “left-hand side”. Of course in this simultaneous equations setup,

there may not be a variable that is uniquely appropriate as “left-hand side” in equation i.

The speciﬁcation insures, though, that whichever variable we think of as on the left hand

side, the time variation in dynamics is one-dimensional, in that it affects all “right-hand

side” variables in the same way. The bar symbol over a

0, j

, d

ij,`

, and c

means that these

parameters are state-independent (i.e., constant across time).

Case I is a constant-coefﬁcient structural equation. Case II is an equation with time-

varying disturbance variances only. Case III is an equation with time-varying coefﬁcients,

as well as time varying disturbance variances.

REGIME SWITCHES 11

We have considered models with Case II speciﬁcations for all equations, with Case II

for the policy equation and Case III for all others, with Case III for the policy equation

and Case II for all others, and with Case III for all equations. That is, we have examined

models with time variation in coefﬁcients in all equations, with time variation in coefﬁcients

in policy or private sector equations only, and with no time variation in coefﬁcients. In all

of these cases we allow time variation in structural disturbance variances of all equations.

The model with time variation in coefﬁcients in all equations might be expected to ﬁt best

if there were policy regime changes and the nonlinear effects of these changes on private

sector dynamics, via changes in private sector forecasting behavior, were important. That

this is possible was the main point of Robert E. Lucas (1972).

However, as one of us has explained at more length elsewhere (Sims, 1987), once we

recognize that changes in policy must in principle themselves be modeled as stochastic,

Lucas’s argument can be seen as a claim that a certain sort of nonlinearity is important.

Even if the public believes that policy is time-varying and tries to adjust its expectation-

formation accordingly, its behavior could be well approximated as linear and non-time-

varying. As with any use of a linear approximation, it is an empirical matter whether the

linear approximation is adequate for a particular sample or counterfactual analysis.

We consider the model with Case III for all equations because we are interested in

whether it ﬁts better than the other models, as would be true if policy had changed within

the sample and Lucas-critique nonlinearities were important. We consider the other combi-

nations because it is possible that coefﬁcients in the policy have not changed enough for the

changes to emerge clearly from the data, or enough to generate detectable corresponding

changes in private sector behavior.

IV. DATA, IDENTIFICATION, AND MODEL FIT

We use monthly US data from 1959:1–2003:3. Each model has 13 lags and includes

the constant term and 6 commonly-used endogenous variables: a commodity price index

Another early paper emphasizing the need for stochastic modeling of policy change is Cooley, Leroy,

and Raymon (1984). More recently Leeper and Zha (2003) have drawn out the implications of this way of

thinking for the practice of monetary policy.

REGIME SWITCHES 12

TABLE 1. Identifying restrictions on A

)

Variable (below) Sector (right) Inf Fed MD Prod Prod Prod

Pcom X

M X X X

R X X X

y X X X X X

P X X X X

U X X

(Pcom), M2 divisia (M), the federal funds rate (R), interpolated monthly real GDP (y), the

core personal consumption expenditure (PCE) price index (P), and the unemployment rate

(U). All variables are expressed in natural logs except for the federal funds rate and the

unemployment rate which are expressed in percent.

The identiﬁcation of monetary policy, following Leeper and Zha 2003, is described in

Table 1. The X’s in Table 1 indicate the unrestricted parameters in A

) and the blank

spaces indicate the parameters that are restricted to be zero. The “Fed” column in Table 1

represents the Federal Reserve contemporaneous behavior; the “Inf” column describes the

information sector (the commodity market); the “MD” represents the money demand equa-

tion; and the block consisting of the last three columns represents the production sector,

whose variables are arbitrarily ordered in an upper triangular form.

As robustness checks, we also used the M2 stock instead of M2 divisia and the CPI (as well as the GDP

deﬂator) instead of the core PCE price index and the paper’s main conclusions remained unchanged.

While we provide no discussion here of why delays in reaction of the private sector to ﬁnancial variables

might be plausible, explanations of inertia, and examination of its effects, are common in the recent literature

(Christiano, Eichenbaum, and Evans, 2005; Edge, 2000; Sims, 2003, 1998). The economic and theoretical

justiﬁcation of the identiﬁcation presented in Table 1 can also be found in Leeper, Sims, and Zha (1996) and

Sims and Zha (forthcoming). This identiﬁcation has proven to be stable across different sets of variables,

different sample periods, and different developed economies.

REGIME SWITCHES 13

In addition to the exact zero restrictions shown in Table 1, we introduce stochastic prior

information favoring a negative contemporaneous response of money demand to the inter-

est rate and a positive contemporaneous response of the interest rate to money (see Ap-

pendix A). More precisely, we use a prior that makes the coefﬁcients on R and M in the

money demand column of A

positively correlated and in the monetary policy column of

negatively correlated. This liquidity effect prior has little inﬂuence on the correlation of

posterior estimates of the coefﬁcients in the policy and the money demand equations, but

it makes point estimates of coefﬁcients and impulse responses more stable across different

sample periods. The instability we eliminate here arises from the difﬁculty of separating

money demand and supply in some sub-periods, and is associated with imprecise estimates

in both equations for this reason. Since a ﬁnding of change in monetary policy across peri-

ods requires some precision in the estimates of policy rule coefﬁcients in those periods, the

liquidity-effect priors are as likely to strengthen as to weaken evidence for changes in the

policy rule. We take up this issue again in discussion of the results, below.

We model and compare the ﬁve speciﬁcations:

Constant: a constant-parameter BVAR (i.e., all equations are Case I);

Variances Only: all equations are Case II;

Monetary Policy: all equations except the monetary policy rule are Case II, while

the policy rule is Case III;

Private Sector: equations in the private sector are Case III and monetary policy is

Case II;

All Change: all equations are Case III.

There are two major factors that make the estimation and inference of our models a

difﬁcult task. One factor is simultaneous relationships in the structural coefﬁcient matrix

). The other factor is the types of restricted time variations speciﬁed in (8). Without

these elements, the shape of the posterior density would be much more regular and more

straightforward Gibbs sampling methods would apply. Appendix A outlines the methods

and brieﬂy discusses both analytical and computational difﬁculties.

The ﬁrst set of results to consider is measures of model ﬁt, with the comparison based

on posterior marginal data densities. The results are displayed in Table 2. For the models

REGIME SWITCHES 14

with larger numbers of free parameters the Markov Chain Monte Carlo (MCMC) sample

averages that are the basis of the numbers in the table behave erratically, and we display “∗”

for these cases rather than a speciﬁc number. Though the estimated marginal data densities

(MDD’s) for these cases are erratic, they remain far below the levels of the MDD’s shown

in the same column above them. In other words, though displaying a single number for

their MDD values might indicate misleading precision, it is clear that the MDD’s for these

cases are very much lower than those of the cases for which we do display numbers.

Note that this is a log-likelihood scale, so that differences of 1 or 2 in absolute value

mean little, while differences of 10 or more imply extreme odds ratios in favor of the

higher-marginal-data-density model. For the upper rows in the table the Monte Carlo (MC)

error in these numbers (based on two million MCMC draws) is from ±2 to ±4. For the

lower rows in each column the error is larger (from ±3 to ±5). These estimates of MC error

are conservative, based on our own experience with multiple starting points for the chain.

Conventional measures of accuracy based on serial covariances of the draws, for example,

would suggest much smaller error bands. When the whole private sector, or the whole

model, is allowed to change according to Case III, the marginal data density is distinctly

lower than that of the best models for a given row of the table and for those versions of

The reason redundant states create this erratic behavior is that our sampling scheme draws sequences of

states as well as parameter vectors. When there is a redundant state, it may be a posteriori likely that the state

did not occur at all in the sample. If we have drawn a state sequence in which one state does not occur, at the

next Gibbs-sampling step, where coefﬁcients are drawn for each state, the data are exerting no inﬂuence at all

on the parameters for the non-occurring state; the draw is from the prior for that state’s coefﬁcients. Since no

draw of the coefﬁcients for that state is likely to improve the likelihood much (the state being redundant), the

MCMC chain can remain stuck with no occurrences of the state for a long time. But the same likelihood value

can also be achieved with this redundant state occurring, but having coefﬁcients matching those of some other

state. Once such a draw occurs, the transition probabilities between the two nearly identical states become ill

determined by the likelihood. This leads to the MCMC chain being stuck for a long time in a mode with all

states occurring, but one or more states nearly identical. All of these draws are likely to imply low posterior

density values, but they can vary quite a bit, depending on the shape of the prior. Thus we obtain MCMC

chains with very strong serial correlation, making estimates of sample means unreliable, but still allowing a

reliable conclusion that posterior weight on the model is low.

REGIME SWITCHES 15

TABLE 2. Comprehensive measures of ﬁt

Log marginal data densities

Constant 12,998.20

Variances Only Monetary Policy Private Sector All Change

2 states 13,345.71 13,383.36 13,280.74 13308.80

3 states 13,434.25 13,446.13 13,380.77 13426.78

4 states 13,466.86 13,480.18 ∗ ∗

5 states 13,455.26 13,400.10 ∗ ∗

6 states 13,510.31 ∗ ∗ ∗

7 states 13,530.71 ∗ ∗ ∗

8 states 13,540.32 ∗ ∗ ∗

9 states 13,544.07 ∗ ∗ ∗

10 states 13,538.03 ∗ ∗ ∗

the model for which we could obtain convergence. The best ﬁt is for the 9-state variances-

only model, though any of the 7 through 10 state versions of that model have similar ﬁt.

The marginal data density for these variances-only models are higher by at least 50 on a

log scale than that for any other model. The best of the models allowing time variation

in coefﬁcients is the monetary policy model with 4 states, whose marginal data density is

higher by at least 50 than that of any other model that allows change in coefﬁcients.

V. BEST-FIT MODEL

There are a number of best-ﬁt models, all of them variances-only models with from 7 to

10 states. Since the results from these models are quite similar, we report the results from

only the 9-state variances-only model. The transition matrix for the 9 states is shown in

Note, though, that the “private sector” and “all change” models may be doing less well because of pa-

rameter count. It could be that more tightly parameterized models of coefﬁcient change in the private sector

would look better in a table like this.

REGIME SWITCHES 16

TABLE 3. Transition matrix for 9-state variances-only model

0.9643 0.0063 0.0117 0.0064 0.0108

0.0030 0.9394 0.0047 0.0070 0.0210

0.0104 0.0159 0.9455 0.0064 0.0046

0.0026 0.0043 0.0042 0.9476 0.0040

0.0058 0.0155 0.0044 0.0068 0.9425

0.0027 0.0056 0.0058 0.0064 0.0051

0.0052 0.0042 0.0081 0.0068 0.0040

0.0033 0.0041 0.0069 0.0062 0.0038

0.0026 0.0046 0.0087 0.0065 0.0042

0.0057 0.0107 0.0095 0.0049

0.0062 0.0061 0.0069 0.0112

0.0063 0.0064 0.0096 0.0057

0.0058 0.0056 0.0062 0.0051

0.0185 0.0058 0.0064 0.0057

0.9406 0.0120 0.0062 0.0050

0.0057 0.9423 0.0062 0.0053

0.0056 0.0054 0.9429 0.0049

0.0056 0.0056 0.0062 0.9522

Table 3. The states appear to behave similarly, and they have a fairly evenly spread set of

steady-state probabilities, ranging from .078 to .19.

The ﬁrst state is used as a benchmark with its variances being normalized to 1. As can

be seen from Figure 1, this state prevails in most of the Greenspan regime and includes

several years in the 1960s. The variances in other states do not simply scale up and down

across all structural equations. Some states affect a group of structural shocks jointly, as

can be seen from Table 4. The 9th state prevails in the Volcker reserve-targeting period,

REGIME SWITCHES 17

TABLE 4. Relative shock standard deviations across states for 9-state

variances-only model

Financial M Policy M demand Private y Private P Private U

First state 1.00 1.00 1.00 1.00 1.00 1.00

Second state 0.95 1.47 1.03 2.07 1.19 1.69

Third state 1.28 1.65 1.84 1.11 1.12 0.91

Fourth state 2.01 2.65 1.93 1.59 1.29 1.37

Fifth state 1.38 2.95 1.24 1.01 0.96 1.17

Sixth state 2.67 2.99 2.32 2.52 0.95 2.13

Seventh state 2.40 4.43 1.21 1.59 2.58 1.05

Eighth state 2.55 4.49 11.44 4.10 10.48 2.67

Ninth state 1.49 12.57 1.53 1.44 1.48 1.44

and primarily inﬂates the variance of the policy shock (Figure 1 and Table 4.) The 8th

state inﬂates the variances of several private-sector equations, and it prevails only for the

two months of September and October, 2001. This is clearly a “9/11” state. The other

states exist sporadically over the 70’s as well as over the period from 1983 to 1987 and

some years in the 60’s. Among these states, the shock variances change irregularly from

state to state. For the 70’s, short-lived states with changing shock variances reﬂect several

economic disruptions (e.g., two big oil shocks) and the ambivalent way monetary policy

was conducted in response to those disturbances.

For this variances-only model, the structural parameters and impulse responses vary

across states only up to scales. Table 5 reports the estimate of contemporaneous coefﬁ-

cient matrix for the 1st state. As can be seen from the “M Policy” column, the policy rule

shows a much larger contemporaneous coefﬁcient on R than on M, implying the Federal

Reserve pays much more attention within the month to the interest rate than the money

stock.

Estimates of the model’s dynamic responses are very similar to those produced by pre-

vious identiﬁed VAR models, so we will not present a full set of impulse responses. The

REGIME SWITCHES 18

5 10 15

FFR

0.0 0.2 0.4 0.6 0.8 1.0

0.00 0.02 0.04 0.06

1960 1970 1980 1990 2000

0.0 0.2 0.4 0.6 0.8 1.0

1960 1970 1980 1990 2000

Time

p9m[, 2:11]

FIGURE 1. 9-state variances-only probabilities; the Fed Funds Rate in up-

per left.

REGIME SWITCHES 19

results are as sensible as for previous models, yet we have a more accurate picture of uncer-

tainty because of its stochastically evolving shock variances. The responses to a monetary

policy shock for the 1st state, together with error bands, are shown in Figure 2.

Note

that, though commodity prices and the money stock decline following a shock that tightens

monetary policy, the point estimates show P declining only after a delay of several years,

and this decline is small and uncertain.

Table 6 reports artiﬁcial long run responses of the policy rate to other macro variables,

as often presented in the literature. By “artiﬁcial” we mean that these are neither an equi-

librium outcome nor multivariate impulse responses, but are calculated from the policy

reaction function alone, asking what would be the permanent response in R to a perma-

nent increase in the level or rate of change of the variable in question, if all other variables

remained constant. The long run response to the level of the variable is calculated as

∑

`=0

∑

`=0

, where

is the coefﬁcient on the `

lag of the “right-hand-side” variable

and

is the coefﬁcient on the `

lag of the “left-hand-side” variable in the policy rule.

The long run response to the change of the variable is calculated as

∑

`=0

∑

i=0

∑

`=0

In Table 6, the differenced (log) variables such as ∆y and ∆P are annualized to match the

annual rate of interest R. Absence of sunspots in the price level will be associated with the

sum of these long run responses to nominal variables (here ∆PCom, ∆M, and ∆P) exceed-

ing 1. For this model the sum is 1.76, well above one, though the error bands on individual

coefﬁcient leave room for some uncertainty.

VI. POLICY REGIME SWITCHES

In this section, we present the key results from the 4-state model with time-varying

coefﬁcients in the policy rule. There are two reasons why this model may be of interest,

despite the fact that it is dominated in ﬁt by the model with only disturbance variances

changing. First, this model’s ﬁt is substantially better than all other models that allow

change in coefﬁcients (Table 2). Second, the model reﬂects a prevailing view that the

endogenous component of US monetary policy has changed substantially since 1960 and

The shape of the impulse responses as seen on scaled plots is the same across states.

REGIME SWITCHES 20

PCOM

Time

resp

0 1 2 3 4

−0.015 −0.010 −0.005 0.000

Time

resp

0 1 2 3 4

−0.004 −0.003 −0.002 −0.001 0.000

FFR

Time

resp

0 1 2 3 4

−0.0005 0.0005 0.0015

Time

resp

0 1 2 3 4

−0.0020 −0.0010 0.0000

Time

resp

0 1 2 3 4

−1e−03 −5e−04 0e+00 5e−04

Time

resp

0 1 2 3 4

0e+00 2e−04 4e−04 6e−04

FIGURE 2. Responses to a Monetary Policy Shock, 9-state Variances-Only Model

Note: Each graph shows, over 48 months, the modal’s estimated response (blackest), the median response,

and 68% and 90% probability bands.

REGIME SWITCHES 21

TABLE 5. Contemporaneous coefﬁcient matrix for 9-state variances-only model

Financial M Policy M demand Private y Private P Private U

Pcom 70.64 0.00 0.00 0.00 0.00 0.00

M 9.21 −130.24 −669.91 0.00 0.00 0.00

R −27.30 688.52 −70.10 0.00 0.00 0.00

y −14.21 0.00 19.85 308.75 −20.77 51.94

P −5.54 −0.00 216.07 0.00 −1061.30 32.38

U 82.37 0.00 0.00 0.00 0.00 766.38

TABLE 6. Long run policy responses in 9-state variances-only model

Responses of R to Posterior peak estimate .68 probability interval

∆ Pcom 0.21 (0.17, 0.73)

∆ M 0.16 (-0.48, 0.44)

∆ y 0.71 ( 0.69, 3.36)

∆ P 1.39 ( 0.45, 2.21)

U -1.01 (-2.80, -0.42)

its simulated results capture some important aspects of conventional wisdom about policy

changes from the 70’s through the 80’s to the 90’s.

Figure 3 shows the implied state-probabilities over time produced by this 4-state model.

We can see that state 1 has prevailed for most of our full sample period and for the entire

period from the late 80’s onward. We call this state the “Greenspan” state of policy, but

of course one needs to bear in mind that this policy regime was dominant in most of the

60’s and in the latter half of the 70’s as well. State 2 is the next most common, occur-

ring most frequently from the early 60’s through the early 70’s (the ﬁrst oil shock period),

though with no sustained periods of prevalence that match those of state 1. We call this the

REGIME SWITCHES 22

TABLE 7. Transition matrix for 4-state policy-only model

0.9627 0.0460 0.0203 0.0334

0.0214 0.9388 0.0195 0.0174

0.0077 0.0073 0.9414 0.0238

0.0082 0.0079 0.0188 0.9254

“Burns” regime, even though it matches up with Burns’s chairmanship even less well than

the “Greenspan” regime matches with Greenspan’s. State 3 prevails during the Volcker

reserve targeting period and nowhere else except one very brief period around 1970. State

4 occurs only for a few isolated months, including 9/11, and seems clearly to be picking up

outliers rather than any systematic change of coefﬁcients.

The estimate of the transition matrix is shown in Table 7. The 4 states behave quite

differently. Nearly half of the steady-state probability (0.49) goes to the Greenspan state.

For the other half, the probability is 0.25 for the Burns state, 0.143 for the Volcker state, and

0.116 for the fourth state. From Table 7 one can also see that the probability of switching

from the Greenspan and Burns states to the Volcker and fourth states is reduced by one half

as compared to the probability of switching the other way.

Differences in the contemporaneous coefﬁcient matrix show up across states as well.

In Table 8 we can see that the Greenspan regime’s contemporaneous coefﬁcient matrix is

broadly similar to that estimated for the full sample with the variances-only model (Table

5). In particular, both policy rules show a much larger contemporaneous coefﬁcient on R

than on M. On the other hand, we see from Tables 9 and 10 that the Burns and Volcker states

both have much larger contemporaneous coefﬁcients on M, with the M coefﬁcient being

relatively largest for the Volcker state. These results are consistent with the observation that

Burns seemed to pay a lot of attention to money growth in the early 70’s and less (more)

attention to money growth (the interest rate) in the last few years of his tenure (Burns, 1987;

Chappell, McGregor, and Vermilyea, 2005) and that Greenspan made the interest rate the

explicit policy instrument.

REGIME SWITCHES 23

year

1960 1970 1980 1990 2000

0.0 0.2 0.4 0.6 0.8 1.0

year

1960 1970 1980 1990 2000

0.0 0.2 0.4 0.6 0.8 1.0

year

1960 1970 1980 1990 2000

0.0 0.2 0.4 0.6 0.8 1.0

year

1960 1970 1980 1990 2000

0.0 0.2 0.4 0.6 0.8 1.0

FIGURE 3. State Probabilities, 4-state Monetary Policy Changing

In the background of each ﬁgure is the time path of the Fed Funds Rate.

REGIME SWITCHES 24

TABLE 8. Contemporaneous coefﬁcient matrix for 1st state in 4-state

policy-only model

Financial M Policy M demand Private y Private P Private U

Pcom 68.03 0.00 0.00 0.00 0.00 0.00

M 34.19 −208.60 −559.30 0.00 0.00 0.00

R −32.62 559.48 −172.64 0.00 −0.00 0.00

y −4.49 0.00 11.87 272.37 −17.51 51.94

P 8.65 0.00 −54.58 0.00 −1029.19 25.45

U 84.70 0.00 0.00 0.00 0.00 705.57

TABLE 9. Contemporaneous coefﬁcient matrix for 2nd state in 4-state

policy-only model

Financial M Policy M demand Private y Private P Private U

Pcom 38.20 0.00 0.00 0.00 0.00 0.00

M 19.20 −221.50 −401.63 0.00 0.00 0.00

R −18.32 188.29 −123.97 0.00 −0.00 0.00

y −2.52 0.00 8.52 206.87 −13.72 42.40

P 4.86 0.00 −39.19 0.00 −806.18 20.77

U 47.56 0.00 0.00 0.00 0.00 576.00

The long run policy responses to macro variables show a similar pattern, as reported in

Table 11. The Greenspan regime shows slightly stronger point estimates of the responses

of the funds rate to money growth and inﬂation than those implied by the variances-only

model (Table 6), but with greater uncertainty because of the smaller effective sample pe-

riod. For the Volcker and Burns regimes the responses of the federal funds rate are, variable

by variable, so ill-determined that we instead present responses of money growth, which

seems closer to the short-run policy target in those regimes. We see that the Volcker regime

REGIME SWITCHES 25

TABLE 10. Contemporaneous coefﬁcient matrix for 3rd state in 4-state

policy-only model

Financial M Policy M demand Private y Private P Private U

Pcom 50.43 0.00 0.00 0.00 0.00 0.00

M 25.35 −393.51 −241.46 0.00 0.00 0.00

R −24.18 136.05 −74.53 0.00 −0.00 0.00

y −3.33 0.00 5.12 235.35 −12.82 41.12

P 6.41 0.00 −23.56 0.00 −753.62 20.15

U 62.78 0.00 0.00 0.00 0.00 558.70

makes money unresponsive to all variables (measured by both point estimates and error

bands). The Burns regime shows a disturbingly high responsiveness of money growth to

inﬂation, though the point estimate is still below 1, which is only partially offset by a neg-

ative response to the rate of change in commodity prices.

Because the Burns regime looks like the most likely candidate for a potential sunspot

incubator, we tried normalizing that regime’s reaction function on the interest rate and

calculating its long-run response to the sum of the coefﬁcients on all nominal variables

— the rate of change in commodity prices, money growth, and inﬂation. This response

is surprisingly well-determined, probably because of collinearity in the sample among the

nominal variables.

The 68% probability band is (.94,3.50), which makes it very likely that

the regime was not a sunspot incubator.

VII. HISTORICAL COUNTERFACTUALS

As a way to quantify the importance of policy change over time, the 4-state time-varying

model makes it an internally coherent exercise to calculate what would have happened if

regime changes had not occurred, or had occurred when they otherwise didn’t, at particular

historical dates. We have run quite a few of these experiments, but the main conclusion

Note that if we calculated long run responses of the interest rate for this regime, variable by variable, we

would get very large, opposite-signed numbers that would have high uncertainty and be difﬁcult to interpret.

REGIME SWITCHES 26

TABLE 11. Long run policy responses in 4-state policy-only model

First state (Greenspan)

Responses of R to Posterior peak estimate .68 probability interval

∆ Pcom 0.09 (-0.19, 0.24)

∆ M 0.23 (-0.46, 2.08)

∆ y 0.43 (-1.28, 0.64)

∆ P 1.99 (-0.09, 2.48)

U -1.29 (-0.91, 0.46)

Second state (Burns)

Responses of ∆ M to Posterior peak estimate .68 probability interval

∆ Pcom -0.24 (-0.50, 0.01)

R 0.09 (-0.02, 0.49)

∆ y 0.18 (-0.43, 0.35)

∆ P 0.92 (-0.17, 1.74)

U 0.05 (-0.025, 0.09)

Third state (Volcker)

Responses of ∆ M to Posterior peak estimate .68 probability interval

∆ Pcom -0.12 (-0.06, 0.05)

R 0.01 (-0.02, 0.20)

∆ y 0.13 (-0.70, 0.64)

∆ P 0.23 (-0.51, 0.28)

U 0.02 (-0.04, 0.06)

is that the estimated policy changes do make a noticeable difference, but not a drastic

difference. In the following we display examples that seem most relevant to the debate on

the effects of monetary policy changes.

VII.1. Suppressing policy shocks. The ﬁrst and simplest of our counterfactual simula-

tions sets the disturbances in the policy equation to zero in the 9-state model. Disturbances

REGIME SWITCHES 27

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

FFR

Year

Actual

Counterfactual

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Actual

Counterfactual

FIGURE 4. Counterfactual paths with no coefﬁcient changes and no policy

shocks, 9-state variances-only model

and coefﬁcients are otherwise set at high-likelihood values, so that if the policy rule distur-

bances had been left in place, the simulations would have shown a perfect ﬁt. As can be

seen from Figure 4, the model leaves the time path of inﬂation almost unchanged. Policy

shocks play a crucial role only in attributing the ﬂuctuations of the funds rate in the late

1970’s and the early 1980’s. The history of inﬂation is attributed almost entirely to non-

policy sources, though of course feeding systematically through a ﬁxed monetary policy

rule.

VII.2. Keeping a ﬁxed Greenspan or Volcker rule in place throughout. If we run a

similar simulation but with the 4-state monetary policy model by placing the estimated

Greenspan rule through the pre-Greenspan period 1961-1987, we obtain the results shown

in Figure 5. This simulation tracks history almost as well as the previous one. Thus the

REGIME SWITCHES 28

1960 1970 1980 1990

FFR

Year

1960 1970 1980 1990

Year

1960 1970 1980 1990

Year

1960 1970 1980 1990

−4

−2

Year

FIGURE 5. Fixed Greenspan policy throughout 1961-87, 4-state monetary

policy model. Each graph shows the actual path (thick line) and the coun-

terfactual path (thin line).

model attributes the rise and fall in inﬂation neither to monetary policy shocks nor to

changes in in policy regime. In particular, the model reproduces the high peak inﬂation

rates of the early 80’s even though the Greenspan reaction function is in place throughout.

With the Burns policy in place throughout this history instead, we obtain the counter-

factual history shown in Figure 6. This simulation also reproduces history very closely,

matching the rise and the subsequent fall in inﬂation. This policy keeps inﬂation slightly

lower in the 60’s and 70’s, but then in the mid-80’s lets the inﬂation level out at a somewhat

higher value.

The modest differences across these policies do not mean the model implies that no

changes in monetary policy could have prevented a rise in inﬂation to near double-digit

REGIME SWITCHES 29

1960 1970 1980 1990

FFR

Year

1960 1970 1980 1990

Year

1960 1970 1980 1990

Year

1960 1970 1980 1990

−4

−2

Year

FIGURE 6. Fixed Burns policy throughout 1961-87, 4-state monetary pol-

icy model. Each graph shows the actual path (thick line) and the counter-

factual path (thin line).

levels. Though the Volcker reaction function is estimated imprecisely because of the short

period in which it prevailed, if we repeat our exercise with the point estimate of the Volcker

policy function in place, we obtain the results in Figure 7. This policy would have kept

money growth much lower, would have kept inﬂation lower by around two percentage

points at its peak, and would have lowered average output growth. Although the output

effect may be difﬁcult to see from Figures 5-7, Table 12 shows the substantial implied

differences in output growth for the three regime point estimates throughout this entire

period.

REGIME SWITCHES 30

TABLE 12. Annual average output growth rates over 1961-86, actual and counterfactual

Actual Burns Greenspan Volcker

3.7206 4.0560 3.2454 2.8956

1960 1970 1980 1990

−5

FFR

Year

1960 1970 1980 1990

−5

Year

1960 1970 1980 1990

Year

1960 1970 1980 1990

−4

−2

Year

FIGURE 7. Fixed Volcker policy throughout 1961-87, 4-state monetary pol-

icy model. Each graph shows the actual path (thick line) and the counter-

factual path (thin line).

These results are not reﬂective simply of the Volcker policy’s focus on growth of mone-

tary aggregates. If we simply double the coefﬁcients on inﬂation in the Greenspan mone-

tary policy rule, while again leaving disturbances in other equations at historical values and

suppressing monetary policy shocks, we arrive at Figure 8. Peak inﬂation is cut nearly in

half, and the inﬂation rate hovers around zero for much of the 1961-1987 period.

REGIME SWITCHES 31

1960 1965 1970 1975 1980 1985 1990

FFR

Year

Actual

Counterfactual

1960 1965 1970 1975 1980 1985 1990

−5

Year

Actual

Counterfactual

1960 1965 1970 1975 1980 1985 1990

−5

Year

Actual

Counterfactual

FIGURE 8. Counterfactual paths if policy response to inﬂation had been

twice as strong as in the estimated Greenspan policy throughout 1961-1987,

4-state monetary policy model.

Without any a priori imposed structure on private sector behavior, the model nonetheless

shows a type of neutrality result. By the 80’s, even though inﬂation is running 4 or 5

percentage points below the actual historical values, with this “inﬂation hawk Greenspan”

policy output is tracking the historical values almost perfectly. The model thus appears to

allow for the public’s learning that a new, lower level of inﬂation prevails. On the other

hand, the tighter monetary policy cuts output growth starting in the early 60’s, and keeps it

well below historical values for most of the 60’s, 70’s, and 80’s. Both of these two policy

rules that lower the inﬂation rate also lower the output growth rate, as can bee seen from

Figures 7 and 8.

The counterfactual simulations that imply lower inﬂation create a marked change in the

stochastic process followed by output and inﬂation. It is therefore quite possible that the

output costs of the stronger anti-inﬂationary policy stance would not have been so persistent

REGIME SWITCHES 32

as shown in the graphs. Our point is not that stricter anti-inﬂationary policy would have had

output costs as great as shown in these graphs. Our point is only that if the Greenspan rule

had been different enough to prevent the rise in inﬂation in the 60’s and 70’s, our model

would have shown the regime change made a difference. In fact, our best estimate is that

the monetary policy regime of the late 80’s and 90’s was not enough different from the

policy actually in place in 60’s and 70’s to have made any substantial difference to the time

path of inﬂation.

VII.3. Distributions of policy functions. Though the policy rules in place before the end

of 1979 and after 1982 are estimated to have similar consequences for the rise and fall

in inﬂation, the estimates leave uncertainty about those policies. Point estimates for both

regimes show, as we noted above, cumulative responses of the funds rate to inﬂation that

imply a unique price level. Nonetheless the Burns regime point estimates are lower, and

the uncertainty about the estimates leaves more probability in the region around a unit re-

sponse than with the Greenspan regime. As might be expected, the model’s simulated time

paths respond nonlinearly as the region with less than unit cumulative response of the funds

rate to inﬂation is approached. As a result, if we conduct our counterfactual simulations

by drawing from the distribution of policy rule coefﬁcients for the Burns and Greenspan

regimes, rather than simply imposing the most likely values, differences between the coef-

ﬁcients become more apparent. In the simulations we describe below the historical shocks

are kept on their historical path, with variances changing with regime according to our es-

timated posterior distribution, but the policy regime distribution is kept ﬁxed in one regime

for all coefﬁcients in the policy equation. This means that the scale of monetary policy

shocks, as well as the coefﬁcients in the reaction function, are being drawn from the distri-

bution corresponding to a single regime.

For the Greenspan regime results are shown in Figure 9, where we see that the median

simulated path displays substantially lower inﬂation than what was historically observed.

It is important to bear in mind that this is not the actual path for any one policy. This is

clear when we look at the median path for interest rates, which is almost uniformly lower

than the historical path. If these median paths were actual paths for any given policy, it

would be a mystery how the policy could lower inﬂation without ever raising interest rates.

REGIME SWITCHES 33

But as can be seen from the graphs for point-estimate policies, policies that lower inﬂation

raise interest rates in some crucial periods, and this is followed by long periods of lower

inﬂation, and hence of lower nominal interest rates. When we display the median path

across many policies that imply periods of tighter policy, but imply different timing for the

periods of tighter policy, we see a uniformly lower path of interest rates.

Note that these simulated draws from the Greenspan policy distribution imply a substan-

tial risk of deﬂation in the 80’s, as well as a risk of output growth below -5%.

A similar exercise with the Burns regime distribution produces the results in Figure 10.

There is little risk of output loss; money growth tends to be higher than the historical path.

The risk of deﬂation is lower, but now there is a substantial risk of no decline at all in

inﬂation in the 80’s, consistent with the conventional view about the effects of the Burns

policy.

VIII. ROBUSTNESS ANALYSIS

In this section we study a number of other relevant models to check the robustness of

our results. The insights from these exercises reinforce the points made in the previous

sections.

VIII.1. The economy with policy changes. We consider an economy with two monetary

policy rules estimated in our 4-state policy-only model: one is the rule associated with the

Burns regime and the other rule is the Greenspan interest-smoothing policy. This economy

consists of the same 6 variables as our actual data and starts with the Burns policy which

lasts for 236 months (corresponding to September 1979 in our sample) and then monetary

policy switches, once for all, to the estimated Greenspan policy rule. At the time of the

switch in policy rules, the scale of non-policy shocks also changes as in our estimated 4-

state model. We simulated 10 samples, each with the same sample length as our actual

data and each with initial values set at the actual data from 1959:01 to 1960:01. For each

simulated data set, we consider four models: monetary policy models for 2 and 3 states and

variances-only models for 2 and 3 states.

Computations for these simulated data are quite intensive. For each model, it takes about a week on

a single processor computer to get the marginal data density. There are a total of 40 models (which would

REGIME SWITCHES 34

1960 1970 1980 1990

−10

−5

FFR

Year

1960 1970 1980 1990

−15

−10

−5

Year

1960 1970 1980 1990

−10

−5

Year

1960 1970 1980 1990

−10

−5

Year

FIGURE 9. The Greenspan policy rule distribution throughout the pre-

Greenspan period, 4-state monetary policy model. Each graph shows the

actual path (thick line), the median counterfactual path (thin dark line), and

the 68% and 90% probability bands (thin light lines).

In 8 out of the 10 data sets, the estimated transition matrix for the 2-state monetary policy

model has one absorbing state, which is of course correct in the simulated data. Thus the

method we have used would have been likely to detect a permanent regime shift if that is

what had occurred.

Figure 11 shows the cdf, across the 10 Monte Carlo samples, of the posterior probability

that there was a change in policy coefﬁcients. In 7 of 10 cases the posterior probability of

be a 10-month computation). We acknowledge the technical support from the Department of Computer

Science at the Georgia Institute of Technology, which designed a Linux-based program called “STAMPEDE"

speciﬁcally for this project. This program allows us to run our jobs efﬁciently on a cluster of computers

simultaneously.

REGIME SWITCHES 35

1960 1970 1980 1990

FFR

Year

1960 1970 1980 1990

−10

Year

1960 1970 1980 1990

Year

1960 1970 1980 1990

−5

Year

FIGURE 10. The Burns policy rule distribution throughout the pre-

Greenspan period, 4-state monetary policy model. Each graph shows the

actual path (thick line), the median counterfactual path (thin dark line), and

the 68% and 90% probability bands (thin light lines).

a change was over .99. In one it was around .2, and in two it was .02 to .03. The log odds

ratio corresponding to the most extreme odds against the policy change (i.e., in favor of

a variance-change-only model) was 3.78. The log odds ratio in favor of variances-only in

our analysis of the historical data is about 60, many times stronger than the most extreme

ﬁnding in these Monte Carlo simulations.

It is also worth noting that the results showed no tendency to favor spurious variance-

change states. The variances-only model with three states had posterior probability less

than 10

−6

in all 10 simulations. The posterior probability on the 3-state model with policy

REGIME SWITCHES 36

0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

cdf of posterior probabilities that policy changed

proportion of draws

probability

FIGURE 11. Posterior probabilities for simulated data.

change (which of course is over-parameterized, but contains the true model) reached a max-

imum of around .04 in one simulation, and otherwise was even smaller than the posterior

probability on the 3-state variances-only model.

These experiments give our methods a stiff test. The estimated Greenspan and Burns

policy rules that we use leave the qualitative behavior of the data very similar in our coun-

terfactual simulations with point estimators. Yet even with these two similar policy rules,

our method is able to detect the switch for a majority of samples.

VIII.2. Other relevant models.

REGIME SWITCHES 37

VIII.2.1. Independent coefﬁcient and variance states. The results so far assume that coef-

ﬁcients and variances switch at the same time. For the monetary policy model, the potential

problem with this approach is that the number of states for the coefﬁcients on the policy

equation must increase with those for the variance state. In a single equation model, Sims

(2001) found that making the transitions of variance and coefﬁcient states independent de-

livered the best ﬁt. In our framework, this can be done by giving special structure to the

transition matrix P. If there are two independently evolving state variables, one indexing

variances and one indexing equation coefﬁcients, and the transition matrices for the two

types of state are Q

and Q

, we get the desired independent evolution by treating each pair

of values for the two states as a single state and setting P = Q

⊗ Q

Estimating a set of models with independent mean and variance states at the same scale

of parameterization as our main models would be a major computational task, which we

have not undertaken. We have instead calculated maximum log posterior density (LPD)

values (“LPD” values rather than log likelihood (LLH) values) for a number of somewhat

smaller scale models of this type which we can label 2v, 2v2p, 3v, 3v2p, and 4v. The “nv”

models are models with n variance states and no policy coefﬁcient changes. The “nv2p”

models are models with n variance states and 2 policy rule coefﬁcient states, evolving inde-

pendently. Because we have only LPD’s, we can’t compute posterior odds, but we can (as

Sims did in his single-equation paper) compare the models by the Schwarz criterion.

The

best of the models by this criterion is the 4v model. With the 2v model as base (therefore

with the zero value), the Schwarz criteria are:

2v 2v2p 3v 3v2p 4v

0.0 11.1 91.7 78.7 127.9

From this pattern of results it appears that a model with just two coefﬁcient policy regimes

is not competitive with variance-only models, even if the variance changes are allowed to

evolve independently of the coefﬁcient regimes.

The Schwarz, or Bayesian Information, Criterion, is usually described as log likelihood minus number

of parameters times log of sample size divided by 2. Under standard regularity conditions it is guaranteed to

be maximal at the model with highest posterior odds, if the sample is large enough. Though here we use LPD

in place of LLH, the same asymptotic reasoning that justiﬁes the criterion based on likelihood applies here.

REGIME SWITCHES 38

TABLE 13. Log marginal data densities for other models

Excluding money in policy rule Permanent regime change

Variances Only Monetary Policy Monetary Policy

2 states 13,330.89 13,347.46 13,154.08

3 states 13,432.88 13,419.88 13,414.53

4 states 13,462.40 13,296.58 13,412.85

Note that these results may explain why previous researchers (Lubik and Schorfheide,

2004; Clarida, Gali, and Gertler, 2000, e.g.) who allow only a single change in residual

variances ﬁnd evidence of coefﬁcient change. Those studies are making a comparison like

our 2v versus 2v2p comparison in the table, which favors 2v2p. It is only when we allow

at least 3 variance states that the addition of a coefﬁcient state ceases to improve ﬁt.

VIII.2.2. Permanent policy shifts. Our experiments with artiﬁcial data suggest that our

speciﬁcation could identify a permanent policy shift if it occurred. Because it is a wide-

spread view that there was a single permanent shift in US monetary policy around 1979,

however, it may nonetheless be of interest to see what emerges if we economize on param-

eters by imposing on our model the requirement that there is an absorbing state — that is,

there is a state that once is entered remains in place for the rest of the sample. This is equiv-

alent to requiring that one column of the transition matrix, which represents the probability

of entering each state conditional on being in this state, is a unit vector with a one at the

diagonal position.

The fourth column of Table 13 displays the marginal marginal data densities of the mon-

etary policy models with permanent changes on the coefﬁcients of the policy equation.

Comparing to the third column of Table 2, we see that the log posterior weight on these

models is lower by at least 60 more than the log posterior weights on the models that do

not impose the absorbing state restriction.

VIII.2.3. Excluding the monetary aggregate. In Section VI, we have shown the importance

of including a monetary aggregate to describe the policy rule under the Burns and Volcker

REGIME SWITCHES 39

regimes. Here we exclude this variable from the policy reaction function to see if this

worsens the ﬁt. The third column of Table 13 reports the measures of ﬁt for a model with

4 states, allowing the monetary policy rule only to change coefﬁcients, and with as usual

variances allowed to change with the state in all other equations. The ﬁt is considerably

worse than the corresponding cases when money is included (see the third column of Table

2), by about 60 in log odds units.

The ﬁt is also worse when we exclude money from the reaction function in the variances-

only model, but the odds ratio is much less extreme. The log odds difference between the

4-state variances-only model and the version of that model with money excluded from the

reaction function is 4.46. This implies an odds ratio in favor of the model including money

of over 80 in unlogged units, but this ratio is much less extreme than the result for the

model that allows coefﬁcient variation in the monetary policy rule. This is not surprising,

since the most salient difference among the three main estimated policy reaction functions

is in the degree to which they give weight to a monetary aggregate. If we shut down this

type of difference among policies, the model with coefﬁcient variation in the policy rule is

penalized much more than the model that ﬁts a single rule to the whole sample. As we have

already pointed out, it seems possible that a model whose prior focused the search for policy

variation in particular economically reasonable directions might be more competitive with

the variances-only model. But the results here suggest that such a model, if it is possible at

all, is not likely to succeed if it excludes money from the reaction function.

IX. CONCLUSION

Monetary policy and its history are complex, and abstract theoretical models that we use

to organize thought about them can hide what was really going on. Explorations of data

with relatively few preconceptions, like this exploration, may bring out regularities that

have been slipping through abstract discussion. In this case, we think this has happened.

Our best-ﬁt model suggests that neither additive disturbances to a linear monetary pol-

icy reaction function nor changes in the coefﬁcients of that function have been a primary

source of the rise and fall of inﬂation over our sample period. Instead, stable monetary

policy reactions to a changing array of major disturbances generated the historical pattern.

REGIME SWITCHES 40

1960 1965 1970 1975 1980 1985 1990

FFR

Year

Actual

Median forecast

1960 1965 1970 1975 1980 1985 1990

Year

Actual

Median forecast

FIGURE 12. Counterfactuals: Greenspan regime in 1962–1987; the Fed

Funds Rate in upper panel and the inﬂation rate in lower panel.

Oil price shocks and the Vietnam war and its ﬁnancing produced disturbances in the 60’s

and 70’s that have not recurred on such a scale since. With such a large role assigned to

“private sector shocks”, it would be useful to consider a model that allows more detailed

interpretation of these shocks. Recent work by Gambetti, Pappa, and Canova (2005) is an

attempt in this direction.

Even if one gives all the prior weight to the four-state policy model, which assumes the

existence of regime changes in monetary policy, our point estimates imply that the impact

on the economy of changes in the systematic part of monetary policy were not as big as

commonly thought. Nonetheless our estimates do imply that a permanent reserve-targeting

policy like that of 1979-82, or a policy that greatly ampliﬁed the reaction of interest rates

REGIME SWITCHES 41

to inﬂation, could have kept inﬂation substantially lower, while exacting a cost in lower

output growth.

In our estimates that enforce changes in policy rule, the strongest evidence for monetary

policy change is that for shifting emphasis on monetary aggregates in the policy reaction

function. This accords with the prominent role monetarism played in policy discussions

of the 70’s. If further research succeeds in ﬁnding clear evidence of changes in monetary

policy behavior in this period, it will most likely be through focusing attention on the

changing impact of monetarism on policy behavior.

Policy actions were difﬁcult to predict, and if there were shifts in the systematic com-

ponent of policy, they are of a sort that it is difﬁcult for us to track precisely even with

hindsight. While our results leave room for those with strong beliefs that monetary pol-

icy changed substantially to maintain those beliefs, it is nonetheless clear that whatever

changes there were of uncertain timing, not permanent, and not easily understood even

today. Models that treat policy changes as permanent, non-stochastic, transparent regime

changes are not useful in understanding this history.

APPENDIX A. ESTIMATION AND INFERENCE

A.1. The Prior. The identiﬁcation speciﬁed in Table 1 is a special case of standard linear

restrictions imposed on A

and D as

nh×1

= U

nh×o

×1

, j = 1,. . .,n,

mh×1

= V

mh×r

×1

, j = 1,.. .,n,







0, j

(1)

0, j

(h)







, d







(1)

(h)







where b

and g

are the free parameters “squeezed” out of a

and d

by the linear restric-

tions, o

and r

are the numbers of the corresponding free parameters, columns of U

are

orthonormal vectors in the Euclidean space R

, and columns of V

are orthonormal vectors

in R

REGIME SWITCHES 42

The prior distributions for the free parameters b

and g

have the following Gaussian

forms:

) = N(0, H

+ j

For all the models studied in this paper, we set H

and H

+ j

the same way as Sims

and Zha 1998 but scale them by the number of states (h) so that the Case I model in (8)

coincides with the standard Bayesian VAR with constant parameters. The liquidity effect

prior is implemented by adjusting the off-diagonal elements of H

that correspond to the

coefﬁcients of M and R for j = 2,3 such that the correlation for the policy equation (the

second equation) is -0.8 and the correlation for the money demand equation (the third

equation) is 0.8. Because we use monthly data, the tightness of the reference prior is set as,

in the notation of Sims and Zha 1998,

= 0.6,

= 0.1,

= 1.0,

= 1.2,

= 0.1,

5.0, and

= 5.0 (see Robertson and Tallman 2001).

The prior distribution for

(k) is taken as

(

(k)) = Γ(

) for k ∈ {1,.. .,h}, where

(k) ≡

(k) and Γ(·) denotes the standard gamma pdf with

being a scale factor (not

an inverse scale factor as in the notation of some textbooks). The prior pdf for

(k) is

N(0,

) for k ∈ { 1, ..., h}.

The prior of the transition matrix P takes a Dirichlet form as suggested by Chib 1996.

For the k

column of P, p

, the prior density is

) =

,. .., p

) = D(

,. ..,

) ∝ p

−1

·· · p

−1

where

> 0 for i = 1,. ..,h.

The hyperparameters

, and

are newly introduced and have no reference values

in the literature. We set

= 1 and

= 50 as the benchmark and then perform a

sensitivity check by varying these values. The prior setting

= 50 is reasonable because

the posterior estimate of

(k) can be as large as 40 or 50 even with a much smaller value

Indeed, a tighter prior on

(k) tends to lower the marginal likelihood for the same model.

REGIME SWITCHES 43

There are two steps in setting up a prior for p

. First, the prior mode of p

is chosen to

such that

= 0.95 and

= 0.05/(h− 1) for i 6= k. Note that

∑

i=1

= 1. In the

second step, given

and

Var(p

) (which is set to 0.025), we solve for

through a

third polynomial and then for all other elements of the vector

through a system of h− 1

linear equations. This prior expresses the belief that the average duration of each state is

about 20 months. We also experienced with different prior values for P, including a very

diffuse prior for P by letting

be evenly distributed across i for given k and by letting the

prior standard deviation of p

be much larger than 0.025. The results seem insensitive to

these prior values.

A.2. Posterior Estimate. We gather different groups of free parameters as follows, with

the understanding that we sometimes interchange the use of free parameters and original

(but restricted) parameters.

p =

{

, k = 1,..., h

}

;







(k), j = 1,... , n, k = 1,.. .,h

, for Case II;

(k), i, j = 1,. ..,n, k = 1, ..., h

, for Case III;

g =

, j = 1,. . .,n

;

b =

, j = 1,. . .,n

;

{

,g,b

}

The overall likelihood function

) can be obtained by integrating over unobserved

states the conditional likelihood at each time t and by recursively multiplying these condi-

tional likelihood functions forward (Kim and Nelson 1999).

From the Bayes rule, the posterior distribution of

conditional on the data is

(

| Y

) ∝

(

)

where the prior

(

) is speciﬁed in Section A.1.

In order to avoid very long startup periods for the MCMC sampler, it is important to

begin with at least an approximate estimate of the peak of the posterior density

(

). Moreover, such an estimate is used as a reference point in normalization to obtain

REGIME SWITCHES 44

likelihood-based statistical inferences. Because the number of parameters is quite large for

our models (over 500), we used an eclectic approach, combining the stochastic expectation-

maximizing algorithm with various optimization routines. For some models, the conver-

gence took as many as 15 hours on an Intel Pentium 4 2.0GHz PC.

A.3. Inference. Our objective is to obtain the posterior distribution of functions of

such

as impulse responses, forecasts, historical decompositions, and long-run responses of pol-

icy. It involves integrating over large dimensions many highly nonlinear functions. We fol-

low Sims and Zha 2004 to use a Gibbs sampler to obtain the joint distribution

(

| Y

)

where S

= {s

,. ..,s

}. The Gibbs sampler involves sampling alternatively from the

following conditional posterior distributions:

Pr(S

| Y

, p,

,g,b),

(p | Y

,g,b),

(

| Y

, p,g,b),

(g | Y

, p,

,b),

(b | Y

, p,

,g).

It has been shown in the literature that such a Gibbs sampling procedure produces the

unique limiting distribution that is the posterior distribution of S

and

(e.g., Geweke 1999).

The probability density functions of these conditional distributions are quite complicated

but can be nonetheless simulated from (for details, see Sims and Zha 2004).

A.4. Normalization. To obtain accurate posterior distributions of functions of

(such as

long run responses and historical decompositions), we must normalize both the signs of

structural equations and the labels of states; otherwise, the posterior distributions will be

symmetric with multiple modes, making statistical inferences of interest meaningless. Such

normalization is also necessary to achieve efﬁciency in evaluating the marginal likelihood

We are still improving our algorithm. Once it is ﬁnished, it is possible that the computing time could be

considerably reduced.

REGIME SWITCHES 45

for model comparison.

For both purposes, we normalize the signs of structural equations

the same way. Speciﬁcally, we use the Waggoner and Zha (2003) normalization rule to

determine the column signs of A

(k) and A

(k) for any given k ∈ {1,..., h}.

Two additional normalizations are (1) scale normalization on

(k) and

(k) and (2)

label normalization on the states. We simulate MCMC posterior draws of

with

(k) = 1

and

(k) = 1

h×1

for all j ∈ {1,... , n}, and k ∈ {1, ..., h}, where the notation 1

h×1

denotes

the h × 1 vector of 1’s. For each posterior draw, we label the states so that the posterior

probabilities of each state for all t ∈ {1,...,T} match closest to the posterior estimates of

those probabilities.

To estimate the marginal data density

) for each model, we apply both the modiﬁed

harmonic mean method (MHM) of Gelfand and Dey 1994 and the method of Chib and

Jeliazkov 2001. The MHM method is quite efﬁcient for most models considered in this

paper, but it may give unreliable estimates for some models whose posterior distributions

have multiple modes. In such a situation, we also use the Chib and Jeliazkov to check the

robustness of the estimate.

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