Drift s a n d Vo lat ilitie s: Mo n e ta ry Po licie s a n d
O utco m es in the Po st WWII U .S.
∗
Timothy Cogley
Arizona State Universit y
Thomas J. Sargent
New York Univ ersity and Hoo ver Institution
Revised: Apr il 2003
Abstract
For a VAR with drifting coefficients and stochastic volatilities, w e presen t pos-
terior densities for several objects that are of interest for designing and eval-
uating monetary policy. These include measures of inflation persistence, the
naturalrateofunemployment,acorerateofinflation, and ‘activism coeffi-
cients’ for monetary policy rules. Our posteriors imply substantial variation of
all of these objects for post WWII U.S. data. After adjusting for changes in
volatility, persistence of inflation increases during the 1970s then falls in the
1980s and 1990s. Innovation variances change systematically, being substan-
tially larger in the late 1970s than during other times. Measures of uncertainty
about core inflation and the degree of persistence covary positively. We use our
posterior distributions to ev aluate the po wer of sev eral tests that have been
used to test the null of time-invariance of autoregressive coefficients of VARs
against the alternative of time-varying coefficien ts. Except for one test, we find
that those tests have low power against the form of time variation captured by
our model. That one test also rejects time invariance in the data.
1Introduction
This paper extends the model of Cogley and Sargen t (2001) to incorporate stochastic
volatilit y and then reestimates it for post World War II U.S. data in order to shed
light on th e following questions. Have aggregate time series responded v ia time-
in variant linear impulse response functions to possibly heterosk edastic shocks? Or
∗
For comments and suggestions, we are grateful to Jean Boivin, Marco Del Negro, Mark Gertler,
Sergei Morozov, Simon Potter, Christopher Sims, Mark Watson, and Tao Zha.
1
is it more likely that the impulse responses to shocks themselves have ev o lv ed ov er
time because of drifting coefficients or other nonlinearities? We presen t evidence that
shoc k variances ev olved systematically o ver time, but that so did the autoregressive
coefficients of VARs. One of our main conclusions is that much of our earlier evidence
for drifting coefficients survives after we tak e stoc h astic volatility int o account. We
use our evidence about drift and stochastic volatility to infer that monetary policy
rules ha ve c hanged and that the persistence of inflationitselfhasdriftedovertime.
1.1 Timeinvarianceversusdrift
The statistical tests of Sims (1980, 1999) and Bernanke and Mihov (1998a, 1998b)
seem to affirm a model that con tradicts our findings. They failed to reject the h y-
pothesis of time-in variance in the coefficients of VAR s for periods and variables like
ours. To shed ligh t on whether our results are inconsistent with theirs, we examine
the performance of various tests that ha ve been used to detect deviations from time
invariance. Except for one, we find that those tests ha v e low power against our partic-
ular model of drifting coefficients. And that one test actually rejects time invariance
in the data. These results about power help reconcile our findings with those of Sims
and Bernanke and Mihov.
1.2 Bad policy or bad luc k?
This paper accu mulates eviden ce inside an atheoretical statistical model.
1
But we
use the patterns of time variation that our statistical model detects to shed light
on some important substa ntive and theoretical question s about post WW II U.S .
monetary policy. These rev olve around whether it was bad monetary policy or bad
luck that made inflation-unemployment outcomes w orse in the 1970s than before or
after. The view of DeLong (1997) and Romer and Romer (2002), whic h they support
b y stringing together interesting anecdotes and selections from go vernmen t reports,
asserts that it w as bad policy. Their story is that during the 1950 s and ear ly 1960s,
the Fed basically understood the correct model (whic h in their view incorporates the
natural rate theory that asserts that there is no exploitable trade off between inflation
and unemp loymen t); that Fed policy makers in the late 1960s and early 1970s w ere
seduced by Samuelson and Solow ’s (1960) promise of an exploitable trade-off between
inflation and unemploym en t; and that under Volck er’s leadership, the Fed came to its
senses, accepted the natural rate h ypothesis, and focused monetary policy on setting
inflation low.
Aspects of this “Berkeley view” receive bac king from statistical w o rk b y Clarida,
Gali, and Gertler (2000) and Taylo r (1993) , who fit monetary policy rules for subpe-
riods that they c hoose to illuminate possible differences between the Burns and the
1
By atheoretical we mean that the model’s parameters are not explicitly linked to parameters
describing decision makers’ preferences and constraints.
2
Volck er-Greenspan eras. They find evidence for a systematic change of monetary pol-
icy across th e two eras, a c hange that in Clarida, Gali, and Gert ler’s ‘new-neoclassical-
synthesis’ macroeconomic model would lead to better inflation-unemployment out-
comes.
But Ta ylor’s and Clarida, Gertler, and Gali’s int erpre tation of the data has been
disputed by Sims (1980, 1999) and Bernanke and Mihov (1998a, 1998b), both of
whom have presen ted evidence that the U.S. data do not prom pt rejection of the time
in variance of the autoregressive coefficien ts of a VAR. They also present evidence for
shifts in the variances of the innovations to their VARs. If one equation of the VAR
is interpreted as describing a monetary policy rule, then Sims’s and Bernanke and
Mihov’s results sa y that it w a s not the monetary policy strategy but luc k (i.e., the
volatilit y of the shocks) that chan ge d between the Burn s and the non-B u rn s periods.
1.3 I nflation persistence and inferences about the natural
rate
The persistence of inflation pla y s an important role in some widely used empiric al
strategies for testing the natura l rate hypothesis and for estimating the natural un-
employm en t rate. As we shall see, inflation persistence also pla ys an important role
in lending relevance to instruments for estimating monetary policy rules. Therefore,
we use our statistic al model to portray the evolving persiste nce of inflation. We de-
fine a measure of persistence based on the normalized spectrum of inflation at zero
frequency, then present how this measure of persistence increased during the 1960s
and 70s, then fell durin g the 1980s and 1990s.
1.4 Drifting coefficients and the Lucas Critique
Drifting coefficients have been a n im por t a nt piece of u nfinished business within macroe-
conomictheorysinceLucasplayedthemupinthefirsthalfofhis1976 Critique, but
then ignored them in the second half.
2
In Appendix A, we revisit ho w drifting co-
efficients bear on the theory of economic policy in the context of recen t ideas about
self-co nfirm ing equilibria. This append ix pro vide s bac kgro und for a view that helps to
bolster the time-in variance view of the data taken by Sims and Bernanke and Mihov .
1.5 Method
We take a Bay es ian perspective and report time series of posterior densitie s for various
econom ically interestin g function s of hyperparameters and hidden states. We use a
Markov Chain Mon te Carlo algorithm to comput e posterior densities.
2
See Sargent (1999) for more about this interpretation of the two halves of Lucas’s 1976 paper.
3
1.6 Organization
The remainder of this paper is organized as follows. Sect ion 2 describes the basic
statistical model that we use to dev elop empirical evidence. We consign to appendix
B a detailed characterization of the priors and posterior for our model, and appendix
C describes a Marko v Chain Monte Carlo algorithm that we use to approximate the
posterior density. Section 3 reports our results, and section 4 concludes. Appendix A
pursues a theme opened in the Lucas Critique about ho w drifting coefficient models
bear on alternativ e theories of economic policy.
2 A Bayesian Vecto r Auto r egr ess ion with Driftin g
Para meters and Stochastic Vola tility
The object of Cogley and Sargent (2001) w as to develop empirical evidence about the
evolving law of motion for inflation and to relate the evidence to stories about c h an ges
in monetary policy rules. To that end, we fit a Ba yesian vector autoregression for
inflation, unemplo yment, and a short term interest rate. We introduced drifting VAR
parameters, so that the law of motion could evolve, but assumed the VAR inno vation
variance was constan t. Th us, our measurement equation w a s
y
t
= X
0
t
θ
t
+ ε
t
, (1)
where y
t
is a v ector of endogenous variables, X
t
includes a constan t plus lags of y
t
,and
θ
t
is a vector of VAR parameters. The residuals, ε
t
, were assum ed to be conditionally
normal with mean zero and constan t covariance matrix R.
The VAR parameters w ere assumed to evolve as driftless random walks subject
to reflecting barriers. Let
θ
T
=[θ
0
1
,... ,θ
0
T
]
0
, (2)
represent the history of VAR parameters from dates 1 to T . The driftless random
walk component is rep resented by a joint prior ,
f(θ
T
,Q)=f(θ
T
|Q)f(Q)=f(Q)
Y
T −1
s=0
f(θ
s+1
|θ
s
,Q). (3)
where
f(θ
t+1
|θ
t
,Q) ∼ N(θ
t
,Q). (4)
Thus, apart from the reflecting barrier, θ
t
ev olves as
θ
t
= θ
t−1
+ v
t
, (5)
4
The innovation v
t
is normal with mean zero and variance Q, and w e allo wed for
cor r ela t ion betwe en the state and measur ement innovations , cov(v
t
, ε
t
)=C.The
marginal prior f(Q) makes Q an inverse-Wishar t variate.
The reflecting barrier w as encoded in an indicator function, I(θ
T
)=
Q
T
s=1
I(θ
s
).
The function I(θ
s
) takes a value of 0 when the roots of the associated VAR polynom ial
are inside the unit circle, and it is equal to 1 otherwise. T h is restriction truncates
and renormalizes the random walk prior,
p(θ
T
,Q) ∝ I(θ
T
)f(θ
T
,Q) (6)
This is a stabilit y condition for the VAR, reflecting an aprioribelief about the
implausibility of explosiv e representations for inflation, unemplo ymen t, and real in-
terest. The stabilit y prior follows from our belief that the Fed chooses policy rules
in a purposeful way. Assuming that the Fed has a loss function that penalizes the
variance of inflation, it will not choose a policy rule that results in a unit root in
inflation, for that results in an infinite loss.
3
In appendix B, we derive a n u mber of relations betw een the restricted and unre-
stricted priors. Among other things, the restricted prior for θ
T
|Q can be expressed
as
p(θ
T
|Q)=
I(θ
T
)f(θ
T
|Q)
m
θ
(Q)
, (7)
the marginal prior for Q becomes
p(Q)=
m
θ
(Q)f(Q)
m
Q
, (8)
and the transition densit y is
p(θ
t+1
|θ
t
,Q) ∝ I(θ
t+1
)f(θ
t+1
|θ
t
,Q)π(θ
t+1
,Q). (9)
The terms m
θ
(Q) and m
Q
are normalizing constants and are defined in the appendix.
4
In (7), the stabilit y condition truncates and renormalize s f(θ
T
|Q) to eliminate ex-
plosive θ’s. In (8), the margin al prior f(Q) is re-weighte d by m
θ
(Q), the probabilit y
3
To take a concrete example, consider the model of Rudebusch and Svennson (1999). Their
model consists of an IS curve , a Phillips curve, and a monetary policy rule, and they endow the
central bank with a loss function that penalizes inflation variance. The Phillips curve has adaptive
expectations with the natural rate hypothesis being cast in terms of Solow and Tobin’s unit-sum-of-
the weights form. That form is consistent with rational expectations only when there is a unit root
in inflation. The autoregressive roots for the system are not, however, determined by the Phillips
curve alone; they also depend on the c hoice of monetary policy rule. With an arbitrary policy
rule, the autoregressiv e roots can be inside, outside, or on the unit circle, but they are stable under
optimal or near-optimal policies. When a shock moves inflation away from its target, poorly chosen
policy rules may let it drift, but well-chosen rules pull it back.
4
These expressions supercede those given in Cogley and Sargent (2001). We are grateful to Simon
Potter for pointing out an error in our earlier work and for suggesting ways to correct it.
5
